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Spin-two fields and general covariance Heiderich, Karen Rachel 1991

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SPIN-TWO FIELDS A N D G E N E R A L COVARIANCE By Karen Rachel Heiderich B. Sc. (Physics) University of Winnipeg  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April, 1991 (c) Karen Rachel Heiderich, 1991  In presenting this thesis i n partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study.  I further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Physics T h e University of B r i t i s h C o l u m b i a 6224 A g r i c u l t u r a l R o a d Vancouver, C a n a d a V 6 T 1W5  Date:  Abstract  It has long been presumed that any consistent nonlinear theory of a spin-two field must be generally covariant. Using Wald's consistency criteria, we exhibit classes of nonlinear theories of a spin-two field that do not have general covariance. We consider four alternative formulations of the spin-two equations. As a first example, we consider a conformally invariant theory of a spin-two field coupled to a scalar field. In the next two cases, the usual symmetric rank-two tensor field,  •jab, is  chosen as the potential. In the  fourth case, a traceless symmetric rank-two tensor field is used as the potential. We find that consistent nonlinear generalization of these different formulations leads to theories of a spin-two field that are not generally covariant. In particular, we find types of theories which, when interpreted in terms of a metric, are invariant under the infinitesimal gauge transformation 7^ —» 7^ + V( V K\ \i ), where K b is an arbitrary two-form field. In c  a  c  )  a  addition, we find classes of theories that are conformally invariant. As a related problem, we compare the types of theories obtained from the nonlinear extension of a divergence- and curl-free vector field when it is described in terms of two of its equivalent formulations. We find that nonlinear extension of the theory is quite different in each case.  Moreover, the resulting types of nonlinear theories may  not necessarily be equivalent. A similar analysis is carried out for three-dimensional electromagnetism.  n  Conventions and Abbreviations  0.1  Conventions  Geometrized units G = c = 1 are used, where c is the speed of light and G is Newton's constant, and, i n general, the conventions of W a l d (1984) [73] are followed. index notation is used throughout.  1  In general, small Greek letters a ,  Abstract  • • •, fi, v, • • •,  etc., are tensor coordinate indices, taking on the values 0 , 1 , 2 , 3 ; small R o m a n letters a, b, c, • • •, m, n • • •, etc., are tensor abstract indices (however, z, j , are reserved for tensor coordinate indices that take on the values 1,2,3); capitol Greek letters r , II, S , T , - - - , etc., are spinor coordinate indices, taking on the values 0,1; capitol R o m a n letters A , B,C, • • •, etc., are spinor abstract indices. Capitol R o m a n letters are also used as labels i n tensor equations. T h e metric g b has Lorentzian signature (-+++) except for spinor equations, where a  the signature is (H  ).  R o u n d brackets around indices (•••) indicate that the indices are symmetrized, + Tba)  ,  (0.1)  T(ab)  ~  \{T b  T^abc)  —  3?(T bc ~\~ Tbca ~\~ Tcab ~f" Tbac ~t~ Tcba ~\~ T b)  a  a  ac  •  ..  It is common when working with tensors to use coordinate index notation, i.e., to choose a coordinate system and to write equations as relations between the components of objects in that coordinate system, e.g., a rank two tensor appears as the sixteen (in four dimensions) quantities T ", which are the components of the ( § ) tensor 'T' in the chosen coordinate system; a vector appears as the four quantities , etc.. An alternative notation is the abstract index notation (see Wald (1984) [73], and Penrose and Rindler (1984) [57]), which can be viewed as a generalization of the 'abstract' notation v for vectors and T for rank-two tensors, to all types of quantities. In abstract index notation, a vector is denoted v , a rank-two tensor as T , a rank (^) tensor as T "' i i ...i , etc.. The abstract indices 'a', V, etc., are not coordinate indices, i.e., v is not a collection of four quantities. They are abstract objects (like '-*') which indicate what sort of quantities V and 'T' are. In practice, abstract indices can be read as if they were coordinate indices. l  M  a  ab  aia2  a,l  )1 )2  a  ill  lm  Square brackets around indices [•••] indicate that the indices are anti-symmetrized, T[ab]  =  ^[abc]  =  \{T b  -  a  Tba)  3~i{T bc  ,  (0.2)  Tbac ~\~ Tbca  a  Tb c  a  ~\~ Tcab  T b) . ac  Vertical bars | - | surrounding a group of indices indicate that the enclosed indices are not effected by the (anti)symmetrization brackets outside the bars. For example, T( \[bK ]\d) a  c  = \{T bKcd  — T Kbd  a  + TdbKca ~ Td Kba)  ac  c  •  (0.3)  The symbol d is used to denote partial derivative: a  T h e L e v i - C i v i t a antisymmetric epsilon tensor, e b d, is taken such that a  eoi23 =  T h e R i e m a n n curvature tensor, R bc , d  a  (V V 0  for arbitrary w . c  •  (0.5)  is defined by  - V  6  V  H = +Rabc wd , d  A  (0.6)  W e use the Einstein summation convention, i.e., summation over re-  peated indices is understood.  0.2  6  +1  c  2  Abbreviations  L T , L G = Lorentz transformation, Lorentz group P T , P G = Poincare transformation, Poincare group L H S , R H S = left hand side, right hand side c c . = complex conjugate M T W = Misner, Thorne and Wheeler, i.e., Ref.[47] Einstein "said in jest to a friend, 'I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over over an index which occurs twice [53]. 2  iv  Table of Contents  Abstract  ii  Conventions and Abbreviations  iii  0.1  Conventions  iii  0.2  Abbreviations  iv  List of Figures  ix  Acknowledgement  x  1  Introduction  1  1.1  General relativity  4  1.2  Linearized general relativity  6  1.3  Summary  2  10  T h e Spin-5 Equations  11  2.1  Spin and" representations of the Poincare group  12  2.2  The spin-s equations  13  2.2.1  Conformal invariance of the spin-5 equations  14  2.2.2  Generalization to non-flat spacetimes - consistency condition . . .  16  2.2.3  Potentials for the spin-5 fields  17  2.3  Tensor equivalent of the spin-one and spin-two equations  20  2.3.1  Spin-one  21  2.3.2  Spin-two  22 v  3  2.4  T h e usual symmetric rank-two tensor potential for a spin-two field . . . .  24  2.5  T h e usual Lagrangian formulation for the spin-two equations  28  2.6  Summary  32  Spin-two Theories and General Relativity  34  3.1  T h e standard argument  36  3.2  T h e gauge invariance approach  44  3.2.1  T h e "spin-limitation principle"  45  3.2.2  T h e "extended G u p t a program"  48  3.2.3  N o r m a l spin-two gauge invariance  50  3.3  4  5  Wald's consistency criteria  52  3.3.1  Consistent nonlinear extension of Maxwell's equations  52  3.3.2  Summary of Wald's consistency criteria  62  3.3.3  Concluding remarks  64  Potentials and Nonlinear Generalization  69  4.1  Divergence- and curl-free vector field  69  4.2  Three-dimensional electromagnetism  73  4.3  Solving for the gauge invariance  75  4.3.1  Vector theory  76  4.3.2  Three-dimensional electromagnetism  81  C o n f o r m a l Invariance - Spin-two Coupled with Spin-zero  85  5.1  Conformally invariant linear equations  86  5.2  Spin-two coupled w i t h spin-zero - conformal K l e i n Gordon equation . . .  87  5.3  Nonlinear extension of spin-two coupled w i t h spin-zero  91  5.4  Solving for the gauge invariance  93  vi  6  5.4.1  T h e general equations  94  5.4.2  T h e linearized equations  96  5.4.3  T h e general solutions  104  5.4.4  Uniqueness of the solutions  108  Nonlinear, Noncovariant, Spin-two Theories  114  6.1  Formulation of the spin-two equations  115  6.2  Nonlinear, noncovariant spin-two theories  119  6.3  O u t l i n e of the calculations  126  6.4  Conclusions  139  Appendices  141  A  Representations of the Poincare G r o u p  141  A.l  T h e Lorentz transformations  142  A.2  T h e Poincare group invariants  143  A.3  Irreducible representations of the homogeneous Lorentz group  146  A . 4 Projection operators  B  C  147  A.4.1  Vector field  147  A.4.2  Rank-two tensor field  148  A B r i e f Introduction to Spinors  150  B. l  A geometric picture of spinors  150  B. 2  T h e curvature spinors  160  Conformal Transformations  164  C. l  Tensor formulation  164  C.2  Spinor formulation  165  vii  Bibliography  167  vin  List of Figures  B.l  (a) A spacetime diagram w i t h one dimension suppressed. A slice at t = 1 represents the future pointing null vectors, (b) The hypersurface t = 1. T h e line L indicates the stereographic mapping from the sphere to the complex plane  B.2  153  T h e geometric concept of a spin-vector as a null flag, (a) T h e flagpole is defined by the null vector k , a  plane generated by l + ak . a  transformation  K  A  ->  a  eK ie  A  and the flag plane is defined by the half(b) T h e hypersurface t = 1.  with 6 -  TT/2,  l -* -l . a  a  With 9  Under the =  TT,  l -+ l a  but K —> — K . Every null flag defines two spin-vectors, K and — K . A  A  A  ix  A  a  . .  159  Acknowledgement  First and foremost I would like to thank my thesis advisor, Professor W . G . U n r u h . Not only did he suggest many of the problems i n the thesis, his insight and way of looking at things never failed to add a new dimension to a problem. He kept me on the right track and was unfailingly patient w i t h my work. Throughout my time as a graduate student, I have have enjoyed discussions w i t h A . Borde, S. F o r t i n , S. Habib, M . Holzer, R . Laflamme, D . Morgan, A . Roberge, J . T h o r n b u r g , ° a n d D . Vollick. I would like to thank my family and friends, M . Holzer i n particular, for providing moral support. M y parents deserve special thanks for their understanding and encouragement. F i n a l l y I would like to acknowledge the financial support of N S E R C and U B C .  x  Chapter 1  Introduction  General relativity is Einstein's theory of gravitation, which he presented to to the Prussian A c a d e m y of Science on November 25, 1915 [53]. "Einstein's gravitational theory, which is said to be one of the greatest single achievements of theoretical physics, resulted i n beautiful relations connecting gravitational phenomena w i t h the geometry of space" [31]. In general relativity, gravity is not regarded as a force field, but rather, gravity is regarded as an integral part of the spacetime structure itself. T h e intrinsic properties of spacetime are described by the dynamical curved spacetime metric [73]. "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve" [47]. T h e precise form of this interaction between matter and spacetime is given by Einstein's equations. Part of the beauty of general relativity lies i n its geometric interpretation. Since its formulation i n 1915, however, there have been many attempts to derive the Einstein equations from non-geometric points of view (for an outline of some of these arguments, see, e.g., M T W [47] p. 416 f.). In analogy w i t h relativistic field theories, such as electromagnetism, physicists have sought to explain gravity i n terms of a field propagating i n a flat background spacetime. In particular, it has been argued, and widely accepted, that the theory of a spin-two field i n a flat background spacetime uniquely gives Einstein's equations  1  (see, e.g., [47], [31], [77], [22]). In this thesis, this claim is investigated. It  This point of view necessarily abandons the geometric interpretation. See MTW [47] table 18.1 p. 431 for a table comparing and contrasting the geometric viewpoint and the field theoretic nongeometric viewpoint. 1  1  Chapter  1.  Introduction  2  is shown that there exists consistent nonlinear theories of a spin-two field that are not Einstein's equations. In order to appreciate and to understand more completely the problems that are studied i n this thesis, it is necessary to first say more about the Einstein equations, the spin-two equations, and the arguments justifying the contention that the spin-two equations uniquely give rise to Einstein's equations. The remainder of this chapter completes a brief introduction to general relativity. This serves to establish notation as well as to introduce useful equations and concepts. In section 1.1, after writing down the E i n stein equations, the Lagrangian formulation of general relativity is given. In section 1.2, the equations of linearized gravity are outlined. In particular, it is remarked that the linearized E i n s t e i n equations are identical w i t h the usual formulation of the spin-two equations. Thus general relativity can be regarded as the theory of an interacting spintwo field. Before being able to discuss interacting spin-two theories, it is important to consider the spin-two equations themselves. Chapter 2 is devoted to a detailed discussion of the zero-rest-mass spin-s equations. It is pointed out that (i) the equations of motion for a spin-s field are invariant under a conformal rescaling of the (flat) metric, (ii) that there are several ways of choosing potentials to describe a spin-5 field, and (iii) that there are severe restrictions on the possible types of interacting spin-3 theories for s > 1. In the case of spin-two, these constraints were believed to restrict the possible types of interacting spin-two theories to Einstein's equations. Particular attention is paid to the usual formulation of the equations of motion for a massless spin-two field i n terms of a symmetric rank-two tensor and the relation of these equations to the linearized E i n s t e i n equations. Chapter 3 is a review of previous work on the problem of spin-two fields and general covariance. Some of the methods used to derive the Einstein equations as the relativistic  Chapter  1.  3  Introduction  field theory of a spin-two field are outlined. One of the clearest and most general procedures is given by W a l d [74], which is reviewed i n section 3.3 i n some detail: In [74], W a l d found that theories which are reducible to the linearized Einstein equations need not necessarily have general covariance but instead, could simply have "normal" spin-two gauge invariance. He conjectured, however, that if the spin-two field is coupled to matter, then generally covariant theories may be the only possibility for consistent nonlinear spin-two theories.  L i m i t i n g the number of derivatives i n the Lagrangian to two, then, uniquely  gives Einstein's equations.  In our analysis, we use the consistency criteria established  by W a l d i n [74]. F i r s t we extend his analysis to include the possibility of conformally invariant theories. Next we consider alternative formulations of the spin-two equations and find theories of a spin-two field which are not generally covariant and which can include coupling to matter. Before we look at the spin-two problem, we consider a more straightforward problem: In Chapter 4 we investigate the effect that the choice of potential has on the nonlinear extension of a linear theory. In particular, we consider the generalization of a divergenceand curl-free vector field. A s i n the case of spin-two, a Lagrangian formulation of this theory requires introducing potentials.  There are at least two possibilities, namely a  scalar field <j>, or an antisymmetric tensor field A . ab  We compare the types of theories  that result from a consistent nonlinear generalization of the vector theory i n each case. A l t h o u g h we have not solved this problem completely, to second-order, the calculations indicate that nonlinear extensions of the different formulations are inequivalent (the linear theories have been shown to be equivalent under a nonlocal transformation). It is worth noting that an entirely analogous situation arises for the theory of a spin-one field i n three dimensions.  Chapter  1.  4  Introduction  In Chapters 5 and 6 we consider the spin-two problem: We investigate the general2  ization of the classical equations of motion for a non-interacting massless spin-two field propagating i n a flat background space-time.  One possible Lagrangian formulation of  these equations is given by the second-order Einstein-Hilbert action. However, there are other ways of deriving the linear spin-two equations from a Lagrangian. In Chapter 5 we include the possibility of conformal invariance by considering the nonlinear generalization of a spin-one field coupled to a spin-two field. In Chapter 6 we investigate the nonlinear generalization of three alternative formulations of the spin-two equations. This leads to consistent theories which are not generally covariant.  1.1  General relativity  In this section, some of the principle equations of general relativity, Einstein's theory of space, time and gravitation, are given. In general relativity, spacetime is assumed to have the structure of of a four-dimensional manifold w i t h a differentiable metric, g b, of a  Lorentzian signature. T h e matter fields of spacetime are described by a stress-energym o m e n t u m tensor, T b- T h e Einstein field equations relate the curvature of spacetime, a  through the Einstein tensor G bi to the matter content of spacetime, according to a  G b — 8irTab ,  (1.1)  a  where the Einstein tensor G b = R b — \§abR, a  Rb  = g R bcd bd  a  a  Rabcd is the R i e m a n n curvature tensor,  a  is the R i c c i tensor, and R = g R b ab  a  is the R i c c i scalar. In general, the  Einstein tensor is a highly nonlinear function of the metric and its first- and second-order derivatives. Note that the Einstein equations are generally covariant, i.e., "the metric, g b, and a  The results of the calculations presented in Chapters 4, 5 and 6 have previously appeared as research articles [43], [44]. 2  Chapter  1.  5  Introduction  quantities derivable from it are the only spacetime quantities that . . . appear i n the equations . . . " ([73] p. 68). In other words, the Einstein equations are invariant under general coordinate transformations. Also, there is no a priori fixed aspect of the spacetime structure, i.e., there is no non-dynamic background geometry such as a given flat background metric - the metric is dynamic. Taking the trace of the Einstein equations, G  ao  gives g G b ab  a  = R b — \g bR a  = —R = 87rT, S O that (1.1) can be w r i t t e n R  ab  = 87r(T -lg T) ab  — 8irT b,  a  a  3  .  ab  Eq.(l.l),  (1.2)  In the absence of matter fields, the Einstein equations (1.1) reduce to  G  = 0 .  ab  (1.3)  Alternatively, from (1.2) the Einstein equations i n vacuum (aka. the vacuum Einstein equations) can be written R  ab  (1.4)  = 0.  Note that the Einstein tensor, G b, satisfies the ((twice) contracted) B i a n c h i identities  4  a  V G  a 6  a  = 0 .  (1.5)  T h e trace-free part of the R i e m a n n curvature tensor, R bcd, is the W e y l conformal tensor, a  Cabcd, which, i n four dimensions, is given by  Cabcd — Rabcd + 2g[b\[cRd]\a] + \Rda\c9d\b •  (1-6)  This is the form that Einstein originally presented them in [53]. It is interesting that neither Einstein not Hilbert seemed to be aware of this identity in 1915 [53]. In Einstein's Nov. 25, 1915 paper, the condition (1.5) was imposed as a constraint on the equations of general relativity in order to ensure that the energy-momentum tensor T is conserved, i.e., that 3 4  ah  VT a  ab  = 0 ([53] p. 256).  Chapter  1.  Introduction  6  A l m o s t simultaneously w i t h Einstein's formulation of general relativity, Hilbert showed that the equations (1.1) can be derived from the action [53] SEH  =  J  dx  (C  4  G  +  a  M  C  M  )  ,  (1.7)  where the gravitational Lagrangian density CG is given by C  G  =  (1.8)  V ^ R ,  and CM is the matter part of the action. aM is a constant and g is the determinant of the metric, g . ab  T h e action (1.7) is called the Einstein-Hilbert action. Variation of SEH  w i t h respect to the matter fields yields the matter equations of motion. Variation of SEH w i t h respect to g  ah  gives the Einstein field equations C O  £ Hab =  = V=9 (G  E  ab  - 8nT ) = 0 .  (1.9)  ab  In this formulation of general relativity, the stress-energy tensor is defined by T  a  f  e  E E - f i - L ^ . 8ir yj-g  6g  ab  (1.10) v  '  In [53], Pais points out that the two formulations (1.1) and (1.7) of general relativity are not quite equivalent: In writing down the equations (1.1), Einstein did not specify the structure of the stress-energy tensor beyond its conservation and its transformation properties. H i l b e r t , on the other hand, gave a definite form to the stress-energy tensor of matter, namely (1.10). In a letter to W e y l (November 1916), Einstein writes "Hilbert's Ansatz for matter seems childish to me." (pp. 257 - 258 [53]). Nevertheless, i n general, i n curved spacetime, the stress-energy tensor T  ab  is usually defined according to the  prescription given by E q . ( l . l O ) , (cf. [47] p. 504, [77] p. 360, [8] p. 87, [73] p. 455).  1.2  Linearized general relativity  In many systems of physical interest, (e.g., our solar system), the gravitational field is weak. For a theory of gravity that is formulated i n terms of a metric, this means that  Chapter 1.  Introduction  7  the metric, i.e., spacetime, is almost flat. In particular, i n the weak field approximation to general relativity, it is assumed that one can write the metric as gab = Vab + lab ,  where 7 ^ is a ' s m a l l ' deviation from the flat metric, -q . B y ' s m a l l ' is meant that there ao  exists a coordinate system where the components of 7 ^ , i.e., |7 | <C 1, (e.g., i n our solar system, |7 | ~ |$| < MI/  MJ/  7 a u /  , are much less than unity, ~ 1 0 ) . Substituting  MQ/RQ  - 6  the metric (1.11) into the vacuum Einstein equations (1.3) and keeping only terms that are linear i n 7 ^ , give the linearized (vacuum) Einstein equations (i)*b  G  =  R  (i)ab _ l *b (i) v  =  <9 <9y c  (  )c  -  Q ^  =  R  | 7 " - ld d j N  5  a  A  -  b  lv  a b  (d d c  - n )= 0,  d  (1.12)  7  l c d  where the superscript (n) refers to the power of the field(s) appearing i n the quantity.  6  A t this order, indices are raised and lowered by the flat metric, r] . Note, however, that ab  gab _ ^ab  ^ab  T h e linearized Einstein equations (1.12), G^  = 0, can be derived from the gravita-  ab  tional part of the second-order Einstein-Hilbert a c t i o n  4  = -(V=^)  2)  =  where 7 = j  = rj j  a  a  a  ab  a  S  (l)ab  ,  + \l  h  and • = d d .  ab  a  -\lab^l  ( 2 )  =  7  c h  d d  -\jd d  a  c  c l a h  Variation of 6 S $  =  G  (  L  )  A  B  SQ*  =  +  d l  c  d  = / d  4  | °7 ,  (1-13)  7  x  ,  Q  hab  The phrase "the linearized Einstein equations" sometimes refers to the linear equations arising from the perturbation of the Einstein equations about any exact solution of the Einstein equation, not just about the flat spacetime solution. Here, we use the the phrase "the linearized Einstein equations" to refer to perturbations about flat spacetime. In order to obtain (1.12) it is helpful to note that in terms of the Christoffel symbols r i, = \g i.d gbd + d g - d g b) the curvature tensor is written R = 2d[ T + 2r% r ] = R . The minus sign appears (cf. Eq.(1.8)), since variation will be taken with respect to j b (and not y , where g = rj - j ). 5  6  c  ci  d  a  b  ad  d  a  a b c  d  b  a  d  a]c  [a  6  d  e  c b a  7  a  ab  ab  ab  ab  Chapter  1.  8  Introduction  gives the linearized Einstein equations i n vacuum. F r o m (1.4), 0 = R^ so that Eq.(1.12), G  ( 1 ) a 6  1  G  a b  c  d lcd  - D , 7  = 0, can be written  RM T h e quantities £ *  = dd  ab  - |D  = dd j  {a b)d  d  7  - \d d j a  a 6  = 0 .  b  (1.15)  (defined i n Eq.(1.14)) satisfy the (contracted) linearized B i a n c h i  identities d Sg a  )ab  = 0 ,  (1.16)  so that the equations of linearized general relativity (1-12) - (1.15) are invariant under the transformation lab  —» lab + <9( A ) , a  =  6  lab + ^C\c7]  ab  ,  (1-17)  where C\c denotes the L i e derivative w i t h respect to the arbitrary (but small) vector field A . Note that this gauge invariance is simply linearized coordinate invariance i n the a  linearized theory: Consider the infinitesimal coordinate transformation X " -> x'" = x» - \?(x ) v  where £  M  ,  (1.18)  is small, i.e., of the same order as 7 &. Under the infinitesimal coordinate a  transformation (1.18), the metric, g , transforms according to ab  ^ M - < , ( ^ ) =~ f ^  •  (1-19)  Taking the linear part of (1.19) gives 7 ^ = 7 ^ + 5 i^C which is just (1.17) w i t h (  a  +  ,  (1-20)  = A , i.e., the gauge invariance (1.17) of the linearized E i n a  stein equations (1.12) is simply invariance under infinitesimal coordinate transformations.  Chapter 1.  9  Introduction  T h e y are, however, not coordinate invariant. Under general coordinate transformations, the metric rj i n the action is transformed. ao  It is standard to simplify the linearized Einstein equations (1.12), G^ m a k i n g the algebraic change of dynamic variables  ab  = 0, by  8  lab ~* lab = lab ~ \r}abl •  (1.21)  T h e n (1.12) becomes G  (  l  = d <9 7  H  (a  6)c  c  Further, since i n vacuum 0 = R  R  c  -  )d  d  ah  =0.  cd  ]  c  d  + f 0 7 , Eq.(1.15), R^  d  cd  {a  - \r d d i  h  =ddj  {1)  =dd f  (1)ah  - \uf  |D7  a 6  +  \v  a b a  (1.22) ab  = 0, becomes  l =0 •  (1.23)  In terms of 7 , the gauge invariance (1.17) of the linearized theory is c d  lab -> lab + <9(A) - lVabd X .  (1.24)  C  c  Eq.(1.12), G ]j = 0, can be simplified further by taking the arbitrary field A a  be such t h a t  9  D A = -2d i .  Then d l  b  0  to  = 0 and (1.22) (or equivalently, Eq.(1.23))  ab  ab  A  a  becomes Q  T h i s choice of A where a  Xa  A  =0•  7a5  (1.25)  does not completelyfixthe gauge: A  = 0 . This residual gauge freedom j  ab  A  is invariant under A  —> j  ab  A  —• A  + Xa  A  -f d' Xb) ~ \^abdcX  c c  a  nD  e  a  to set 7 = 0 i n regions where T = 0 ([73] pp. 80, 186). ab  Thus, the linearized Einstein equations i n vacuum, G(  1)ab  = 0, after a change of  dynamic variable and gauge fixing, can be written •7 j  8  ab  B t  =0,  <9 7 = 0 ,  is called the trace reverse of j b since j a  a  7 =0.  a6  = —j.  This gives the inverse transformation  lab ~ \rjabl-  This condition can always be satisfied to first-order in ~y i, ([53] p. 280).  9  a  (1.26) j i, = a  u s e (  i  Chapter  1.  Introduction  10  These are precisely the equations that were written by Fierz [32], and Fierz and P a u l i [33], i n 1939, for the equations of motion for a spin-two field, where the spin-two field is represented by  1.3  j. ab  Summary  T h i s completes our brief introduction to general relativity. T h e Einstein equations are given by E q . ( l . l ) , G b = 8irT , which are derivable from the Einstein-Hilbert action (1.7). a  ab  T h e quantity, G b, which appears on the L H S of the Einstein equations, is called the E i n a  stein tensor and, as a consequence of the Bianchi identities, is identically divergenceless, \7 G a  ab  = 0. T h e linearized approximation to gravity was obtained by perturbing the  metric about flat spacetime, g  ab  under the transformation j  ab  = rj + 7a6, E q . ( l . l l ) . The linearized theory is invariant ab  —> j  ab  + d( \b), Eq.(1.17). F i x i n g A , one can write the a  a  linearized E i n s t e i n equations i n the form given i n Eq.(1.26). Since this corresponds w i t h the equations of motion for a spin-two field written by Fierz [32], general relativity may be regarded as the theory of an interacting massless spin-two field ([73] p. 76). However, there are many ways of formulating the equations of motion for a spin-two field. In the next chapter we examine some of these possibilities.  Chapter 2  The Spin-5 Equations  In order to be able to determine the types of theories that result from the consistent nonlinear generalization of the spin-two equations of motion, it is first necessary to carefully consider the non-interacting theory itself. In this chapter, the equations of motion for a massless spin-5 field are given and various properties of the equations are discussed. Particular attention is paid to the potentials that can be used to describe the spin-s fields, and various possibilities are mentioned. T h e specific examples of spin-one and spin-two are investigated i n more detail. In section 2.4, i n order to clarify the relation between the linearized E i n s t e i n equations and the equations of motion for a spin-two field, the usual formulation of the spin-two equations i n terms of a symmetric rank-two tensor potential is derived. T h e chapter ends w i t h a discussion of the usual Lagrangian formulation of the spin-two equations as originally derived by Fierz and Pauli i n 1939 [33], and their argument is reviewed. For the general discussion of the spin-5 equations the spinor formalism is used. T h e p r i m a r y reason for this is to make contact w i t h much of the literature on massless spin-5 fields.  A l s o , since the spinor form of the spin-s equations are linear, it is often more  straightforward to deduce general properties of the spin-5 equations i n the spinor form a l i s m rather than i n the tensor formalism [57] (see A p p e n d i x B for more on spinors). M u c h of the material on spinors that is included i n this chapter can be found i n Penrose and R i n d l e r (1984) [57], (1986) [58]. See also W a l d (1984) [73] as well as other references given i n A p p e n d i x A and A p p e n d i x B .  11  Chapter 2.  2.1  The Spins  Equations  12  Spin and representations of the Poincare group  T h e laws of physics i n M i n k o w s k i spacetime are believed to be invariant under translations, rotations and boosts of spacetime, i.e., under the proper Lorentz transformations (LT).  1  For a physical theory that is defined on M i n k o w s k i spacetime this means that  these spacetime transformations induce transformations of the physical states of the theory [73]. These transformations are represented by linear transformations of a vector space, i.e., by matrices, and the elements of the vector space, called the representation space, are then the mathematical representations of the states of the theory. T h e possible physical states of a relativistic theory then have well defined transformation properties under a representation of the Poincare group ( P G ) . There are two P G "invariants", m  2  and S , w h i c h can be interpreted physically as the squared mass and the squared angular 2  m o m e n t u m about the centre of mass [73]. In any irreducible representation, these invariants are multiples of the identity and hence characterize the irreducible representation. A physical field of mass m and spin s is then an element of the vector space that is acted on by an irreducible representation of the P G , i.e., a spin-5 mass-m field is a quantity that transforms according to the irreducible representation of the P G characterized by m and s. B a r g m a n n and Wigner (1948) [4], have given for each irreducible representation a differential equation the solution of which transforms according to that representation. These differential equations are consequently called the spin-5 equations. There are many equivalent ways of writing the spin-5 equations. M u c h of the literature uses the spinor form of the equations given i n the next section.  See Appendix A for definitions, more detail, and references.  1  Chapter 2.  2.2  The Spins  Equations  13  The spin-s equations  A pure zero-rest-mass field of spin-s i n flat spacetime can be represented by a totally symmetric spinor field  2  w i t h n = 2s indices that, for s > 0, is a solution to  $A A ...A X  2  VL  ([25], [4], [59], [56]) d '^A ...A ^0.  (2.1)  MA  lA2  n  These are the free field equations for a massless s p i n - | field i n flat spacetime. Equiva3  lently, Eq.(2.1) can be w r i t t e n  4  dB<B$A A ...A x  2  = dB'A\(f>BA2...A •  n  (2.2)  n  Note that the s p i n - | equations (2.1) i m p l y that the field <j>A A ,...,An is also a solution to 1  the wave e q u a t i o n  5  Vcf> A ...A Al  where • =  2  2  =0,  n  (2.3)  There are three additional features of the s p i n - | equations (2.1)  3AA'Q '• AA  that are important to point out: (i) the s p i n - | equations (2.1) can be regarded as being conformally invariant [58], (ii) there is no "natural" generalization of these equations to  The simplest kind of spinor K , a spin-vector, is an element of a two-dimensional complex vector space called spin-space. The complex conjugate of K , K is denoted K . Indices can be raised or lowered by the antisymmetric spinor epsilon (.AB- A spin transformation A B, i-e., n —* A gK , is a unimodular two-dimensional complex matrix. More complicated spinors may be built up from a spinspace analogously to the way tensors are constructed from a vector space. Tensors in a Minkowski vector space are a special case of spinors. The association between tensors and spinors is made via the Infeldvan der Waerden symbols <r (which are usually taken to be proportional to the Pauli matrices). For example V T = <r a BB ^'AA'T . For a more detailed discussion, see Appendix B. See Wald (1984) [73] Ch.13 for a detailed discussion of how the invariance of a physical theory under proper LT can be used to derive/motivate the type of quantity that can represent physical fields and equations that the physical fields obey. To see this, write Eq.(2.1) as e V ^<t>A A ...A = 0 and take into account the antisymmetry of the spinor epsilon e . To see this, note that dc'A^B is antisymmetric in AB, sinceflatspace derivative operators commute: d 'Ad ' = d ' dAC> = -dc> d ' = d >[Ad ] '• Then by (B.18), d -Ad ' = \e ^- Thus, acting on (2.1) with d ' gives d ' d '^ A ...A - \& D^.^-.i, = \^<t>BA^...A = 0. 2  A  A  A  A  A  A  A  AA  a  AA  b  b  1  BB  a  a  3  4  ALB  A  1  2  n  AlB  5  C  C  C  B  c  c  B  B  C  A  c  A lB  A lB  c  B  c  B  AB  Ai  AlA  Ai  2  n  n  B  Chapter 2.  The Spins  14  Equations  non-flat spacetimes [73], [57], (iii) there are many possible ways to describe a spin-5 field i n terms of potentials [56],[58].  2.2.1  Conformal invariance of the spin-s equations  The field equations, JF = 0, for a field <f> are said to be conformally invariant if under the conformal transformation 9ab -+ 9ab - ^ 9ab ,  (2-4)  2  (j)-^4> = n <j>,  (2.5)  '  w  the field equations, T = 0, maintain the same form; more precisely, the field equations are conformally invariant if the following statement is true: T = 0 if and only if T = 0, where T = J-{g bi <f>), and T = !F(g b, (f>). Another way of stating this is to say that there a  a  exists a real number u>, called the weight of the quantity, such that f  = Q^f.  Here,  fl is an arbitrary real, positive, scalar field. T h e s p i n - | equations w i l l be conformally invariant, then, if V^<^ ... 2  A  =0  n  where <f>A A ...A — Q <f>A A ...A f ° w  1  2  n  1  2  (2.6) is true, by expressing V  V ^4> ... AlA2  An  One can show that the statement  ' i 4 > A . . . A i n terms of (j> ,„ Al  2  (2.6)  = 0 ,  A  some real w.  r  n  Alj4  <-  AlA2  n  and  An  Referring  S7BB'-  to A p p e n d i x B on spinors and A p p e n d i x C on conformal transformations, the spinor equivalent to the metric conformal transformation (2.4) is given by E q . ( C . 6 ) , e  AB  e  AB  = fl- e , x  —>  so that  AB  v '4MA ... MA  2  An  =  v ^ i  =  e  A l S  (n <f> ... ) w  AlA2  An  e^ 'n- (O V B  2  =  w  B B  ^  l ^ 'V B,{Sl <t>MA ...A ) MB  B  W  B  l A 2  ...^ + V  2  B  S  ^ ^  M  2  . . .  n  A  n  )  , (2.7)  where the derivative operator only acts on the quantity immediately following unless brackets indicate otherwise. T h e relation between two derivative operators is given by  Chapter 2.  The Spins  Eq.(C.ll), (2.7)  VAA'TB  =  15  Equations  VAA>TB  ~  where from (C.9),  ^A'BTA,  = ^ V 1  T A> A  y t i 4  /0.  Thus  becomes  V  A  ^  A  M  2  . . . A  =  N  ^ t ^'Vl- [n {V B4MA ...A -^B' ABA ...A B A  2  W  B  -  ^B'A <t>A BA ...A 2  L  Z  n  )  1  A  2  N  2  1  2  N  L  •  1  BB  N  B'A 4>A A ...A - B)  N  + ion - S7 ,ttcj A A ...A \ w  2  N  (2.8)  Note that a l l but the first two and the last terms are zero since they are symmetric i n A and  B but contracted w i t h the antisymmetric e .  Substituting i n (C.9), £l~ 'V >Q  AlB  1  BB  :  =  Eq.(2.8) becomes  TBB'I  y  A  L  A  ' ^  A  A  L  2  . . . A  N  =  ft-  {V ^cj> A ...A  2  +  A  M  2  N  T ^ci> A ...A  +  A  B  2  N  wT '^ <}>BA ...A ) A  B  2  N  •  (2.9)  Taking w = — 1, the last two terms cancel so that (2.9) reduces to  V  A  L  A  ' ^ M A  2  . . . A  N  =  V l - ^  M  A  ' ^ A  L  A  2  . . . A  N  T h i s shows that the massless spin-5 equations, \7 'i(f>A A ...A spinor  (f>A A ...A 1  2  N  2  (2.10)  N  = 0, w i t h the symmetric  ,  (2-11)  AlA  1  •  a field of weight w = — 1, .  4> A ...A AI  2  =  N  ft~ <f>A A ...A 1  1  2  N  are conformally invariant. Note that the wave equation (2.3) satisfied by the spin-5 field 6  is also conformally invariant.  7  Here, we only consider conformal transformations between flat metrics so that  6  r >r , AB  [58] p. 124.  BAI  VAA'TBB'  —  In [11], Bracken and Jessup discuss the conformal invariance of the wave equation (2.3). They show that "wave equations satisfied by free massless fields are not in general locally conformally-invariant". Additional conditions need to be satisfied. However, here we do not regard the wave equation (2.3) as being the defining equation for a spin-n particle, but rather the massless spin-n equations (2.1). Then the wave equation (2.3) for a massless spin-n particle, i.e., a field (f> i— i that satisfies (2.1), is conformally invariant. For example, Bracken and Jessup show that in order for the wave equation for some field AiA jformally invariant, then T must be a spin-one field, i.e., it must satisfy Maxwell's equations. 7  A  A  T  2  0  con  I  A  2  A  Chapter 2. The Spins  Equations  16  If the Einstein equations i n vacuum hold, i.e., i f R of the B i a n c h i identities,  A  = 0, the spinor equivalent  B  = 0, is given by Eq.(B.43),  V[ i2&c]de a  ^  A  = 0, (cf.  ' ty ABCD  A  A p p e n d i x B ) . These correspond to the equations of motion for a spin-two field, (2.1), so that the W e y l conformal spinor, equations  V 'tyABCD  represents a spin-two field. However, the  0 are not conformally invariant since  —  AA  tyABCD-,  conformal weight zero, i.e.,  tyABCD  = ty ABCD  tyABCD  is a spinor of  (see A p p e n d i x C ) . In other words,  tyABCD  may not be the most general possible representation of a spin-two field.  2.2.2  Generalization to non-flat spacetimes — consistency condition  Relativistic equations can usually be generalized to curved spacetimes by replacing the flat metric, r] b, by the curved metric, g , and by replacing the flat derivative operator, a  ao  <9, by the derivative operator, V , associated w i t h g b, [73]. However, this procedure of a  a  a  "naturally generalizing" the flat s p i n - | (2.1) does not i n general yield satisfactory field equations for s > 1 ([12], [13], [61], [57], [73]). This is due to an algebraic consistency condition that relates the field  to the W e y l curvature spinor  4> ...A„ Al  tyABCD-  the algebraic constraints arise, act on the "curved" s p i n - | equations withV  A i  F r o m (B.32), B  (f>A ...A 1  B  bmce A  =  0  X  ^  A  Q the  fi  5  =  rst  X  B  N  =0  ^  A  w  o  ABA ...A  A  A <j>A A BA ...A 1  2  2  n  s  + x  + •••+ X  i  n  H  h  Contracting  D  A  A  ^  > 1, (2.12) becomes  ^  B  B  j M . . . A  A  AAMBM..^  n  ^ B  terms on the R H S are zero. Therefore  AlA2B  3  B  (2.12)  + §A'B'C £AB)UD-  D  A  T h e n , for  A J M A . B M . . ^  t  (X BC £A>B<  D  X  .  A  = XABC ^D-  VA'(A^'B) W  +  A  ... J= V ^ V ^ K ^ . . . ^ =  V ^ V ^ K ^ , . . ^ =  X (BC)  A l  (VAA'^BB'-^BB'^AA')^C  gives  e'' A  AlA  ,  A 2  0 = V^(V^</>  with  \7 'i  To see how  X  ALA2B  An^Ai-.An-iB  -  (2.13)  Chapter  2. The Spins  17  Equations  =  fiA A2B(A ...A X  A)  I  N  3  Since only the symmetric part of XABCD the W e y l curvature spinor ty  ABCD  =  (2-14)  •  ALA2B  1  appears i n (2.14),  XABCD  c  a  n D  e  replaced by  This gives the following condition for  X(ABCD)-  <t A ...A )  1  N  0 = <f>A A B{M..jJ*A*)  •  AlMB  l  i  ( -!5) 2  T h i s algebraic condition is sometimes referred to as the Buchdahl-Plebanski constraint ([12], [13], [61]). It can be shown that as a result of the constraint (2.15), few i f any solutions to the natural curved spacetime generalization of the spin-s equations (2.1) exist ([73], [57]). However, for a spin-two field, there is one well known solution to (2.15), namely the vacuum E i n s t e i n equations. T h e B i a n c h i identity for the vacuum Einstein equations is S7 'tyABCD AA  =  0, E q . ( B . 4 3 ) . Since tyABCD is completely symmetric, this means that it  represents a spin-two field. Taking tyABCD for <I>ABCD i n (2.15), the consistency condition (2.15) becomes  tyABC(DtyE)  ABC  —  0, which is automatically satisfied. Thus the vacuum 8  Einstein equations can be thought of as a (consistent) curved spacetime spin-two theory. See also Refs.[2], [3], [72], [42] and [30] for more specific difficulties that arise when attempting to construct, i n particular, interacting spin-two theories.  2.2.3  Potentials for the spin-s fields  It is often necessary to introduce potentials to describe the spin-s fields, for example, i n order to be able to formulate the equations i n terms of an action principle. Since the solution  °f the s p i n - | equations represents a pure s p i n - | field, it is a  4>A A -A X  2  N  gauge independent quantity. However, gauge transformations arise when potentials are introduced. There are many possible potentials that can be used to describe a spin-s field. 8  To see that  •*ABCE-*D  ABC  ^ABCD^E  + *ABCE*D  ABC  ABC  = -^ BCE^AD A  BC  = ^'CB*ABD°  = 0,  note that the second term can be written  = -*  A  B  C  ABCD •  Chapter  2.  The Spins  Equations  18  Penrose (1965) [56] has shown that one can find, at least locally, a completely symmetric spinor field 4 ' '" ' ,A 1  A k  such that  AI...A„  d ''iJ> i"- '*A ...A AtA  A  = 0,  A  l  n  (2.16)  and such that the field (f> ...A defined by Al  n  <j>A ...A =d ...d ^--- '* ...  ,  A  1  n  AlA[  AkA  Al  An  (2.17)  is completely symmetric and a solution to the s p i n - | equations (2.1). Thus is a potential for the s p i n - | field (f> ... Al  pletely symmetric, rewrite (2.16) as e '' ' e A  dB'Bip ' "' ' A ...A A 1  - d i i> ^- 'k „,  A k  l  A  n  B  B  Al  AlB  d itp ^^ ' A  A k  BB  Al  ...  An  A 1  An  A k  A  A  Al  An  so defined is com-  = 0. T h i s implies that  An  = 0, so that d ' A il> ' "' ' Ai...A  A  B Al  T o see that <f) ...  An  ij) 'i- 'k „,  is symmetric i n A  A k  k  n  k  and Ai and hence all unprimed indices. Since flat space derivatives commute, repetition of this argument shows that field <j> ... Al  defined by (2.17) is indeed completely symmetric.  An  To see that i t satisfies the s p i n - | equations, note first that (2.16) shows that if; 'i-~ 'k ___ A  A  Al  satisfies the wave equation ^tp ' '" ' ... A 1  d ^<f> ... A  Al  = d ^d A  An  A l B l  ...d  0. Thus, the field <j> ... Al  — 0. A c t i n g on (2.17) w i t h d * then gives  A k  Al  A k A  ^  B , A  AlA  An  '^ ' A  k A l  ...  = \d  A n  defined by (2.17) w i t h • ^ ' ' " ' A  An  An  1  A  k Al  M A l 2  ...  An  ...d  A k A  ,n^---K ^ A  A n  =  given by (2.16) is a s p i n - f  field. T h e definition (2.17) is not effected by certain gauge transformations. T h i s is because the potential ip "' Al  Ak Al  ...  describes a pure spin field i n terms of a mixture of spins, i.e.,  An  transformations of ip 'i"' ' ... A  A k  Al  An  which effect the lower spin parts may not effect (2.17).  For example, for s > 1, the definition <j> ... Ai  the transformation ij} '^ A ...A A  2  ~^ "fy ' A ...A A x  n  2  n  = Q A\i> ' A ...A  is unchanged under  A x  An  +d  Al  A<1  2  A ^A ...A i 2  z  n  n  where A ^ . . . ^ satisfies the  spin-(s — 1) equations. Let us consider some specific examples of potentials defined by (2.16) and (2.17). A spin-one field (j)  AB  can be described i n terms of the potential ip B A>  by d 'ip 'B BB  A  = 0,  Chapter 2. The Spins  where  19  Equations  This is not effected by the transformation  <J>AB — QAA ^'B1  where DA = 0. A spin-two field  d ' ^, A  B  (J>ABCD  such that  A  ^ 'BCD  <J>ABCD — ^AA'i>  + d '  A  A  0, where D  where  B  =  where  A  c  where  \ c,  4>ABCD  d ' A ' , A  D  d  d  B B  'k c  B  D  A  A  A B C  s  c  d B>A ' '  B  AA  d  D  ij) 'BCD '^ ' ' D A  B  C  d '^ ' ' ' AA  B  —>  A  A  A  B  c  B  = +  = 0,  c  D  —>  ip ' ' 'u A  A A  =  A  —> i> ' 'cD  B  such that  C  with the gauge invariance  4>ABCD = dAA'dBB'dcC^ ' ' 'D  a % A ' ' , where  such that  A B  0, (iii) and by ^ ' ' '  =  d ' ip 'BCD  The gauge invariance is ip ' 'cD  A B  A  'K '  0, (ii) by ^ ' 'CD  =  B  d A>^BB'i> ' 'CDD D  A  These definitions are unchanged under  BCD-  A  ~^ ^ 'B +  A  can be described in terms of potentials  that satisfy (2.16) and (2.17) in three ways: (i) by ij) 'BCD 0, where  ip 'B  ip ' ' 'D  +  A B c  = 0.  B C  B  A spin-s field can also be described in terms of potentials that are more general (but more complicated to work with) than the potential (2.17) [58]. For example, let ip 2--- 'n A  d * '^ '- ' A  A  A  Al  n  =0 .  A n)  Al  Then  A  be a completely symmetric spinor such that  A  Ai  A  defined by (2.16) and  ip 'i--- 'k ...A i  (2.18)  defined by  4>A!...A„  <f> ... Al  = d  An  A {A  *  - -.d i>A )A>...A>  ,  K  An  1  N  (2.19)  is a spin-1 field. The symmetrization in (2.18) provides a less stringent condition on the potential  than its unsymmetrized counterpart (this will become clear below.)  tpA '" ' A2  A n  1  For example, (2.18) and (2.19) are conformally invariant (in flat space) while (2.16) is not. Note that Eq.(2.19) is not effected by the transformation d ( ' h ' '" ' * A 2  A 3  A  To see that  MA  AL  A 2  —> ipA ' '" '  A n  A 2  1  A n  1  for arbitrary symmetric A. 3~- 'i.  A n  Al  d '^ ...A  ipA ' "' '  $A ...A X  a  s  N  defined in (2.19) satisfies the s p i n - | equations consider  = \d ^ A  N  + ••• + d ' A  A  2  A  [d ' A  A  2  - - • d '^ A' ...A>  - - - OA^/'^A^A'..^  + d '  A  An  Al  2  A  N  --•  2  A  + d ' ... A  Al  2  d '^A A>...A' A  AI  d „ ' i>A Ai ...A> ] A n  An  n  2  n  2  N  ,  +  Chapter 2.  (  -  i  The Spins  )  n  -  1  a  ^  .  .  20  Equations  d  .  A  n  d  K  A  ^  ' ^  A A  l  - \ u e ^ . . . d <-^ A  lnd 'id <  K  ...d _ '^ _ ^ ...  A  An  ,... _^  A 2 A  A  A  An  2  An  lA  Ali  K  - \ u d ^ ... d _ '^ ^ , ... \  . (2.20)  A  A  An  An  A  3  Aln  Consider now just the last term: - \ud *>. M  ..d _ '^ '^ ,... , A  = -\d >.  A  An  An  A  A  Substituting § E < ^ = d ' d >  A  n  -cV> • • •  (2.21)'  d ^ - i d j ^ d * ^ ^ ^ . ^  = {-\) - d n  .  into (2.21) and relabeling gives  AlA n  AnA  . . d ^ n s j y ^ ^ ,  A  n  .. . d  l  A2K  A n K  d  ^  A  A A  '^-  '^  A  .  (2.22)  Repeating for similar terms i n (2.20), Eq.(2.20) becomes d '^ ... MA  Al  An  =( - l ) " "  1  ^ .. . d  , d ^ /^- ^ A  A n A  A  which equals zero from (2.18). Thus the field ip 'f" ' A  ,  A  n  (2.23)  is a potential for a s p i n - | field  A n  Al  4>A ...A 1  N  Moreover, Eq.(2.18) is conformally invariant: ^MA[^ K...K) A  =  ^  =  rr-V  i B  {  W  T B  {  A  H A  - f T B c ^ ^ ^  1  '  A  0  2  '  B  1  B  K  ^  )  W  ^  A  A' ...A'  + V  2  [ B  A  N  H A  )  A  ^ - ^ + • ••+T  ^ B B , ^ - " ^ ]  +  * ~  B  K  C  )  ^  -  ?  ^  A  { A  I  ^  A  ' ^ B  ^  )  A  0  ,  > -  '  \  K  )  , (2.24)  where a l l but the first three terms i n the last equality are zero. Taking w = — 1 the first and t h i r d t e r m cancel showing that the equations (2.18) are conformally invariant.  2.3  Tensor equivalent of the spin-one and spin-two equations  H a v i n g established some general properties of the spin-s equations i n terms of spinors, it is important to translate some of the results into tensor notation.  Following, the  equivalent tensor expressions for the spin-one and spin-two fields are given.  Chapter 2. The Spins  2.3.1  Equations  21  Spin-one  To find the tensor equivalent to the symmetric spinor §AB-, referring to ( B . l l ) and the discussion preceding (B.24), multiply § B by 1A'B> (to obtain the paired (tensor) index A  combinations  AA',  and add to  BB')  <J>AB"^A>B>  its complex conjugate. This gives the real  spinor =  FABA'B'  Referring to (B.22), F  4>AB~eA'B<  +  <j>A'B' AB E  is an antisymmetric tensor and  ab  <"» Fab •  (2.25) =  FABC' ' C  2<J>AB-  TO find the  tensor expression for the spin-one equations of motion, d ' 4>AB = 0, take the divergence AA  of  and its dual,  FABA'B'  =  *F BA'B> A  d 'FAA>BB>  = d ,<i>AB  AA  b^'*F . B-  A  B  + d '$ ,  <- d F ,  A  B  A  Setting (2.27) to zero gives  a  dF .  A  =  d B'4'AB A  (2.26)  ab  A BI  = -id B><l>AB + idB '4*B>  AA B  gives d '(j)AB  ^FABB'A',  [a  (2.27)  bc]  Taking i n addition (2.26) set to zero  9B '<f>A'B'A  = 0. Thus one sees that the tensor equivalent to the spin-one equations  A  B  are M a x w e l l ' s equations dF  =0  d[ F  =0  ab  d ' (j>AB = 0 ^ {  a  •  AA  a  bc]  .  (2.28)  F r o m (2.18), a spin-one field can also be described i n terms of the potential where d ^ 'ipA '* A  potential  A  = 0, and  B  IJ)AA>,  <J> B — ^ \ ^ A)B>B  A  )  B  one need only add to  A A<  =  A  I\)AA'  ip i AA  <"»• a ,  (2.29)  A  which is the familiar electromagnetic vector potential. Substituting into (2.25) (and making use of d ^ 'ij)A '* A  gauge invariance  ip A' A  —» 4>AA>  A  B  + d A'h A  A  TO obtain the tensor equivalent to the  its complex conjugate  ^AA> + ^AA'  ij) A',  <f> B A  = 0) then gives the usual F a  B  b  —> A + d \. a  d '\B^A)B'  ~ 2d[ A ]. T h e  ab  corresponds to A  =  a  a  T h e potential  Chapter 2. The Spins  I\>AA>  22  Equations  that is defined by (2.16),  (/>AB =  and (2.17),  d B^PAB', B>  d 'ipA ' AA  =  B  0, for a spin-  one field, can now be shown to be less general than the corresponding potentials w i t h symmetrizations: Lowering the B' index of d referring to (2.29), corresponds to d A  AA  'ipA '  =  B  0, gives  = 0, which,  d 'ipAA' AA  = 0, so that the unsymrnetrized potential (2.16)  a  a  corresponds to electromagnetism i n the Lorenz gauge. Hence, also DA = 0.  2.3.2  Spin-two  Similarly, the tensor equivalent to the symmetric spinor  4>ABCD,  (2.17) w i t h n = 4, can be found by multiplying  by IA'B'^C'D'  4>ABCD  defined by (2.16) and and then adding  this to its complex conjugate: =  KAA'BB'CC'DD'  ^ABCD^A'B'^C'D'  Kabcd  + <f>A'B'C'D' ABCCD E  F r o m A p p e n d i x B , this is precisely the spinor equivalent to a rank-four tensor has the symmetries ^(J>ABCD-  d  K( ) ab  =  cd  K  )  =  abl<cd  K[abc]d = K  = 0, where  a b a d  •  (2.30) which  K bcd, a  =  K BCDA' 'C ' A  C  A  Taking the divergence of (2.30) gives  AA  KAA'BB'CC'DD  =  1  Q B'<f>ABCD^CD'  + &B  A  ^A'B'C'D^CD  A  d K a  ^  a b c d  (2.31)  .  E q u a t i n g this to zero (considering parts symmetric and anti-symmetric i n CD) give the spin-two equations of motion d '<i>ABCD A  =  B  0  <->  d K  (2.32)  = 0.  a  a b c d  Note that, unlike i n the spin-one case, there is only one tensor equation corresponding to the spin-two equations. first note that 9  To see that  -^abc e  f 9 h m  d {\c^  K  m  s  h m  de  = 0 *->  d[ K ]  d K  g h d e  bc  a  bc  de  = -\e  f  g h d e  )  = ±et> d K , hm  u  gh]de  d *K  abcd  dj{\e^ K [57].  a  a  *K  c  de  a b c d  =  g h d e  )  = -\e  a b c m  a  a  0, where  d *I<i s  = 0. T o see this,  d K bcd  = 0, consider the following:  a  m  a b c m  = 0  d[ I(h }  = 0 is equivalent to d  d[ K \ a  T h i s is because  m d e  *K bcd a  d[ K ] a  .  bc  de  =  \e  a b e l  K  e j  = 6{ 6 6^djK  c d  .h  9  =  9  a  b  Conversely d *Rf f  m de  ghde  =  Chapter  2.  The  Spins  23  Equations  Then o\ Kbc]de  = — ^B'^ABCD^C'D'  * KAA'BB'CC'DD'  a  E q u a t i n g this to zero is equivalent to equating  +  <f>A'B'CD' CD €  to zero.  d 'KAA'BB'CC'DD' AA  Thus a spin-two field can be represented by a rank-four tensor symmetries  K( b)cd =  K b{cd)  a  equivalently,  a  d[ Kb ]de a  —  c  =  K[abc]d =  K  a b a d  tensor,  R bcd, a  4>ABCD  =  a  0- For example, the W e y l conformal tensor,  c  dK d  =  a  abc  0, or  can represent  C bcd, a  — 0, the R i e m a n n curvature  ab  also represents a spin-two field.  To obtain the tensor equivalent to the potential and  which has the  Kbd  = 0, and which satisfies  a spin-two field. If the vacuum Einstein equations hold, R  (2.33)  •  where  i)AA'BB'i  from Eqs.(2.16) and (2.17), add to  dAA'dBB'ty ' 'CD-, A B  d'  ifi ' ' CD  cc  A  B  = 0  its complex  ipAA'BB'  conjugate h-AA'BB' = IpAA'BB' + ty AA'BB'  where with  h^  b  is a symmetric traceless rank two tensor. gives  IA'C'ZB'D'I  ^i^AA'BB'  —  0, D^a6  = 0.  d 'tyccDD' cc  ^  ^Ib •>  (2.34)  Contracting  10  T h i s corresponds to  dh" a  a  b  d'  ty ' '  cc  A  B  = 0.  CD  —  0  Also, since  0- Thus, the tensor equivalent to the spin-two equations where  =  the spin-two field is represented by the potential  according to Eqs.(2.16) and  ty A'BB> A  (2.17) d d  AA  <PABCD  =  c  c  ' V  ty>AA'BB' +  = o  ' c D  d«h  , , <j>ABCD =  where  B  0 A  CD  B  ab  = o  {  J  {  . ° K  B  =  (2-35)  0  This is the form of the equations of motion for a spin-  ^ab-  AA<BB>  QAA'QBB4  }  T  two field that were written by Fierz i n 1939 [32]. They are precisely the gauge fixed It is clear that the tensor t corresponding to a spinor TAA'BB that is symmetric T(AB)(A'B') must be symmetric and traceless. Conversely, to show that the spinor T >BB that corresponds to the symmetric and traceless tensor t must be symmetric, note that: t = tba *~* AA'BB' = TBB'AA'- Contracting the spinor part with e gives e T IBB' — —£ AB<BA' so that e T >BB' is antisymmetric in A'B'. However, in order that g t b = 0 <-> e e T I I = 0, e T A'BB' must be zero. This implies that AA'BB' is symmetric in AB [57]. 10  1  ab  J  AA  T  ab  ab  AB  AB  ABT  AB  AA  ab  T  AA  AB  a  A  B  AB  AA  BB  A  Chapter  2.  The  Spins  24  Equations  equations for linearized gravity (1.26). F r o m the discussion of section 2.2.3, it is clear, however, that there are more general choices for potentials to describe a spin-two field. To find out how to write dAA'dBB>ty ' 'cD  + d 'dBB'tp  A B  aoca  AB  CD  AA  consider the spinor  i n terms of  K >  =  b  since  d A'dBB'h ' 'CD, A B  =  A  =  = 0. Now  d "if> 'D> cc  (J>ABCD  AB  C  defined by  KAA'BB'CC'DD'  ^AA'BB'CC'DD'  from (2.34) note that  h^ ,  ~[pAA'^DD'^CC  + dBB'dcC'hDD'AA'  ~  BB< ~  ^BB'^DD'^CCAA'  <-> ~4<9[ <9|[ /l^| ] ,  QAA'^CC'^-DDCBB')  a  d  (2.36)  b  which, due to its symmetries, can be written as, (cf. Eqs.(B.22) and (B.30)), KAA'BB'CC'DD'  where  —  XABCD  =  —  0(A  =  9{A  A'ct  9  A  is simply  B B  h D)A'B>  ,  ty>CD)A'B'  +  C  the tensor potential  h^  T h e transformation to  2.4  hl  b  ->  h% + d( \ ), b  a  h  b  gives  c D  0  A  A  =  B  tycD)A'B'  B  dAA'dBB'ty '  A B  =  CD  &ABCD  (2.38)  •  i n (2.30)), shows that  Therefore, from (2.36),  Kabcd =  ipAA'BB'  d(  4>ABCD  KAA'BB'CC'DD'-  Thus, the potential  (2.37)  ,  E  B'r  OB  C o m p a r i n g this w i t h Eq.(2.30) (putting KAA'BB'CC'DD'  X.A'B'C'D' AB^CD  A B  C  o XABCD  +  Contracting (2.37) w i t h e ' 'e ' ',  \KABCDA 'C ''• A  XABCD^A'B'^C'D'  -4<9 d| /^ [a  [d  .  w  (2.39)  defined according to Eqs.(2.16) and (2.17), corresponds to  related to the spin-two field —• TAA'BB'  TAA'BB  1  where  D\  b  =  0 and  + QAA dX  A  A  1  K  ^-BB  1  a b a  i  according to (2.39).  , where  d ' AA  KAB  1  0, corresponds  —  = 0.  The usual symmetric rank-two tensor potential for a spin-two field  Let us return to the question of potentials for a spin-two field from the tensor point of view. F r o m (2.32) a tensor  K bcd a  which has the symmetries  K( b)cd = K b( d) a  a  c  = K[ bc]d — a  Chapter  2.  The  Spins  25  Equations  = 0, and which satisfies d[ Kbc]de = 0>  K°bcd  c  a  be taken to represent a spin-two field. We  n  a  have already seen i n section 2.3.2 that one way of choosing potentials showed that K bcd a  could be written i n terms of a traceless symmetric tensor h^ as K bcd = b  ~^d[ d\[ h^ ^  a  a  d  b  Eq.(2.39). It is useful to see how this expression arises from the tensor point of view and to find the resulting spin-two equations of motion. To represent a tensor K bcd which has the symmetries K ^ a  and which satisfies d[ K ] a  K(ab)cd  bc  —  de  c  d  = K b(cd) — K[abc}d a  = 0,  0, consider each of the symmetries of K bcd i n turn: a  = 0 and d[ K b]cd — 0, imply that there exists a rank-three tensor T bc such that e  a  a  Kabcd  since d[ d Tb]cd e  a  [a  ,  b]cd  (2.40)  = 0. Note that Eq.(2.40) is invariant under the transformation T b —> a  a  a  a  c  0, give the following two equations, respectively, that the potential T  K abc]d —  c  where v b is an arbitrary rank-two tensor. The symmetries K b( d) — 0 and  T 6 c + d Vbc a  = dT  abc  must  satisfy dialed)  = 0 ,  (2.41)  d r  = 0 .  (2.42)  [a  bc]d  W r i t i n g T ;, i n terms of its symmetric and its antisymmetric parts, T b a  c  a  where S bc = T ( j ) , and A a  a  d[ T c] a  b  = \(d[ Sbc]d  d  a  c  bc  = ^(5 6 + a  C  A b ), a  c  = T[ &] , and substituting this into (2.42), gives 0 = a  abc  c  Since d[ S ]  + d[ A ] ). a  c  d  a  dA [a  bc  is zero, this gives  d  = 0 ,  bc]d  (2.43)  so that one can introduce the potential E](, defined by a  Aabc = d E [a  ,  b]c  (2.44)  since d[ db]^ d = 0. Note that 3 b is invariant under the transformation a  c  a  Eab -> S  a ! )  + d\  a b  ,  (2.45)  Chapter  2.  The Spins  Equations  26  where A is an arbitrary one-form field. a  Returning to Eq.(2.41), (2.41) implies that one can introduce the symmetric rank-two tensor fl  by  ao  T (6c) = a  since d[ d ]fl d a  b  =  c  d Cl a  ,  bc  (2.46)  0. E x p a n d i n g out Eq.(2.46)  -d n a  = -\(T  bc  +T ) ,  abc  (2.47)  acb  w r i t i n g cyclical permutations  d n c  = l(r  a b  + r  c a b  c b a  (2.48)  ),  and adding the preceding three equations, yields  db^ca + d flba — d flbc c  Writing T  abc  i n terms of S b a  a  and A  c  = T( ) + T[ ]j — 6c  a  (2.49)  T[ba]c •  ca  gives  abc  Sabc = dbflca + d flba — d Vlbc + A bc c  a  a  + A j, .  (2.50)  ac  C o m b i n i n g results, from (2.40)  Kabcd  =  \{d Tbcd  — dbTacd) ,  —  \{d Sbcd  — dbSacd + d Ab d  a  Substituting i n (2.50) for S b a  Kabcd  =  c  4 (d d flbd a  c  a  c  — d A d) b  ac  (2.51)  .  gives  + dbddftac — d ddflbc a  + d Abcd + d A c a  a  b  ad  + d A bc a  d  — dbd Cl d c  + dA a  dcb  a  - d Adac - d A ca) b  b  d  •  (2.52)  Chapter 2. The Spins Equations  W r i t i n g this i n terms of ^  27  v i a Eq.(2.44), gives  a o  Kabcd = d[ d\[ Q, ]\d\ a  c  l(d[ d\[ Eb]\d]  b  a  Since "E b only appears i n the form 'EL(ab),  o  n  ec  a  (2.53)  + d[ d\[ Zd]\b]) .  c  a  c  define the symmetric rank-two tensor  n  a  field jab by - E  jac = -2{fl  ac  ( a c )  ) .  (2.54)  F r o m Eq.(2.45), (2.54) is invariant under Jab -> Jab + 9(ah) •  (2.55)  = 2d[ dp7 ]|(,] ,  (2.56)  T h i s gives Kabcd  a  c  -fi^&L' f °  (cf. (2.39)). Note that the R H S of (2.56) is just the linearized R i e m a n n tensor, the metric g b = i] b + j ba  a  also be traceless, K bad  —  a  K  where j — j field.  a  = r] Jabab  a  However, i n order to represent a spin-two field, K b d  a  a  r  must  c  0. Setting the trace of (2.56) to zero gives  c a c b  = d d jb) d  d  [a  -  \^lab -  \d d j a  =  b  0,  (2.57)  These are the equations that are usually taken for a spin-two  Eq.(2.57) is just the linearized vacuum Einstein equations, R ]j = 0, Eq.(1.23). A t a  this point, as discussed i n Chapter 1, one often fixes the gauge and makes an algebraic change of dynamic variables to obtain the equations (cf. Eqs.(1.26), (2.35)) dj a  ah  =  0,  aj  a b  =  0.  (2.58)  In summary, i n trying to find a potential for a tensor K b d,  where K( b) d = K b( )  =  = 0, we were led quite naturally to the potential j b related to K b d  by  a  K[abc]d = K b d c  c  c  a  a  c  a  cd  a  c  Eq.(2.56). T h e equations of motion for the spin-two field, then, are the linearized Einstein equations (2.57). Let us investigate this result further: A s previously mentioned, the W e y l  Chapter 2. The Spins Equations  28  conformal tensor, C bcd, can be taken to represent a spin-two field. It has the symmetries a  C(ab)cd — C { ) ab  V  A j 4  '4>ABCD  cd  —  =  C[ ]d abc  =  C°bcd  0, is W C bcd  = 0. T h e n the spin-two equation corresponding to  = 0. T h e R i e m a n n curvature tensor, R bcd, can also be  a  a  a  taken to represent a spin-two field provided that the vacuum Einstein equations hold. Rabcd  has the symmetries R( b)cd = Rab(cd) — R[abc]d = 0, and V[ R ] a  a  bc  de  = 0 is identically  satisfied. T h e spin-two equations are then taken to be the vacuum Einstein equations, R°bcd  —  0. Let us compare these two formulations: T h e W e y l tensor and the R i e m a n n  tensor are related by  (2.59)  Cabcd = Rabcd + 2g[b\[cRd]\a] + \Rga[c9d}b •  Taking the divergence of (2.59) and using the Bianchi identities (1.5), gives  V Cabc d  d  |V  =  t 6  (R  a]c  -  ,  \g ] R) a  c  Setting this equal to zero, gives the spin-two equations V C bcd a  a  (2.60) — 0.  However, i n terms  of Rabcd, the spin-two equations are R b = 0. W h i l e R b = 0 implies V C bcd a  a  a  a  —  0, the  converse does not hold. In other words, the two formulations are not equivalent, and, as already discussed, there are more general representations of a spin-two field than that given by (2.57).  2.5  T h e usual Lagrangian formulation for the spin-two equations  There is one remaining hurdle to overcome i n order to complete our discussion of spin-two fields, namely, a Lagrangian formulation of the equations needs to be f o u n d .  11  Fierz and  P a u l i (1939) [33], were probably the first to t r y to find a Lagrangian formulation for the  "As a rule of thumb about theories of physics", Feynman remarks that: "Theories not coming from some kind of principle, such as Least-Action, may be expected to eventually lead to trouble and inconsistencies." [31] p74. 11  Chapter 2. The Spins  29  Equations  (massive) spin-two equations of motion, which they took to be, as written by Fierz i n  1939 [32], °7 6 = r  a  ^ l l ,  ^ l l =  0,  (2.61)  where 7 ^ is a symmetric, traceless rank-two tensor field and K is a non-zero constant. To obtain b o t h of these equations  (2.61) from  a variational p r i n c i p l e , Fierz and P a u l i 12  introduced an auxiliary scalar field $ (see also Chapter 5 ) . A general ansatz was then made for the Lagrangian that was a functional of both jj tions were required to give  (2.61) as  well as  $ = 0.  and  b  T h e derived field equa-  These requirements give conditions  on the unknown numerical coefficients of terms appearing i n the general Lagrangian density. T h e zero-rest-mass spin-two equations were then obtained by setting K = 0. T h i s argument led Fierz and P a u l i (though not uniquely) to the second-order Einstein-Hilbert action by identifying 7 = $ . T h e starting point for many discussions of interacting spin-two theories is the L a -  [67], [28], [29], [20]).  grangian found by Fierz and P a u l i , (e.g.,  For this reason, it is  useful to outline their calculation. They took the general second-order Lagrangian (in our notation) C%% = where jj  b  2 K  (  C  6  7j 7 6  T  a  6  +c $ )2  2  C  5  7j n7 6  r  a  6  -  C  l  7  T  c  ^ 47j a  -C4$a ^7j -c $D$ c  6  d  3  = j b — | ^ a b 7 for some arbitrary symmetric rank-two tensor field j . a  ab  , (2.62)  (In [33],  F i e r z and P a u l i began by setting c = C5 = CQ — 1. O f course, one of the parameters can 4  be fixed without loss of generality.) Variation of Spp = / d xCpp 4  w i t h respect to 7 ^ and  Fierz and Pauli were interested in quantizing the equations of motion for particles of arbitrary spin (as derived by Dirac in 1936 [25] and rewritten by Fierz in 1939 [32]) interacting with an electromagnetic field. Their strategy was to add extra terms to the equations of motion that would vanish as the interaction went to zero. Rather than trying to directly derive these additional terms, they introduced auxiliary fields and then required that all the equations be derivable from an action principle and that in the absence of interaction the auxiliary fields vanish (they referred to this procedure as an artifice). As a preliminary step, they sought a Lagrangian formulation for the free field equations. 12  Chapter 2. The Spins Equations  30  gives the equations of motion  u  =  lab  2c n i 2  =  {a  1  ^  Jab  v  v Jd c  - 2c d d i  Tab  6  -f^(i rfrl  Jed  v  ^ 7 d  C a  T c  - 2c D7  Th)c  c  T a 6  5  - c d d <S> a  b  4  , + ic n$) = 0, 4  = 2c «: $ - 2c D$ - c d d 2  2  3  4  a  = 0.  Tab bl  (2.64)  To find conditions on the values of the Cj's, Fierz and Pauli noted that the double d i vergence of (2.63) taken together w i t h (2.64) form a linear homogeneous system. B y requiring that the determinant of the coefficients is not equal to zero, equations (2.61) and $ = 0 can be satisfied. T h e divergence of (2.63) is dS a  = 2c K d  l) ab  2  F P  6  - (  T a b a l  C l  + 2c )c9 D7 5  - \  T6c  c  d d d b  C l  c  -  T c d d l  \c Ud ®  .  b  4  (2.65)  Note that i f C\ = —2c$, the second term vanishes and (2.65) becomes d £&  = 2c K d  ab  2  a  6  which implies that d S a  = dj  x) ah F  Tcd  P  - d  T a b a 7  c  (\  b  C l  d d c  T c d d l  + |c n$) 4  = 0 if $ = 0 and d d j c  Tcd  d  , .  (2.66)  = 0. Taking the  divergence of (2.65) gives dd£ a  b  where T = d d -f . Tcd  c  d  = (2  1) ab F P  « -( 2  C 6  C l  + 2 5)n-i C  C l  n)r-fc4nn$,  .  (2.67)  Eqs.(2.67) and (2.64) form a linear homogeneous system for $ and  T. If the determinant of the coefficients of the system is never zero, then $ = 0 and r = 0. T h i s determinant A is det A = 4  [C C K 2  -  4  6  K  2  (c (c + fci) + c c ) • + (c (c + | 2  5  3  6  3  5  2  (2.68)  )- ^c = 0.  (2.69)  C l  ) - ^c ) ••] .  In the massive case, K / 0 so that det A ^ 0 if c (c + fci) + c c = 0 , 2  5  3  6  c (c + | 3  5  2  C l  Chapter 2. The Spins Equations  31  If c i = — 2c , Eq.(2.69) reduces to 5  C l  c  2  = -4c c 3  6  ,  cic = ^c . 3  (2.70)  4  For AC 7^ 0, i f (2.69) is satisfied, then (2.63) and (2.64) i m p l y (2.61) and $ = 0. Fierz and P a u l i chose the values c = c = c = 1, c\ = —2c , c = — | , and c = — | . However, 4  5  6  5  3  2  note that other choices are possible. T h e equations for zero rest-mass were then obtained by setting K — 0. It then no longer follows that the equations corresponding to (2.63) and (2.64) i m p l y (2.61) and $ = 0. However, the massless equations are invariant under  lib  -  lib + 0 ( A ) - lVabd X ,  (2.71)  C  c  $  $ + dX ,  (2.72)  c  c  where X is an arbitrary one-form field. T h e n one can choose the gauge such that d i a  0  r a h  =  0, Eq.(2.61) a n d $ = 0. Identifying $  =  7tT& =  7,  (2.73)  lab - \Vabl ,  (2.74)  the equations (2.63) and (2.64) (with K = 0) become  jrW  _ (i)  =  j R  =  0  (2.75)  ;  the linearized Einstein equations i n the absence of matter. T h e Lagrangian (2.62) becomes Cf  P  = -A(^-gR)  (2)  .  (2.76)  (Variation of (2.76) w i t h respect to 7^ of course leads not to (2.75) but to the linearized Einstein equation i n their usual form, Eq.(1.12).)  Chapter 2. The Spins  32  Equations  In summary, Fierz and P a u l i began w i t h a general ansatz for a second-order L a grangian i n terms of the fields -jj and $ , Eq.(2.62). T h e coefficients were fixed by b  demanding that the equations of motion must give $ = 0 and (2.61). In the massless l i m i t , however, the gauge invariance of the theory must be exploited i n order that the equations of motion give $ = 0 and (2.61). In this manner, then, Fierz and P a u l i were perhaps the first to derive the equations (2.62) from a variational principle. In [31], Feynman also arrives at the second-order Lagrangian (2.76). Feynman wanted to find the equations of motion for a symmetric rank-two tensor h  ab  starting from the  Lagrangian density C  =4  F  +h T  2)  ,  ab  ab  (2.77)  where Cp* is a general ansatz for a second-order Lagrangian i n h , and T  ab  ab  is the stress-  energy tensor. Variation of CF gives the equations of motion  £  F = ^  L  oh  =  £F {  +T  ab  AH  = 0.  (2.78)  ab  T h e n the condition that the stress-energy tensor, T , is conserved, i.e., that d T ab  ab  a  =0  gives that Sp* is the second-order Einstein-Hilbert action (1.13), i.e., Cp* = Cp* , and H  that Sp*  2.6  ab  is the linearized Einstein equations, (1.12), i.e., Sp*  ab  =  EQ* '. 0,1  Summary  T h e important points to remember from this chapter are the following: field is represented by a completely symmetric spinor equations d '' AA  4>ABCD  <J)ABCD  A spin-two  that satisfies the spin-two  = 0. In terms of tensors, a spin-two field can be represented by  the rank-four tensor K , abcd  where K( ) ab  cd  = K b( d) = K[abc]d = K a  c  c  = 0 and V K a  hcd  abcd  0. W h i l e there are several possible potentials that can be used to describe a spin-two field, the usual choice is a symmetric rank-two tensor field, j . ab  In addition, the usual  -  Chapter 2. The Spins  Equations  33  choice of Lagrangian is the second-order Einstein-Hilbert action. Since severe consistency constraints arise when one attempts to find an interacting theory of a spin-two field, and since there is one known consistent nonlinear spin-two theory, namely general relativity, many arguments have been put forward to show that general relativity was indeed the unique nonlinear spin-two theory. discussed.  In the next chapter, some of these arguments are  Chapter 3  Spin-two Theories and General Relativity  General relativity is a geometric theory of gravitation i n which the structure of spacetime and the matter content of spacetime are inorexably intertwined. The spacetime itself is viewed as a "dynamic participant i n physics" - the curvature of spacetime "tells matter how to move" and the matter content of spacetime "tells spacetime how to curve." T h e geometric nature of general relativity (aka. geometrodynamics) contrasts sharply w i t h other field theories which describe physical forces such as electricity and magnetism i n terms of d y n a m i c a l fields on a "God-given" flat background spacetime. T h e desire for a uniform description of nature led to many attempts to understand, reconcile and eradicate the "philosophical" differences between the two "types" of theories [46], [39]-[41], [31], [77], [22]. O n one hand, the success of general relativity i n the description of nature led physicists to try to interpret field theories such as electromagnetism from a geometrical point of view [82]. M o r e predominantly, on the other hand, the success of field theories led physicists to t r y to interpret gravitation from a field theoretic point of view [63], [64], [46], [67], [83], [31], [77], [22]. It is this latter endeavour that has led to "derivations" of Einstein's "no prior geometry" theory of gravitation from the field theory of a spin-two field on an immutable flat background spacetime. T h e c l a i m that a consistent spin-two theory is necessarily Einstein's equations has a venerable history. Originally it was argued that any consistent spin-two theory was nonlinear due to its coupling to the stress-energy tensor ( G u p t a [40], Feynman [31]). T h e n there were many attempts to extend this argument by choosing various ways to  34  Chapter 3. Spin-two Theories and General Relativity  35  define the stress-energy tensor and to actually derive the Einstein equations ( K r a i c h n a n [46], Feynman [31], T h i r r i n g [67], Wyss [83], and Deser [22]). These methods were i n general based on the divergencelessness of the stress-energy tensor and concentrated on determining the complete Lagrangian for the interacting theory. A n alternative approach, pursued independently by Ogievetsky and Polubarinov [51], Fang and Fronsdal [28], and W a l d [74], (see also Wyss [83]), was to eliminate the matter model entirely and to concentrate on the gauge invariance of the interacting spin-two theory. Yet others, sought the low-energy limit of a quantum theory of gravity, (e.g., Boulware and Deser [9]). A l m o s t all of these arguments concluded the uniqueness of Einstein's equations. W a l d [74] however, found another possibility - i n addition to generally covariant theories, consistent nonlinear spin-two theories could have normal spin-two gauge invariance. In this chapter, the various arguments claiming that any consistent theory of a spintwo field is generally covariant are reviewed. In section 3.1, first the standard argument originally proposed by G u p t a [40] and Feynman for nonlinearity of the spin-two equations is given. T h e n it is shown how various extensions of this argument (Feynman [31], T h i r r i n g [67], W y s s [83], and Deser [22]), have been used to arrive at the Einstein equations. Not only is this of historical interest, it also provides insight into the question of spin-two fields and general relativity. B y a careful examination of the assumptions that are made i n these arguments, it may be possible to find strategies that can be employed to find counter examples to the assertion that nonlinear spin-two theories are generally covariant. In section 3.2, the alternative approaches of Ogievetsky and Polubarinov [51], Fang and Fronsdal [28], and W a l d [74] are discussed i n some detail. Since i n this thesis, we use the consistency criteria established i n [74], particular attention is given to this work, and an in-depth example of these arguments is presented i n section 3.3. Throughout, general observations are made regarding the problem and the various methods of attack. In the discussion of the different procedures, there is some repetition for the sake  Chapter 3. Spin-two Theories and General  Relativity  36  of clarity.  3.1  The standard argument  In studying Einstein's theory of gravitation, analogies are frequently made w i t h Maxwell's theory of electromagnetism.  W h i l e the theories have striking similarities, they have  equally striking differences, perhaps the most prominent being that while Maxwell's equations are linear, Einstein's equations are highly nonlinear. Maxwell's equations were known to correspond precisely w i t h the equations of motion for a spin-one field.  In  1939 [33], the observation was made that the equations of motion for a spin-two field were "identical" to the linearized Einstein equations. Hence, the full nonlinear Einstein equations could be viewed as a theory of an interacting massless spin-two field [73]. F r o m a field theoretic point of view, then, the differences between Maxwell's equations and Einstein's equations would be related to this difference i n spin. G u p t a [39]-[41] used this difference i n spin i n an attempt to try to understand the nonlinearities of Einstein's equations without founding it on a geometrical interpretation: He proposed that any theory of a spin-two field must be nonlinear since it couples to the stress-energy tensor of all matter. T h i s argument has now become textbook material ( M T W [47] p. 424, p. 181 f.). (In Refs. [67] and [77], it is also pointed out that Feynman made similar arguments at the C h a p e l H i l l Conference i n 1956. Feynman's point of view was somewhat different (cf. [31]). He argued that gravitational forces result from the v i r t u a l exchange of a spin-two particle called the graviton (see also [77], [9]).) T h e "standard" argument ([47], [39]-[41], [67], [31], [83], [22], [28]) that any consistent theory of a spin-two field must be nonlinear (in contrast w i t h theories of a spin-one field which have a consistent linear formulation) is based on a Lagrangian formulation together w i t h the four principle assumptions: (1) the spin-two field is represented by a symmetric  Chapter 3.  Spin-two  rank-two tensor field,  Theories and General  Relativity  37  (2) the spin-two field obeys the linearized Einstein's equations  (referred to as the equations of motion for a spin-two field i n this context) which are derivable from the second-order Einstein-Hilbert action, (3) the spin-two field couples directly to the total stress-energy tensor of matter including the spin-two field, T , ab  (4)  the stress-energy tensor either (i) is required to be derivable from the action by variation w i t h respect to the flat metric n  ao  (Deser [21]), or (ii) is defined canonically ( G u p t a [40]),  or (iii) is specified precisely ( T h i r r i n g [67] and Feynman [31] take particle matter, W y s s [83] takes the canonical K l e i n Gordon stress-energy tensor), or (iv) is any other rank two tensor that is symmetric and divergence-free (Fang and Fronsdal [28]). Cases (ii) and (iv) are used to argue the nonlinearity of any spin-two theory, while (i) and (iii) are used to argue that a consistent spin-two theory must be general relativity. T h e argument for nonlinearity is as follows: T h e source for the spin-two equations (the linearized Einstein equations) is the stress energy tensor for matter, excluding the spin-two field, which appears on the R H S of the equations of motion. Since the L H S of the equations of motion are divergenceless, the R H S must also be divergenceless. This means that the R H S must be the total energy-momentum tensor of matter including the spin-two field, since it is conserved. Therefore, the stress-energy tensor of the spin-two field must be added to the R H S . Now, i n order to derive these extended equations from an action, higher-order terms must be added to the action. However, these higher-order terms i n t u r n give rise to new terms i n the stress-energy tensor. In order to get these new terms from an action principle, the action must i n turn be further extended. However then new terms arise i n the stress-energy. A n d so on. Thus the assumption that the linearized Einstein equations couple directly to the stress-energy tensor which is specifiable somehow from the action leads to the conclusion that any physical consistent spin-two theory must be a nonlinear theory. Consistency is seen here to mean conservation of the stress-energy tensor. (This does not happen i n the case of a spin-one field (represented by a vector)  Chapter 3. Spin-two Theories and General  Relativity  38  since a spin-one field is assumed to couple to a current and the uncharged (photon) vector field does not contribute to the current. A spin-two field, on the other hand, itself has energy and thus contributes to its source, the stress-energy tensor.) It is further argued that if some prescription is given to define the stress-energy tensor, (e.g., cases (i) or (iii) above) then this line of reasoning leads to Einstein's equations. Let us consider this argument and the assumptions i n more detail and see to what extent the various procedures can be regarded as proofs. The starting point is taken to be the gravitational part of the second-order Einstein-Hilbert Lagrangian, C and the linearized Einstein equations, SQ and (2)).  According to M T W [47] p.  =  G  \ Eq.(1.13),  = 0, Eq.(1.14), (assumptions (1)  181, "The choice of Lagrangian is dictated by  the demand that "fab be a 'Lorentz covariant, massless, spin-two  field.'  • • • see, e.g.,  (2l  • • • Feynman (1963)" [31]. Feynman [31] chose C  G  by demanding that it give rise to  divergenceless equations of motion (see Chapter 2). Others, (e.g., T h i r r i n g , [67], Fang and Fronsdal [28]), chose C  G  since it is the Lagrangian determined by Fierz and P a u l i (2^  [33] (the first to write down C  A  , [47], see Chapter 2).  However, as pointed out i n  Chapter 2, the Lagrangian found by Fierz and P a u l i is not unique (nor do they claim it to be). Indeed, it was remarked i n Refs. [28] and [67], that any equations related to SQ * by a transformation of the form j b —* lab + c>labl, are equivalent, where c is a real AB  a  number. Thus, it was argued, that one could, without loss of generality, formulate the spin-two equations i n terms of a rank-two symmetric tensor field, j b, that satisfies the a  (2)  linearized Einstein equations, £ * , where £ * a b  G  G  a b  =  N e x t it is asserted that to the spin-two equations, £ * G  symmetric source term, T^ , ab  for the spin-two field, j b, a  a b  = 0, must be added a  (where for the moment, the  superscript (n) indicates the powers of j b appearing i n the term). a  T h e source term,  T ( ° ' , is constructed from the other fields with which the spin-two field interacts. T h e af>  Chapter 3. Spin-two Theories and General Relativity  39  equations for the spin-two field are then £$  ah  = 8wT^ ,  (3.1)  ab  which are derivable from the modified Lagrangian density  +  OLMCM  ,  (3.2)  by variation w i t h respect to 7^, where CM = C * — ^i bT^ M  and O J M = 167T. However,  0)ab  a  since the linearized Einstein equations are divergenceless, d £G*  ab  a  must also be divergenceless, d T^  0)ab  a  = 0, the R H S of (3.1)  = 0, (indeed this is the criterion that Feynman [31]  used to arrive at CQ* (see Chapter 2)). It is then shown that the theory as it stands is not consistent. T h i s is most clearly illustrated by choosing a specific matter model, say scalar matter as described by the K l e i n - G o r d o n equation. In the noninteracting theory, Q^rp(o)at - p p t i o n a l to the matter equations of motion. Hence, the conservation of g  T'( ) 0  ab  r 0  0 r  follows from these equations.  However, one finds that i n the interacting theory,  the matter equations have additional terms arising from the interaction so that T^°*  ab  is  no longer proportional to the matter equations and, hence, is no longer conserved as a consequence of these equations [83], [31]. It was argued that this inconsistency resulted from the failure to include the spin-two field i n the stress-energy tensor. T h i s problem could then be resolved by having on the R H S of Eq.(3.1) the total stress-energy  tensor  of matter i n c l u d i n g the spin-two field. T h i s is assumption (3). Before proceeding, note that there are alternative methods to obtain a divergenceless R H S without it being the complete stress-energy tensor of matter and the spin-two field. For example, the R H S could be an identically conserved symmetric rank-two tensor field constructed from the matter fields [74]. A s another example, i n [21], Deser and L a u rent were interested i n constructing interacting spin-two theories without self-interaction. Consistency of (3.1), i.e., conservation of the R H S , could be achieved by taking the R H S  Chapter 3.  Spin-two  Theories and General  Relativity  40  of (3.1) to be the divergenceless part of the matter stress-energy tensor. T h e nonlocal projection operator P  ao  = r] i — n~ d db was used to project out the divergenceless part of 1  a >  a  the stress-energy tensor (which was taken for the specific example of particle matter (as i n [31])). T h e quantity that appears on the R H S of (3.1) is then the divergence-free tensor J  = (P Pbd+pPabVcd + qPabPcd)T . cd  ab  ac  T h e constants p, q were taken to be p = q = - 1 to  agree w i t h Newtonian gravity. T h e theory so defined is a highly nonlocal nongenerally covariant spin-two theory without self-interaction which depends on the choice of matter model. B y introducing auxiliary fields, locality can be restored. Thus, assumption (3) can also be relaxed. However, i n this example, the price is nonlocality or auxiliary fields. R e t u r n i n g to the principal discussion, since the various arguments differ slightly depending on the criterion chosen to define the stress-energy tensor (assumption (4)) for the sake of preciseness we consider case (i), where the stress-energy tensor is defined by variation of the action w i t h respect to the flat metric, rj b, [21], so that a  (3.3) (the argument using the canonical definition of the stress-energy defined by T  c  — r] £ , ab  M  b  =  where 4> represents the fields, is identical [40]). Therefore, consistency  where  (3.6)  Chapter 3.  Spin-two Theories and General  Relativity  41  T h e n variation of (3.6) w i t h respect to j b gives (3.4), (the superscript (2) on T^  ab  a  here that T^  ab  is composed of two powers of the fields i n general, i.e., T^  means can also  ab  include the matter field, and not just the spin-two field, although the latter is sometimes assumed). W i t h respect to assumption (4), i n [28], Fang and Fronsdal remark that i n the 1962 thesis of E . R . Huggins (California Institute of Technology) that Huggins  "attempted  to construct [the third-order term i n the action C^ ) by requiring • • • that ^ 1  be the  energy-momentum tensor of the spin-two field. He concluded that this is a highly i l l defined procedure that can succeed only when the answer is known i n advance. T h e idea that the spin-two field should be coupled to the energy density is attractive, but 'the trouble is that we need an extra condition to define the energy.' " Completing the m a i n discussion of the standard argument, variation of (3.6) w i t h respect to r} now gives an additional term i n the stress-energy tensor, namely ao  j i ( 0 ) a 6 _j_  rp(2)ab  _|_  ji(3)afc  ^ ^ 8-K 8j] ' ab  Therefore, the Lagrangian (3.6) must be further modified by adding the higher-order term C\ M  so that variation of the augmented action w i t h respect to i]  ab  gives (3.7). A n d so on.  T h i s reasoning leads to the conclusion that "any self-consistent theory of an interacting massless, spin-two field i n flat spacetime is as nonlinear as Einstein's equations" [28]. A c c e p t i n g this reasoning (and the assumptions), the challenge was then to show that the spin-two equations were indeed Einstein's equations. To see how this was done, we consider two examples i n particular, the arguments of Deser [22], and of Feynman [31]. In [22], (see also [9]), Deser uses the above reasoning i n the absence of matter, (i.e., rp(o)ab _ equations.  to argue that any consistent theory of a spin-two field must be Einstein's In particular, consistency is defined by the conservation of T , ab  and T  ab  is  defined by the variation of the action w i t h respect to r] b, i.e., consistency is realized by a  Chapter  3. Spin-two Theories and General  42  Relativity  the condition [9] SCdq^Tlcd)  _  <XMSC{lab,Vcd)  aM __ Tab  Sjab  2  2  ^fy  6r)  ab  For £ ( ' Deser takes the linearized P a l a t i n i action C \  (The P a l a t i n i Lagrangian Cp  2  P  for the full E i n s t e i n equations takes the metric density, ^/—~gg , and the connection ab  coefficients, T , as independent variables (see M T W [47] p. 491 f.). V a r i a t i o n of Sp a  bc  w i t h respect to T  a bc  gives the usual expression for the Christoffel symbols i n terms of  the metric. V a r i a t i o n of Sp w i t h respect to \/~-gg  ab  linearized form by substituting \f—gg  = rj -\-h ,  ab  ab  and keeping terms of second-order i n h  ab  the tensor r  ab  =  then gives R  ab  —  Combining the equations that Deser  a  bc  AB  — 0. Deser obtains the  ab  and T .)  —KT ,  ab  where h is a tensor density, into Cp  ab  obtains from variation of Cp* w i t h respect to h  gives R  ah  and T  a  where  K  bci  gives R^ — 0. Adjoining  is some constant, which Deser  asserts can be derived from the full Palatini action. Thus, Deser argues, any consistent theory of a spin-two field must be Einstein's equations. However, this argument depends strongly on the particular prescription for defining the stress-energy tensor. Feynman [31] pursues a slightly different line of reasoning, (see also M T W [47] p . 179 f.). He considers the specific example of particle matter so that S^* = / dru u a  T  ab  b  and  = u u , where u is the particle four-velocity and r is the proper time. T h e matter a  b  a  part of the action can then be written SM = j dr{  + )T  and the total action is given by Eq.(3.2), S  +O:MSM-  Vab  by Eq.(3.1),  £Q*  AB  =  STrT^ . ab  G  ab  lab  ,  (3.9)  T h e spin-two equations are given  However, Feynman points out that since the gravitational  field itself has energy, its exclusion from the R H S of (3.1) leads to internal inconsistencies, e.g., the conservation of T  ab  which follows from (3.1) says that a particle moves on a  straight line i n contradiction w i t h the equations of motion derived from SM- Therefore, (3.9) needs t o be modified by adding terms involving higher powers of the fields t o  Chapter 3. Spin-two Theories and General Relativity  43  the action. A t this point, Feynman makes a simplifying assumption, namely, that the higher-order terms involve only the spin-two field, ~f b, and that, therefore, the matter a  only couples to the spin-two field i n the form stated i n Eq.(3.9), (rj b + j b)T . ab  a  a  This gives  h i m the complete gauge invariance of the theory. He is then able to determine the form of £ ( ' . Constructing invariants of the infinitesimal transformations, he finds that the 3  complete action is the Einstein-Hilbert action, SEH, evaluated w i t h respect to the metric gab — Vab + lab- W a l d [74] has shown that the (implicit) assumption that the complete matter Lagrangian is given by C * + C * M  M  — g T , ab  ab  is equivalent to the assumption  that the matter part of the action is generally covariant, and hence, assuming that the equations of motion are satisfied, that the complete Lagrangian is generally covariant. T h e n , l i m i t i n g the number of derivatives i n the action to two, one obtains the EinsteinHilbert action. W a l d points out that the assumed linear coupling of matter to the spintwo field is a very restrictive assumption. For example, that such a coupling leads to a consistent theory is peculiar to the case of particle matter. Moreover, he points out that F e y n m a n obtains the result that the equations are generally covariant w i t h respect to the combination g  ab  variables such as j  = i]ab + 7a6 but not even for a metric related to g  ab  by a change of  —• Tab + ^ JcaJbd- In other words, it is certainly possible to relax cd  ab  many of the assumptions that are made i n [31]. We briefly mention the work of Weinberg [77], and Boulware and Deser [9], who approach spin-two theories from a special relativistic quantum particle theory point of view. Weinberg argues that the Lorentz invariance of the S-matrix leads to (3.1). However, he then refers to Feynman to conclude that (3.1) leads to the complete Einstein equations. Boulware and Deser [9] argue that the methods of perturbative quantum field theory show that the low frequency limit of any (presumed to exist) theory of quantum gravity must give Einstein's equations. T h e y assume the Lorentz invariance of the S-matrix together w i t h (1) and (2) above and that the forces are transmitted by the v i r t u a l exchange  Chapter 3. Spin-two Theories and General Relativity  44  of gravitons. Fang and Fronsdal [28] and W a l d [74] note that presumably a translation to classical theory could be made and that while the results of [9] are related to their results, additional assumptions are made i n [9]. For example, W a l d points out that the "low-energy l i m i t " of a theory of quantum gravity is obtained by limiting the number of derivatives appearing i n the action to two. Fang and Fronsdal [28] remark that: "It is possible, if all the assumptions made were carefully sorted out, and if all the references to quantum theory expurgated, that it would be found that Boulware and Deser anticipated some of our results [see next section], but to do so is a project i n itself. We prefer to deal w i t h the classical problem directly, by methods that avoid quantum extensions that are irrelevant to i t " [28].  3.2  T h e gauge invariance approach  In the preceding section, a number of interesting approaches to the problem of interacting spin-two theories and general covariance were discussed. Various limitations of these arguments were pointed out, and some objections that were raised i n the literature were mentioned. A common feature of these methods was to introduce a stress-energy tensor which was defined according to various specific prescriptions. However, it seemed to be generally believed that the conclusion that a consistent nonlinear spin-two theory must be general relativity, d i d not depend on the particular form taken for the stress-energy tensor. A n alternative approach, i n which it was not necessary to introduce a matter model, was pursued independently by Ogievetsky and Polubarinov [50], Fang and Fronsdal [28], and W a l d [74], (although coupling w i t h matter was also considered [74]). These works concentrated on the gauge invariance of the theory. W h i l e there are certainly similarities between the different arguments, there are also important differences. Following, their  Chapter 3. Spin-two Theories and General Relativity  45  work is presented chronologically. A g a i n there is some repetition for clarity.  3.2.1  The "spin-limitation principle"  In a series of papers ([49]-[52] and references therein) Ogievetsky and Polubarinov investigated interacting fields of definite spin. In the case of a (massive) spin-two field [51], they sought the possible types of local interacting theories of a symmetric rank-two tensor field 7 (, that excluded spin-one. T h e equations were to be derivable from an action a  principle and to be compatible w i t h the condition m (d j 2  a  ab  + ad ) b l  (3.10)  = 0 ,  where a is an arbitrary number and m is the mass of the field j .  They call (3.10) the  ao  spin-limitation p r i n c i p l e . A key assumption that Ogievetsky and Polubarinov make is 1  that they allow only coupling constants of dimensionality m a s s . - 1  2  Using (3.10) Ogievetsky and Polubarinov show that the massless equations of motion must satisfy some general divergence identity. This divergence identity is reinterpreted in terms of an infinitesimal gauge invariance of the action. Assuming that there are no other identities that the equations of motion must satisfy, the infinitesimal variations 3  must form a group, i.e., the commutator of two of the infinitesimal variations must also be a similar variation, (otherwise, the commutator would give an additional identity). general, a symmetric rank-two tensor field, y b, (16 components) contains a spin-two field (5 components), three spin-one fields (9 components) and two spin-zero fields (2 components). The condition (3.10) serves to exclude the spin-one fields and a single spin-zero field from -jat, (see Appendix A). They work in natural units where h = c = 1 so that energy, momentum and mass have the units mass, and time and length have the units mass . Coupling constants of dimensionality mass"" for arbitrary nonnegative n are allowed provided that such coupling constants are of the form c" where c is a coupling constant of dimensionality mass ™ so that setting any one coupling constant to zero reduces the equations to their linear form, i.e., eliminates all interactions. They refer to this assumption as the "principle of minimality". This has the effect of limiting the number of derivatives appearing in the action to two. I n component form, the divergence identity gives four conditions. In order to eliminate the three spin-one and one spin-zero degrees of freedom, only four identities are needed. Hence they assume that there are no additional identities. a  2  -1  -  3  Chapter 3. Spin-two Theories and General  Relativity  46  T h i s argument leads Ogievetsky and Polubarinov to an equation that is very similar to the integrability condition derived by W a l d i n [74]. (However, the decision to restrict the dimension of the coupling constants limits the equations to first-order interactions.) To solve this equation, Ogievetsky and Polubarinov write out the most general form for the variations. Due to the l i m i t a t i o n of the dimension of the coupling constants, the general form for the variations is restricted to zeroth- and first-order quantities. Substituting i n these general expressions they find, up to a change of variables, that interacting theories compatible w i t h the spin l i m i t a t i o n condition are generally covariant. Ogievetsky and Polubarinov also include interactions w i t h scalar, vector, and spinor fields. However, they do not allow the fields to couple to one another (since they restrict the dimension of the coupling constants). In order to better compare the arguments of [51] w i t h that of [28], and [74], it is helpful to outline the calculation of Ogievetsky and Polubarinov i n [51] i n our notation. T h e y took the general Lagrangian  C p = C p{j b, 0  0  a  ddab,  ddj , d  c ab  (j>, d <f>, • • •), where '• • •' a  represents other fields which we shall not consider. W r i t i n g the Lagrangian as COP = C p  - lm (j j 2  0  ah  ab  + 0^77) ,  (3.11)  ) =0  (3.12)  the equations of motion can be written cab  OP — f  - m ( 2  C  + m/  a f e 7  a 6 7  where *a6  OP —  dS a  a OP  = d £$ a  - m (d 2  P  84>  hab  ab al  + a# 7) = 0 • 6  (3.13)  (3.14)  Chapter  3.  Spin-two Theories and General  Therefore, if d 8 a  is identically zero, or zero as a consequence of the equations of motion,  op  then (3.10) follows. d £ a  d [B £^  where  B  cd  = 0 if the following generalized divergence identity holds  op  + D F p)  ah  a  = C\ Z  ah  P  0  d  is a function of j  a b c d  , rj ,  a b  ab  tabcd] D  is a function of <f>,  r) ,  (o)ab  s 8  a  ab  B  a  cd  b  [c  =  47  Relativity  + pr] r) , ab  d)  cd  ab  C^  c d  + AT  d 0  and t  and e  abcd  =  2  abcd  ; C  ab  b  (3.15)  is a function of d j  a c d  c  ab:  and A is a function of d <j>,  ;  0, D^  OP  ,  + ad j)  - m {dal  b  P  b  a  ab  r) , ab  rjab,  and  and  e  abcd  = 0, and A^ = 0. In the case of zero b  mass the last term in (3.15) vanishes. Equivalently, COP must be invariant under the following infinitesimal gauge transformations Sx lcd  =  B d Xb  8J  =  D dX  a  X  +C X  ab  cd  cd  a  a  + AX  ab  (3.17)  .  b  b  (3.16)  ,  a  a  b  Restricting the dimension of the coupling constants, the most general expression for the tensor fields are  4  B  + pr,  = 8\c8 d)  c d  + J2b B\  ab  b  a b  1)ah  Vcd  t  cd  ,  i=0  C\  d  =  J2  c£\ cd  ,  1)e  where the precise form for the Bf  b cd  ,  D  ab  = d D^  ,  )ab  3  A" = a A^ 2  b  ,  (3.18)  etc., are written in Eqs.(5.48). Note that Eqs.(3.18)  are the complete expressions for the tensor fields, not just the expressions up to first order (as Eqs.(5.48) are). Also, since the dimension of the coupling constants is restricted, (3.16) does not involve </>, and (3.17) does not involve -f , i.e., there is no coupling ab  between different fields, (cf. Eqs.(5.30) and (5.31)). The condition that (3.16) form a group gives - 8^ 8 )j a  9a  =8  ab  Xa  j  ab  ,  (3.19)  In terms of the parameters used in [51], 26i = b, 2bj = d, 263 = /, 64 = e, 65 = c, bg = a,60 = —g, 2ci = b', c = c', c = e', c = a', 2c = / ' , 2c = d', c = -g'. 4  2  3  4  5  6  0  .  Chapter  3.  Spin-two Theories and General  Relativity  48  where \a is some function of 9 and tp , (cf. Eqs.(5.36) - (5.38)). Substitution of the a  a  assumed form for the exact gauge invariance, 8\ j a  and (3.18), and assuming that Xa = xi°\ known coefficients.  = ^lab  ab  + £J^7a&, given by (3.16)  gives a system of 17 equations for the un-  Ogievetsky and Polubarinov are able to show that this system of  equations has seven solutions which, up to a change of variables, gives, (1) a theory w i t h no interactions or, (2) a theory w i t h the gauge invariance 8\cJ b = \C\cg  .  (3.20)  T h e condition that the infinitesimal transformations  (3.17) form a group is given by  a  ab  (3.19), where now Vg is a vector defined by (3.17). Ogievetsky and Polubarinov find a  84 X  3.2.2  = \C ^ X  .  (3.21)  T h e "extended G u p t a program"  In [28], Fang and Fronsdal are interested i n extending the standard argument proposed by G u p t a [40] for the nonlinearity of any spin-two theory (cf. section 3.1). E l i m i n a t i n g the matter model, they seek a nonlinear generalization of the linearized Einstein equations i n terms of a formal power series i n the coupling constant K , which they call the " G u p t a program".  T h e (consistency) condition that the equations of motion linear i n  K are divergenceless allows them to find the term i n the Lagrangian that is linear i n K (restricting the number of derivatives). Assuming that the complete theory must be i n variant under a L i e algebra of infinitesimal transformations, Fang and Fronsdal are able to show, using deformation theory, that the determined linear gauge invariance leads to the E i n s t e i n equations. In more detail, the complete Lagrangian density is written  UF  =  4  2)  oo  +E n=3  * ~ £FF(n) n  2  ,  (3.22)  Chapter 3. Spin-two Theories and General Relativity  where the  CpF(n)  a  r  e  49  assumed to be polynomials in j  and its first-order derivatives,  a 0  and CFF{3) has no constant term and no term linear in j . The equations of motion are ab  then cab — >FF  _ (l)ab  0L  C  —  FF  where the S ^ FF  <?FF(I)  c  °lab  G  ab  has no constant term. To order b  F  of motion. Thus one can write  C  (o o q \ [o.ZO)  J  K,  and its first and second-order derivatives, and the fact that the linearized Einstein equations are  = 0 identically or as a consequence of the equations  d £p ( ) a  n-l cab FF(n)  K  n  are polynomials in j  divergenceless implies that  -t- 2_j -2  C  —  2  d £p p( ) b  a  2  f ° some r  = C d£c^ b  cd  c  C db  c  Fang and Fronsdal  prefer to write this schematically as 4^  =0  _>  d €^ {  a  ab F  = 0.  (3.24)  In order to determine  £j?F(3)  three, i.e., that  = £^ '• Writing out the most general expression for  £FF(3)  they assume that it is a homogeneous polynomial of order  3  C^ \ 3  they find,  up to a change of variables, that either £( ) = 0 or that £ ( ' = £Q\ (note that they have 3  3  restricted the number of derivatives appearing in C^ *). Comparing their results with the 3  work of Feynman [31] (and Wyss [83]) who used the condition for particle matter) to find that £( ) = 3  CQ\  C d b  c  (obtained  = T^ d  Fang and Fronsdal also adopt  b  c  C d b  c  =  r' ) ^. 1  6  This gives them an expression for the linear gauge invariance of the theory. In order to find a solution to all orders, they follow a suggestion of Wyss [83], namely that the complete theory must be invariant under a Lie algebra of infinitesimal transformations which may be fixed from the first-order expression, for the gauge invariance. Fang and Fronsdal are able to make this precise, and in terms of the mathematical theory of deformations they are able to show that from the determined linear gauge invariance, the complete theory must be generally covariant. The assumption that the number of derivatives in the action is limited to two then gives the Einstein equations. Fang and Fronsdal point out that although their calculation leads to the Einstein  Chapter  3.  Spin-two Theories and General  Relativity  50  equations, this does not "automatically solve the problem [that any nonlinear theory of a spin-two field is generally covariant]". T h e paper concludes w i t h a number of suggestions on possible generalizations of their calculation. For example, they suggest that one should allow for the possibility of higher derivative theories.  3.2.3  Normal spin-two gauge invariance  In [74], W a l d found a counter-example to the statement that any consistent nonlinear theory of a spin-two field must be generally covariant. He considered the extension of the linearized E i n s t e i n equations and found two possible types of consistent theories: (i) theories that were generally covariant, and (ii) theories that had normal spin-two gauge invariance, (the gauge invariance of the linearized theory). A "key assumption" that is made i n all of the preceding arguments is to limit the number of derivatives appearing in the Lagrangian to two. It is the relaxation of this condition that leads to the counterexample to the "folk theorem" found i n [74]. W a l d ' s argument is as follows [74]: Linear equations of motion that are derivable from an action principle can be nonlinearly generalized by adding terms of third-order or higher i n the fields to the second-order Lagrangian. For example, the linearized Einstein equations can be generalized by adding arbitrary nonquadratic terms to the second-order Einstein Lagrangian. However, the resulting equations may not necessarily be consistent in the sense that, i n a perturbation expansion, each linear solution is tangent (in the space of functions) to an exact solution. T h i s implies that if the original linear equations satisfy a linear identity, (e.g., the linearized Bianchi identity) then the complete nonlinear equations w i l l be consistent if they satisfy a (to be determined) generalized identity. T h i s identity, or equivalently, infinitesimal gauge invariance of the action, w i l l arise from an exact gauge invariance if the commutator of two of these infinitesimal variations is again such a variation, (i.e., if the space of variations is involute). T h e resulting equation is  Chapter 3. Spin-two Theories and General Relativity  51  referred to as the fundamental integrability condition i n [74], and can be solved order by order for the generalized identity. In this manner, the most general gauge invariance of a consistent nonlinear generalization of the original theory may be found. W a l d also considers coupling to matter. He shows that it is possible to find consistent theories of a spin-two field that is coupled to matter which are not generally covariant. However, if an additional "quite natural" assumption is made about the coupling, namely that i n lowest order j \, couples to the stress-energy tensor T , where T ab  ah  a  is determined  — r] b + 7a&, he conjectures  from the action by variation w i t h respect to the metric g  ab  a  that this would eliminate the possibility of finding theories having normal spin-two gauge invariance. However, he is only able to show this i n certain cases. As "perhaps the simplest" example of a theory with normal spin-two gauge invariance, W a l d suggests Cw = CG* + (R^*) , where n > 2. O f course, i f the further decision is n  made to l i m i t the number of derivatives i n Cw to two, then it is no longer possible to find nonlinear spin-two theories w i t h normal spin-two gauge invariance. In [19], Damour and Deser agree that while normal spin-two gauge invariant theories are possible, it is asserted that "they are highly artificial and not obviously quantizable because of the higher derivatives. • • • this reinforces the uniqueness of Einstein's equations." However, 5  the point of W a l d ' s result is that alternatives to generally covariant spin-two theories are indeed possible. T h e y may be ruled out by further restrictive assumptions, as W a l d points out, such as specifying how 7^ couples to matter, or the decision to not considerhigher derivative theories. Moreover it is important to be aware that alternatives exist It should also be mentioned that in [19], the term (R^) = £ in Cw is reinterpreted as j T , where T = = • P ( i j W ) - = (r] a - 5 5 ) ( i 2 W ) - , (OP is a local projection operator). This "type of coupling" is then referred to in [19] as "non-minimal" or "projection-operator based", referencing the paper by Deser and Laurent [21]. However, in [21], only interacting theories of a spin-two field without self-interaction were considered, which were constructed by projecting out the divergenceless part of the stress-energy tensor for particle matter. This resulted in theories that were highly nonlinear. 5  n  ab  j n i  n  ab  a i  1  n  ab  a  6  ab  1  ab  Chapter 3. Spin-two Theories and General Relativity  52  and to know when they can be used. In any case, i n Chapter 6, we exhibit nonlinear spintwo theories that are not Einstein's equations, that are not necessarily higher derivative theories, and which are "obviously quantizable" [68],[66], [44]. In the next section, Wald's argument is reviewed.  3.3  Wald's consistency criteria  Since i n this thesis, we employ the consistency criteria of W a l d [74] to investigate a number of questions regarding nonlinear theories, i n this section, these arguments of are discussed i n detail. W e choose, as an illustrative example of Wald's consistency criteria, the theory of a spin-one field, since the equations are more straightforward (fewer indices) and the algebra is considerably less involved.  Thus, i n this section, the problem to  determine the types of theories that arise from the consistent nonlinear generalization of the equations of motion for a spin-one field is discussed. T h i s is a problem that W a l d investigated i n [74]. He found that a consistent generalization of Maxwell's equations has the usual spin-one gauge invariance. In the following, we repeat this calculation, w i t h particular emphasis on the basic criterion for consistency that is presented i n [74].  3.3.1  Consistent nonlinear extension of Maxwell's equations  T h e equations of motion for a massless spin-one field i n Minkowski spacetime are given by Maxwell's equations, (cf. Chapter 2), dF  =  0,  (3.25)  dF  =  0,  (3.26)  ab  a  [a  where F  ab  bc]  = —F . Eq.(3.26) implies that there exists a one-form field A such that ba  a  F  ab  = 28 A [a  b]  ,  (3.27)  Chapter 3. Spin-two Theories and General Relativity  53  since d[ d A ] = 0. Note that this specification for F , (3.27), is unchanged under the a  b  c  ab  transformation A -> A + d X , a  a  (3.28)  a  where A is an arbitrary scalar. W i t h the choice of potential A , (3.26) is an identity while a  (3.25) are the equations of motion which can be derived from the action «S|^ S ?M = - \ Jd xF F , {  4  (3.29)  ab  ab  by variation with respect to A , a  6S jr(i)a = _ A o . oA {2)  2d d A [b  =  = dF  a]  b  b  ba  = 0,  (3.30)  a  where, the numerical superscript (n) indicates the power of the field(s) (in this case A ) b  appearing in the term. Note that (corresponding to the gauge invariance (3.28)), satisfies the identity d^  = 0.  1)a  a  (3.31)  In order to determine the types of consistent nonlinear extensions of the linear theory (3.27) - (3.31), consider the action S  AA  that is obtained by adding terms of third-order or  higher in A to C^l • There are no restrictions on the types of terms added to C^l except a  that the number of derivatives at each order are to be bounded by a (fixed) integer N. Variation of S  AA  6  with respect to A gives the complete nonlinear equations of motion, a  r- = -r^r SA  = F  {1)a  +f  {2)a  +^  ( 3 ) a  + • • •= 0,  (3.32) y  J  a  which we want to be local partial differential equations involving a bounded number of derivatives.  denotes the quadratic part of T JF' ' a  3 0  the cubic part, etc.. Given any  Lagrangian density, it is always possible (theoretically) to write down the corresponding The analysis of Fang and Fronsdal [28] was limited to polynomial Lagrangians, and the analysis of Ogievetsky and Polubarinov [51] was limited in order. 6  Chapter 3. Spin-two Theories and General Relativity  equations of motion, T  a  54  = 0. However, since the linear term, J-^ , a  i n the equations  of motion is identically divergenceless, i n general, as discussed i n the previous sections, there are severe constraints on the complete equations to ensure that they are consistent. In order to find a general way to avoid any potential consistency problems, it is helpful to understand the nature of the consistency problems encountered and to try to determine how they might i n general be avoided (rather than take one particular prescription which, while being valid, is perhaps more restrictive than necessary).  One way to quantify  the consistency problem is as follows: Consider solving the equations (3.32), F  — 0,  a  perturbatively, i.e., try to find a one-parameter family of exact solutions A (e) to T  a  a  = 0  such that  4,(0)  =  0,  (3.33)  0.  (3.34)  For example, take (3.35) where A^* = 0, A^* is the linearized solution, A^* is the second-order perturbation, etc.. T h e first-order equation i n this perturbative expansion is ^[A ] l)  0  = 0.  (3.36)  Solution of this linear equation (3.36) gives the linear perturbation A^\  which should  be a reasonable first approximation to A . Solving the second-order equation b  f("'rf l )  + ^rfU< ] = o 1,  I  for A  should give a better approximation to A , where J ^ [_,  of T '.  However, recall that the first-order quantity  b  a  :  b  a  (3.37)  _] is the quadratic part  is identically divergenceless,  Chapter 3. Spin-two Theories and General Relativity  d F^  a  a  55  = 0, Eq.(3.31). Taking the divergence of (3.37) and using (3.31) then gives  «9 ^ [4 4 ]=0a  1)  1)  a  (3-38)  5  Note that this is a condition for solving the second-order equation (3.37) which only involves the linear perturbation A^\ i.e., although (3.38) arises at second-order i n perturbation theory, i t gives additional conditions that the (already solved for) linear perturbation  must satisfy. This is the consistency problem. O n l y the solutions of the  linear equations (3.36) that are also solutions to the second-order equation (3.38) are allowed. However, i n general, not all solutions (if any) of (3.36) will also be solutions to (3.38) - the two equations are, i n general, incompatible [74]. Similar problems occur at higher orders. For, example, taking the divergence of the third-order equation ^ [A ] a  + J*V [A?\  { 3)  a  b  4 ] + J* >[A \ 2)  2  2  C  A ] + ^ [ A ^ \ A?\ A ] = 0 , ( 1]  a  (3.39)  1]  b  d  gives  dajM'iA ),  4 ]+d^[A?\ 2)  1  A^} + d ^ l A ^ , ^ ,  A ] = 0.  (3.40)  1]  d  Eq.(3.40), a third-order equation, gives further conditions on the first- and second-order solutions A^ \ A^ *. In general, higher-order equations i n the perturbation expansion 2  b  w i l l impose restrictions on lower-order solutions. This quantification of the consistency problem allows one to see clearly how the consistency problems originate and, perhaps, how they can be generally avoided, making as few assumptions as possible. Thus, one sees that i n a nonlinear extension of J-~^ , consistency problems may a  arise since  is divergenceless, i.e., since the linear quantity J ^ 7  obeys the linear  divergence identity (3.31). T h e strategy employed i n [74] to avoid these pathologies is to find a generalization of the divergence identity (3.31) such that if the quantities J- satisfy a  the generalized identity then the equations (3.38), (3.40), etc., would be satisfied as a  Chapter 3. Spin-two  Theories and General  Relativity  56  consequence of this identity together w i t h the lower-order perturbation equations - if one generalizes the equations of motion, one should also generalize the associated identities and corresponding infinitesimal gauge invariances. W a l d shows that it is sufficient that T^  l)a  satisfies an identity of the form dT  = CT  a  a  + B\d T  a  + D \d d r  h  a  + ••• ,  a  a  a  h  (3.41)  where the s u m on the R H S is finite, to ensure consistency i n the perturbation solutions. T h e tensors C , B ,  and D  a  a  b  and (in four dimensions) e  are locally constructed from A and its derivatives, rj b  ab c  b  ; at zeroth-order C^ = 0, B^  = 0, D^  a  abcd  a  0)ab  b  c  = 0, (so the  first-order part of (3.41) is just (3.31)). For example, to see that Eq.(3.41) together w i t h the first order perturbation equation, Eq.(3.36), imply that the second order perturbation equation is divergenceless, i.e., that Eq.(3.38) is satisfied, consider the second-order part of the general identity Eq.(3.41): dT  = C *F^  {2)a  l  a  +B ^ \ d ^  a  a  )  + D^ d d F  b  ab  a  + •••.'  {x)c  c  a  b  (3.42)  Evaluating (3.42) at the linearized solution A^ * gives b  d J^ [A \ a  a  = Cl ^] 1  Ai *]  { x  1  h  & [AP] + BW\[AW]  + D^ [A ] ab  d d ^ [A ]  1]  c  d^iAi *}  1)a  c  d  a  b  1  + ••• = 0 ,  { 1] b  (3.43)  where the last equality follows from the linear perturbation equation (3.36). Thus the linear equation (3.36) and the identity (3.41) i m p l y that (3.38) holds. Similarly, to show that (3.40) follows from the first and second-order equations, (3.36), (3.37), and the identity (3.41), evaluate the second-order part of (3.41) at A^ * and A^ \ 2  b  a ^ ' [4i ,4 ] + d ^ [A^\A ] 2 a  2)  1)  a  a  + ClVf] ^  [4 ] + B^MA, *}  + t)^ [A^}  a  b  ab  ^ [4 ] + BM' [AP] (1)a  2)  c  2)  b  e  d T^ [A^] b  a  2  + B^\[A ]  b  d T^ [A^]  2)  a  + b^ [A ]  2)  d  F^lAi *}  1  d ^ [A^}  1  c  c  = CW[AP]  a)  d d T^ [A ]  ab  = C^IA, *}  {2)  a  (1)a  b  b  d  a  d d T^[A ]  + •••  + b^ [A[ \  d d ^[A ]  1]  a  b  d  ab  c  l)  (3.44) 2)  a  b  d  ,  Chapter 3. Spin-two Theories and General Relativity  57  (where the last equality follows from (3.36)), and the third-order part of (3.41), at  5 ^ )-[4 ,4 ,4 ] = CW[AN a  3  1)  1)  1)  + B^KiA^} + D ^ M ? }  d  a  ^  ^ > [ A b  l h  \ A ^ }  + bW* [A£\AP]  a  2)a  iy  + b^ [A } ab  c  d ^ [A^] h  a  B ^ A ^ ]  d  d a d t ^ i A ^ , A ? ]  ( 1] b  ^ [ A ^ ]  d a d ^ i A i  c  CP[Al \ & [Al ,AW] + iy  + CW[A?\AU]  + BW\[A$\AP\  [ A ^ \ A ^ ]  d d^\A$\AW) =  [  A[^\  a  T ^  b  1  * }  [ A  + •••  +  {  x h  •••  \ A ^ \  (3.45)  .  Using the second-order equation (3.37), (3.44) becomes  d  a  ^  a  [ A \  -B^MA^]  2  \ A ^ ]  +  d J* *[AWAl ] = -Cf ) [4 1 ) ] 2  2)  a  d*?W*[A£\ A?]  A d d i n g (3.45) and (3.46) gives (3.40).  - D ^ M P ]  ^ [4 4 ] (2)B  d d T^ [A[ \ a  a  b  l  1)  >  1)  A?]  •  (3.46)  Similarly, at higher orders, the divergence of  the nth-order perturbation equation w i l l be satisfied as a consequence of the lower-order equations together w i t h (3.41). Indeed, "the identity (3.41) is sufficient to ensure that the divergence of the nth-order perturbation equation contains no new information beyond what is available from the lower-order equations" [74]. This consistency criteria is at least as general as any discussed i n the previous sections. A g a i n , it is possible that there are theories which may not satisfy the divergence identity (3.41) but which can nevertheless be deemed to be consistent by some other criteria, (i.e., if within this framework only generally covariant theories are found, the general problem is not automatically solved). A t this point an additional assumption is made to ensure that the exact nonlinear equations of motion, T  a  = 0, are local partial differential equations that involve a  bounded number of derivatives. This requirement can be achieved by limiting the number of derivatives appearing i n the generalized identity (3.41) to at most one, i.e., take only the first two terms on the R H S of (3.41) and allow no derivatives i n B  A B  and only  Chapter 3. Spin-two Theories and General  Relativity  58  one derivative i n C . W i t h this restriction, an equal number of derivatives occur on each a  side of (3.41). Otherwise, including say the term D d d Jah  c  perturbation expansion, a single derivative of J ^ 7  tive of J ( r  n _ 1  )  c  a  on the R H S of (3.41), i n a  c  0  would be related to a double deriva-  0  which would i n turn be related to a double derivative of ^ (  on, i.e., "successive J ^ 7  0  n _ 2  )  c 5  and so  would contain a larger and larger total number of derivatives",  (unless the higher derivative terms conspire to cancel or vanish). W a l d points out that there m a y be other ways of ensuring that the equations of motion, T  a  = 0, are local  partial differential equations involving a bounded number of derivatives. Indeed he refers to this assumption as "probably the most significant loophole" i n his analysis. However, his procedure is still at least as general as all of the previously mentioned procedures and is manifestly more general than most. dT  7  T h e restricted form of the identity (3.41) is then  = CT  a  a  a  a  + B\d T  .  h  a  (3.47)  In order to ensure that a generalization of Maxwell's equations is consistent, then, it is sufficient that the quantities T  a  satisfy an identity of the form (3.47). It remains now to  solve for the unknown tensor fields C and B . a  a  0  It is convenient to rewrite (3.47) as follows 8 (B\F ) b  a  where B^  a b  8  = CT  ,  a  a  (3.48)  = S , and C °* = 0, but are otherwise the same as their hatted counterparts a  b  a  (limiting the number of derivatives). Since the equations are assumed to be derivable from an action principle, this identity can be reinterpreted as an infinitesimal gauge invariance of the action: 0 7  =  jd x\[d (B F )-C F A  a  a  h  b  a  a  For example, in [51], the identity corresponding to (3.47) is restricted to the form d T = Where we have made slight changes from [74] to be consistent with later calculations. a  a  8  cT A . a  a  Chapter 3. Spin-two Theories and General Relativity  Jd x A  [B d X  + C X] T  a  b  59  a  ,  h  b  (d x [B d x + c x] 4f 4  a  s  b  Eq.(3.49) is the statement that the action S  x  = B dX  + CX .  a  b  b  (3.49)  b  is invariant under the infinitesimal variation  Aa  8A  a  a  (3.50)  b  T h i s infinitesimal gauge symmetry will arise from an exact gauge symmetry if the commutator of two of these infinitesimal variations is again such a variation, i.e., if the space of variations is involute. In order to see this, W a l d gives the following argument: Let AA be the manifold of field configurations, AA — {A }a  T h e n the action S is a scalar function  on A4, S : AA —> 3£, and the infinitesimal variation 8\A ,  (3.50), defines a vector field on  b  AA. Let W  p  be the subspace of the tangent space V at p e AA that is generated by the p  collection of vector fields at p that satisfy (3.49), i.e., W is the vector space composed of v  the infinitesimal gauge symmetries of S. Let W denote the set of W , W = {W \p p  p  e AA}.  These subspaces W are said to be integrable if there exist submanifolds M < AA such that the tangent space of each point i n Af coincides w i t h W. called integrable submanifolds.  9  These submanifolds are  If the submanifolds can be found, then one can obtain  an action S w i t h the infinitesimal gauge symmetry (3.50) by taking <S to be the scalar field which is constant on the submanifolds. Otherwise, if the submanifolds cannot be found, i.e., the subspaces W are not integrable, then there is no exact gauge symmetry corresponding to (3.50). (However, there could exist a larger gauge group). Therefore, if I t is helpful to r e c a l l the definition o f integral curves, w h i c h are a special case of i n t e g r a l submanifolds: Let v be a s m o o t h vector field o n the m a n i f o l d M, (i.e., W = v\ a n d W = v). T h e n there exists a unique f a m i l y of curves M < M f i l l i n g M such t h a t for each point p e M, the tangent to the curve passing t h r o u g h p is v\ (i.e., the tangent space o f each p o i n t coincides w i t h W = v). In the case of a vector field the subspaces W = v (of the tangent spaces) are one-dimensional, a n d the integral s u b m a n i f o l d s , w h i c h are curves, can always be f o u n d . If d i m W > 1, then the submanifolds can be found if a n d o n l y i f (by F r o b e n i u s ' s theorem [73]) the c o m m u t a t o r o f two elements of W lies i n W. O t h e r w i s e , it is possible t h a t the ' W - p l a n e s 'twist a r o u n d ' " a n d cannot be s m o o t h l y 'added u p ' so t h a t a s m o o t h scalar f u n c t i o n cannot be defined. 9  p  p  p>  7  Chapter 3. Spin-two Theories and General Relativity  60  the subspaces are integrable, then the infinitesimal gauge symmetry (3.50) corresponds to an exact gauge symmetry. A consistent generalization of S^ \ 2  <S, is then invariant  under (3.50). B y Frobenius's theorem, the subspaces are integrable if and only if the commutator of two of these variations is again such a variation, i.e., if for all scalar fields 9 and ty there exists a scalar field x such that (3.51)  X 1  where V#, V ^ , and V  represent infinitesimal gauge directions, given by (3.50), i n the  x  manifold of field configurations. Substituting (3.50) into (3.51) gives  S (B d ty + C ty) - <v {B\d 0 a  e  b  a  b  a  + C 9) = B\d b  +  aX  c ,  (3.52)  bX  where Sg denotes the linear change i n the quantity induced by the variation (3.50). Since B  a b  involves no derivatives of A , and C , at most one derivative of A , Eq.(3.52) can be a  a  b  written  dB -—^8 A d ty oA  dC + -^-8 A ty dA  a  e  c  a  e  c  c  dC d 8 A ty od A  +  b  a  c  a  -{9 <- ty) = B\d  e  c  c  +C  aX  bX  ,  (3.53)  which, using (3.50), can be further specified: b  dA  (B d 9 d  c  d  + C 9) d ty + ~ - (B d 9 d  c  a  c  d  + C 9) ty c  r  dC  b  + dd A, a  •d (B d 9 d  a  c  d  + C 9) ty - (6> <- V) = B\daX + C c  bX  Eq.(3.54) is an integrability condition for the unknowns B , a  b  (3.54)  •  C , and x- In order to solve b  (3.54) then, one must find a scalar field x, which depends on 0 and ty (and A , i] , etc.,), a  and expressions for the tensor fields B , a  b  ab  C , (which are independent of 9 and ty). This is b  simplified somewhat by observing that, by definition, there is some freedom i n specifying  Chapter 3. Spin-two Theories and General Relativity  the fields B  a b  61  and C : First note that since the field A i n Eq.(3.50) is arbitrary, (3.50) is b  unchanged by the rescaling A-  /A ,  (3.55)  where / is an arbitrary function that is locally constructed from A , rj , and (in four b  dimensions) e  ab  , w i t h / ^ ° ' = 1. Correspondingly, the tensor fields B , C transform as a  abcd  b  b  follows: r>a  C A l s o , expressions for B  a b  r r>a  D'a  b  JJ  —•  ti  -  C' = B d f  b  6  — JfJ  ,  b  + fC .  a  b  b  a  (3.56)  b  or C , that are related by a change of variables are considered b  to be equivalent. Under an algebraic change of dynamic variables, A -» A (A ,r] ,e ) b  the quantity T  a  b  b  cd  ,  abcd  (3.57)  transforms according to  ^ r  = ^ = ^ £ £ = **££. 8A dA  8A  a  b  dA  a  (3.58)  a  Substituting this into (3.48), one finds that the tensor fields B , C , transform under a a  b  b  change of variables (3.57) according to  B\  -+ B\ = B  dA  a b  c  dA ' b  C  — 8A -> C = C r f . oA  c  c  b l  (3.59)  b  Thus one seeks expressions for B  a b  and C up to a rescaling (3.56) and a change of c  variables (3.59). A t this point, rather than attempting to directly find a solution to the complete integrability condition (3.54), it is pragmatic to expand the tensor fields i n a power series  Chapter 3. Spin-two Theories and General Relativity  62  in A , b  B\ =E  B ( n ) a  Cc = E Ci  » >  X= E X  ,  n)  n  ra  W  ,  (3-60)  n  and to first try to solve the zeroth-order part of the integrability condition (3.54)  (0 *-»  =d  bX  {0)  (3.61)  •  T h e most general linear expressions for the tensor fields (restricting the number of derivatives) are given by B^  a b  where c\ is an arbitrary constant.  = 0,  =A Cl  ,  c  (3.62)  Substituting these first-order expressions into the  zeroth-order equation (3.61), gives  d  ( W -  ed ty) = d c  (0)  bX  •  (3.63)  Taking the curl of (3.63), gives C i = 0. Thus, the first-order expression for the infinitesi m a l gauge invariance (3.50) is 8^A = 0. Indeed, i t can be shown by induction [74], a  that the general solution of the complete integrability condition (3.54) is B b = 8 b, a  C  c  a  = 0, so that a nonlinear generalization of Maxwell's equations has normal spin-one  gauge invariance, 8 A x  3.3.2  a  = d xa  Summary of Wald's consistency criteria  In summary, W a l d ' s argument is as follows [74]: Consider a nonlinear extension of the equations of motion for a spin-one  field,  = 0, (3.30), that is obtained by adding  terms of quadratic and higher order i n A to the second-order action SJfjh, Eq.(3.29). a  A l t h o u g h one can, i n principle, obtain some sort of equations of motion from such an extended action, i n general, since the linear quantity  is divergenceless, consistency  problems may arise i n a perturbation solution. Such problems can be avoided, however,  Chapter 3. Spin-two Theories and General Relativity  if the generalized quantity T  a  63  satisfies a generalization of the linear divergence identity.  T h i s generalized identity can be interpreted i n terms of invariance of the full action Se,Mi  under infinitesimal variations of the field, 8\ A . b  These infinitesimal variations  a  w i l l correspond to an exact gauge symmetry i f the space of variations is involute. T h i s requirement yields an integrability condition, the solution of which gives the infinitesimal gauge symmetry of the complete nonlinear theory. Thus, one is able to determine the gauge invariance of a consistent generalization of Maxwell's equations. T h i s consistency criteria has been applied to a number of problems. In [74], i n addition to considering extensions of Maxwell's equations, W a l d investigates extensions of a collection of spin-one fields as well as extensions of the linearized Einstein equations. In the former case, the generalization of a collection of k spin-one fields, {Aa ,...,  A }  1  obeying the linear equations J~^ = 2dbd^ A \ a  b  infinitesimal gauge invariance, S^Aa*  =  1  a n d c^^c"^  —c^irv,  space { A } ) .  a  D \^ a  k  a  = 0, was found to have Y a n g - M i l l s  = 0 A + c' ^A A , M  a  l  K  v  where  y  a  = 0, (i.e., & ^ defines a L i e algebra on the (internal) vector V1  earized E i n s t e i n equations, G^  ab  = 0, can have normal spin-two gauge invariance or  general covariance, (see Chapter 6).  In [18], Cutler and W a l d find that the nonlin-  ear generalization of a collection of spin-two fields, {jab^}, obeying G^ the infinitesimal gauge symmetry, 8\ ^ ^ b  r &"„ = ^a" 0(g- ) (2d j / a  1  0  a  ^u[-K p]\ a0  —  In the second case, W a l d finds that a nonlinear extension of the lin-  M  c  = {A/},  cda  l/  = 0, (i.e.,  ia b)  ab  - dd-fab' ), 3  — V  ( a  A  / i 6 )  ~  = d( hf a  g^ = 7] 8^„ + a^^j b , P  bv  ab  a  ab  ^ ab vK , c  ll  and  defines an associative commutative algebra).  10  v  = 0, has where  = a" ^), (l  In [75], these  infinitesimal gauge invariances are interpreted precisely i n terms of algebra-valued tensor fields and diffeomorphisms on algebra manifolds. 10  In [60], Reuter constructs a six-dimensional gravity model which has this new gauge symmetry.  Chapter 3. Spin-two Theories and General Relativity  3.3.3  64  Concluding remarks  The major problem investigated in this thesis, concerns consistent nonlinear spin-two theories. In Chapters 5 and 6, the consistency criteria outlined in section 3.3.1 is used to determine the nonlinear generalization of a number of formulations of the (linear) spin-two equations. In particular, we find theories that are not generally covariant. In Chapter 4, the effects that the type of potential chosen to describe a field theory have on possible interactions, are investigated. Again, there may be other criteria by which a theory can also be deemed to be consistent (as emphasized by both Fang and Fronsdal [28], and Wald [74]). For example, it may be possible to relax some of the assumptions that are made, (e.g., the decision to limit the number of derivatives appearing in the infinitesimal gauge invariance). Consequently, it may be possible to find yet more general classes of consistent theories. However, since we were able to find a counterexample to the assertion that any consistent nonlinear theory of a spin-two field be generally covariant within the present framework, we do not explore that possibility here (beyond the assumptions that were relaxed in our calculations, which were, however, not concerned with the basic consistency arguments presented in [74]). In [1], rather than adopt the criterion that, in order for the infinitesimal symmetry (3.50) to arise from an exact gauge symmetry, the space of variations be involute, Anco derives an equation for the induced variation of the field equations by an infinitesimal local symmetry.  11  Application of this equation to a candidate infinitesimal symmetry  A n c o is in particular interested in relaxing the assumption that the commutator of two "infinitesimal symmetries that implement a gauge invariance" is also an "infinitesimal symmetry that implements the gauge invariance". The motivation seems to be as follows: Consider a Lagrangian theory. Let Q be the group of infinitesimal local symmetries of the Lagrangian. The space of symmetries is involute, i.e., Q has Lie algebra structure. The gauge symmetries of the theory are obtained by evaluating the infinitesimal symmetries on the solution space of the theory, {$}, i.e., the gauge symmetries are the set ^|{^}, which is also involute. Let K, be the normal subalgebra of Q that gives rise to the trivial gauge symmetries, i.e., fC <J Q is such that k e fC implies thatfc|{<£}is simply the trivial symmetry <f) —• <f>. (In other words, K, is 1 1  Chapter 3. Spin-two Theories and General Relativity  65  (3.50) and to the commutator of two of the infinitesimal symmetries (3.50), (however, without specifying the form that the commutator takes), then gives restrictions on the arbitrary tensor fields (the case of a spin-two field, j ,  coupled to a spin-| field, (j> , is, A  ao  in particular, considered). The equation for the induced variation of the field equations by an arbitrary infinitesimal local symmetry 8<f>A, (where (f> represents generic fields, and the index A, represents A  generic indices), is derived in [1] as follows, (which is essentially identical (with minor modifications in notation) to what appears in [1]): L e t  12  <j>^ be a smooth two parameter  family of field configurations such that: (i) for each fixed f3 > 0, /3rj, (f>^° is a one-parameter family of local symmetries of the Lagrangian, where (a), 8<j> =  ,  A  a  n  d  ( )> $<t>A = b  HA[4>B\  (ii) (f) = •§p<f>^ \ =o=[3 is an arbitrary field variation of compact support. 1  A  a  First note that the mixed partial derivatives, with respect to the parameters a and (3, of the action is zero,  a=0 8=0  /3=0  "  the kernel o f the the h o m o m o r p h i s m f r o m Q to <?|{0}-) C o n s i d e r the symmetries g e Q t h a t are related up to a n element k e fC - these are the (left) cosets, g + IC = {g + k\k e K-}, oiQ w h i c h p a r t i t i o n Q. T h i s set o f equivalence classes is called the quotient set a n d is denoted QjK, = {g + K\g e Q}, w h i c h is i s o m o r p h i c to N o w , take a representative f r o m each element oiQ/K.. T h i s collection need not have L i e a l g e b r a s t r u c t u r e . However, i t gives rise to the gauge symmetries of the theory. F o r e x a m p l e , the c o m m u t a t o r of any t w o such representatives m a y have terms p r o p o r t i o n a l to the equations of m o t i o n . Unless evaluated o n the s o l u t i o n space {(j)}, the c o m m u t a t o r m a y fail to lie i n the collection o f representatives oiQ/K,. I n other words, the c o m m u t a t o r of two i n f i n i t e s i m a l l o c a l symmetries of the L a g r a n g i a n t h a t " i m p l e m e n t the gauge i n v a r i a n c e " m a y not also be one. However, i t is not clear t h a t the class of s y m m e t r i e s (3.50) w o u l d constitute such a restricted representation o f the infinitesimal s y m m e t r i e s . In any case, rather t h a n take the c o n d i t i o n t h a t the space of v a r i a t i o n s be i n v o l u t e , i n [1], a n alternative e q u a t i o n , n a m e l y (3.68), for the i n d u c e d v a r i a t i o n o f the equations is used. I n this discussion, the letters a and'/?, are used as parameters, a n d the s y m b o l ' 0 ' ('nought') is used as a l a b e l . T h i s deviates f r o m the i n d e x convention used elsewhere i n this thesis. 1 2  Chapter 3. Spin-two Theories and General Relativity  66  since for fixed ft, 8(j) ^ is a local symmetry of the action S. Now consider the following: A  da  I  dy  r  d a = 0 8=0  d  - 1  r s s i r A d f / ' a = 0 /3=0  A ^°A U  d{3  a J  Q = 0 8=0  d£ a,f3 dftdaj  da d/3 A  d(3  +  {S  where (8E)  A  — ^^-\ -p.  £)  e  + S  A  8=0  d{8fy-  (8£) j>A+[£ A  a = 0  A  a = 0 8=0  (3.65) j  8{8fydf/ 8f/  B  S  - ^  89 A  dp  (3.66) a = 0 8=0,  (3.67)  Since <f) is assumed to be an arbitrary field variation of  a=Q  A  compact support, this series of equations shows that  0 = (8S) + £ A  8(8<j> )  B  B  (3.68)  which is a condition that a candidate infinitesimal gauge symmetry (3.50) must satisfy. Let us consider the conditions (i) and (ii) regarding the two-parameter family of field configurations, (f)^, i n more detail. In particular, i n going from (3.65) to (3.66) (where both the a a n d f3 dependence i n 8(j> ^ are retained) A  d  8{8f/)  IQ,(3 Q = 0 8=0  use of the condition (ib) was made.  d£_ dp  a = 0  (3.69)  8=0  In other words, it was necessary to assume that  Scj)^ has no explicit dependence on the parameter ft, i.e., that the dependence of 8(f)  P  A  S^AI^B'^}  o  n  fl i  so n  =  l y through the intermediate cf)^, (as stated i n condition (ib) above,  however, it does not seem necessary to assume that there is no explicit dependence of 8§  a A  on a). It is not immediately obvious that this condition is compatible w i t h the  Chapter 3. Spin-two Theories and General Relativity  67  condition that <f>^ is an arbitrary field variation, (condition (ii) above). W h i l e there are certainly two-parameter families of field configurations which do not obey both of the conditions (i) and (ii), it seems plausible that one can always find a two-parameter family which does. To illustrate, first, from (3.69), it is assumed that S(f> ^ depends explicitly on A  <j>jf, i.e., <f> [a, (3,  4>s^i and hence, so must  A  3  <t>c'^\i  s  o  w  assume this as well. Consider  e  the following smooth two-parameter family of field configurations:  (3.70)  + <*PX [<I>B] + <XPXA<1>B ], 13  A  where n , £ , and XA, A  a  r  e  A  arbitrary functions of (f>%' ; and rj , £ , and XA, are arbitrary B  A  A  functions of 4>s- T h e n k°>>  0  6f/  da  (3.71)  '  which, for constant f3, f3 , is required to be a one-parameter family of local symmetries 0  of the Lagrangian, 6<f> . A l s o , from (3.70), 8  A  df/ d(3  = &[*B]  a=0 B=0  (3.72)  + U[<l> B  a a=0 6=0  w h i c h is required to be an arbitrary field variation of compact support, (j> . F r o m (3.71), A  a,B^ dfx  d  d/3  V  da  (S [f/}df/' VA  a = 0  6=0  V  6f/  da  +  XA a=0=B  +  A[?B  d4>\B  b/  da  a  S=0  XAW/]  a =  B=0  0  (3.73)  a=0=P  In going from (3.70) to (3.73), one sees that i n order that (3.69) holds, it is necessary that the last three terms i n (3.70), EAI^B' ], XAI^B], and XAWB ]-) are not included i n 13  13  the expression (3.70) for <f> , (otherwise, there is explicit dependence of [3 i n (3.71) and 0  A  Chapter 3. Spin-two Theories and General  Relativity  68  hence (3.69) would involve additional terms so that (3.68) would not follow), i.e., the two-parameter family, (f> , (3.70) must be restricted to the form 3  A  f /  + *VAIM  =  + ^  A  [f/\  + p&fo]  .  (3.74)  (Note that if the term rj [(j)^] was not included, the condition (3.69) would be identically A  zero, (which is the reason why 8<f> is written vsft<f> )-)T h e question is whether, w i t h 3  A  A  these restrictions, that <f> is still an arbitrary variation. F r o m (3.74), (3.72) reduces to, A  d/3  a=.0 /3=0  = &[M  •.  (3-75)  Since £°4 is an arbitrary function of 4>s, it indeed seems to be possible that one can always find a two-parameter family of field configurations, <j>°/\ such that the conditions (i) and (ii) hold, and that 8(f)  8 A  does not depend explicitly on (3. Consequently, i n  our calculations, having solved for the gauge symmetries of the theory, we investigate the conditions that the equation (3.68) puts on these gauge symmetries. We find that i n some calculations, the condition (3.68) enables us to eliminate or put further constraints on some of the parameters appearing i n the expression for the first-order gauge invariance, parameters which we were not able to include i n the general solution for the gauge symmetry.  Chapter 4  Potentials and Nonlinear Generalization  In Chapter 2 it was pointed out that there are many different potentials that can be used to describe a spin-5 field. Certain potentials clearly provided a more general description of a given theory than others. O n the other hand, it is also possible to have equivalent descriptions of a theory i n terms of different potentials. For example, a tensor and its dual can give equivalent descriptions of a theory yet be different types of tensors. W h e n two different potentials give equivalent formulations of some linear theory, the question naturally arises whether the possible types of interactions depend on which potential is used. In other words, how does the choice of potential effect the nonlinear generalization of the theory? In section 4.1 we investigate the nonlinear generalization of the theory of a curl- and divergence-free vector field i n two of its formulations. We find that although the linear formulations are equivalent, the possible types of interacting theories may not be equivalent. In section 4.2 we point out that a similar situation arises for nonlinear generalizations of the theory of a spin-one field i n three dimensions.  4.1  Divergence- and curl-free vector field  T h e theory of a divergence- and curl-free vector field can be described i n terms of a scalar field,  or, i n terms of an antisymmetric tensor field, A . ab  The two descriptions  have been shown to be equivalent up to a nonlocal transformation.  In addition, these  two classically equivalent formulations lead to completely equivalent local quantum behaviour [69]. Here, we investigate how the choice of potential effects the types of classical 69  Chapter 4. Potentials and Nonlinear Generalization  70  nonlinear theories which arise from a consistent generalization of this vector theory. We shall find that nonlinear extension of the vector theory is quite different in each case. It is not straightforward to determine if the types of nonlinear theories arising from the two equivalent linear theories are equivalent. However, to second order, our calculations indicate that the types of nonlinear theories need not be equivalent. We consider a vector field V in four space-time dimensions that satisfies the equations a  d[ V a  dV  = 0,  b]  = 0,  a  a  (4.1)  where [•••] denotes antisymmetrization. There are four ways of introducing potentials to describe this theory: Defining V = d (f>, one can write the theory in terms of the scalar a  a  field (j). Then d[ V ] = 0, and variation of the action a  b  S? = -yd*xd <l>d </>, )  (4.2)  a  a  with respect to <f>, gives the equations of motion 6S EE —i-  1  (2)  = •  = dV  = 0,  a  a  (4.3)  where the numerical superscript denotes the power of the field appearing in the expression. (Note that Eq.(4.3), J F A  ab  (1)  = dV a  a  = 0, implies that V  a  _ ba^ Alternatively, by defining V = d A , a  =  b  ab  locally, where  one can write the vector theory  ab  A  = dA  b  (4.1) in terms of the rank-two antisymmetric tensor field A . ab  Then, d V a  a  = 0, and  variation of the action  Z = -\$d x d A 8 A  S{  A  b  ac  c  b  a  ,  (4.4)  with respect to A , gives the equations of motion ab  £c(2)  4V = j~ f b  = d dA  throughout, boundary terms are ignored.  c  [a  b]c  =d V =0. [a  b]  (4.5)  Chapter 4. Potentials and Nonlinear Generalization  (Note that Eq.(4.5), A ^ = d V a  [a  71  = 0, implies that V = d <f> locally, for some scalar field  b]  b  b  <j).) These equations are unchanged by the transformation ab  A  where A  a 6 c  _^ ab A  g^labc} ^  +  ^  is an arbitrary rank-three tensor field. In four dimensions the gauge transfor-  mation (4.6) can be written ab  A  ab  A  ^bcdQ^  +  where A is an arbitrary one-form field, and e  abcd  c  ^  ^  is the L e v i - C i v i t a totally antisymmetric  tensor. A l s o , A \* satisfies the linear identity a  Vfil^O. ^ .  (4.8)  Equivalently, the vector theory (4.1) can be described i n terms of the duals of (f> and A , ab  *<f> and *A . ab  Let us now see how the choice of potential used to describe the vector theory (4.1) effects nonlinear generalizations. In the formulation of (4.1) i n terms of a scalar field, the quantity  Eq.(4.3), does not satisfy some tensor identity. Hence, there are no  constraints, of the k i n d discussed i n section 3.3, on the types of terms that can be added to the Lagrangian. T h i s means that any non-pathological addition to £ ^ w i l l result i n a complete consistent nonlinear theory, described by the Lagrangian £ $ . In other words, the Lagrangian  = £f + V(</>) ,  (4.9)  where V(<f>) is an arbitrary nonquadratic potential, w i l l give consistent equations. F o r example, the cubic potential v(<f>)  is one possibility.  =  ij>d m, c  (4.io)  Chapter 4. Potentials and Nonlinear Generalization  72  O n the other hand, i n the formulation of the vector theory (4.1) i n terms of A , ab  the quantity A^  satisfies some tensor identity, namely the identity (4.8). This imposes  b  restrictions on the types of terms that can be added to the Lagrangian. In particular, to assure consistency i n a perturbation expansion, adopting the consistency criteria of [74] (see section 3.3), we consider only full nonlinear equations that are invariant under the generalized infinitesimal gauge variation (see section 4.3.1) S A  =D  cd  where the unknown tensor fields D e  abcd  and rj  ab  d\  +E X  [cd]ah  Xa  a  and E  abcd  ,  a[cd]  a b  (4.11)  are locally constructed from the fields A ,  abc  ab  (the M i n k o w s k i metric), with D^  =e  abcd  abcd  , E^  = 0. The requirement  abc  that the space of variations be involute, then gives an equation that can be solved for these unknown tensor fields. One solution for D  and E  abcd  abc  gives, for the most general  gauge invariance of the extended theory, S A  =e  cd  Xa  cdab  d\ a  In theories w i t h this type gauge symmetry, A  cd  form d A . ac  c  .  b  (4.12)  can only appear i n the action i n the  It would not be possible, for example, to construct a theory that would  be equivalent to that described by C^ of Eq.(4.9) with V(<f>) given by Eq.(4.10). If (4.12) was the only allowed gauge invariance for consistent extensions of  , then the  types of nonlinear theories resulting from the generalization of the vector theory (4.1) would depend on the potential chosen to describe the linear theory, i.e., equivalent linear formulations of (4.1) would not give rise to equivalent types of nonlinear theories. There may, however, be solutions other than (4.12). We find that the most general first-order infinitesimal gauge invariance can be written 8 A { 1]  x a  cd  = J (4 e ^ U a  1  c  a  e  +A e 6 a  + {d - 2d ){A t d \ eba[c  x  7  ab  e  d]  c d a e  a A ) 6  e  +  + d A 7  a f e  e  c d a 6  A e dX) cdbe  ae  a  b  a A e  e  Chapter 4. Potentials and Nonlinear Generalization  + e (d A e ^ X [c  1  d  + dA e X)  abe  ab  b  e  a  a  c  + ed A e  acde  ba  + (ei - 2e )(d A e ^ X ^ 7  73  a  e  7  + dA e X)  d  bcda  be  a  e  be  X  becd  be  a  ,  (4.13)  where d\, d , &\ and e are arbitrary constants. (Eq.(4.12) is just the special case d\ = 7  d  7  = ei = e  7  7  = 0.)  Higher-order calculations may put further conditions on these  parameters, however, i n light of the complexity of (4.13), it is not clear how one could obtain a solution to all orders other than i n the special case d\ = d  7  = e\ — e  — 0,  7  (4.12). It is difficult to see if one could now construct a theory (having the complicated gauge invariance (4.13)) that would be equivalent to that described by Eqs.(4.9) and (4.10). Generalization of the vector theory (4.1) is quite different i n the two cases:  SJf* can  be extended arbitrarily whereas there are fairly strict conditions on possible additions to S^ A  . Indeed, the resulting types of nonlinear theories are not necessarily equivalent even  though they arise from equivalent descriptions of the same linear theory.  4.2  Three-dimensional electromagnetism  It is interesting to note that a precisely analogous situation arises for the theory of a spin-one field i n three dimensions. T h e equations of motion for a massless spin-one field in M i n k o w s k i spacetime are given by Maxwell's equations, (see Chapters 2 and 3), 3F  = 0,  ab  a  where F A  a  ab  d F [a  bc]  = 0,  (4.14)  = —F . One usually formulates this theory in terms of the vector potential  by defining F  ba  ab  — 2d[ A y T h e n d[ F ] = 0 and variation of the action a  b  a  bc  Sl% = -iJ<r xF F* i  ,  b  ttb  (4.15)  w i t h respect to A , gives the equations of motion a  8S  (2)  jrU)» =  2d d A ^ [b  oA  a  =  b  a  = dF  ba  b  = 0 ,  (4.16)  Chapter 4. Potentials and Nonlinear Generalization  (which i m p l y that F  = dG  ab  74  locally, where G  abc  = & ).  abc  c  abc]  These equations are  invariant under the transformation A  -» A + d X ,  a  a  where A is an arbitrary scalar. Also, J ^ 7  satisfies the identity  0,  dF  (4.17)  a  = 0.  {1)a  a  (4.18)  Alternatively, Maxwell's equations can be formulated i n terms of the completely antisymmetric rank-three tensor field G , by defining F abc  = dG .  ab  abc  c  Then d F a  ab  = 0 and  variation of the action (4.15), w i t h respect to G , gives the equations of motion abc  6S  {2)  = (which i m p l y that F  = \^d G  = \d F  d  ah¥  [c  ah]  =0,  (4.19)  = d[ A^ locally). These equations are invariant under the trans-  ao  a  formation gabc _^ gabc  where  A\  +  Q^abcd]  ^  is an arbitrary rank-four tensor. Correspondingly, J F ^ ' satisfies the identity  i  abaI  0[e*£]]  = 0•  (4.21)  In four dimensions these two descriptions of Maxwell's equations are identical, since Qabc  =  e  abcd^  Qabc  a  n  d  ^  are duals of each other). However, i n three dimensions, Qabc  _  e  a6c,0  b i t r a r y scalar field ty, and A  = 0. It is not difficult to see,  [abcd]  5  for  s  o  m  e  a r  then, that i n the formulation of (4.14) i n terms if), (i.e., G ), the linear equations (4.19) abc  no longer have a gauge invariance and J ^ does not satisfy some tensor identity. There 7  are therefore no restrictions on possible additions to S  G  .  O n the other hand, i n the description i n terms of the vector-potential A , the identity b  ( 2)  (4.18) constrains the possible extensions of C  Aa  to theories that are invariant under the  Chapter 4. Potentials and Nonlinear Generalization  75  infinitesimal transformation (cf. (3.50)) 6A x  where B  a b  and C  a  b  = B\d X  + CX ,  a  (4.22)  b  are locally constructed from the fields A , r] , and e d (in four dib  ab  mensions), w i t h BW  = S  b a  abc  and C f ) = 0, (cf. section 3.3 and Ref.[74]). In [74], (cf.  b a  section 3.3), it was found that the most general solution for Eq.(4.22) i n four dimensions simply gave the usual gauge invariance (4.17).  In three dimensions, however, a more  general gauge symmetry may be possible. To second order we find that the most general gauge invariance resulting from a consistent nonlinear generalization of Maxwell's equations i n three dimensions can be written  6A x  a  = d X + kXe d A c  a  b  abc  + 2k XA d A 2  b  [a  b]  + ••• ,  (4.23)  where k is an arbitrary constant (the special case k — 0 simply gives Eq.(4.17)). A p p l i cation of the equation (3.68) shows that k = 0. Then, the gauge invariance is given to all orders by the usual invariance (4.17). Paralleling the discussion given for the vector theory (4.1), the two equivalent formulations, S * and SQ* , of the theory of a spin-one A  field i n three dimensions, need not necessarily give rise to equivalent types of nonlinear theories.  T h i s means that the types of nonlinear theories arising from two equivalent  formulations of a linear theory may be inequivalent.  4.3  Solving for the gauge invariance  In this section the calculations leading to the expression (4.13) for the first-order i n finitesimal gauge invariance for the vector theory (4.4), and the expression (4.23) for the second-order gauge invariance for a nonlinear generalization of Maxwell's equations i n three dimensions (formulated i n terms of the vector potential A ) a  are outlined. Briefly,  we adopt the criteria of [74] i n order to ensure that the nonlinear theory SA  AB  arising  Chapter 4. Potentials and Nonlinear Generalization  f r o m t h e g e n e r a l i z a t i o n o f S *,  76  (4.4), is c o n s i s t e n t .  A  T h i s e n t a i l s , as d i s c u s s e d i n d e t a i l  i n s e c t i o n 3 . 3 , f i n d i n g a g e n e r a l i z a t i o n of t h e l i n e a r i d e n t i t y (4.8), r e i n t e r p r e t i n g t h i s i n (2)  t e r m s o f a n i n f i n i t e s i m a l gauge i n v a r i a n c e o f S ^ A  v a r i a t i o n s is i n v o l u t e .  a n d r e q u i r i n g t h a t t h e space o f these  :  T h u s o n e a r r i v e s at a n i n t e g r a b i l i t y c o n d i t i o n for a n u m b e r o f  u n k n o w n t e n s o r fields. It is e x p e d i e n t t o e x p a n d these u n k n o w n s i n a p o w e r series i n terms of t h e field A  ao  S^ A  b  a n d first o b t a i n a n e x p r e s s i o n for t h e l i n e a r i z e d gauge i n v a r i a n c e o f  w h i c h c a n b e s i m p l i f i e d b y e q u i v a l e n c e u n d e r a c h a n g e of v a r i a b l e s a n d t h e i n h e r e n t  a r b i t r a r i n e s s i n t h e d e f i n i t i o n o f t h e u n k n o w n t e n s o r fields. I n t h e case o f t h e v e c t o r t h e o r y ( 4 . 4 ) , t h i s leads t o t h e e x p r e s s i o n (4.13) for t h e i n f i n i t e s i m a l l i n e a r gauge i n v a r i a n c e . H o w e v e r , i t is u n c l e a r h o w t o f i n d a s o l u t i o n t o a l l orders.  I n t h e case o f e l e c t r o m a g -  n e t i s m , t h i s leads t o t h e e x p r e s s i o n (4.23) for t h e second-order gauge i n v a r i a n c e . I n t h i s case, w e are f u r t h e r a b l e t o m a k e use o f t h e i d e n t i t y (3.68) t o show t h a t (4.23) r e d u c e s t o n o r m a l s p i n - o n e g a u g e i n v a r i a n c e (4.17).  4.3.1  Vector theory  T h e l i n e a r v e c t o r t h e o r y (4.1) c a n b e g e n e r a l i z e d b y a d d i n g powers of t h e field A  ab  of  o r d e r t h r e e a n d h i g h e r t o t h e s e c o n d - o r d e r a c t i o n S * , E q . ( 4 . 4 ) . T h i s gives rise t o t h e A  n o n l i n e a r a c t i o n SA  b  a n d t h e g e n e r a l i z e d e q u a t i o n s of m o t i o n A b,  ab  a  B y t h e a r g u m e n t s o f [74] (see s e c t i o n 3.3), these g e n e r a l i z a t i o n s o f S * A  w i l l be consistent  i f t h e q u a n t i t i e s A b satisfy a g e n e r a l i z a t i o n of t h e l i n e a r i d e n t i t y 5[ «4-o6] = 0> E q . ( 4 . 8 ) , a  c  w h i c h is o f t h e f o r m d (D A )  = E A  cdab  a  where D ^  a b c d  = t  , E^  abcd  a b c  ab  (4.25)  = 0. T h u s , t h e i n f i n i t e s i m a l gauge i n v a r i a n c e of t h e n o n -  linear theory is taken t o be S A  cd  Xa  ,  bcd  ab  = D^  a b  d \ a  + E ^X , a  b  a  Eq.(4.11). This infinitesimal  Chapter 4. Potentials and Nonlinear Generalization  77  gauge invariance w i l l arise from an exact gauge symmetry if for all one-form fields A , 7r a  a  there exists a one-form field Xa such that [VxM where V\ , a  V ,  and V  Vb  Xc  = V  ,  Xc  (4.26)  represent infinitesimal gauge directions, given by Eq.(4.11), i n  the manifold of field configurations. Substituting (4.11) into (4.26), the condition that the space of variations be involute, gives the integrability condition QJj[ef]cd  D (d X d 7r  - 8 ir 8 X )  klab  8A  a  kl  +  b  c  (*A £ Q  d  cte/1  a  K  b  c  - (<^ £  + E (X 8 n  c [ e / ]  a  f  c  d  ) A = D^ 8 cd  c  w8 X )  -  fkl  d  f  c  d  + E ^Xd  ,  d  cXd  (4.27)  where 6\ etc., denotes the change i n the quantity induced by the infinitesimal variation a  Eq.(4.11). Before attempting to solve (4.27) for D  8 A , ab  Xa  a b c d  , E  a b c  ,  and Xa, note that  (a) the infinitesimal gauge variation (4.11) is unchanged by the transformations jQ[ab]cd  ^  c[ab]  where f  a b  rjlab^eQ^c^  _^  E  j~)[ab]ce jd  is locally constructed from A ,  +  E<  i[ab]jc^  r) , and e  ab  ab  abcd  ?  , with f ^  a  = 8 , and that (b) a  b  b  under the change of dynamical variables A  a b  ^A  a b  (A  c d  ,  V e f  ,e  g h e f  ),  (4.29)  the tensor fields transform according to jj[ab]cd  dA  ah  ^ j-yefcd  dA*f ' jj<c[ab]  £cef8A  _^  ab  (4.30)  8A f e  F i r s t , consider the zeroth-order part of (4.27), which involves zeroth- and first-order quantities, e 8 7r 8 X " klab  c  d  a  b  AU  + e \88X kla  8A  kl  c  d  a  {  ^  88 A d  kl  Chapter 4. Potentials and Nonlinear Generalization  - (A  78  <-> T ) = e d efab  C  c  .  0) aXd  (4.31)  The most general expressions for the first-order tensor fields are, (limiting the number of derivatives), (l)[ab]cd  edf[b a]c  =  D  die  v  + de  A  ahef  7  (i)f[ab]  =  E  e i t  ^d  faabed^  +  Aef  +de A  cd  ecf[a  efV  A  a ]  8  cdab  abec  + e 8A f  f  dabc  cd  8  [a  d  b]f  5  +e e  f  c  e  +ed A  ab  4  ^ dA  d e c  f  9  ^ [ a ^ c  +  = £d Z>|  d  g  dc  +ee dA  + e e dA 7  + de A  b]d  d  2  ^ [ y ] *  +  efV  +ee dA cabf  c d  ^ab^cd  +  d  ce  i  '' ,  +  edA ^  1 ) [ a 6 ] c  c[a b  6  c  V  =£ e ^ f  )  /  , (4-32)  M  i=l  where the d{ and the ej are arbitrary constants. (The terms d ecd[b a] 3€  A  e  a  n  d  e3€  cfd[b a] dcA  d  are omitted since they are not linearly independent from the other terms.) At this point we also note that the most general expression for f^ f b (1)a  = hA  +fe A  a  a  b  2  is  a b  .  cd  bcd  (4.33)  Substituting (4.33) into the first-order part of the transformations (4.28) shows that i=9 D  _  (l)[ab]cd _^ >{l)[ab]cd D  ^ dD  +fD *  i:X)[ab]cd  l  l  1  - 2  [ah]cd  g  f D^ ^ a  Cd  2  i=l t=9  — 4:f D^ ^ a  d'.£)( )^  EE ^  Cd  2  1  cd  i=l (l)f[ab]  E  '(l)f[ab]  E  _  ^  e i  EJ  ~  1 ) f [ a b ]  /iE^  1 ) / [ a 6 ]  -  2f E{  1)f[ab]  2  t'=l  -4/ 4  1)/ta6]  2  ==Ee;.4  1)/[o6]  ,  (4.34)  i=l  i.e., fi effects e and d , and / effects e , e , c?4, and <i. However, the most general 8  9  2  4  second-order change of variables, A^ * 2)e  terms in the expressions (4.32).  6  5  = a e ^A \A , abc  3  e  bc  does not effect any of the  Chapter 4. Potentials and Nonlinear Generalization  79  Substituting the expressions (4.32) into the zeroth-order part of the integrability condition (4.27), Eq.(4.31), gives [2(d - d )d ir d X [e  2  1  f]  + 2{d -d  a  a  x  + d + d' )dKJ \\ l  2  8  + 2{d + d' - 2d )d TT dl \K a  2  9  7  + d e ^d nd X 6  ab  e]  a  a  9  - 2(d + 4 ) d r d A [ e  f  8  + d' t d ir d \  e  A  ejah  c  c  a  +  b  + 2{e + e' - 2e )Tr d d \ a  a  1  8  7  - 2{e' + e )^ n\^ 8  efab  4  c  a  f]  a  b  e )^d d X e]  9  abc  5  - ( A ^ ;r ) = e d efab  a  c  a  a  ee Kd dX  +  b  ab  5  8  c  / ]  d' e *d K d \  - 2(e' +  f]  + e' e 7r d d X  e  9  [e  a  | a |  a 7  a  c  f]  a  b  • (4.35)  { 0) aX b  Note that the parameters e and e' do not appear i n (4.35). In order to find conditions 6  2  on the parameters di and e;, one can take the divergence of (4.35) so that the R H S disappears.  T h e n , choosing various cases for the fields A and 7r , gives conditions for a  a  the d and e;. T h i s task is somewhat simplified by combining terms on the L H S into l  divergenceless groups, which i n addition to vanishing when acting on (4.35) w i t h d , give e  possibilities for the form of x i ° ' -  Combining techniques (see Chapter 5 and Ref. [43] for  more details about this approach), we find di = d ,  d = d-  2  2d ,  1  8  de = e = 0, 5  xi  0)  = [(di + 4 - 2d )\ e d ir a  d  7  where d[ k ] = 0. a  b  b  abcd  \d' = d' = e ,  e = e - 2e ,  7  9  x  4  s  7  4  d' + d — 2d7 = e' + e — 2e7 , 9  2  8  + d' (ir d \ 4  d  d  - r d X )} d  c  d  c  t  - (ir ~ X ) + k , a  a  c  (4.36)  In the special case where A and ir are constant one-forms, the a  a  first-order part of the integrability condition (4.27) does not involve any second-order quantities. Checking at this order under these circumstances gives the further conditions  e i  = e ; 2  ei = 0  or  e' = 0 . 6  (4.37)  F r o m the first-order expressions (4.32), the conditions (4.36), and (4.37), the first-order  Chapter 4. Potentials and Nonlinear Generalization  80  gauge invariance of a nonlinear generalization of the vector theory (4.1) can be written W ab  g  =  A  D  W W c d  d  c  X  £(i)/M  +  d  + e A dX) abed  c  ednb  ab  d  c  6  c  e/  + d f e ^ ^ U r f A / + e d A X) cabJ  d  dc  + e e d A X ,+ cdab  7  where the fx and f  f  cd  f  f  ee dA X dabc  8  f  c  d  e i  there is indeed a best way to choose fx and f . 2  d e fA d \ abe  7  x  + d' d A X f  4  ef  ab  f  a6ec  7  +  x  c  A'' 0 A e  c  (J  e' d A X c[a  6  ,  b]  7  d  b]  c  + {e - 2e )e X d A dec[a  f  c  7  - 2e - d + 2d )e  have as yet not been specified.  2  +  b  c  + (di - 2d )e t°a A U + (4 + ec/  ef  + \A^d \ ^)  c  4  7  a]  d  + d' {A d \  c  e  dx{e d X A  A/=  ce  (4.38)  It is not clear how best, or if  T h e complexity of (4.38) seems to defy  determination of the gauge invariance to all orders, except i n the special case where all of the remaining nonzero parameters are set to zero, which gives (4.12) as a general solution. It is possible that Eq.(3.68) would give further restrictions on the parameters i n (4.38). However, it is reasonably tricky and involved to check this. (We have checked i n the special case where A is a constant field, so that only the e; terms remain i n (3.68). a  Beginning w i t h no conditions on the e;, (3.68) confirms that eg = ex — 2e and ex = 7  e .) 2  In any case, the form of (4.38), (which includes the special case (4.12)), seems to indicate that the possible nonlinear extensions of the vector theory (4.1) depend on the choice of potential used to formulate the theory. Since we are able to illustrate the contention that the choice of potential is indeed important i n the example of three-dimensional electromagnetism, we do not pursue the effects of (3.68) on (4.38) i n a more general scenario. Choosing (somewhat arbitrarily) e' = 0 from Eq.(4.37), using f 6  2  to set d& — 0, and  fx to set d' + dx — 2d = 0, Eq.(4.38), the most general solution to the zeroth-order part g  7  of the integrability condition (4.27), can be written as Eq.(4.13) w i t h  xi°*  — k-  taking the special case dx = d = ex = e = 0, gives the gauge invariance (4.12). 7  7  c  Again,  Chapter 4. Potentials and Nonlinear Generalization  4.3.2  81  Three-dimensional electromagnetism  T h e formulation of this problem is given i n [74], and discussed i n detail i n section 3.3. Summarizing, the consistency arguments of [74] require that generalizations of Maxwell's equations have the infinitesimal gauge symmetry S\A  = B bd \ a  b  a  + C(,A, Eq.(4.22). T h e  condition that the space of variations be involute, gives the integrability condition given by Eq.(3.54). Under the rescaling, A —» / A , Eq.(3.55), the tensor fields B ,  Cb, transform  a  0  according to Eq.(3.56).  Under a change of variables A  B b, Cb, transform according to Eq.(3.59).  b  —> Ab(Ab,r) d,e b d), c  (Note that A^  a  = ri\A , c  a  c  and A^  Eq.(3.57), = 0.) In  four dimensions, the most general expressions for the first-order tensor fields are given by Eq.(3.62): CJ. * = C\A , and B^ b 1  — 0. In three dimensions, however-, it is possible  a  C  to write the more general expressions B^  =  b a  C^  WA e , c  b  a c  =  Cl  A  + ce dA c  a  2  .  b  abc  (4.39)  Substituting (4.39) into the zeroth-order part of the integrability condition (3.54), (which is Eq.(3.61)), gives 2b e d tyd e b  1  +- (il>d 6 - 0d ty) = d  c  abc  Cl  a  a  {0)  aX  •  (4.40)  Taking the curl of this equation shows that ci = h = 0 , and that x^°*  1S  a  (4.41)  constant, which, checking at second order i n the case of constant ip,  9, is zero. However, c is still completely arbitrary. Because this problem is not too 2  complicated, we can go to the next order. T h e most general second-order tensors are BW\  =  bAA  Ci  =  cAdA  2)  +  b  2  3  a  c  a  c  bS AA, b  3  c  a  c  + cAdA+ b  4  a  b  cAdA, b  5  b  a  (4.42)  Chapter 4. Potentials and Nonlinear  82  Generalization  and the most general expressions for / ( ) and A are 2  a  f  = fAA  {2)  x  A  ,  a  a  (4.43)  = nAAA  3)  .  b  a  2  a  b  (4.44)  Under the second-order rescaling, (4.43), one gets  B  W\  _  C[  B'W  -> C f  2)  = bAA  b  )  + (b +  b  a  2  a  f )8 A A , b  3  1  = (c3 + 2 / ) A 5 A + c  1  c  c  a  c  c y l a A  a  4  o  6  +c A 5 A ,  (4.45)  6  5  6  6  0  and under the second-order change of variables, (4.44), one gets  B  i(2)b  ^7(2)^  a  )A A  + (6 + n ) O A  6  =  ( & 2 +  2 n 2  a  3  2  ,  c  c  (4.46)  where b = 63 + /1. 3  Substituting (4.42) into the first-order part of the integrability condition (3.54), gives  [(62 - 2b' )d tyd 0A 3  b  + c tyd 0d A a  5  + (c + c  b  4  + (4 + c )tyd d 6A  b  a  b  5  2 2  a  )^5 ^ A 6  6  + (4 -  b  b  a  c )tyd 0d A 2  2  b  b  a  + c tyO0A } - (0 <-•tf>)= 8 4  a  ,  (1) aX  (4.47)  where b = b + 2n , b = b' + n , and c' = c + 2 / i . 2  2  2  3  3  2  3  3  must be of the form A (0d ty — tyd 8). A d d i n g and subtracting  B y inspection,  b  b  b  (for example)  (b - 2b' )[tyd 8 0A 2  3  a  b  b  + tyd 9d A ] - (0 <-»ty),  (4.48)  b  b  a  to (4.47), gives  "(62 - 2b' )d (A 0d ty) b  3  a  b  + (4 + c + b - 2b' )tyd d 0A 5  + (c - c % - 2b' )tyd 08 A  b  3  2  3  b  a  2  3  a  +  b  b  + (c + c )tyd 0d A 2  5  2  b  b  •  a  -(9~ty)  c tyd 0d A  b  4  +  a  b  c tyU8A 4  = d  [l) aX  a  ,  (4.49)  Chapter 4. Potentials and Nonlinear Generalization  83  Taking the curl of (4.49) one finds 4 + c + b - 2b' = 0 , 5  and  =  2  2b'3)A (0d ty b  ( 6 2  -  c + c  3  5  2 2  = 0,  c = 0 ,  - tyd 9). F r o m Eq.(4.43), we choose f  b  b  6' = 0, and from (4.44), we choose n 3  = -\b  2  (4.50)  4  = -63  x  to set b . (Then also dbx^  2  2  to set  " 2  -  = 0.) W i t h  these choices, the second-order expressions (4.42) for the tensor fields reduce to 5  (  2  )  6  a  Ci  = 0 ,  2)  =  2c 22A cd[aAc] .  (4.51)  U p to second order, then, this gives the infinitesimal gauge invariance 6\Aa  k\eabcd cA b + 2k 2xA bd[aAb], first-order  Eq.(4.23), where  k=  = d aX +  c , as a solution to the zeroth- and 2  parts of the integrability condition (3.54) with dax^  — 0. However, i n this  example, we find that the identity (3.68) gives the further condition that k = 0. For a spin-one field, the equation (3.68), for the induced variation of the quantity T ', a  (4.16), by an arbitrary infinitesimal local variation of the Lagrangian (4.15), is  0 = S\J~ a + J~ b —r—8\Ab , oA  a  +  (  T h e zeroth-order part of (4.52) is simply  4  '  5  2  )  Sx°*J r( 1* a — 0, which, by definition, is true.  first-order part of (4.52) is 0 = Substituting pW , a  4 ^ +^ 0)  (2)a  -8  1 ) a  C  from Eq.(4.16), and S^A , a  ^ A - ^ g U A ^  .  (4.53)  from Eq.(4.23), into (4.53), and taking A  constant so that the quantity ,7-*( ) drops out, gives 2 a  0 = kX(e cd^ ad b]dbdbA -e bacdcd dd[dAb]) , c  =  kXDF e  abc  cb  .  (4.54)  The  Chapter 4. Potentials and Nonlinear Generalization  Thus, (4.52) shows that k = 0.  Chapter 5  Conformal Invariance — Spin-two Coupled with Spin-zero  In Chapter 3, many procedures to determine the possible classes of consistent field theories resulting from the nonlinear generalization of the equations of motion for a massless spin-two field were discussed. In general, the starting point for these calculations was the linearized Einstein equations. necessary.  However, this assumption is more restrictive than  In particular, it implies that the gauge invariance of the linear theory is  lab - » 7a6 + d( \ ). a  0  It was noted i n Chapter 2, on the other hand, that the massless  spin-5 equations, (2.1), can be regarded as being conformally invariant. T h e requirement that any interacting spin-two theory reduces to linearized gravity, however, eliminates this as a possibility for a candidate gauge invariance for the interacting theory. In order to incorporate the possibility of theories w i t h conformal invariance, a more general formulation of the spin-two equations is necessary. One method of describing the spintwo equations i n terms of a symmetric rank-two tensor, j b, such that the equations are a  invariant under a conformal rescaling of the metric, is to introduce an auxiliary field (for more on auxiliary fields, see [33], [34], [16], [65], [35], [7], [36], [29]). Indeed, this is the approach that Fierz and P a u l i [33] employed to derive the equations (2.61) from a variational principle (see Chapter 2). Here, we employ this strategy to try to extend the analysis of section 3.3 to include conformally invariant theories. In particular, we consider the nonlinear generalization of a linear conformally invariant theory of a symmetric rank-two tensor field, 7 ^ , coupled to a scalar field, </>.  85  Chapter 5.  5.1  Conformal Invariance - Spin-two Coupled with Spin-zero  86  Conformally invariant linear equations  T h e massless spin-two equations, V '(J>ABCD = 0, Eq.(2.1), with 4>ABCD AA  field of confor-  a  m a l weight w = —1, are conformally invariant, (cf. section 2.2.1). Indeed, the conformal invariance of the equations is believed to have some importance for the future prospects of quantization [7], [36]. However, the conformal invariance of the spin-two equations is not necessarily respected by the equations for the potentials that are chosen to represent the spin-two field, 4>ABCDI (cf. section 2.2.3). In order to be able to find interacting spin-two theories that may be invariant under conformal rescalings of the metric, it is necessary to first find a promising linear theory. Choosing a symmetric rank-two tensor field as the potential, this means a linear theory that is invariant under the transformation  7afc -> lab - ttrj  ab  ,  (5.1)  where f l is an arbitrary real scalar field (cf. A p p e n d i x C ) . One way of finding such a theory is by brute force: W r i t e out the most general second-order action i n ^  a b  (assuming  second-order derivatives),  £ M  G  =  I  a  l 7 a b  a  a b 7  + a  c b 2 l  d d  + a d8i  a  +  d  c  c l a b  3l  cd  a  4  7  n  ,  7  (5.2)  where the a, are arbitrary constants, and derive the equations of motion, c 0(2)  = ~ ^ — = 2a d c/( 7 a  2  6 ) c  c  OJab  + 2a D 1  7  o 6  + ad0i a  3  b  + n (a d d ab  c  3  + 2 a D ) = 0 . (5.3)  d  4  lcd  7  F i n d conditions on the arbitrary constant coefficients by demanding that the equations of motion are traceless (and hence conformally invariant), •no.bS.Mal =  ( «2 + 2  ±a )d d i  ab  3  a  b  + (2a + a + x  T h i s gives the conditions that 2a + 4et = 0 and 2a 2  3  3  x  8 a  4  + a + 8a 3  ) n  4  of preciseness, take the one free parameter to be, say, a , and fix a 2  7  =  0 .  = 0. x  =  (5.4)  For the sake — 4  - T h e n the  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  87  Lagrangian density (5.2) w i l l yield equations that are invariant under the conformal transformation (5.1) provided that a  2  In Chapter 6 we consider the case a are conformally invariant.  a = ! (a - |) .  = -|a ,  3  4  (5.5)  2  = | , and find interacting spin-two theories that  2  However, i n this type of formulation of the equations, the  1  "normal" spin-two gauge invariance, 8~f  ab  = d( ip ), °f the equations is lost. a  b  2  One way  to formulate a conformally invariant spin-two theory i n terms of 7 &, but to maintain A  the "normal" spin-two gauge invariance, i.e., to find a linear theory that has the local symmetry hab = d( \ ) - tt'q b , a  is to introduce, along w i t h j , ab  b  (5.6)  a  an auxiliary field, the simplest one being a scalar field, <f>.  In the next section, we consider the linearized massless conformal K l e i n - G o r d o n equation i n curved spacetime, as a candidate linear theory.  5.2  Spin-two coupled w i t h spin-zero - conformal K l e i n G o r d o n equation  First let us consider the (full nonlinear) conformally invariant theory described by the action St = - | J d xy/^(g W ^V ^ 4  + eft$ ) ,  ab  a  2  b  (5.7)  I n [26] (see also [5]) the Lagrangian that corresponds to a = | is written down. F r o m (5.3), the requirement that d£$£ = 0 fixes all of the a* parameters to be those of the second-order Einstein-Hilbert action (cf. section 2.5). This excludes (5.5). 1  2  2  b  Chapter 5.  88  Conformal Invariance - Spin-two Coupled with Spin-zero  and the equations of motion resulting from variation of (5.7),  3  / = ^ ( V V $ - Cm) = 0 ,  (5.8)  \y/=g{Z&G + e ) = o,  (5.9)  a  a  ab  og b  ab  a  where,  Q  ab  ^  =  _ l ) ^ ^ y $ V $ + (1 - 2 £ ) V $ V $ + 2 £ # $ V V $ - 2 £ $ V V $ . (5.10) c  a  6  a b  c  c  a  Taking £ = | , Eqs.(5.7) - (5.9) are conformally invariant [54].  Under a conformal  4  —*• u> g b, the equations transform according to T% —• Tt, —  rescaling of the metric, g  2  ab  a  u>~ J-£, and 8£ —»• Sf' = u>~ ££ , and the action 3  6  c  6  b  is equivalent to §t up to a boundary  b  term, which we ignore (since throughout we assume locality). T h e quantities Tt, and £ £ satisfy the identities b  V £f-lg  a  a  b  V $Tt a  l<f>Tt-g €f ab  =  0,  (5.11)  =  0,  (5.12)  (where we are taking £ = | ) . Correspondingly, the equations of motion are invariant under the  transformations gab  $  ->  gab  + V( A ) + a  -» $ + l  A  ug , 2  b  ab  ^ +w * ,  (5.13)  1  T h e equations (5.8) are sometimes referred t o as the massless conformally coupled K l e i n - G o r d o n equations i n c u r v e d spacetime. T h e massless K l e i n - G o r d o n equations i n M i n k o w s k i spacetime are < 9 3 $ = 0. T h e u s u a l p r e s c r i p t i o n for generalizing equations to curved spacetimes is s i m p l y to m a k e the " m i n i m a l s u b s t i t u t i o n s " , rj —> g , a n d d —• V , [73]. T h i s yields V V $ = 0. However, V V <J> — £.R<I> = 0, where the constant £ is arbitrary, is another p o s s i b i l i t y consistent w i t h the " m i n i m a l s u b s t i t u t i o n s " p r e s c r i p t i o n [73]. T a k i n g £ = | makes the equations c o n f o r m a l l y i n v a r i a n t . 3  a  a  a  ab  a  ab  a  a  a  a  4  P a r k e r [54] has a neat argument to show that the equations (5.9) are c o n f o r m a l l y i n v a r i a n t : O n e  G = G b + 6 6 & / w , where G b = G b(g b) - G b(u> g b), (see, e.g., [73]). T a k i n g = $ gives Ga {§ gab) = Gab(gcd) + 60 j(ffcd, $ ) / $ . One then also has G b(&g b) =jS b(g b) + 6@ab(® ,gab)/&- Since $ g = &g , ( t a k i n gJ > = w ~ ^ $ ) , equating G (^ g ) = G ($ g ) gives Gab(gcd) + 6@ab(gcd,®)/$ - Gab(g b) + d>Qab(^ ,gab)/^ > w h i c h is precisely the statement that t h  c a n w o r k out t h a t  2  2  ab  a  a  a  2  LO  a  a  a  2  h  2  a  a  o  2  a  2  ab  ab  ab  2  2  a  equations (5.9) are c o n f o r m a l l y i n v a r i a n t .  2  a  2  ab  ab  ab  a  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  89  where A is an arbitrary one-form field. a  Let us now linearize this theory to obtain a linear theory of a spin-two field, 7^, coupled to a spin-zero field, (f>, that is conformally invariant. T h e second-order part of the action (5.7) is second-order action S f  =  -ljd x[d cf d ^  =  - \ Jd x  4  +( ( ^  a  )  a  { 1 )  \d <t>dj + ([w  4  + \i)(d d  a  c  (i.e., substitute g ° = rj b, g$ — -) bi a  b  a  R ^ + R^+2RWcf>)] , +Yd d  - dd )  d  c  l c d  (5.14) la  cl  b  [a7c]  *}}  ,  = 1, and $( ) = <j> into (5.7) and retain terms x  a  that are second order i n the fields). Variation of (5.14) gives the equations of motion s s P  77(1)  ?  where G^  ab  (l)ab  = d d <f> - £ ( d d a  6<f> _  SS.(2)  c  _ 1  £ [G  ( 1 ) a 6  - d d  d  a  + 2{r)  )  a  l c d  d d j)  ab  a l  = o,  - d d <i>)\  c  a  c(  h  =0 ,  is the linearized Einstein tensor. T h e quantities J ^ and £ ^ 7  (5.15) a  b  satisfy the  linear counterparts to (5.11) and (5.12), (I)a6  \ff  ]  ' Vabe^  = 0,  (5.16)  = 0  (5.17)  (recall that £ = | ) . Thus, the linear gauge invariance of the theory (5.14) is lab  -> lab + <9( A ) - VlTjab a  6  (5.18) Alternatively, we could have obtained the linear theory (5.14) - (5.18) by brute force: W r i t e out the most general second-order action i n 7^ and <f>. ' M G  2  =  «i7a6 Vl  + a  ab  + a <f)d d iab a  5  b  c h 2 l  d d a  c l  ab + a  + ae^O-f  d d c  3 7  d  + a 4>U(j> , 7  l c d  + a  4 7  D7  (5.19)  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  90  where the a; are arbitrary constants, and derive the equations of motion, c  (l)ab  cWn.  MG *A  _  vo  ^  2)  2  ==  c  —  (l)ab  .  = t '  s  M  o a o i j  + a0  Gi  5  ,  n •<£ = 0 ,  a&nJ.  a <p + a ri 6  lab  JJC(2)  (5.20)  8<f>  Find conditions on the arbitrary constant coefficients by demanding that the equations of motion satisfy the identities, (in order to be able to get the gauge invariance (5.18)), =  °a^MG  2  '  U  K  f MG  (5.21)  = U,  VabO  2  MG  where k is an arbitrary constant. This gives a = —2a,i, « 3 = 2aj, a = — a , fcas = —4oi, 4  2  x  &a = 4ai, fc a = —6ai. For preciseness, fix a = — ^ . These conditions give 2  6  7  x  Scc  [2)  = -\j  d x [C(y/=^R)W - l(R U 4  - ^ 0 ° ^ ] ,  il  (5.22)  together with the equations of motion  8(f) rn(2) r  (l)a6  _  °^>CC _ i >  G^ -\(r d d <f>-& ^'<f>) ab  ab  c  }  (where £ = | ) . The quantities T  l  c  (5.23)  and £cc, satisfy identities  x  a6  c c  p, p(l)a6 oc a  =  0  (5.24)  c  (5.25) and the linear gauge invariance of the theory is lab d>  ->  lab + 0( \ ) - Q,T) , a  b  —> ^ — &f2.  W i t h A; = — | this gives precisely (5.14) - (5.18).  AB  (5.26)  Chapter 5.  5.3  Conformal Invariance - Spin-two Coupled with Spin-zero  91  Nonlinear extension of spin-two coupled with spin-zero  Thus we have a linear theory of a spin-two field, ^ , coupled to a spin-zero field, <f>, that a b  is not only conformally invariant, but is also invariant under the "normal" spin-two gauge transformation. We would like to find the types of theories that result from a consistent nonlinear generalization of the linear theory (5.22) - (5.26). In particular, we seek to find nonlinear theories of an interacting spin-two field that are conformally invariant. Thus, the goal is to find a complete action, Sec, and exact equations of motion, £ Tec  d c c  = 0 and  = 0, _  cd  P  8 S  cc  _  _ SScc _  n  9  by adding higher powers of the fields, 7 j, and <f>, to the action S . a  The linearized form of  cc  the equations (5.27) are given by Eqs.(5.23), £ c <  = 0 and Tec  cd C  =  _.  0- Following precisely  the same reasoning as i n section 3.3, consistency of the nonlinear equations is ensured if the exact quantities £  1  c d  c  c  C  d (B £ ah  a  cd  and TQ satisfy the two generalized identities + D Tcc)  d  = C £ +A T c  ab  c c  H£ ab  b  c c  b  d  cd  + GT  =  CC  c c  ,  b  C  (5.28)  0.  (5.29)  These identities are imposed i n order that the higher-order equations do not further restrict the solutions of the lower-order equations.  Equivalently, the complete theory  must have the infinitesimal gauge symmetry  Sled  =  6<f> where B^)  ab cd  =6 6 , a  b  {c  d)  B d\ ab  cd  +C  a b  = D dX  = -h ri , 0  ab  c d  \ + a  nH ,  (5.30)  cd  + A X + VLG ,  ab  a  a  (5.31)  b  b  b  D^  = 0, G^ = g , C^\  ab  0  To obtain the linear theory (5.15) - (5.18), h /g 0  0  to obtain the linear theory (5.22) - (5.23), h /g 0  0  d  = 0,  = 0.  — 2, (e.g., g = |, h = 1), a n d 0  Q  — —1/k, (e.g., g = —fc, ho = 1). 0  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  T h e tensor fields rj d,  (f),  c  B d,  H ,  ab c  ab  G,  D  a n d (in four dimensions)  a  ,  b  a  and A , are locally constructed from 6  C d a c  eb -  a  7^,  and A can include single derivatives of b  C d  cd  92  c  lab and 4>. (Note that the second term i n (5.31) can be absorbed into the t h i r d term by setting f l becomes B  a  b  c  d  =  =  c  a  b  (  c  d  + 0 . T h e n (5.31) becomes  B dd Xb  +  b  6f d B  -A X /G  )  ,  b  ab c  H  a  =  ab  H  {  a  b  )  ,  (C  a  and  c  d  — A H d/G)\ a  C \  d  c  cd  C \  c  d  D  a  b  d  a  \  b  +  f l G and (5.30)  Also, note the symmetries  + CtH .  a  =  S</> =  y )  We find that the condition that the infinitesimal symmetries (5.30) and (5.31) arise from exact symmetries, restricts the form of the infinitesimal gauge invariances to f>9cd 6<f>  where V  a  =  V A ) -  g hn ,  =  lg X V 4>  + g Sl  ( c  ab  r f  b  a  cd  (5.32)  0  0  ,  is the derivative operator associated w i t h the metric g  Eqs.(5.89) a n d (5.90)).  (5.33)  ab  = n  ab  + qj  ab  (cf.  A l t h o u g h a theory having the symmetries (5.32) and (5.33)  appears to be conformally invariant, there exists a change of variables such that (5.32) and (5.33) reduce to Sg b a  where, g  =  V A  =  -\g %V 4>  ( a  a  a  — e ° g , <j> = —(j), X = X e ° 9  ab  g  ab  w i t h the metric g . ab  b  6 )  b  ,  (5.34) + g& ,  , and V  a  (5.35)  is the derivative operator associated  In other words, the gauge invariance (5.32) and (5.33) is simply  general covariance. I n the next section, we outline the calculations which lead to this conclusion. Thus, we were not able to find a conformally invariant theory of a spin-two  field  by formulating the equations i n terms of a symmetric rank-two tensor field, j b , w i t h a  "normal" spin-two gauge invariance, coupled w i t h a scalar field, <f>. Indeed, it is somewhat interesting that a nontrivial trace identity could not be established between j  a  b  and (f>.  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  93  In the next chapter, we w i l l drop two of the principal assumptions made i n the analysis of this chapter: we consider, on the one hand, formulating the spin-two theory i n terms of a symmetric rank-two tensor field, 7&, which does not have "normal" spin-two gauge a  invariance, and, on the other hand, formulating the equations i n terms of a traceless, rank-two tensor field, -fj b  I* both instances we find theories which are not generally 1  covariant. We are also able to find theories that are conformally invariant.  5.4  Solving for the gauge invariance  A c c o r d i n g to the consistency arguments of section 3.3, the nonlinear extensions of  S  GC  (cf. Eqs.(5.22) - (5.26)), See-, w i l l be consistent if See is invariant under the infinitesi m a l gauge transformations  (5.30) and (5.31).  In order that these infinitesimal gauge  symmetries arise from an exact symmetry, one must exhibit, for all pairs of one-form fields and functions, 0 and ipb, Co  a n <  a  i ^> ©  a  n  d T , pairs of one-forms and functions, \c  and IT, r and A , <j>b and T, respectively, such that a  [Ve ,V^}  =  V  [Vc.,V ] =  V  [Ve,V ]  a  a  r  =  + V,  (5.36)  + V,  (5.37)  V+ + V ,  (5.38)  Xc  n  Tc  A  e  r  where Vg , etc., are vector fields on the manifold of field configurations defined by a  Eq.(5.30) and (5.31).  5  T h e calculation is considerably more involved than in [74] since it involves the coupling between two fields. Although both j i, and <f> were included in the analysis of [51], the coupling between them was not considered. In [29], the generalization of the Fierz-Pauli Lagrangian (2.62) written in terms of h t, and (j) fields is investigated. 5  a  a  Chapter 5.  5.4.1  94  The general equations  Rewriting via  Conformal Invariance - Spin-two Coupled with Spin-zero  (5.30)  Next,  i n terms of the tensor fields B ,  and  = 0 , and F = 0 , since  c  C , a  cd  [VQ, VT]  = 0.  ab  d ty (B d e  cd  9h  a  +  that cj>  (5.38),  b  ab  i n component form, can be written  Eq.(5.36),  dB  ab  cd  we find first, from  (5.31),  H , G, D , A , and  ab  (5.36) - (5.38)  dC  b  ef  g  + c  h  h  ab  h  e f  +c\  ty d (B efdaeb  cd  9  g  s  e )  o  a  )  -^d ty  +  g  a  (D d e ef  b  e  + jfijWe  +  f  Ae e  {D d e + A e ) 9h  9  g  dd <j>  h  g  e  +  dC  BC  b  h cd  + C\ B )  ef ty (B dJ ab  h  ef  b  s  -  {o  + -Qf-tyb  a  <- ty ) =  B  c  c  a b c d  d  (D d 6  +  9h  g  h  + c \  a X b  d  X  A9) 9  g  + n#  a  (5.39)  cd  and, BT)  BD + ~^d ty  ab  ab  8 ty (B d 0  + C 9)  ef  d% cd  a  b  cd  e  h  f  cd  h  a  (D d 6 c  + ty 8 A ] - (0 <-> ty ) = D d b  b  Finally,  Eq.(5.37),  fdB  + C< (j)  e  c  +^  cd  + EG .  c  aXb  (D< d ( d  c  8B cd G d<f>  cd  -d C  c  C  Xc  (5.40)  i n component form, becomes  {B^ d Q cd  C  C  d  +A  ab  6c  + A8)  cd  b  d  + A%c)  ab  abrj  I\ —: 9%  ii f +  ef  e  - CdS C ab = B 8 T d  +  cd  a  ab  C  D  C T D  AB  D  +  AH  ab  ,  (5.41)  and, BC ^—(B 8 ( ab  c  BC + -7r7(D d (  + C ()  cd  d  d  ab  died  ab  d  a  - (6 A b  d<p  + A() a  b  b  Q  a  =D Bn ab  + Ar  + GA  a  a  a  (5.42)  T h e notation 8^ is used to denote the linear change i n the quantity induced by the b  =B  variations 8^  ab  °TpbV " ')  =  ab  d tp + C ?p and a  cd  a  b  —hb'Jab + ab  cd  a  ™  dd i c  8^ <f>  o^Ociab a b  b  H  = D d ip -f A ty , ab  a  a  b  a  ^7~ 4> P + a d  d<f>  (  b  a  dd 4> c  / 8^ O <p , b  c  (5.43)  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  95  and SQ is used to denote the linear change i n the quantity induced by the variations A  fi7a6 = &>H and Sa<f> — O G , ab  djab  dd j c  d<t>  ab  dd cf> c  In order to determine the gauge invariance of the complete theory, one must solve the equations (5.39) - (5.42), the integrability conditions, for the unknown tensor fields B d, H , G, D , A , C d, and Xa, IT, r , and A . This is tractable order by order i n -f ab  ab  c  b  ab  a  c  0  ab  and <f>. Note first that there is a certain arbitrariness i n the definition of the tensor fields, which can be seen by considering that, since ip and f l i n (5.30) and (5.31) are arbitrary, b  the equations (5.30) and (5.31) are unchanged by the transformations ip —> f ty b  a  f2 — > eO, where f  a b  and e are locally constructed from j , ab  e  <j> and t] w i t h f^°* ab  a  a  arid = S  a  b  and e^ ) = 1. Correspondingly, the tensor fields are defined up to the transformations 0  nab . nlab nae n cd —> rs d = rs j C  .  cd  a  rjab  file  e  rb , re  ,  s~tb  fe  U cd —> U cd — rS cdOai 6 + ^ cd) 6 5 ^a6  ^ j-ylab £)  A*  _> A' = A f  H  ab  G  e  b  J  ae  e b  —> H' — eH ab  ab  b  +  D df , ab  e  a  b  ,  - • G' = eG.  (5.45)  Also note that, under a change of variables <j> 7a6  -»• ^ ( ^ , 7 i ) , 0  -» 7o6(^7cd),  the tensor fields transform according to  (5-46)  b  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  O-jcd  _  b A  ^ =  rr  96  0(f>  C ^ - f A ^  T r _ r r ^7crf . p ^7crf Ofab 0(p V =H * 1 2 - + G § .  G  (5.47)  T h i s rescaling freedom (5.45), and the equivalence up to a change of variables freedom (5.47), can be used to simplify the expressions for the tensor fields. Since there are a number of equivalent ways of using these freedoms, we choose the one which gives the nicest (in our opinion) final result.  5.4.2  T h e linearized equations  T h e next step is to expand the tensor fields i n a power series i n the fields 7 & and <j), (e.g., 0  B  ab cd  = 5TJn B^ * , n  etc.) and to substitute the zeroth- and first-order expression for the  ab  cd  tensor fields into the zeroth-order part of equations (5.39) - (5.42). T h e most general linear expressions for the tensor fields are B^  a b c d  = b  e  o7e{c  + Wr)  abe d)  + b8  b  5  + b j rj  ab  2  + b 8\  a [cl d)  +b  ab  lcd  c l  7  +  cd  b 8\ 8  b  Al  c  d)  + b <f>8\ 8 = £ biB?  ab  d )  ri  ab  3lV  + b ^ r,  b  6  cd  b  cd  8  c  ,  )ab  d)  cd  j=0  C  = c d -f e  {l)e cd  Q  a  b{c  +cd^  abe  e  d)  x  cd  + c d -y 2  {c  e d)  + c 8 d -y + e  3  {c  d)  cdj 8 b  4  b  e  {c  d)  8  + c d jr]  + c d&7 Vd + c d 4>r] + c 8 d <f> = ] T c C\  e  5  e  e  cd  6  7  (i)ab  A  ( 1 ) 6  =  =  d l  r,  a b 7  +  d r +d 2  1  b  2  3  a7  ab  8  {c  l)e  d)  x  i=0  (l)ab  cf> = J2d Dl  ab  3V  a d -f + a d <f>-ra d b  cd  3  b  D  e  1)ab  i  t=l  = t'=lJ2^  (1)6 i  A  5  ,  cd  ,  Chapter 5.  H  Conformal Invariance - Spin-two Coupled with Spin-zero  = hir] j +  {1) ab  ab  h fab 2  + h ri b<f> = E 3  97  >  a  i=l  G  =  ( 1 )  gi-y  92</> =  +  J29iG i (  (5.48)  •  1 )  To see how the terms i n (5.48) are effected by the transformations (5.45), write the most general expressions for f^  and  b e  32 /  (  1  )  6  e  =  / l 7 ^ e + /27 e + / 3 ^ e = 2  () 1  e  =  E / ^  6  i  ei  e2<l>j=i= Y e EV l  +  (  1  ,  e .  t=l  .  i  (5.49)  Substituting (5.48) and (5.49) into the linear part of (5.45), gives at first-order e/  5  E  =  /(1)ab  i=0  bM \ 1)a  d + hB^  = E c.-C?^ + /aC7i  C'W/  +  ab  1)6 e/  eS  f BP  ab  2  + f B^  ab  ef  3  + / <7<  8  1)6 e/  2  + / G< 3  1 ) b  e /  EE £  ab  t=i  i=l  ab  1)ab  cJCp* ,  (5-50)  E^+ffoeiG^+^G^EiE^. D^ ,  =  D'W  KBl  t=0  j=i  and  =£  i=0  i=0  -  ef  i-l  5l  1)a6  A  = A * ) , i.e.,  / ( 1 ) b  1  modifies the terms  6  G|  e/ and  1 ) 6  e /  ,  (i.e., / i :ffects b and C 3 ) and can therefore be used, for example, to eliminate one of 4  these terms; F\  affects the terms B Q* f and <  e  ab  e  G P , (i.e.,  e i  C^^e/?  e  the terms BQ * j 1  ab  and G ^ ^ , (i.e., f 6  affects / i i , 51); and  3  affects 6, 8  (i- -5 e  c ) ; E^ 8  h affects  6,  6 c ) ; i * ^ affects 1  2  affects the terms  and  affects the terms H^} and G ^ , (i.e., e affects / i , d  2  #2)- However, at this point, it is not clear how best to exploit these parameters.  3  We  could simply state that, w i t h the benefit of hindsight, we choose certain values for /  1 ?  / 2 , / 3 , e i , and e However, rather than limiting our choices at this point, we will simply 2  suppose that the values of /1, f , f , 2  3  e i , and e are fixed, without stating explicitly their 2  cd  ,  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  98  value. We then denote the modified values of the coefficients of the terms effected by the transformation (5.45), b , c , 6 , c , b$, c , hi, h , gi, and g , by a prime: b\, c' , b' , c' , 4  3  6  2  8  3  2  3  6  2  b' , c' , h[, h' , g'i, and g' . 8  8  3  2  To see how the first-order expressions (5.48) are effected under a change of variables according to (5.47), note that the most general change of variables (up to second order) can be written  7a6  =  m  + rn r] <j> + m yr)  i7ab  2  + m j"f 7  <f> =  ab  3  + m ^ <f)  ab  4  +  ab  ra rc 677 5  ™el<x 1bc  +  a  + msj4>r] + m (f>(j>T} + m r] f -<(  C  ,  cd  ab  ab  ni<f> + n j  +  2  n  9  AB  w  3 # + " 4 7 7 + 5<^7 + " 6 n  ab  •  7cd  c r f 7  cd  (5.51)  Substituting (5.48) and (5.51) into the zeroth-order part of (5.47), gives B  = mi8\ 8  { 0 ) a b  + m3r)efr b ,  b  e f  e  s)  i°d = (-mih  H  + g rn  0  0  G  3  cd  ,  b  2  = h r] 0  ,  cd  0  0  2  (5.52)  0  C^°* f = 0, A'°) = 0. T h e form for the zeroth-order gauge invariance (5.18) gives b  6  e  mi = 1, m  2  and  - kh m )in 0  =nf  {0)a  = g rii - 4 / i n = g ,  (0)  and,  2  D »  - 0, m  = 0, ni = 1, n  3  = 0. A t this point we take mi — 1, m  2  3  = 0,  n = 0 but leave m and n\ unspecified. W i t h these choices, substituting (5.48) and 2  2  (5.51) into the first-order part of (5.47), one finds that the expressions for the first-order tensor fields are effected by a change of variables as follows: B  W a b e f  =  + ^M  J2b'M \ 1)a  d  + ( 2 m + m di) B ^ 5  + (m + m d )B^ D  W a b  =  2  rn J2 diD\  1)ab  »'=!  3  ab  c d  7  + m B^  ab  8  cd  + m B^  a b  2  +  1)ab  cd  4  + 2n D[ 4  1)ab  cd  + 2n D 6  w  +  m 2  + m 2# 6  d )B^ 2  )ab cd  +  b c d  m B^ 6  ab cd  = E M ^ c - ,  ab  cd  (2m  { 1)ab 2  + nD  { 1)ab  5  3  = ]T d'iD?  )ab  i=l  ,  Chapter 5.  ef  Conformal Invariance - Spin-two Coupled with Spin-zero  2-s i°i i=0  -  C  cd +  m  2 1^5 a  i=l  11$  e  / +  2  m  3 C  G  6  e  mdL  +  2  2  f — 2_^ ^i i=0  cd '  C  e  7  t=l  J2KHS + (m g[+g m -ho(8m  =  /  99  2  0  s  +m  5  + 2m10))H[l d )  7  i=l  + {gom - h (im 4  0  + 2m )) iJ^cd  7  6  {2gom - h (m  +  9  0  + 4 m ) + m g' ) H^]]  4  8  2  2  3  =  ^  h'iH  ,  icd  i=i 2  ni E  =  =  ^  + {g n - h {Sn - 2 n ) ) 0  5  0  4  + ( 2 n - 4/i n ) G  6  5 o  3  0  5  X ) 2  E ^ G f ,  (5.53)  «=i  and, C^ f  — C^ f,  b  b  e  e  A^  = A ^ ) , i.e., m affects the terms w i t h coefficients b' , / J , and  h  6  4  / i ; ms affects 63 and h[; m 3  6  and h ; m 2  8  2  affects b , b' , (in particular the combinations, 65 = \(b -\-b' )), 5  6  5  6  affects fei, b' , (in particular, the combination, b' = |(fri + & ) ), ^ i , and / i ;  7  4  affects 67, /I'J and / i ; m 3  9  x  affects / i ; and m 3  1 0  4  ^8  2  affects b and Z^; 723 affects g' \ n 2  2  4  affects d\  and g[; n affects c/ , ^ and g ; n affects d and g[. W h i l e we can unambiguously choose 5  m  3  2  6  2  = — (m g' — /to(4m + m ) h' )/2g J  9  2  2  8  4  r  3  0  so that h' = 0, and n = — (ni<7 — 4:h n )/2g , it 3  3  2  Q  5  0  is not immediately clear what is the best choice for the remaining parameters. However, we view t h e m to be fixed at this point. Quantities effected by a second-order change of variables are denoted by an overline. F i n a l l y , note that the R H S of the zeroth-order part of (5.39) is and the R H S of the zeroth-order part of (5.40) is g U . 0  {0)  d(cxd°* —  Substituting  rj h I[( \ 0  cd  0  n<°> from (5.40)  into (5.39), the zeroth-order part of the equations (5.39) and (5.40) can be combined to give the zeroth-order integrability condition corresponding to (5.36) dB^ — -dctydde0f +dC^ —-tp dd9 cd  d  ab  ab  d  c  e f  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  h /<9D + Vab— —z  dA^ d ip d 9j + —  (1)cd  ^ ipdd d 6f  d  0  c  d  e  c  - (e  e  100  <-ty )= a g> .  c  (5.54)  (BX  c  90 V lef OdcJef , Similarly, the R H S of the zeroth-order part of (5.41) is <9( T^ ' — T] hoA^°\ and the R H S d  0  ab  a  of the zeroth-order part of (5.42) is go^°\ Substituting  from (5.42) into (5.41) one  gets the zeroth-order integrability condition corresponding to (5.37) dn%>  - dB^ H riefh — C7  _ #  d  i  u  ah  0  c*7cd  dB^ wd</>  d  0  e /  h rj \— — — g \ df  ab  h 1- rjn^hn—; h- „ d^cd  ab  0  cd  ab  0  Q£)(l)ab  -9o-  - Q  8<f>  ^  K  b - V*%C*S§ AM ]  =  d(aT  . (5.55)  V>  Now substitute the expressions (5.48) for the zero- and first-order tensor fields into the zeroth-order part of the integrability conditions (5.39), (i.e., (5.54)) and (5.40). T h i s gives, [\bo  (d e d ty  abe  Hd  lel  + \cod d e a  {c  + \c ty n0 4  {c  c)  a  ty  c)  e  2  + ( +|  d)  le[  a  b  + \c' ty d d e  ahe  m)  + d e d ty ) + (h - b' )d e d ty  +§(& -  a  b  C5  C 6  e  c  +(  e  d  +f{ o  4  {c  d)  a  + \c' )i> d d e 2  e  (c  a  6  + (4 +  s  C l  b' )d 0 d ty  5  d)  e  3  e  4  6  3  -(^^Vc) =  [c  d)  W ^  a  a  lal  d)  \c )ty d d e  + | a ) ) Vcdty d d 9 + § ( c + | a )  ai  {c  e  e  ]  6  a xg ,  (5.56)  .  (5.57)  )  (c  and, [(ax + | a ) ^ 5 5 r e  3  a  e  +  \a ty ne ]  - (6 <-> 0 ) = g Ii  e  3  i0)  e  C  C  (Note that b , b , 2&i = b + 6 ' , 265 = b + 6 ' , 6 , 6 ' , J 2  3  x  4  5  6  7  8  0  1 ?  J , d , do not appear i n 2  3  (5.56). T h i s means that there exists a change of variables such that these quantities can be set to zero (cf. (5.99) - (5.110)). Also, c , c' , a , h'i, h , h' 7  appear.)  8  2  2  3y  g' , and g' do not x  2  Substituting the expressions (5.48) for the zero- and first-order tensor fields  into the zeroth-order part of the integrability conditions (5.41), (i.e., (5.55)) and (5.42) gives, SlVcddaC  [h\ + h (h + b + 46 ) - g b 0  2  3  0  7  +I  (g\  + h {Ad + d ) Q  x  2  g d )\ 0  3  Chapter 5.  +  Conformal Invariance - Spin-two Coupled with Spin-zero  ttd (  (h + h (4b~U + b + b'e) - gob's) + ( d n  {c d)  2  + r] ( d n  [h (  e  cd e  5  0  0  5  0  7  0  g c' )  + 4c + c ) -  2  3  ^ (ft (4ai + a ) -  + 4 c + ce) - g c +  0 Cl  (h (c'  d)  {c  101  g a )] 0  3  0  4  8  = d^rff , (5.58)  2  and, (g\ + K{4d  + d )-  gods) nd ( a  2  x  + (h (A  a  0  + a) - ga) 3  ai  0  ( d ft  2  = c?oA  e  e  (0)  .  (5.59)  (Note that the parameters h'3 and g' do not appear i n the integrability condition (5.58), 2  so that there exists a change of variables such that they can be set to zero.) There are many ways to solve the equations (5.56) - (5.59). For example, to solve (5.56) for Xd°\  n o  ^  that, by inspection, x °^  e  m  u  t be a linear combination of the terms  s  d  ty d 0  a  d  a  ty d e  a  d  a  -  e d r,  -  edr  d  ty d e a  a  a  a  ,  d  -  d  e  abe  d  e d ty , a  a  d  (e d ty - ty d e ) . e  a  b  a  e  (5.60)  h  Therefore, adding the term (which is zero) . (4 + \c )d e d ty {c  A  -{  d)  a  - (c + \c )d e d ty  a  [c  d)  a  + (  a  + \b d d e \e ^  + §4 W ^ i - A  Cx  4  3  Q  a  {c  [b  d)  abe  C l  \c' )d e d\ \ty  +  2  [c  a  a  d)  - \b d d e\ \e ^ , 0  e  a  {c  b  d)  abe  (5.61)  e  to the L H S of (5.56), and rearranging, gives d(c  (\c rd B 2  d)  a  +  ( + \c' )Vd\ \6 Cl  2  a  +  d)  + b d e e d ty 1 2  + ( | ( 6 - b'e) + 5  Cl  0  e  ic  d)  abe  + \c )d e d ty 2  {c  a  H  a  (4 + \c )ty d B  b  A  d)  + (l\ -  +  a  a  Eu +  + \{K + c )d d 0 e ty  d)  o  a  + ( c + \ce + |(«i + I « ) ) r)cdtyed d 0 5  e  3  a  a  {c  w  d)  abe  \b e Q  4+  abe  l  8 d ty ) e  d)  b  c )d B d ty a  2  i  (c  d)  \c^ n0  +  e  a  (c  a  d)  + \ ( c + | a ) r, </>D0] e  6  3  c d  e  - ( * c ~ </0 = 5 x i ) • ]  (c  (5.62)  F r o m (5.62), one can identify  xd 0) = [(4 +  1 2  c )ty d e 4  + i6 e/ o  d  6 e  a  a  + (ci +  |4)^°a ^ + |40 ^° a  ^ 0 a ^ ] - (0c *-> 0) C  •  o  (5.63)  Chapter 5.  102  Conformal Invariance - Spin-two Coupled with Spin-zero  T h e n (5.62) reduces to } bod e e d ty  + (b\ - E> + ' + lc )d e d ty  ahe  e  2  {c  d)  a  4  c 3  (c  4  + ( | ( 6 - h) + c + \c' )d e d ty x  (c  2  ]a]  d)  + (c + i c + f { s  6  Q  a  + \c ty m  a  5  d)  [c  4  c )d d e  + i(6o +  a  b  0  a  + \ (<* +1«3)  d)  (c  m)  ty  ahe  r/ ^ °0 C (  e  e  e  + | a ) ) W « 9 < 9 0 ] - (6» ^ Vc) = 0 . (5.64) e  ai  3  e  a  a  c  It is now relatively straightforward to see that the coefficients of each of these terms must be zero. For example, take ip to be a constant one-form. T h e n (5.64) becomes a  i(bo + c )d d e e ty {c  a  lb]  d)  \c ty ne  +  abe  0  e  + (c + |c6 + | ( a  {c  A  + | a ) ) Vcdtyed d 6 e  5  x  3  + \ (c + | a ) / / ^ ° ^  a  a  Choosing 0 such that n# = 0 and d 0 a  a  d)  a  a  6  3  c  =0.  e  e  (5.65)  = 0, reduces (5.65) to \ (b +c )d d( 8^e ) ipe  =  abe  0  0  a  c  d  0. T h i s implies that bo + c = 0. Returning to (5.65) (with b + c = 0), taking 0  but d 0  0  = 0  0  0, shows that we must have c + \c% + ^"-(ai + \a ). Now only the second  a  5  a  3  and the fourth terms remain i n (5.65). Taking ?/># = 0 but 9 otherwise arbitrary, gives a  a  c = 0, and so, finally, c + ^ a 4  6  [\bod e e d ty {c  d)  a  = 0. Eq.(5.62) has been reduced to  + (by ~ VA +  ahe  e  3  b  a  c' )d 8 d ty a  3  {c  d)  + {\{h - b'e) + ej + ! c ) 0 0 0 | | ^ ) ] a  2  Taking, i n (5.66), d 9 a  b  — A  ab  — A[ ] ab  (c  o  a  - (0 ~ ^ ) = 0 . C  (5.66)  c  to be a killing vector field and d ty = S a  b  ab  = S( ) ab  to  be symmetric, give (|(6s — b' ) + C\ + | c ) = 0. This leaves the first two terms remain i n e  (5.66). T a k i n g d tp = A a  b  ab  2  = A[ ] and 0 arbitrary, gives bo = 0. Finally, we have that ab  a  (b\ - b' + c' ) = 0. 4  3  Alternatively, one can get conditions on the coefficients i n (5.56) by acting on (5.56) (or (5.62)) w i t h dfd and antisymmetrizing over / and c, g and d. This eliminates all g  the symmetrized derivative terms, i n particular the R H S , d^x^- Taking special cases  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  103  for the arbitrary fields 9 and tp give conditions on the remaining terms. a  This gives  a  equations that the coefficients of the tPs, c's and a's must satisfy. In summary, the information from the zeroth-order integrability conditions (5.56) and (5.57) is h - h + c + |c = 0, 3  goc + hai = 0 ,  gc + ha  5  0  i ( 6 - h) +  4  6  0  3  5  = 0,  C l  + |4 = 0 ,  (5.67)  c =c =b =0. 0  A  (5.68)  0  Similarly, from (5.58), one finds that ^  = (M4  ( 0 )  + 4c + c ) - g c ) ^ 3  4  0  ,  8  (5.69)  and h'i + h (bi + h + 46 ) - goh + |  (g\ + /*o( ^ + 2cl ) - g d ) = 0 ,  (5.70)  h (ci + 4c + ce) - g c + ^ (/* (4ai + a ) - g a ) = 0 ,  (5.71)  0  4  3  0  5  2  0  7  0  0  3  3  0  2  h + h (4V + b + b'e) - gob's - h (c' 4c' + c ) + g c = 0 . 2  0  4  5  0  2  3  4  0  8  (5.72)  In order to simplify the conditions (5.67) - (5.72), it is now opportune to take advantage of the transformations (5.45) and (5.47) to fix the values of some of the parameters appearing i n the equations (5.49) and (5.51). Note that there are a number of equivalent ways of choosing these parameters. We choose fi = - c -> c = 0 , 3  ei = (-#1 - gon + h (8n s  0  4  f  3  + 2n ))/go 6  8  6  m  m  9  7  =  -(  &1  2  + ^4)  ^  '  4 =  = - ( / i - / t ( " i + 4m ) + m g' )/2g 3  0  4  8  2  2  0  5  -»  5  8  4 = 0,  m = - | ( 6 + m c?i) —> 6 = 0 ,  8  m = -b  = - c ->  -> g\ = 0 ,  m - -b' -> b' = 0 , 4  3  m  ->  3  2  6 = 0,  6' =  (4 -  65) ,  (  + 24) -> h  = 0,  5  §  3  =  _  & 7  6  A' = 0 , 3  m  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  n  3 = -(g' ~ 4:h n )/2g 2  raio  =  0  5  -* g = 0 ,  0  2  + " ^ 2 ^ 2 ) — • &2 = 0 ,  ~\{b  2  n  n = -dz -> J = 0 , 5  104  =  4  - » J i = 0 ,  n — -\d  3  6  —> cl = 0 .  2  (5.73)  2  (Note that e is redundant since e affects only g and h , while mg affects only h' , and 2  n  2  2  3  3  affects only g' .) Combining results, we have  3  2  b = b\ = b = 63 = b' = b = b's = c = c ' = c = c ' = 0 , 0  2  4  7  0  3  4  8  J i = d = d = / ? i = h' = g\ = g> = 0 , 2  3  3  (5.74)  2  and h + h (b' - c ) = 0 , 2  0  <7 c + h ax = 0 , 0  5  6  - | 6 ' + c + |c' = 0 ,  2  6  gc +ha = 0 ,  0  0  e  0  x  2  g c + h (a -  3  0  7  0  2  C l  )= 0,  (5.75)  where we have not yet specified the parameter f from (5.49). 2  5.4.3  T h e general solutions  One obvious choice for f is to take f — b — b so that b' = b and then, from Eq.(5.73), 2  2  5  b' = 0. (Equivalently, one can choose f 6  6  6  5  = —c so that c' = 0. T h e n b' / 0.) This  2  2  2  6  leaves the four arbitrary parameters c' , a , a , and a , from (5.75), i n terms of which the 2  x  2  3  remaining non-zero parameters, C5, c , c , h 6  7  can be written  2l  ci = - | c ' ,  ft  2  goes = -ho^ ,  gc 0  6  = -/t a 0  3  = c' h ,  2  2  0  5oC = -h (a  ,  0  7  +| ' ) .  2  c  (5.76)  2  Substituting (5.74) and (5.76) into the first-order expressions for the tensor fields, gives C \ {1)  d  = \c' {2d 2  A  ( 1 ) 6  - d )  e  = ai5  - r/ ,|  e  {cld)  lcd  6 7  +ad 3  (aia  c  ab  a7  + a d cf> , b  2  e 7  +  adY 3  b  b  + (a + § 4 W ) 2  = c' h 2  o7ab  ,  , (5.77)  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  and B~'  = 0, D  {1)ab  = 0,  {1)ab  cd  JW"  = 0. Defining  -i(o a 7 fc  =  105  +  1  a a.7  B6  3  +  a ^)-^c aV, 3  (5-78)  3  Vab ~ C 7 b ,  9ab ~  (5.79)  a  2  the most general first-order solution (5.77) to the integrability conditions (5.39) - (5.42) can be w r i t t e n c ^ \  A™  = - r w  d  = -goJW  e c  , +  V c d  l  ,  0  H$ = -gUho  - |c' 5V , 2  ,  (5.80)  together w i t h , X  ( 0)  n r  d  ( 0 )  (0) 6  A* * 0  where T^  e cd  =  \c'Mad 6 -6M )  =  j- l(  =  h c' tt(  =  j;(h {U  a  ,  a  d  o  0  + la )rdedaO 3  a i  2  ,  e  3  e  c  c  ,  b  0  + la ty n6 ]-(0 ~iP )  a  a )-g a )Cd n,  +  1  3  2  0  (5.81)  e  is the linearized Christoffel symbol for the metric (5.79). These equations  give the first-order gauge invariance S hcd  =  (1  8W<f, i.e., SW-fcd = ( V A ( c  Qgo)^.  -T =  {1)e  0  ( 1 ) e  c  h  gQ  + g hXJ cd  + r/ ^oJ  e  A + e  c' h n , 2  0lcd  - jM \ -\c' \ d ct>,  e  (5.82)  b  b  -  e  d )  X  cd  2 h  gASl)™,  and 6^<f> = (-\d g V (j>\ - g J X + ab  b  2  a  0  One can show by direct substitution that a complete solution to the integrability  conditions (5.39) - (5.42), is (dropping the primes and bars except on h and g ): B 0  S S%D a  b  ab  {c  C\  0  ab cd  =  = 0,G = g , and o  d  = -Y\  d  + g' h J cd  0  e  ,  A = -g J b  0  b  - \c g V ab  2  a  cf> ,  H  ab  = -g h ab  0  ,  (5.83)  b  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  106  together w i t h ,  Xd  =  n  where V and J  e  a  ty°vA - 9 v Va)  (  + c r> j e  a  \c2  d  2  e  z  a  a  c  d  hoC tt(b  A  j- (^ (4a! + a ) - g a ) ( V f i ,  c  - {6 <- 0 )  a  ah  z  n  2  - e j ty ) ,  c  + \a g rne ]  + \a ) i/> VeV 0  fll  c  d  C  C  ,  (5.84)  e  0  0  3  2  e  is the derivative operator associated with the metric (5.79),  gb a  —  ^ab ~  c  27af>,  is arbitrary. T h e n the most general gauge invariance of the complete theory is  ^7cd  =  V ( A ) + g dh J \  H  =  -\c g \ V <f>  c  rf  0  b  c  a  g n-g J X . b  +  ab  2  a  (5.85)  — tthog d ,  a  c  0  0  (5.86)  b  Let us rewrite this by making the following substitutions: let q = —c (so that (5.79) 2  becomes g  ab  = rj + ? & ) , m u l t i p l y (5.85) by q, let 0 —• tt/q + J X , and let X —» X /q. d  ab  7o  d  d  d  T h e n (5.86) can be written {*9cd =  V( A ) c  =  ^  where, from (5.52), / i = ho(l — g m ), 0  = ho/go  0  g dh n c  (5.87)  0  A V ^ + ^ft  b  (5.88)  b  g  = <7o i) and i n order to obtain the linear  = —1/qk  (where this latter equations allows us to  Q  0  equations (5.23), h /g  a  d  2  0  n  absorb q i n t o go)- Thus, to obtain the linear gauge invariance (5.26), we take m = 0 6  2  and n i = 1 so that  = ho and g = go, and thus also h /go 0  0  = ho/go  =  —  1/?^ which  gives the original linear equations. W i t h these considerations, Eqs.(5.87) and (5.88) can Indeed, we could have begun by including q explicitly in the equations (cf. the constant c in [43]): in S(?Q, ^cc ^ ^cc°''> k setting yai, — > qyab- Then variation of S^QQ with respect to qyab gives 1 ,b ab Sccah• ^ identities, (5.28) and (5.29), would than be modified by replacing £c Q by q£c^ • The infinitesimal gauge symmetries, (5.30) and (5.31), would be the same, however instead of ho/go = —1/fc 6  a n <  n  v  e  we would have gotten ho/go — —l/qk.  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  107  be w r i t t e n  Sged 6<f>  V A ) - g dh tt  =  =  ( c  c  t i  lg V <j>\  (5.89)  + g n.  ab  a  ,  0  (5.90)  0  b  However, there exists a change of variables that shows that the gauge invariance (5.89) is simply general covariance, namely, ftp l  gb a  -»• g b = e*> gab,  (5.91)  j> = -(f>.  (5.92)  a  tf, F r o m (5.89), one gets  = e » " I V A ) + h-g V <f>Xd) ,  ^6  (5.93)  d  ( a  b  ab  and from (C.4) and (C.5)  Sg where V \  b  a  ab  =  (V A (a  6)  + ^ A  (  6  V  a  )  ^  ,  (5.94)  is the derivative operator associated w i t h the metric (5.91). T h e n , defining  = A e ^ , (5.89) and (5.90) become 6  Sg b a  =  V A  H  =  -\g \V 4> + g h.  ( a  6 )  ,  (5.95)  ah  A  0  (5.96)  In other words, the gauge invariance that we have exhibited is simply general covariance. We see that this corresponds to the choice m,2 = ho/go which sets ho = 0 and which, i n the linear case, shows that the gauge field <f> can be absorbed into 7^ by a redefinition of the fields. However, it was not evident a priori that this would hold to a l l orders.  Chapter 5.  5.4.4  Conformal Invariance - Spin-two  Coupled with  108  Spin-zero  Uniqueness of the solutions  Let us remark on the uniqueness of the solutions (5.95) and (5.96). First, we find conditions on the possible changes of variables, i  and (j), such that one is able to set  cd  B  = S ( 5 ), a  c d  H  b  c  d  = -{h rj  ab  0  + q>y ), D  ah  ab  = 0, and G = g :  ab  0  (i) F r o m Eq.(5.47), there exists a change of variables such that the tensor B d  can be  ab  c  set to S ( S d), a  b  c  i.e., B  a b  if there exists a 7  = S t 6 d), a  c d  b  c  died  _ g-lab  d  "  lab  cd  ^7led  _ j^-labjjef  c d  ~  e f  such that  d<f>  (5.97)  •  Substituting this into the condition d %d  &%d  dfabdjed  dicdd^a  2  = 0  (5.98)  and taking the zeroth-order part, gives cd V  +  dfab  dicf  where, from (5.51),  # 7 cd  (0)  ' Q£){l)ab  d<f>  $7 cd  dE-Wef  d% ef  (5.99)  = 0,  dfab  (0)  d<t>  (5.100)  m r] d 2  c  Eq.(5.99) is the zeroth-order condition that there exists a change of variables j  cd  that B  a b c d  =  6 (c6 ). a  b  d  (ii) Similarly, there exists a change of variables such that H  ab  exists a i  cd  such  = —(h rj + ?7a&), if there 0  ab  such that died d(j>  =  _ -^l G  oT1ab +  q j a b )  _  Q-^HJ ^ 1  d-Jab  (5.101)  Substituting (5.101) and (5.97) into Plef  d% 2  d  d<f>d~f d dfcdd(f> c  = 0  (5.102)  Chapter 5.  109  Conformal Invariance - Spin-two Coupled with Spin-zero  and taking the zeroth-order part gives (dB^  a  h  b e  f  Vef  d<f>  +  0  d B ^  c  d e  f  \  r.  go  + rn ri f 2  'dD^  e  1  dG™  go  Olab  -  0  cd  Vcd  d<f>  Olab  ~^f~Ji— (h  hodD^ '  ab  go  dfab  a —6 (eS /) = 0 . go  h) 0  go dfab  T h i s is the zeroth-order condition that there exists a change of variables j B d ab  c  = 6 ( 8 d) and H a  b  c  ab  = -(h r] b 0  +  a  cd  (5.103) such that  qi )ab  (iii) F r o m Eq.(5.47), there exists a change of variables such that D  ab  = 0, if there exists  a <b such that d ~  B  a  b  (5.104)  d<t>  D  lcd  Substituting (5.104) into  8 (j> 2  d  2  Ofabd-Jcd  (5.105)  = 0,  dfcddfab  and taking the zeroth-order part, gives the condition (o)  = 0  d<f>  (5.106)  where, from (5.51), dl  (0)  (5.107)  d<f> (iv) F i n a l l y , there exists a change of variables, cf>, such that G = go, if  —  — go^r  — Lr  flabT.  •  (5.108)  d"1ab  d(j>  Substituting (5.108) and (5.104) into d <f> 2  (5.109)  0,  djcdd<j> d<j>d~fcd  and taking the zeroth-order part, gives godGW  9o  dj  a  fdD^ I—— V 9<f> ab  + i n  h Vcdgo  0  d D ^ d<y  ab  (5.110)  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  110  E q . ( 5 . 1 1 0 ) i s t h e z e r o t h - o r d e r c o n d i t i o n t h a t t h e r e e x i s t s a change o f v a r i a b l e s <f> s u c h = 0 a n d G — go-  that D  ab  N o w , l e t us c o n s i d e r t h e s p e c i a l case o f (5.83) w h e r e c  2  = 0, w h i c h is ( o f course)  a l s o a s o l u t i o n o f t h e z e r o t h - o r d e r i n t e g r a b i l i t y c o n d i t i o n s (5.54) a n d (5.55). o r d e r t e n s o r fields c a n t h e n b e w r i t t e n C ^  B^ cd = 0, D^ ab  —g J^ Xb-  ab  = 0, a n d  e c d  T] h J ,  A^ = -g J ,  {1)e  =  cd  b  0  {1)b  {l)e  cd  lcd  = 0,  0  = r] h J \ ,  = 0, so t h a t , 8^  T h e first-  0  and  e  6^<f> =  T h e n a c o m p l e t e s o l u t i o n t o t h e f u l l i n t e g r a b i l i t y c o n d i t i o n s , (5.39) - (5.42),  b  0  is  B  = 8 ( fi d)  ab  a  cd  C cd =  ?  b  c  e  G =g ,  rj h J cd  e  0  ,  H b = —hoVab ,  A = -g J , b  0  D =0 , ab  a  d  b  0  {cXd)  =0 ,  d T (C  D)  =  0,  (5.111)  a n d , LT, A a r e g i v e n b y (5.81). T h i s gives t h e gauge i n v a r i a n c e for t h e c o m p l e t e t h e o r y  hcd  = d{ X ) + r] ho\ J - rj h $l , e  c  cd  d  e  cd  0  S(f> = g tt - goJ h . b  (5.112)  Q  O n e c a n s h o w b y i n d u c t i o n t h a t (5.111) is t h e u n i q u e s o l u t i o n ( u p t o a c h a n g e o f v a r i a b l e s (5.47), a n d t h e rescaling freedom c  2  (5.45)), corresponding t o t h e first-order  conditions  = 0, ( 5 . 7 5 ) , a n d (5.74) [73], [43]. W e h a v e s h o w n t h a t E q . ( 5 . 1 1 1 ) is t r u e for n = 1,  ( i . e . , f o r B^  ab cd  ,  e t c . , w i t h n = 1). N o w a s s u m e t h a t (5.111) i s t r u e u p t o a r b i t r a r y n.  T h e n t h e nth-order integrability conditions, using the inductive hypothesis, are the same as t h e z e r o t h - o r d e r i n t e g r a b i l i t y c o n d i t i o n s , (5.54) a n d (5.55), w i t h t h e (l) r e p l a c e d b y ( n + i ) ( h o w e v e r , w i t h o u t y e t m a k i n g u s e o f t h e t w o freedoms (5.47) a n d (5.45)).  Rearranging,  these equations are  •/d ^ B  A  ab cd  ^7e/  g B("+l)e/ J  dlab  e  \  J  h()  / (n+l)ab dD  ^ go \  D  Olef •  + [ty S^C^ a  Q (n+l)ef  a c d  datybd 0j e  df  ab  - (0 <-ty )}= d ^ e  c  {eX  ,  (5.113)  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  111  and, 'QB^)  h dB^ ^ \  ab  +1  cd  ~xl  go  A v c d ^ -  where C^ ^ n+  -  = C^ ^ n+1  cd  a  J  0~f f e  'ef  g V  i ^ S ^ l (-g nd ( ) 0  Olab  c  d  90  - < ^C^  d  d  a  b  O^ef = d Ttf  , (5.114)  lc  J  '  + h r] A^ ^/go. 0  a  o<P  0  ( n + 1 ) a t  0  cd  In order to show that Eqs.(5.113) and  +l  cd  /i 9D  ab  0  +' Vcd— 'l  90  <?7ab  a  n+  cd  ~ 3  0(p  go  h fdD( V  b  0  Vef 'ef  cd  (5.114) imply that (5.111) holds for (n +1), first let ty and ( be constant one-form fields a  a  so that the first terms in both (5.113) and (5.114) vanish. Then, operating on (5.113) and (5.114) with d dh, and antisymmetrizing over g and c, h and / , the RHS's of (5.113) g  and (5.114) are eliminated, and these equations can be written  C^tA^^ic] =0,  (5.115)  0.  (5.116)  4%%C  (  n  +  1  )  °rf||c] =  Eq.(5.115), is the statement that the quantity <9[ <9|[/ C ( ) ']| ] is invariant under the , ra+1 a  9  transformation j  —> ^  ab  (n+  a  + d( \ ). In order for this to hold, f  ab  a b  expression d[ d\[ C ^ ^ g  l  c  ab  can only appear in the  in the form d[ d\[ j \]d]. However, while d[ d\[ C  a  (n+1)a  h  a  c  g  b  an (n + l)th-order quantity and thus contains (n + 1)  7a6' > s  h  ]\ ]  d  is  c  it is assumed to involve at  most three derivatives. Consequently, the sum of all terms with more than one ~f must ab  cancel. Considering terms with only one j , for n > 1, it can be seen that (5.115) can ab  only hold if these terms also vanish. Thus, we restrict our attention to terms consisting solely of </>'s. While such terms satisfy (5.115), however, (5.116) can only hold if these also vanish. These considerations show that d[ d^h,C^ " ^ n+1  g  (j(n+i)d^  b m u s  t kg  a  >a  = 0, which implies that  symmetrized derivative, i.e., that there must exist a tensor  g^ ^ n+1  e d  such that C^ Taking g^ ^ n  e d  =  — /'  n  +  1  ) d, e  d a b  = dJ ^ (  n+1  e  one sees that the C^ ^ n+1  freedom (5.45). Substituting C +^ (n  d ab  .  d)  d ab  (5.117)  can be set to zero by the rescaling  = 0 into (5.113) and (5.114), operating on the  Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero  112  resulting equations w i t h d dh, antisymmetrizing over g and c, h and / , and taking special g  cases for the fields ty , 6 , ( , J), (cf. [73]) gives the equations a  /dB( ^ n+  V  a  a  h dB( V \  ab  n+  cd  0  d<f>  V  e  f  g  h fdD^ ^  ab  n+  cd  d*f  0  h  ab  0  J  ef  Vcd  g  \  d<f>  ho  d G ^  %  Ol  0  —oVcd^  V e f  g  dD( V \ n+  0  d  0  7 e  /  )  1 8 H ^ ~  g  ab  ab  =  0  • (5-119)  Ofab  0  C o m p a r i n g these equations w i t h (5.99) - (5.110), one sees that, from (5.99), Eq.(5.118) is the condition that indicates that the tensor field B  ab  can be set to 8 ' 8 ) a  cd  up to order n  b  c  d  by a change of variables (assuming the inductive hypothesis), provided that r n = ho/go, 2  i.e., that there exists a change of variables such that B  ab  = <5 ( c; d), H a v i n g made this a  cd  6  c  choice, referring to (5.103), Eq.(5.118) becomes the nth-order condition that D can ab  be set to zero by a change of variables. Similarly, Eq.(5.119) is the nth-order condition (5.99), w i t h q = 0, that there exists a change of variables that allows one to set both B d  and H  = 6(5 )  ab  a  c  b  c  to —horjab-, (again, provided that m  ah  d  2  = h /go which, from (5.52) 0  gives ho = 0). Finally, w i t h these choices, Eq.(5.119) becomes the condition that G can be set to g (together w i t h D  ab  0  = 0). Thus, if m = ho/go (and consequently h = 0), then 2  0  (5.111) is the unique (up to a change of variables and the rescaling freedom) solution to the integrability conditions (5.39) - (5.42) w i t h the first-order conditions c = 0, (5.75), 2  and (5.74). T a k i n g m = h /go gives h = 0, and Eq.(5.111) becomes 2  B  ab cd  = 8\ 8 c  w h i c h gives 8^  cd  b d )  —  0  , C\  d  = 0, H  0  ab  = 0,  D  a b  =:0,  d( \ ), and 8<j> — g (p, — J Xb). b  c  d  0  8^cd 6<f>  =  G = g , A = -g J b  b  0  0  ,  (5.120)  L e t t i n g S) —> 0 + J Xb, we get b  d( X ) , c  d  = g Sl • 0  (5.121)  Chapter 5.  Conformal Invariance - Spin-two Coupled with Spin-zero  113  This argument can immediately be extended to the case where c 7^ 0, [74], [43]: 2  Define AB^ ,  AH \  ah  etc., to be the difference between the nth-order parts of two  ab  arbitrary solutions to the integrability conditions (5.39) - (5.42) having the same firstorder solutions (5.75) and (5.74), i.e., AB^  ab  preceding inductive proof, one can set AB^  )ab  AC^%  d  = AC^  e cd  + h g AA^ /g e  0  cd  0  = 0, AH$ = 0, AH$  = 0. By a repetition of the = 0, AD^  ab  = 0, AG  (n)  = 0,  = 0, V ( c A X i ) _ 1 ) = 0, and V ( c Arj ) n - 1 ) = 0. Thus  the solution (5.95) and (5.96) is unique (up to a change of variables and the rescaling freedom).  Chapter 6  N o n l i n e a r , N o n c o v a r i a n t , Spin-two Theories  It has long been presumed that any consistent nonlinear theory of a spin-two field must be generally covariant ([46], [67], [31], [83], [77], [28], [51], [22], [48], [9], [20], [74], [47], [23], and references therein. See also Chapter 3.). For example, the assertion that string theory is a theory of gravity is based i n part upon this belief ([10], [23], [38]). Recently, however, W a l d [74] found another possibility, namely, theories which are reducible to the linearized E i n s t e i n equations (1-12) could simply have "normal" spin-two gauge invariance. However, if the spin-two field interacts with matter, a "quite natural assumption" about the coupling of 7 & to matter seems to eliminate this possibility. a  In this chapter, we exhibit classes of consistent nonlinear theories of a spin-two field which are not generally covariant and which remain viable when coupled to matter, including coupling i n the "natural" manner specified i n [74]. In Chapter 4, we found that different choices for the potential used to describe a linear field theory, could possibly lead to different types of consistent nonlinear theories. We now employ a similar strategy to find consistent nonlinear theories of a spin-two field which are not generally covariant. We consider three different formulations of the linear spin-two theory. In the first two cases, the usual symmetric rank-two tensor field, 7 &, is chosen as the potential. In the a  t h i r d case, however, a traceless symmetric rank-two tensor is used as the potential. We find types of theories which, when interpreted i n terms of a metric g b, are invariant under a  the infinitesimal gauge transformation y b —> ~f b + V( V A"| |b), for any two-form field c  a  a  a  c  K b. W e also find classes of theories that are conformally invariant. Thus, one sees that a  114  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  115  the types of nonlinear spin-two theories resulting from the generalization of the linear spin-two equations, depends crucially on the formulation of the linear theory. Indeed, our calculations indicate that there are many more possibilities for consistent nonlinear spin-two theories than those illustrated here. In section 6.1, we discuss the three alternative formulations of the spin-two equations that are investigated in this chapter. In section 6.2, the infinitesimal gauge invariances of the nonlinear extensions of each of these theories are given. The calculations are outlined in section 6.3. The final section is a brief summary.  6.1  Formulation of the spin-two equations  The usual formulation of the massless spin-two equations,  V '<J>ABCD AA  = 0, Eq.(2.1), is  in terms of a symmetric rank-two tensor potential 7^ satisfying the linearized Einstein equations, (1.12), which are derived from the second-order Einstein-Hilbert Lagrangian, (1.13), (see Chapter 2). In general, this is taken as the starting point for calculations determining the nonlinear extension of the spin-two equations. However, in Chapters 2, 4, and 5, it was shown that not only was it possible to formulate the theory of a spin-two field in terms of different potentials, but also that, in terms of the usual symmetric ranktwo tensor potential, alternative second-order actions were possible. We pursue these possibilities here. First, let us recall that, in seeking nonlinear generalizations of, say, the linearized Einstein equations, G$ — 0, (1.12), the nonlinear equations were severely constrained. This was in order to avoid consistency problems that could be encountered since the linear equations were divergenceless. In Chapter 5, additional constraints arose since the linear equations were also traceless. In other words, it is the identity that the linear equations satisfy which dictate the constraints placed on the consistent interacting theories. This  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  116  motivates one to find alternative formulations of the spin-two equations which yield equations of motion that satisfy certain conditions, other than, or i n addition to, being divergenceless and/or traceless. In Chapter 5, we wrote down the most general second-order Lagrangian i n 7 &, a  £ M  G  I  ai7 a7  =  a 6  a Y d d -j b  -)-  a 6  + a -yd d "/ c  a  2  c  3  ab  d  cd  + a -yD^, Eq.(5.2), and the correspond4  ing equations of motion, S ^Q = 0, Eq.(5.3). T h e condition that the quantity  , be  M  (identically) divergenceless, gives the linearized Einstein equations. The requirement that S ^Q be traceless, gives the following restrictions on the a - parameters: 2a + 4 a = 0, M  t  2  3  and 2a + 0 3 + 8 a = 0, (5.5). Now, consider the requirement that £ ^Q satisfies (idenB  x  4  tically) b^ dh£ MGi c  M  = ®- This condition gives the relation a = —2a . W i t h this choice,  i  2  x  the equations of motion that are obtained for 7^, are invariant under the transformation lab —*• lab + d( d K\ \b)- After an appropriate change of dynamic variable and choice c  a  c  of gauge, the equations of motion can be written n~y  ab  = fj, d J  = 0, Eq.(1.25), the  AB  A  equations of motion for a spin-two field. This is the starting point for the alternative formulations of the spin-two equations studied i n this chapter.  Following, are the L a -  grangians considered, together w i t h the corresponding equations of motion, identities, and symmetries: (o) F i r s t , by way of comparison, the usual formulation of the spin-two equations is outlined.  T h e Lagrangian density is given by the gravitational part of the second-order  Einstein-Hilbert action Eq.(1.13),  4? = - h - * 7 ° + b d ddab - \id d D  6  cb  a  c  d lcd  +i n 7  7  ,  (6.1)  variation of which gives the linearized Einstein equations (1.12) co( ) 2  g  (l)ab =  =  Q (a b)c _ l^ab _ iQagb^ _ ^ab^gd^ cd  7  _ ^  =  Q ?  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  where £ ^  a  b  117  obeys the linearized contracted Bianchi identities Eq.(1.16), d e§  (6.3)  = o.  )ab  a  Correspondingly, the equations of linearized gravity are invariant under the transformation (cf. E q . (1.17)) lab  * lab + d \b) [a  •  (6.4)  (i) A s a first alternative to the usual formulation (o), we consider the Lagrangian obtained from  , Eq.(5.2), by fixing the proportionality between  = -\kilab^  + \hl  a b  c h  d d  + k  a  c l a b  and a to be a = — 2a\, 2  d d c  2 l  d l c d  +  2  fc 7°7  ,  3  (6-5)  where k\, k , and k are arbitrary constants, (where we have set a\ = —\k\, (k ^ 0), 2  3 — k, a  — k ).  a  2  3  4  x  T h e subscript K is simply a label which identifies the quantities  3  associated w i t h the Lagrangian (6.5), (l< refers to the constants " & " ) . Variation of C$ gives the equations £ o(2)  [i)ab  e  = _K_ 0"fab  =  where £ ^  a  b  k i d c d  (a b) 7  c  _ l  k l  a^b  k d &i  + r, {k d d  a  +  ab  2  c  2  d lcd  + 2^Q ) -  0 ,  7  (6.6)  satisfies the identity  d d £K [c  (  = o,  )a]b  b  (6.7)  (2)  and, therefore, C  K  is invariant under the local infinitesimal transformation lab  where K  ab  lab + d d K\ \ c  {a  c  b)  ,  (6.8)  is an arbitrary two-form field. (Another way of viewing the gauge invariance  (6.8) of C$ is to say that C$ is invariant under the "normal" variation 8i where ( , rather than being completely arbitrary, is divergenceless, d ( a  b  words, (  a  = dK , a  b  ab  for arbitrary K  ab  —  —K ). ba  ab  =  d( ( ), a  b  — 0. In other  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  118  (ii) As a second alternative formulation, we consider the special case of C  K  potential & appears only in the form i  — \r]abl,  1  7a  4 = -\l*^ 2)  ab  where the  ab  + \l d d ch  - \id d  a  c  clah  d  +  lcd  £  7  n  7  ,  (6.9)  where the subscript c is simply a label identifying the quantities pertaining to the theory given by (6.9), (c refers to the conformal invariance of the theory). Variation of C ' c  gives the equations  4  j,c(2) 1H  where £ ^ 1  c  = Y^  "lab  ab  =  ^  DCD{A  ~  B)C  - b^^-rcd  -  -|°7),  (6-10)  satisfies the two identities d d Sg [e  )a]b  b  Vab£  C  = 0,  (6.11)  -  (6.12)  0 ,  and, consequently, C$ is invariant under the transformations lab ->• 7a6 + d(ad K\ \ + O n ,  (6.13)  C  c  b)  a6  where Vt is an arbitrary scalar field. (iii) As a final example, we consider formulating the theory (1.26) in terms of the symmetric traceless potential 7 ^ , and take the Lagrangian 4  2)  = -Hb*i  + \i d d il.  Tab  Tcb  (6.14)  a  c  The sub/superscript T, in addition to being a label identifying the quantities associated with the spin-two theory (6.14), indicates that the potential ij  b  is traceless (r stands  T h a t is, c o m b i n e the d e m a n d t h a t the q u a n t i t y (5.3), be traceless, w h i c h gives the c o n d i t i o n (5.5) o n the parameters a,, (i.e, 2 a 4- 4 a = 0, a n d 2 a i + a + 8 a = 0), w i t h the d e m a n d t h a t 1  2  3  3  4  d^db^MGhL = ^ ' h i c l i further specifies that a = — 2a\. T a k i n g ai = — 1 (for preciseness), gives (6.9). N o t e t h a t this is precisely the L a g r a n g i a n t h a t F i e r z a n d P a u l i found [33] w i t h the a r b i t r a r y scalar field replaced b y 7. w  2  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  for traceless).  (This Lagrangian can be obtained from C  dynamic variable 7JJ,, to be the traceless combination i  , by taking the fundamental  — ^// i7- Alternatively, C^  can  by setting the field 7^ i n Eq.(1.13) to be traceless.) V a r y i n g  C?\  a  ao  be obtained from C$  c  119  subject to the constraint that 7^ remains traceless, gives the equations /;c(2)  = where R^  ab  and Rj^  - i ^ ! ? = 0 ,  4  1)a6  (6-15)  are the linearized R i c c i tensor and linearized curvature scalar for  7^, respectively. T h e quantity £ ^  a b  , satisfies the same identities as £ ^ , a b  c  d d£ [c  { 1]a]b  b  T  Tj £^  )ab  ab  =  0,  =  0,  (6.16)  but the equations are invariant only under ll -  lab + d d K c  {a  m  .  (6.17)  T h i s completes our listing of the alternative linear formulations of the spin-two equations. Note that all of these linear theories are fully Poincare invariant since taking K b a  to be an arbitrary linear or quadratic function of the coordinates gives the full Poincare group.  6.2  Nonlinear, noncovariant spin-two theories  N e x t , the nonlinear generalization of each of these formulations of the theory of a spin-two field is investigated. According to the consistency criteria of [74], discussed i n section 3.3, we begin by identifying the most general possible expression for the infinitesimal gauge invariance of the full theory that is required to ensure consistency i n a perturbation  Chapter 6.  Nonlinear, Noncovariant, Spin-two Theories  120  expansion. T h e condition that the space of variations be involute, gives an integrability condition which we t r y to solve order by order i n the fields to find the infinitesimal gauge invariance. H a v i n g determined the first-order gauge invariance of the complete theory, because higher-order calculations are extremely messy, we then t r y to find a solution, i n special cases, to all orders by inspection. We verify the validity of our choice by direct substitution. Following, are the results of the calculations. T h e details are outlined i n the next section.  (o) I n [74], a consistent nonlinear generalization of the the standard formulation of the spin-two equations, CQ\ CG, is shown to have the infinitesimal gauge s y m m e t r y 6« x  where B^  a  — ^(JJctyb,  bcd  a  n  = B\ d \ cd  7cd  +C X  b  a  ,  b  bcd  2  (6.18)  d C^°] = 0. T h e condition that the space of variations be  involute, gives an integrability condition, the first-order solution of which can be w r i t t e n  S^lab  = Cl(A 5 a6 + 2 a  a^)A ) C  c 7  7 c (  ,  3  (6.19)  where ci is an arbitrary constant. W r i t i n g (6.19) i n terms of the metric g  ab  = rj b + 2ci~f b, a  a  one gets  = \ [Cx g }  S^hab  a  {1)  ab  (6.20)  ,  = ci(A d 7 b + 2 y ( db)\ ) is the linearized form of C\ g  where | [C\ag ]^  ,  a  c  ab  c  c  a  a  a  ab  the L i e  ,  derivative of g b w i t h respect to the vector field A . Note that i n the expression a  a  indices are raised and lowered w i t h the metric g  ab  2  3  ab  so that within the square brackets  indices are raised and lowered by g . However, once C\ag ab  C\ag ,  ab  is linearized, i.e.,  [C\ag b]^\ a  W h e r e t h e f o r m o f (6.18) differs s l i g h t l y f r o m the corresponding equation i n [74]. E q . ( 6 . 1 9 ) is equivalent t o the corresponding expression i n [74] up t o a change o f variables a n d a  rescaling A —+ f b\ . c  c  b  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  121  indices are raised and lowered by the flat metric 7] . U p to first order, then, the infinitesab  i m a l gauge invariance of a consistent nonlinear extension of linearized gravitation can be written &\«lab  =  d( \ )  =  1  a  + ci(A d 7 c  b  c  2(X d g c  c  o 6  -I- 2 7  d ) A ) = \C\ag  ,  c  c ( a  6  ab  + 2g d \ ).  (6.21)  c  ab  cib  a)  Moreover, (6.21) is a complete solution to the integrability condition (6.49), 5\ y b a/  a  = V( A ) ,  = \£>\°-g b  a  a  (6.22)  b  which can be verified by direct substitution. Also, it is the unique solution corresponding to the first-order solution (6.19), [74]. Taking the special case c\ = 0, shows that normal spin-two gauge invariance, 8\a-j = d( \ ), is also a possibility for the gauge symmetry ab  a  b  of the complete Lagrangian CGRecall that a theory is said to be generally covariant i f "the metric, g , and quantities ab  derivable from it are the only spacetime quantities that can appear i n the equations of physics" [73]. In the context of a theory of a spin-two field, j ,  i n flat spacetime, the  ab  theory is said to be generally covariant if there exists a change of dynamic field variable, 7afe  — •  gab,  such that the original theory can be made to be independent of the choice  of flat background metric, r] , [74]. In [74], W a l d has shown that any theory w i t h ab  the infinitesimal gauge invariance (6.22) is generally covariant and conversely, that any generally covariant theory must have the-gauge invariance (6.22).  Thus, a consistent  nonlinear generalization of the linearized theory of general relativity is either generally covariant, or invariant under the "normal" spin-two gauge transformation. (i) Extensions of C  K  , CK, are ensured to be consistent if they have the infinitesimal gauge  symmetry fe 7e/ ac  =  d d K b  d  a c  D  a c b d ( e f )  + d K b  a c  B  a c  \  e f )  + K  C \ a  a e  e f )  ,  (6.23)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  where D^  = Tj S 8 ,  acbd  ad  ef  c  B^  = 0, and C^  acb  b f  e  ef  122  = 0. We find that the solution of  ac ef  the zeroth-order part of the fundamental integrability condition (Eq.(6.66) below) gives, for the first-order part of Eq.(6.23),  +d r d Kj)a b  +  b  +  (e  b  1  d  2  a  i d  {  e  K  ]  a  l  f  d  [  e  +  )  V  c  )  d  a  d  7  m  K  m  d  u  a  +  m  K \  l  m  ]  e  \ d  a  +  )  l  e  j  d  K  m  b Vefd d K a  n  c  7  m  +  a  d  d  m  [  e  l  m  a  ,  ca  K  j  )  a  )  (6.24)  •  where b, b , & n , and c are arbitrary constants. Interpreting this in terms of the metric x2  7  9ab  -  Vab -  bjab  (6.25)  ,  Eq.(6.24) can be written xi )  (1)  1  + (b  ^(a^ K\c\b)\ C  + ( c - \b)d d^K^  ~ §&)3 0(a#| |&) c  12  7  + b r] d -id K  c  7  C  d  b)  11  ,  c  ab  cd  (6.26)  where [V V /v c  ( a  | c | 6 )  ]  ( 1 )  =  [g V  V  =  -b ( d d K  cd  ( a  { d [  K }  + d d\ \K  ab  7  {e  ]al  + d dK + ± d  and  V  a  {e  a  i d  {  [el  + d  s)a  e  K  + ^\ d  ab  f)b  ab  bl  ( 1 )  m  a  a ( e l f )  d K  f)b  m  a m  + \d  + \d d K\ )  )  ul  s )  l e f  d K  a  b a  m  m a  + +  dd K b a  {e  d d m  ,  m  W  K  e  m]e)  ab  ab  c  c  )  V r „ V A i \c\b) c  (i)  K  f ) a  A s noted after Eq.(6.20),  w i t h i n the square brackets indices are raised and lowered w i t h the metric g . a  m a ( e l  ab  (6.27)  is the derivative operator associated with the metric g .  once V ( V A ' | | ( ) is linearized one gets  7 f)  However,  so that i n Eq.(6.26) indices are  raised and lowered by the flat metric rj . Using the equation (3.68), i n the case where ab  k = —\ki, 2  (6.26) simplifies to C:7«6 = [ V V A | | ] C  ( a  c  6 )  1  +  hlVabd^FKai  (6.28)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  and i n the case where k ^ — \k\,  (6.26) simplifies to  2  V V / v , c|6) c  (1)  (c - \b) [d  (a  where a = (\k\  +  123  (d jK )  + a^d^d'Ka]  c  (a  m  + 4fc ). In the special case (c =)  /C2)/(^1  612 =  2  ,  (6.29)  c — \b, (which is the 7  case when k = ~\k\), and b\\ = 0, we are able to show that 2  &K lab  (6.30)  V( V A| | ) C  =  ac  a  c  f e  is a solution to a l l orders. O f course, if we set b — 0, we get the t r i v i a l solution (!>A' 7a& — QC  d  d K , Eq.(6.8). c  { a  m  (ii) In addition to being invariant under the infinitesimal variation (6.23), extensions of are also required to be invariant under the "conformal" variation (6.31) where H ° a  b  = n . In this example, we find that the solution of the zeroth-order part of ab  the fundamental integrability condition can be written  til™  h (r d d K  + d d\ \K  b  {e  lal  +d d K  + d)  [e  {el  + d  ab  bl  f)a  (c d d r K c  Vef  b  6  + d^^d^Kba  ah  f)b  c  a  a ( e l f )  s)b  d K  a m  - bdrdK c  ab  + \d  m  b  9  a  a  d K  m a  - b drdK w  c  b  d d r Kab b  {e  f)  +  m  l e }  b  cb  +  dd K ) ma  m  j)a  ~ b d ^d K ) a  ca  {el  n  c  ac  hnj .  1  ,  (6.32)  ef  In the special case where c = bg — bw = bn = 0, we find the complete solution 6  hab where V  a  = V V / \ | | ) + flg c  ( a  c  6  ab  ,  (6.33)  is the derivative operator associated with the metric  gab = Vab + h~f  ab  ,  (6.34)  Chapter 6.  and g d g ab  c  ab  Nonlinear, Noncovariant, Spin-two Theories  = 0, i.e., g  ab  124  is a constant-determinant metric. T h e condition that  7 a  j only  appears i n the combination •j  — \rj -f at linear order holds to all orders i n this special  case. Also, h = 0 gives 8^  a  ab  ab  = d( d K\ \ ) c  ab  c  b  + £lr] , Eq.(6.13), as a general solution. ab  (iii) In this last example, because J(, is traceless, extensions of eft are limited to theories 7  w i t h the infinitesimal gauge invariance  <^ 7 /  =  P \f(d d K D  ftryj  =  P \^H  ac  T e  g  +dK B \^  aM  b  d  ac  9  g  h  (6.35)  ac  gh  b  ac  h  ,  (6.36)  where P ef  = 6° J  9k  {  -  h f)  (6.37)  projects out the trace. W e find that the solution of the zeroth-order part of the fundamental integrability condition can be written C/7a 6 r  WlL  =  d( J d d K c  7  +j d dK  d  {a  Tc  b)d  +d^dK )  d  ia  b)  c  dc  d  b  dc  = 0.  ,  (6.38) (6.39)  T h i s infinitesimal gauge invariance can be interpreted i n terms of a constant-determinant metric i n the following way: Let g  ab  be a symmetric field defined i n terms of the traceless  field J by 7  6  9*b = e ^ \ T  a  C  r,  M  e^ ) T  d  ,  b )  (. ) 6  40  where the indices have been raised b y the flat space metric r) , and / is an arbitrary ab  constant. Since rj ^ ah  b  = 0, det <7b is a constant. Then, we can see from the zeroth-order a  gauge transformation (6.17), that  =  Jd d K c  (a  m  .  (6.41)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  125  A l s o , from the first-order infinitesimal gauge transformation (6.39), we have 7  =  fd [jJ d d K c  + \f  T  (a7b)c  + ~f d d K  d  (a  Tc  Tc  b)d  [i d d K Tc  h)  b)  dc  - ^ d ^ K ^ ]  d  (a  +  d  {a  cd  d^ d K ] T  ab  d  dc  .  (6.42)  ,  (6.43)  Taking d = \ f, this becomes [S g }  = f {i \ d d K  {1)  Kac  2  ab  T  + ld l d K )  d  a  b)  c  dc  d  7 b  dc  which is just [SK g ] ac  where £  c«.  a  = |V A c  c a  , and  {1)  ab  =  /[fc(.V  =  M?],  b )  V^]  ( 1  \ (6-44)  denotes the L i e derivative w i t h respect to the vector field  A l t h o u g h we can write the solution Eq.(6.39) succinctly i n terms of g , ab  the highly nonlinear relationship between g solution, 6j^ , b  because of  and 7j , it is difficult to find a complete 6  ab  by inspection. However, we do know of a theory that has the linearized  gauge invariance (6.44) and whose linearized equations of motion are given by (6.15), namely S = Jd xR(g )  ,  4  ab  where R(g ) ab  (6.45)  is the curvature scalar for the constant-determinant metric g . ab  T h e equa-  tions of motion for (6.45) are R  ab  - \R g  = 0,  ab  (6.46)  and the gauge invariance of the theory is ^K gab ac  =  C^g  =  g (aV V K  ,  ab  dc  c  b)  d  ,  (6.47)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  where R  ab  126  is the R i c c i tensor for the metric g . Despite the fact that we cannot write 8^ ao  i n closed form, we can infer that generalization of C  can lead to classes of consistent  T  theories of a spin-two field w i t h the gauge invariance (6.47) where g is related to 7j by 6  ao  Eq.(6.40). T h e theory (6.45) was first considered by Einstein i n 1919 [27]. It is particularly interesting since any solution of Eq.(6.46) is a solution of Einstein's equations w i t h cosmological constant ([55], [6], [81], [84], [45], [14], [15], [79], [68], [70], [66]). Most recently it has been considered as a candidate for a quantum theory of gravity ([68], [70], [66]).  6.3  Outline of the calculations  In this section we give an outline of the calculations leading to the gauge invariances given i n section 6.2. A g a i n , for comparison, we include the example of linearized general relativity, a calculation already carried out i n Ref. [74].  Due to the complexity of the  remaining examples, only the principal equations are mentioned. T h e method of solution is sketched i n more detail i n Chapter 5. (o) T h e requirement adopted for consistency of the theory given by Eqs.(6.1) - (6.4), is that extensions of CQ\ CG, yield equations of motion 8 , which satisfy the generalized G  d  divergence identity da (B\ £ ) D  CD  G  = C £ bcd  .  c d G  (6.48)  In other words, the action SG has the infinitesimal invariance 8\a^  cd  — B b dd X a  c  a  h  + Cb dX ', b  c  (6.18), where B bcd and Cbcd are functions of y b, Vab, and e cd- T h e tensor Cbcd can a  a  ab  include derivatives of 7^. T h e n , for all vector fields 0 and tp , i n order that the space of b  a  variations be involute, one must exhibit a vector field x°, such that [Vg , V^b] = V c, where a  x  V$a, etc., are vector fields on the manifold of field configurations defined by Eq.(6.18). R e w r i t i n g this commutator i n terms of the tensor fields B b a  cd  and Cbcd v i a Eq.(6.18) gives  Chapter 6.  Nonlinear, Noncovariant, Spin-two Theories  127  the integrability condition ^^d^  [B\ d 6  b  h  ef  *l> (B dJ h  +  g  a  +^Va, (B\ dj  c 6} a  ef  +C 9)  b  - (6 <- V ) = B d C  a  bef  +  b  gef  c  a  bcd  aef  c  e ) a  a e f  +C  b aX  •  a  acdX  Before proceeding to solve (6.49) order by order in the unknown fields B ,  C , and  a  bcd  note that the infinitesimal gauge invariance is unchanged by B  —• B'  a  and C  -> C  ecd  = B df a  ecd  bcd  a  +C f ,  b  bcd  =  bcd  a X  ,  B f , a  e  ecd  b  where f(°) = 6 , (cf. the first two of (5.45)).  b  e  bcd  a  bcd  (6.49)  b  e  b  a  In addition, the tensor fields transform under a change of variables according to B f a  — •  be  B  bef  B bcd a  =  —  =C  Cb f —> C f  ,  e  be  f  be  —— ,  (6.50)  (cf. the first two of the Eqs.(5.47)). To solve the zeroth-order part of (6.49), note that the most general linear tensor fields are given by the first two of the Eqs. (5.48), with the appropriate indices lowered, and without the terms involving the scalar field </>, i.e., with bi = b  &  =  C7 =  B  C7  = 0: =  {x)a bcd  +  b j (c£d)b  ea  0 e  + b rj (cYd) 5  Cell  =  ab  + c d jr] e  b  cd  + b j6 ri  b i 7] a b  2  3  + C d^ X  cd  + ced j ri b  cd  b  e  cd  + C d( f )e 2  c  4  c  d)b  b  3  e  c  4  d  b  c  d  e  (6.51) e  2  a  cd  . = fifS  e  b  3  b  C rj ( d )-y + C d f ( T/ )  +  d  The first-order part of the rescaling freedom f , f ^ affects 64 and c , and f  + b y6 \ rj  a  cd  a  oda-yb(c£d)e 5  +  a  + M (c7d)6 ,  b  c  hS y  b  e b  -f  /276 , s  is such that /1  affects 6 and c . The most general change of variables up to 6  2  second order is lab - Vllab  + P2Vabl + P 3 7 ( a ° 7 b ) c  This gives at zeroth order B^ f a  be  +  = Pi  PlVabU  + Psllab  +  + PeVabl^Jcd  P2^ bi]efa  I  and cfjj  •  (6.52)  = 0, so that the  zeroth-order gauge invariance is S^lab  =  A)+ p 5 A 2  c  c  .  (6.53)  Chapter 6.  Nonlinear, Noncovariant, Spin-two Theories  128  Since we are always free to make such a transformation, let us retain the parameters pi and p i n the equations for the moment, rather than setting pi = 1, and p = 0. A t 2  2  first order, we find that p affects the term w i t h coefficients 65 and b' (in particular the 3  6  combination, (65 + b' )); p affects 63; p affects 61 and b' (in particular, the combination, 6  4  5  4  (61 + 64)); and pe affects b . A t this point we can take p = — |(pi&3 + P2(W + 46 + b' )), 2  4  so that b — 0, and p = -\(p\b 3  6  3  4  + p (Ab + 65 + b' )), so that b = 0.  2  2  2  6  2  Substituting the expressions (6.51) into the zeroth-order part of the integrability condition (6.49), gives [\b (d 6 e d i>  +d e e  abe  0  ic  e  d)  a  b  (c  ]e]  +l(h-b' )d e d ^ +( +(c  5  +  {c  la  d)  a  + \c' )^ d d e 2  e  lc ) iP d dJ e  6  Vcd  + (b\ -  b  (c  a  d)  3  +  e  {c  n  abe d)  ^  A  (c  d)  b',)d e d ^ a  {c  d)  a  + \c' i> d d e  e  2  + (c' + \c )^ d d 9  e  Cl  a  + \cod d e t  a  6  d i; )  abe  d)  +  e  e  e  c  \c ^ ae A  e  d  (c  d)  lc iP ae ] - {0 <- Vc) = d x$ • e  6Vcd  e  (6.54)  {c  e  Proceeding as i n Chapter 5 (or as i n [74]), one finds that b\ - b\ + c' + \c = 0 ,  (6.55)  5(6 -6 ) + c + | c = 0,  (6.56)  c = c = c = c = 6 = 0,  (6.57)  3  A  7  5  0  = [c' ^ dj  0 )  3  d  1  4  +(  a  X d  6  Cl  +  2  5  6  0  \c' )rd e a  2  + \c'^ d e \ a  d  a  d  - [0 <- </>) . c  C  (6.58)  Taking fi = —c sets c' = 0. T h e change of variables t e r m p can now be used unambigu3  3  5  ously: takeps = - | p i ( 6 i + 6 ). T h e n bi-*b\ = | p i (61 4  —  6^) and 6 -> b' = - | p i ( & i - & ) , 4  4  4  i.e., b' = —bi. T h e n (6.55) becomes 61 — 6' = 2\ = 0, so that bi = 6' = 0. There are 4  4  4  still the two freedoms p , f , as well as Eq.(6.56). We mention two ways of choosing the 3  2  remaining parameters: (1) Taking f  2  — b — b gives that b' = 65. T h e n taking p = —b gives 65 = 0 and 5  6  6  3  5  b' = 0. T h e only two remaining parameters are c\ and c' which are related by Eq.(6.56) &  2  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  d + | c ' = 0. T h e n B'^  = 0, and C'^ = c (d ' lcd — 2d( -f ) ). The first-order solution  a  2  129  y  bcd  d  1  e  c  d  e  to the zeroth-order integrability condition (6.49) is then w r i t t e n  6 £fab = c (d -2d e  {  1  xi  =  0)  lab  e {alb)  )X ,  (6.59)  E  (e d r-^d B ).  (6.60)  e  Cl  e  a  a  T h e R H S of (6.59) is the linearized Christoffel symbol —T^ rj + 2ci7 t so that (6.59) can be written S^^ ab  4  a  for the metric g  e ab  = [V^A;,)]^ ' where V  is the derivative  1  ab  =  ab  Q  operator associated w i t h the metric g . W a l d shows that the unique (up to a change of ab  variables and the rescaling freedom) complete solution to the integrability (6.49), w i t h the first-order part (6.59), is #A 7ab a  With  Xa =  Cl(0 V 0 e  E  A  = V A ) , ( a  (6.61)  b  - V>eV 0 ). e  a  (2) A n o t h e r way of choosing the final two freedoms f and p , is to take f 2  3  2  — —c , so that 2  c' = 0, and p% = —65, so that 65 = 0. The only two nonvanishing parameters are then 2  related by ci - | 6 ' = 0, so that B'^  = 2c 6 ^ ,  )a  6  and C'^  a  bcd  1  d)b  = cdf .  d  x  e  cd  This gives  the following first-order solution to the integrability condition (6.49), = c {\ d  4-7ai (°)°  c  clab  lc{a  b)  = c {rd e -e d r) a  x  = \\C^g f  +2 d \)  a  x  1  e  e  e  ,  )  ab  = c [c^e f  (6.62)  ,  a )  1  where C^a denotes the L i e derivative w i t h respect to the metric g  ab  (6.63) —n  ab  + 2ci^ . ab  This  is a complete solution of (6.49), h*fab = \£\«gab  = V( A a  6 )  ,  a X  = CiC^9  a  .  (6.64)  Note that p is still arbitrary. Alternatively, we could have started with the term &oi?7cd6 i in B(°> , so that 6^j = d \d) + b r) d \ . 4  a  2  a  bcd  ed  {c  0i  cd  a  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  130  Consider now the interaction of CG with matter fields, 4>, (where ^ is a generic symbol for any matter field), so that the complete Lagrangian can be written, C  A  OLCM-I  where £M is the matter Lagrangian, and a is a constant.  = CG —  Assuming that the  matter equations of motion hold, i.e., that ^j^- = 0, according to the results given above, C  w i l l either be generally covariant, or invariant under normal spin-two gauge  A  transformations.  In [74], examples of Lagrangians, C , having normal spin-two gauge A  invariance are constructed.  However, it is also pointed out that, assuming that  couples directly to the stress-energy tensor T  at lowest order, i.e., C ^ =  ab  and assuming that T  ab  J^ T , ab  M  is defined according to some prescription, say T  ab  j ab  is obtained by  ab  variation of CM w i t h respect to the flat metric, this possibility seems to be eliminated. T h i s is illustrated using the example of a Klein-Gordon scalar field, S$ = f d <j>d <f). T h e a  a  problems encountered are precisely those discussed i n Chapter 3 (in particular Ref. [83]). (i) W e limit the possible generalizations of £ $ (cf. Eqs.(6.5) - (6.8)) to theories with the infinitesimal gauge invariance 6 ^ Kac  ef  = d d K b  d  a c  P  a c b d ( e /  )  + dK B ( ) ac  K C ( ), ac  +  acb  b  ef  ac  e}  Eq.(6.23). T h e n , for all two-form fields K , and L , we must exhibit a two-form field ab  M, ab  a b  such that [VK^VLJ  = V »  (6.65)  ,  MA  where V f t , etc., are vector fields on the manifold of field configurations defined by ai  Eq.(6.23). R e w r i t i n g (6.65) i n terms of the tensor fields D  a b c d e f  ,  B  a b c e f  and C , ab  ,  ef  via  Eq.(6.23) gives the integrability condition dD ( f) a  dP  acbd  e  H  n  m  n  g  h  m n 3 h  kl  U  [d d L (d KmnB ki d  ac  { e f )  b d  L>  kl  d d L d d K„  + K nC ki)  mnp  b  S  d  ac  g  — {K  mn  p  b  m  h  <-> L )]  ac  ac  Olkl +  \d L 8 B ( f)  + L S C ( f)  acb  b  ac  Kab  = d d M b  d  a c  -  ac  e  ac  P  a c b d { e f )  Kac  + d M b  e  a c  B  a c b [ e f )  (K ac «-> + M  a c  C  L )] ac  a c ( e f )  .  (6.66)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  In order to solve (6.66) for D  abcd e  f, B  abc e  f, C f,  131  and M , note that the arbitrariness i n  ah e  ac  the definition of the tensor fields, (the freedom to rescale K — » f b Kcd)i  is  cd  ab  r\[ac](bd) mn f r> (ef)Ja I ac  r)[mn](bd) (ef) ^  u  B C where fj$  -  [mn]b (ef)  mn  e  /  mn  ac  )  =8 8 . m  a  n  c  ef)  ac  5  + 2d f  b {ef)  -* d \ f  H (  fac B^  d  +df  mn  a  b  ac  D^ \ ,  mn  [a  B^  b {ef)  ac  mn  bd  (6.67)  ef)  +ddf b  d  D^ \  mn  ac  bd  ,  ef)  In addition, the tensor fields transform under a change of vari-  ables according to D  -  )  M ( H ) W  B^\  ef)  (-/)  D^ \ ^-, bd  gh)  -  B^\ ^-,  -  ^  (6.68)  gh)  (  ,  ^  -  First we consider the zeroth-order part of the integrability condition (6.66), which involves at most first-order quantities. W r i t i n g out the most general linear tensor fields  £>  ( 1 ) [ a c ] ( M )  (e/)  BWM\  ef)  =  d^  =  WdHfV  d  ( e V  b  ^  6)[c  a  c  k  + b d^ 9Vef  M[ca  ef)  e  + 47  + b d r\eS [c  f)  1 (e  3  + bed y 6^ ^ k  b f)  + b ri d V K  b  + C \  a]  (/  b [c  2  + bd ^ 6\ 6 k  2  + b d 8^  c]b  s  + rl c^ 7 <5 e)  f)  w  ef  k(e  f)V  (e*  a l  f)V  + drf  M  +  b f)  e  e  7  n  V  e  f  6*>  f )  b^\ 8 h f)  +bd ^  b  +b  B  K7  [ c  {e  d ^  s )  b  ^  c  b  hd i H*  +  b  +  b  {e  b d 6\8  f)  [c  12  a  7  b  e  f)  b d 8^ ^ , b  13  {el  f)  = c^^f) + c $\ aY f) 2  + c 6^ d d 5  {e  f)  k7  a]k  e  + c d d^ )  ]  3  + ceri d d ef  k  [a c]k 7  {e  +  a] f  + c d d y6 [c  7  (e  c 8^ d^d A  a]  s)  (e  ,  k 1})k  (6.69)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  132  and substituting these expressions into the zeroth-order part of the integrability condition (6.66), gives  \d d d K d d L h  m  + \d d d K d d L  a  2  m  a  {e  a  })b  m  m  + §(&4 + h)d L d d d K b  a  bU  m  e)  a  a  + lhod K d L m  a  {e  a  e  m  7  e  m  a  m  ab  a  a  2  4  a  b  a  m  9Vef  b  m{e  m  ac  10  ef  })a  bc  - (K where d'[ = \(d\ c  7  + d ) and d 4  do not appear.)  = \(d\  4  — d ).  m  e)a  a  ]m]  J)  m  a{e  a  f)  m  m  n  6  m{e  Hf)  ddK L ) m  a  (e  lml  f)a  lc r, Dd d K L a  6  ab  m  c  ef  m  ac  = d d M$  ,  a  (e  f)  (Note that d", d , d , bn, b ,  4  ba  lb d K ad L  ~ L)  ab  lml  nL d d K )  -  c  m  a  (f  +  m  1  - $b ri nd K &'L  c  m  a  f)a  - \b d d d K d L  f)a  + \c {d d K L  m  m(e  + +  b  m[e  + i(c2 + c )ad d K L 4  m  b  na  m  5  b  f)m  - \(b + b )d d d K d L  ]m]f)  {e  m  a(e  n  (e  lhd d d K d L  -d (nL d d K 3  f)n  c )d Dd K L  +  na  1  n  m  7  n  m  a  lmal  lb )d d d K d L 2  a  s  +  b  }  m  {e  + |(c +  b  m  m  f)a  d';d d K d d L  + (&x -  a  e)  + \c d d d d K L 3  J)b  + \b d d d L d K  a  m  im  (e  m  - \{b + h)d L d d d K 2  +  b  2  5  6  i2  (6.70) b  13  and  To get conditions on the terms appearing i n (6.70), we note that  the R H S of (6.70), d d M$ ,  can be eliminated in two ways, (1) by taking the trace  a  {e  f)  of (6.70), and (2) by acting on (6.70) w i t h d dh and antisymmetrizing over g and e, h g  and / . Additionally, the L H S can be grouped into traceless pieces and pieces that are symmetrized derivatives. T h i s gives two equations that the coefficients of the c's, d s and :  fe's must satisfy. Proceeding as in Chapter 5 (cf. [43], [44]) we find  d = d'l, d  3  —  b = ci = b = b = 0 , 6  2  b = b = c = - & = -b 3  3  C  5  7  4  = -d  2  =b , 8  EE C = - C 4 , c = -(6 + c), b = (d2  5  b = b 9  10  = ce = 0 .  t  \b) ,  (6.71) (6.72)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  In the special case where K  and L  ao  133  are constant two-forms, the first-order part of the  ao  integrability condition (6.66) (which w i l l only involve first-order quantities i n this special case) gives the added condition that c = 0. Further simplification can be obtained using the two freedoms (6.67) and (6.68). T h e most general second-order change of variables is Tab = 3l(a lb)c  + n rj b-fj + n 77a& + nerjabj -y .  c  n  (6.73)  0  4  5  a  cd  We find that n can be used to eliminate d'{; n , d ; and n^, d . T h e most general linear 3  5  expression for f b  6  5  is  cd  a  If fi = — fi = f[, then f[ affects d , d , b , b , 67, 6 , c and c . f affects c/ , bi , 613 and 2  c.  4  3  8  b  3  W e use the two freedoms i n (6.74) to set d = 6  7  i 3  5  6  3  2  = 0. This gives the linear gauge  invariance (6.24) (or (6.26)), K Jab  =  S  a  [V +  ( a  V^ |  (c  | c  6 )  ]  + (fe  ( 1 )  - \b)d d K (al  -|fe)5  c 7  %/^)  + bnVab&i&Kd  c  7  1 2  m  ,  (6.75)  which is a solution to the zeroth-order part of the integrability condition (6.66), where M °j is given by a  M $  = b [K d d Lj  + K dd L  m  b[a  b  f]  + \d K d L \  m  b[a  m  ]mlJ]  In the special case when by — C7 = | 6 , and 6 2  n  - (L  m  m[j  a]m  <- K )  ac  7ab  ac  a  (6.76)  = 0, a solution to the complete integrabil-  ity condition (6.66) having the first-order part given by Eq.(6.24) is 8fc Eq.(6.30), where V  .  ac  = V( V /f| |6), c  a  C  is the derivative operator associated w i t h the metric g b = rf^ — b^ab, a  Eq.(6.25), and M f is given by a  M  af  =  b [K V V L m  b[a  - {L  f]  ac  + K VV L  b  b  m  b[a  <-* K ) ac  •  m  + iV /, m  [mlf]  V L M  m [ /  Q ] M  (6.77)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  134  T h i s can be shown by direct substitution. It is possible that there are other solutions to the integrability condition (6.66) taking different conditions on b, bn, b\ and cj. 2  T h e equation (3.68) gives further restrictions on the constants 6 , bn, and c , namely, 12  c= b  12  b (h  =c ,  (6.78)  7  + 4fc ) = (c -  n  7  (\k  2  + k) .  x  (6.79)  2  Thus, the four arbitrary parameters b, 6 , bn, and c , are reduced to the two arbitrary 12  7  parameters b and c. In the case where k = — \k\, bn remains arbitrary, but c =  This  2  gives the infinitesimal symmetry S^l 7ab = [ V ( V i f | | ) ] c  c  a  c  6  ( 1 )  + bnr} bd fd K , d  cd  In the case where k ^ —\k\, bn = a(c — \b), where a — (\k\ + k )/(ki 2  gives 6£lj  ab  2  = [ V V A | ] ( ) + (c - i6)[a (a ^|c|6)) + a^d^Ka], c  ( a  1  | c  6 )  (a  c  + 4& ). This 2  Eq.(6.29).  7  W h e n considering additionally the interaction of  Eq.(6.28).  c  a  w i t h matter, direct extensions of  the arguments for nonlinearity of the spin-two equations discussed i n Chapter 3, seem to eliminate the possibility of a theory w i t h the trivial gauge symmetry Sx^-Jab =  d( d K\ \b), c  a  c  (6.8). We also remark that the criteria (2.70) that Fierz and P a u l i gave for second-order  (\ 2  Lagrangians i n f , ab  applied to C  , would give the following restriction on the arbitrary  K  ki coefficients h = \h + \k  2  k  2  + fy , fci  (6.80)  1  so that one parameter would still be completely free. (In cases (ii) and (iii), c = c = 0 3  4  so that (2.70) is automatically satisfied.) (ii) For extensions of £ £ \ (cf. Eqs.(6.9) - (6.13)), we are only interested i n theories that have the infinitesimal gauge invariance (6.23) and the conformal invariance (6.31). T h e n , for a l l two-form fields K b and L b there must exist a two-form field M a  a  ab  and a scalar  field T such that [V ,V ] Kab  Lab  =V  Mab+  V. T  (6.81)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  135  Substituting the infinitesimal gauge variations (6.23) and (6.31) into Eq.(6.81) gives the integrability condition (6.66) except that the R H S is now ddM D \  +dM B \  acb  h  d  ac  +M C  ac  e})  h  ac  + rH  ac  el)  ac  {ef)  ef  (6.82)  .  Furthermore, for all two-form fields N , and scalar fields fi, there must exist a two-form ab  field Q , and a scalar field A , such that ao  [VN ,V ] = V ab  Q  + V.  Qab  (6.83)  A  Substituting the gauge invariances (6.23) and (6.31) into this equation gives a second integrability condition  n,d d N b  d  ac  — V Olkl  OH, tt^(d N B  +  kl — tiki  # 7 'ki  +N C )  acb  b  ac  acbd (ef)  ac  kl  ac  ddQ D \  kl  d  ac  Q  ac  ef)  +Q C  ac  e})  -  ac  ac  b  + dbQ B \  acb  b  - dN 6 B \ ef)  ac  ac  (ef)  N 6C  ac  ac  Q  + AH  ef  (ef)  .  (6.84)  T h e arbitrariness i n the tensor fields is given by Eq.(6.67) together w i t h H  ab  (6.85)  —• cH b a  where <r' ' = 1. T h e first-order expression for a, o^  = 017, affects h .  0  2  In addition,  under a change of variables the tensor fields transform according to Eq.(6.68) together with H b —• H<. cd a  Olab  (6.86)  cd  W e use n , n , n and n from Eq.(6.73) to set cl", h , d and d to zero respectively. Sub3  4  5  &  2  6  5  stituting the most general expressions for the linearized tensor fields, Eq.(6.69) together with H^Kb = hjab  + h ji]ab 2  ,  (6.87)  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  136  into the zeroth-order part of the first integrability condition, Eq.(6.66) w i t h the R H S given by (6.82), gives the zeroth-order equation (6.70), except that the R H S is now + T% .  d d M$ a  ie  (6.88)  ef  })  Substituting the first-order expressions (6.69) and (6.87) into the zeroth-order part of the second integrability condition (6.84), gives the additional equation {hi -d  + di-  4  4:d )ad( d N 6  e  3  12  4  c  / )  + (h -be-br-  f)a  c  m  4  + (6 - 6 - 6 - 4b )d nd N 5  + (c + c - c - 4 c ) d d O i V | |  c  (e  W)  ~ W - b - Un)rj d nd N  + (b  w  ef  9  5  7  c  a  3  (e  ca  ( e  ih )d fld N  = d d Q$  c  a  3  a  {  f)  {e  + A<%  e /  .  (6.89)  Solving the first zeroth-order integrability condition, (6.70) w i t h the R H S given by (6.88), as i n the previous example, we get (6.71). Solving (6.89), we get h = hi ,  &!3 = -6x2 = -\(b+h)  ,  .  c =-\c+\(b+h) 7  (6.90)  A higher-order calculation shows that c = 0. Using f[ from Eq.(6.74) to set d = 0, and /  3  to set b + h = 0, Eqs.(6.71) and (6.90) give the linear infinitesimal gauge invariance  (6.32) as a solution to the zeroth-order part of the integrability conditions, w i t h M °j a  given by (6.76), and (°)  T  =  [\b d d d K d L h  a  m  — (L  m  a  m  bc  <-» K )  ac  + \b nd K d L  c  9  ac  w  =  2hVLN  A  =  (610 - \h - b - Abn)d ttd N  ( 0 )  a}  a  w  ac  +  \c Od d K L \ a  &  m  c  m  ac  , 9  In the special case b  a  ,  Q } 0 a  c  m  = b — ce = 6 9  c  ca  n  .  (6.91)  = 0, a solution to the integrability conditions (6.66)  (with R H S (6.82)) and (6.84), w i t h the first-order solution (6.32), is given by Eq.(6.33),  137  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  where M j is given by Eq.(6.77), and a  T  0,  =  2hftN  Qaf  ,  af  A  (6.92)  -|Av nv jv , c o  a  c  where V . is the derivative operator associated with the constant-determinant metric a  (6.34). This can be verified by direct substitution. Again, there may be other solutions. Equation (3.68) gives no additional information in this case. ( 2)  (iii) A generalization of C , (cf. Eqs.(6.14) - (6.17)), is required to have the gauge T  invariances (6.36). This yields the fundamental integrability conditions  (  a Jjacbd _ _ l l  dD  c  +  D  n r  m  P  s  a r\mnop _ ±  t  \ a  D  c  M  ^ ^ ^ J ^  \  cbd  efd d L  Hi  b  d  [(d K B  + KmnC  mnp  ac  p  mn  st  mn st  ) — (K <-» L )\  + P h  (dbL 6 B \ j) + L 6 C \ j))  = P  b  ef  ac  g  ef gh  ac  d  ac  e  e  +dM B  acbd  ac  Kac  b  ac  ac  +M C  acb  ef  ac  eI  ac  - (K <-> L )  a  Kab  (d d M D  ac  ac  ef  ac  + rH )  (6.93)  ,  ef  and, pef  pst  3D acbdef  H, V %i d  T  djki  (d N B b  ac  st  = P  kl  +  acb  ef gh  NC ) ac  {d d Q D b  P h(d N 6 B  ac  d  ef  st  9  ac  acb  ef  b  +  acb  u  + d QaeB  acbd  ac  b  ef  +Q C  ac  ef  ac  ef  N SC ) ac  ac  Q  + AH ) ef  ef  .  (6.94)  These equations are the same as Eq.(6.66) (with RHS (6.82)) and Eq.(6.84), except for the two projection operators P  ab  : one projects out the overall trace, and the other  cd  arises from the variation with respect to jj , i.e., b  fc)  hab  =  M  fc)  dj  ab  =  6jJ  d  pab  cd  hl  (6.95) d  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  138  The most general expressions for the linear tensor fields are the same as (6.69), except 5  that they do not include any of the terms involving rj f (since P / 7/ / = 0) or the trace e/  e  g  l  e  Of Jab, {l)[ac}{bd) (ef)  D  + f/47 B  (l)[ca)b  (ef)  T t C  (e»  )  + b d ~f (e6 f)  hdWe}^  =  ^ /),  a I ( 6  b  + b d^ \ 6  a]  T[C  Ta  2  + hd <y 6 \ 6 k  T[a  c  k  e  +b  b f)  e  3  6  d  k 7  ^  c f ) V  ^  +  b f)  +  b  hd  645[C7 { e  j  T [ c f )  T | i > l  ^  (/^ e) 1  b  + bd w, Tb  8  Q(1)[C  (e7  f)  ci^[c(e7Ta]/) + c S  (ef)  [ c  2  + =  c^  h  T  c { e  d d f )  T a ] k k l  { e  D^  f )  + c  3  d  {  e  d ^  f  )  +  c^ d d% c  a]  {e  k  , (6.96)  .  xl ab  We have the usual arbitrariness i n the definition of the tensor fields given by Eqs.(6.67) and (6.85). A t linear order however, the only possibility for / ^ | ° ^ is f[6[J j ^.  Also, a  c  b  change of variables jj  —> T"J , induces  b  pe  6  f [ac)(bd) kiD  P iB^ ef  pef r^  {ef)  pef  b  k  p>[ac]b  (ef)  pef (~<[ac]  P kiC \ ef  r)[ac](bd) J-s ((ef)  m.Ti  [ac  ef)  P kiH  -T  1  dj (ef)  (ef)  dj  T  ki  Imn  dlli  pef rr HhL r ab-tlef 7 T 1 T mn  ef  ef  U p to second order, fab = ™l7a6 +  mn^  ®1kl  2{lJ lI)c ~  m  C  a  (6.97)  (6.98)  llW^ab)  We have also considered terms involving the antisymmetric Levi-Civitaepsilon tensor, e bcd, however, we find that by the allowed freedoms (the change of variables (6.97) and the freedom to rescale the arbitrary gauge parameter, Eqs.(6.67) and (6.85)), together with the integrability condition, that these terms can be set to zero. 5  a  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  139  allows us to set d" to zero. Substituting the expressions (6.96) into the zeroth-order part of the first integrability condition (6.93), gives Eq.(6.70), except that the bg, b  and c  w  6  terms do not appear and included on the L H S are the terms \{bs - 6 - h)d d d K d L a  4  m  b  b  + i(6 - h -  c  mc  a  h)Ud K d L m  5  + | ( c - c - c )Dd d K L ] a  4  5  m  3  n  n  a  .  c  m  a  m  ca  (6.99)  Substituting the expressions (6.96) into the zeroth-order part of the second integrability condition (6.94), simply gives nhid d N {e  = d d Q\%  a  {e  W)  a  ,  }  (6.100)  which tells us immediately that hi = 0. Solving, we find Eq.(6.71) and, at the next order, that c = 0. Using the rescaling parameter f[ to set b — 0 we find that the linear gauge invariance (6.39) is a solution of the zeroth-order part of Eqs.(6.93) and (6.94) w i t h  Kf] Q[° where k  6.4  = 2dd K d L m  ] b  =  + k  n  m[a  f]n  [aJ]  ,  0,  (6.101)  is a two-form field such that $( d fc| |/) = 0. a  ab  e  a  Conclusions  In summary, we were able to find classes of nonlinear theories of a spin-two field that were not generally covariant by considering alternative formulations of the linear spintwo equations.  In particular, we considered the three formulations (i) C , Eqs.(6.5) K  - (6.8), (ii) C c\ Eqs.(6.9) - (6.13), and (iii)  , Eqs.(6.14) - (6.17). In the special  (  (2)  cases described, we found that generalization of (i) C , could lead to theories w i t h K  the infinitesimal gauge invariance 8j  = d( d K\ \ ), c  ab  a  ( 2)  c b  Eq.(6.8), or 8j  = V( V A'| |i ), c  ab  a  c  )  Eq.(6.30); that generalization of (ii) C , could result i n theories w i t h the symmetries c  Chapter 6. Nonlinear, Noncovariant, Spin-two Theories  hab  = d( d K\ \ ) + flrjab, Eq.(6.13), and 6j  = V( V A"| |  c  a  c  c  ab  b  140  a  c  & )  + ftg , Eq.(6.33); and ab  that generalization of (iii) C?\ could lead to theories having the infinitesimal gauge invariances 6jJ — d( d K\ \ ), Eq.(6.17), and 8g c  b  a  c  ab  b  = C^g , Eq.(6.47). Moreover, there ab  may be additional classes of allowable theories. We have only succeeded i n finding general solutions i n certain special cases. T h e requirement that at linear order the dynamical variable j b (lab) a  c o u  pl  e  directly to the stress energy tensor of matter, may eliminate  the possibilities of interacting theories having the linear gauge invariances, (6.8), (6.13), (6.17), (see Chapter 3). In some sense, we have really only considered some of the simplest possible alternative formulations of the spin-two equations V '4>ABCD = 0, Eq.(2.1). Other formulations, i n AA  terms of different potentials that were, say, not symmetric rank-two tensors, would also likely lead to interesting types of theories that were not generally covariant. It should also be mentioned that new types of gauge invariances would undoubtedly be found by considering collections of spin-two fields satisfying some of these alternative linear theories, or b y considering interactions w i t h other fields. These projects are certainly computationally challenging. Moreover, they may provide insight into the quagmire of quantum gravity.  Appendix A  Representations of the Poincare Group  Relativistic field theories are based on invariance properties under the proper Lorentz transformations ( L T ) . Consequently, the possible types of relativistic wave equations for a physical system can be determined from the representations of the Poincare group ( P G ) . Corresponding to every irreducible wave equation is a system of differential equations. T h e irreducible representations of the P G (and hence also the (elements of the) representation space and associated differential equations) can be characterized by the two P G "invariants", mass and spin. Thus given the mass and the spin of a field, the transformation properties of the field under an irreducible representation of the P G are known. Conversely, one can determine the mass and the spin of a field from its transformation properties. In this appendix, after briefly defining a L T and stating properties of representations of the L G , the infinitesimal Lorentz transformation (ILT) are considered. In particular, the invariants mass and spin which characterize the representation are constructed from the generators of the I L T . Also, an irreducible representation is given and it is shown how to determine the spin content of a field. The continuous unitary representations up to phase of the P G were obtained by Wigner i n 1939 [80], and the associated differential equations (as well as a summary of [80]) are given i n B a r g m a n n and Wigner (1948) [4]. This appendix only states some of the results of these papers which are relevant to the discussion of spin. See also Weinberg (1962) [76].  141  Appendix A. Representations of the Poincare Group  A.l  142  T h e Lorentz transformations  A relativistic system defined on the spacetime (M,g b) is invariant under translations, a  rotations and boosts of the spacetime, i.e., under the (active) proper L T . In terms of coordinates, a (passive) L T from one system of spacetime coordinates system of coordinates x  to another  is given by  m  x  M  x "  = A  1  V + A" ,  (A.l)  where A is a constant four vector and the matrix A is such that 77^ = A M  7r  A r y ^ . The p  fi  t/  p  set of a l l L T form a group called the the Poincare group ( P G ) or the inhomogeneous Lorentz group. For example, performing a second L T on ( A . l ) , one gets  A ^ x " + ^,1  =  (A.2)  where A , i = A A i and A , i = A A i + A . This means that any representation D(A, A) of 2  2  2  2  2  the P G must satisfy the group multiplication property /J(A A )D(A ,A ) 2 >  The  set of L T w i t h A  M  2  1  1  = D(A A ,A A + A ) . J  2  1  2  1  2  (A.3)  = 0 is called the homogeneous Lorentz group or, simply, the  Lorentz group ( L G ) . A L T is called proper (or sometimes called restricted) if det A = +1 and A ° > 1, i.e., no space or time inversion. Since the symmetries of nature are believed 0  to be the proper L T , i.e., the translations, rotations and boosts of the spacetime, i n what follows, only the proper transformations are considered (however, the adjective "proper" w i l l , i n general, be dropped). Under a L T , a field \P transforms according to = D(A,A)tf  .  (A.4)  Appendix A. Representations of the Poincare Group  143  For example, considering homogeneous L T , for $ a scalar field one sees that D = 1, for a vector field D = A , and for ty a second rank tensor field D = A A . If the states ty of a system are represented by vectors w i t h a unit norm and are determined up to an arbitrary phase (as i n a quantum theory for example), then the P G is represented by unitary matrices [/(A, A) which need only satisfy the group multiplication property up to sign [80] C / ( A , \ )U{A 2  2  Xi) = ± C / ( A A , A A  U  2  1  2  X  + A ) . 2  (A.5)  T h e representations up to sign of the P G correspond to the group of translations and linear transformations of unit determinant on a two-dimensional complex vector space.  1  T h i s infinite dimensional representation is a double-valued representation of the P G ,  2  and can be written completely i n terms of irreducible pieces. Elements of the representation space are called spinors. See also W a l d (1984) [73].  To find the invariants which  characterize the irreducible representations, it is convenient to consider the I L T .  A.2  T h e Poincare group invariants  A n infinitesimal L T , i.e., a transformation that is infinitesimally close to the identity transformation, is given by -> x" where the six real parameters eters e  M  describe translations.  1  =  + LO\)X" + e" ,  (A.6)  = u ^ „ ] describe rotations, and the four real paramT h e unitary matrices corresponding to the infinitesimal  transformations (A.6) can be written U{1 + w, e) = 1 + \u J ah  ab  - ie  a  aP  + 0(2) ,  (A.7)  T h i s group is denoted ISL(2,§) ("I" for inhomogeneous, " 5 " for special, (i.e., unit determinant), "L" for linear, "2" for 2-dimensional, "(J" for complex). The homogeneous transformations are then represented by the group of two-dimensional complex matrices of unit determinant, SL(2,$). M o r e correctly, ISL(2,$) is a representation of the (twofold) covering group of the P G . :  2  Appendix A. Representations of the Poincare Group  where J  ab  =  144  and p are called the generators of the transformations .and correspond a  to the total angular momentum and the four-momentum of the system. In the case where 3  the field ty i n (A.4) is a vector field, the generators of rotations and boosts J  ab  are the four-  by-four matrices (J" )^ = —2i6 [^St „}. T h e n (A.7) reduces to (A.6). T h e total angular 13  momentum J  a  3  can always be written i n terms of the orbital angular momentum L  ab  ab  and intrinsic angular momentum S . The generators satisfy the following commutation ab  relations  [Jab, Jed] ~ ^9[b\\cJd\\a\ , b e Jed] = 2ig [bPa] ,  (A-8)  [p ,Pb] = 0 .  c  (A.9)  a  (These commutation relations can be brought into a more familiar form, by identify.  ing K  %  = J  .  .  .  .  .  and J = t j J 1  l  2  —*  —*  where K is the generator of boosts, and J , the an-  3  k  gular m o m e n t u m three-vector, is the generator of rotations (i,j,k take on the values 1,2, 3.) T h e n one may rewrite (A.8) as the three commutation relations [J\ J ] =  ie^ J ,  J  [K\K>] = -ie^ J , k  k  \J\K>\ =  k  k  ie* K .) k  k  The angular momentum JA — Sx x p of a particle at point B with momentum p about an arbitrary point A where 6x is the position vector from A to B, can be written as = 28x^ ^ where = J^eit and i,j,k range from 1 to 3. In a relativistic system this relation becomes Jjj" = 28x p"^, the angular momentum of a particle at event B with four-momentum p^ about event A, where 8x^ is the fourvector from event A to B. Then J , J , J are seen to be the 1-, 2- and 3-components of the angular momentum vector J. (There is no clear physical significance for the J° components (Weinberg (1972)[76]).) For a system described by the (conserved) energy-momentum tensor T^ , the T ° are the components of the momentum density so that the (conserved) total angular momentum of an isolated system about A is given by J^" = J d x (Sx^T" - 8x"T' ) where the integration is over the spacelike hypersurface £ of constant time. See Goldstein (1980) [37] section 9.5 for a discussion of angular momentum as the generator of rotations, and momentum as the generator of translations. Note that J ^ " is not invariant under a change of reference point: — J^ — —8a p -\-8a p where 8a = a^ — a^ and p = f (PxT . One can define the intrinsic angular momentum four-vector 5 by Sn = -^t p p J^ which is invariant under a change of reference point, where m — —p^p - Note that in the rest frame of the particle S = J and 5° = 0. Also, in any frame 5 has only three independent components since S^p = 0. Thus can be regarded as the internal angular momentum, i.e., the spin of the particle. Then the total angular momentum of the system can be written in terms of its orbital L " and intrinsic 5"" parts as Jf" = + L» where S» = ^p S e ^ and = 2y^p^ where y» is the perpendicular displacement of the event A from the centre of mass of the world line of the particle. See also M T W [47] pp. 152 - 159. 3  1  tJ  i,i  23  3 1  21  l  v  3  0  l  i0  s  tl  M  v  u  11  11  110  a  M  2  a  1  ltl  11  M  11  l  v  v  a  a  p  v  /  Appendix A. Representations of the Poincare Group  145  A group "invariant" is a nonlinear function of the generators which commutes w i t h all the generators, (i.e., a Casimir operator). Defining the P a u l i - L u b a n s k i vector W by a  W  = -\t p S  a  b  one can show that [J , W ] = 2ig [ W ], ab  c  m  2  c  a  = -  P  cd  (A.10)  and [p , W ] = 0. Thus one sees that  b  a  ,  a a  ,  ahcd  P  b  W =WW 2  ,  a  a  (A.ll)  commute w i t h a l l the generators and hence are P G invariants. It can be shown that the P G has only two invariants. Thus the values of m and W characterize an irreducible 2  2  representation. Here, we only mention the two types of representations which are believed to have physical significance, namely representations with (i) non-zero rest mass, a n d representations w i t h (ii) zero-rest-mass w i t h integral or half-odd-integral spin: (i) T h e irreducible representations w i t h finite mass are characterized by the values of m W , with W 2  2  = mS 2  2  = m s(s + 1), where S 2  and the spin s = 0, | , 1, • - -  2  > 0 and  is the squared spin angular momentum  2  . The space of states that transform according to this  representation can be given by the set of completely symmetric spinors (f> ^- " that are A  solutions to the differential equation ( • -\-m )<f> 2  Al  = 0, where • = dAA'd '.  An  AA  A  (ii) T h e  irreducible representations w i t h zero-rest-mass and integral or half-odd-integral spin are characterized by the values of m — 0 and, since W = 0, W = sp where the helicity 2  2  a  (spin)  a  s — 0, ± | , ± 1 , • • • , is the component of the spin S parallel to the momentum.  4  T h e space of states that transform according to this representation can be given by the set of completely symmetric spinors (j) i- n A  A  that are solutions to the differential equation  C U J A J ^ " " " = 0. See also Chapter 2. 1  4  More accurately, the magnitude of the helicity is called spin. The two possible values of the helicity correspond to the two states of polarization. 4  Appendix A. Representations of the Poincare Group  A.3  146  Irreducible representations of the homogeneous Lorentz group  A n y matrices satisfying (A.8) w i l l provide a representation of the L G . Note that by defining the matrices S  % +  and S'_  Si  = K!eV ± iJ ) ife  ± iK )  = \{J  01  l  {  .  (A.12)  the commutation relations (A.8) can now be written as the commutation relations for two independent sets of angular momentum matrices [S , S{] = id\S>l l  +  '  SL] = ie S . ij  ,  k  k  [SI, St} = 0.  (A.13)  T h e representation of angular momentum matrices J i n terms of irreducible matrices denoted (here) by J  2  = J • J and J  z  is well known: The standard representation is one i n which are diagonal. This infinite dimensional matrix can be written as  the direct sum of irreducible submatrices, (i.e., matrices that can not be written i n block diagonal form) each of which is labeled by a positive integer or half integer j and is (2j +1) dimensional where j is the largest eigenvalue of J and J z  2  — j(j + l ) . T w o commuting sets  of angular m o m e n t u m matrices, Ji and J , can be added to give the set of "total" angular 2  m o m e n t u m matrices J = J\ + J which can be written i n terms of irreducible submatrices 2  J =  jfa+Ja)  ©  j(h+32-\) © . . . © j(\h-h\) ( f  or  details, see any quantum mechanics text,  for example, Cohen-Tannoudji, D i u and Laloe (1977) [17] pp. 644-660 and pp. 10031042).  Thus the pair of angular momentum matrices S± provide a representation of  the (homogeneous) L G labeled by the pair of positive integers or half integers ( s , s _ ) . +  Defining S = S  +  + S-, from (A.12) we see that S = J so that the representation labeled  by ( s , s-) describes particles of spin s = ( s +  +  + s_), (5+ + s_ — 1), • • •, | s — +  It is reasonably straightforward now to determine the transformation properties of various quantities and thus their "spin content". For example, for a scalar field, D — 1 (where D is defined i n Eq.(A.4)) so that (s+,s_) = (0,0), i.e., a scalar field describes  Appendix A. Representations of the Poincare Group  147  spin-zero particles. For a vector field, D = A which gives ( s , 5 _ ) = ( | J | ) -  Thus a  5  +  vector field describes particles of spin-one and spin-zero. For a rank-two tensor field, D = A A so that (s ,s-)  = (\,  +  |) ® (|,  \) = (1,1) @ (1,0) © (0,1) © (0,0). Thus a  rank-two tensor field describes one spin-two particle, three spin-one particles and two spin-zero particles.  A.4  P r o j e c t i o n operators  It is somewhat more complicated to isolate the pure spin part of a tensor field. Following, the operators that project out the specific spin part of a vector and a rank-two tensor field are given (see [34], and [51]).  A.4.1  Vector field  A vector field has four components, three of which describe a spin-one field and one of which describes a spin-zero field (see the end of section A . 3 ) . Thus one can write A  = A  a  + A  a  = PA  a  1  a  0  1  h  b  + PA a  0  b  b  ,  (A.14)  where the pre-subscript («) indicates the spin content of the quantity, and the P s  a b  are the  vector projection operators that project out the spin s part of the field. The projection operators i P % and oP b are given by a  r  P\  =  0  The operator \ P  a  \ - ^ ,  6  P\  = -If  (A.15) .  (A.16)  is the transverse projection operator, i.e., p (iP A ) a  0  a  b  b  = 0, and the  operator oP b is the longitudinal projection operator. a  F o r example, substituting (J )^„ — -2z'(5 [ <$ „] into the expressions for 5 ± (A.12) gives S± • S± \l = s±(s± -f 1) which implies that (s ,.s_) = ( | , | ) . 5  a<3  a  /3  /J  +  Appendix A. Representations of the Poincare Group  148  For a vector field, the generators of rotations and boosts J matrices (J )^  = —2i8 ^8^ ^. The squared Pauli-Lubanski vector W is then  af3  a  2  (W y  = -2p  2  b  T h e action of (A.17) on P A a  1  {W )\  and P A  b  a  b  0  2  ^8\ - ^  .  =  m s(s + l){ P A )  (VK )% ( i ^ V H  =  0,  6  c  c  2  (A.17)  is given by  b  b  (iP A )  2  2  a  1  = 2m { P A )  b  2  b  a  1  a  b  (A.18)  and spin s = 0 for the  c  c  longitudinal part,  ,  b  b  (A-19)  which gives spin s = 1 for the transverse part of A , iP A ,  A.4.2  are the four-by-four  oP A b  c  c  Rank-two tensor field  A rank two tensor h  ab  has sixteen components: five components describe one spin-two  field, nine components describe three spin-one fields, and two components describe two spin-zero fields. One can write  hb  \h  = 2^a6 +  a  + h  ab  Q  where the projection operators P b  cd  s  tensor, the generators J  a  ab  =  ( 2 ^ 0 6 ^  (A.20)  1  project out the spin-s part of a (o) tensor. For a (o)  are given by (J ) d  ab  iPab^ + oPab^hab  +  ab  c  ef  = -2i(r) y 8 } 8j -f r) [ 6 ] 8 ), so that e  c  a  f  b  d  a  b  e  c  the squared P a u l i - L u b a n s k i vector is (W ) 2  Cd ab  = -±p 8 8 2  C  a  d b  + 2p ( abV ~ Sa 8 ) + 8p J>% > - 2(7 abp p + V^PaPb) • (A.21) 2  Cd  2  cd  a  a  T  ab  T  C  d  b  Note that the action of (W ) b (p h J = 0, n h J  d  V  C  (o  d  1  on a tensor field h^ that is transverse and traceless T  b  = 0) is {W ) h J 2  cd  ab  T  = m s(s + l)h 2  T  T ab  = -6p h J 2  T  ,  (A.22)  Appendix A. Representations of the Poincare Group  149  so that the spin of a transverse and traceless (o) tensor is s = 2. T h e operator which projects out the spin-two part of a (o) tensor is (see [34], [49]) 2Pab  Cd  T h e n (W ) 2  = 6j 6 Q  cd  ab  C  b  ~ 1  + j>(jlPaPbP p  ^  C  (2-Pc/^e/) — 6 m (2P b h d) 2  Cd  {C  cd  c  2  (A.23)  •  d)  and the rank-two tensor P b h d  cd  a  + V PaPb) ~ ^J{a Pb)P  d  a  c  is trans-  verse and traceless. T h e projection operators for the spin-one part are given by iP  = 6a 8b [c  c d a b  - p PbP p  d]  c  ,  + fA* P*)V  d  {c  a  d)  (A-24)  which can be written i n terms of three orthogonal parts,  -  i cd p cd . Pab — \"\ab +  p cd . p cd \*2ab. + \"Zab i  (\ t)r\ \A.ZO)  where iPiab  =  ^ ( ^ ^ P ^ - w y ) ,  (A.26)  lP2ab°  =  SaV  (A.27)  =  $6 l p p4  cd  d  iPsab  cd  [a  c  ~ ^  P  b  ]  p V ,  ,  b]  (A.28)  (the last two operators are zero on a symmetric tensor field). The spin-zero projection operator oPab is given by cd  0  Pab  Cd  = § (VabV  + ^PaPbpY  Cd  ~ ^ ( w V + PaPbV*))  ,  (A.29)  which can be written i n terms of two orthogonal parts, dPab  Cd  = oPlab + O^afe^ ,  (A.30)  Cd  where iPiab iP2ab  cd  = | (-a« r/ a 6  c d  cd  ((1 + a)  = lt  d  Vab  + f,PaPbP p  where a is an arbitrary real number.  c  d  - (1 + ± a ) ±  - jt(riabP p c  d  )  ,  (A.31)  ~ 4av PaPb))  ,  (A.32)  cd  P  a  P  b  Appendix B  A B r i e f Introduction to Spinors  P h y s i c a l quantities are most commonly described by tensor fields and satisfy tensor equations.  In a system that is locally like 9ft and which has a Lorentzian m e t r i c 4  1  (for  example, the spacetime i n which we live), there is another mathematical formalism available, namely the two-spinor calculus. Spinor calculus encompasses tensor calculus and, i n many ways, is simpler. T h e concept of a spinor was briefly introduced i n the footnote on page 13. Here we take the opportunity to motivate and to expand on the statements made there and to introduce the material necessary to understand the equations i n the m a i n text. T h e approach here follows roughly that of Penrose and Rindler (1984) [57] (to w h o m the reader should refer for further discussion, details and proofs) and is not intended to be complete.  B.l  2  A geometric picture of spinors  Rather than just giving the abstract definition of a (two) spinor, let us first (following Penrose and Rindler) motivate the concept using a simple geometric picture. see that associated w i t h a spin-vector K  A  3  We shall  (the simplest type of spinor) is a unique future  When working with spinors, the signature of the metric is taken to be (+ ). This has no physical significance, but avoids complications in the sign conventions for raising and lowering spinor indices. There are many different ways of introducing spinors and developing the spinor calculus. See also Wald (1984)[73] Ch. 13, and Pirani (1965)[59] and references therein for alternative approaches and discussion. Although the concept of spinors and spinor algebra does not rely on this geometrical viewpoint and can be developed independently, a geometric spacetime interpretation can be conceptually helpful. 1  2  3  150  Appendix B. A Brief Introduction to Spinors  151  pointing M i n k o w s k i null vector k , which can be interpreted as a "flagpole", together 4  a  w i t h a two-dimensional half-plane orthogonal to k , which can be interpreted as the "flag a  plane". T h e flagpole together w i t h the flag plane give the geometric picture of a spinvector as a "null flag" i n a Minkowski vector space. However, under a rotation by 2ir, while the null flag rotates into itself, the spin-vector goes to its negative. Thus every M i n k o w s k i null flag corresponds to two spin-vectors. First, we briefly discuss Minkowski vector spaces. Let V be a M i n k o w s k i vector space, i.e., a four dimensional inner product space w i t h a metric of signature (+  ). For example, i n special relativity, a Minkowski vector  space is the set of position vectors originating from an arbitrary origin of the set of points AA. which make up the Minkowski spacetime.  O n a curved spacetime manifold,  5  M i n k o w s k i vector spaces are the tangent spaces of points. Denote an element of V, a M i n k o w s k i vector, by v . a  of its components  =  In a basis { e ^ } = { e , e i , e ° , es ], v is written i n terms a  a  0  a  u , u , v ) as v = u e ° — v°e 1  2  3  a  a  2  M  M  a 0  +ve 1  1  a  + ve 2  2  a  +  ve , 3  a  3  where ' V is a label indicating which component of v or which of the four basis vectors a  { e ^ } , and V is the (abstract) vector index. In the standard representation, each point a  i n AA. is represented by a quadruple of real numbers, and V consists of the set of vectors originating from the point (0,0,0,0). T h e components of v  a  (t,x,y,z).  are then the coordinates  This is the conventional way of representing Minkowski vectors i n terms of  M i n k o w s k i coordinates. T h e concept of a spin-vector arises from a different way of representing Minkowski vectors by coordinates, namely from a coordinatization of the null vectors of V (which The null vectors, i.e., vectors n such that n n = 0, are taken to be future pointing by convention. B y Minkowski spacetime is meant the flat spacetime of special relativity, i.e., a four dimensional manifold (roughly a set of points that is locally like 9£ ) with a Lorentzian metric. By spacetime manifold is meant a set of points with the local structure of a Minkowski spacetime. 4  a  a  a  5  4  Appendix B. A Brief Introduction to Spinors  152  span V) i n terms of complex numbers. Consider the future pointing null vectors originating from an arbitrary origin. T h i s null cone can be represented i n a coordinate system (t,x,y,z)  by any time slice, say t — 1, which is the hypersurface x  2  + y + z 2  = 1 (see  2  Figure 1). T h i s sphere can be mapped to the extended complex plane using stereographic projection, i.e., the sphere can be coordinatized by the complex numbers  x + iy  together w i t h the point / = oo. B y defining the pair of complex numbers ( / c , ^ ) (not 0  1  both zero) by / = £ . the point /  (B-2)  = oo w i l l correspond to some finite label, say (1,0).  T h i s pair ( K ° , K ) , X  coordinatizing the future pointing null vectors of V, can be regarded as a coordinate representation of the spin-vector K . The spin-vector, or one-index spinor, K is written A  A  in terms of these components « , T = 0,1, as ( « ) = (K°, re )", where "r" is a coordinate r  r  1  index indicating which component of K , and A" is the (abstract) spinor index. F r o m A  U  this geometric construction, it is clear that associated w i t h any spin-vector K is a future A  pointing null direction. Note, however, that since the time slice taken as a representative slice of the null cone was arbitrary, this association does not distinguish between K  A  XK , A  and  where A is a non-zero complex number. A useful way of writing the relationship between the coordinates of M i n k o w s k i vectors  and the components of spin-vectors can be obtained by inverting the stereographic mapping ( B . l ) . After a rescaling by A (so that now only n  A  and e K , L9  for 0 real, correspond  A  to the same n u l l vector) one finds  y = 7 7 = ( K V - a * ') , 1  0  z = -j= ( « ° K ° ' - K * ') 1  1  ,  (B.3)  Appendix B. A Brief Introduction to Spinors  153  Figure B . l : (a) A spacetime diagram with one dimension suppressed. A slice at t = 1 represents the future pointing null vectors, (b) The hypersurface t = 1. The line L indicates the stereographic mapping from the sphere to the complex plane.  Appendix B. A Brief Introduction to Spinors  where ( « ' ) =  154  K ') denotes the complex conjugate of n , K . Rearranging (B.3), one  r  1  A  A  gets  72 ^  t +z  I  x + ly  x — iy  t—z j  AC  /C  „.0—l' K K  K  K  K K  t  \  —0'  (B.4)  — 1'  Equations (B.3) and (B.4) can be written i n terms of the quantities o \ a  called the  AA  Infeld-van der Waerden symbols, as follows:  k* = a^rr'K n t  k where (k ) = (t,x,y,z).  o  -i  a  i 0  have been chosen here to be  6  (cr , J , r r  \/2  AA  rr  1 / 1 0  0  (B.5)  ^  M  proportional to the P a u l i matrices  '\  K K  T h e components <7 ' of u '  11  rr  —  ,  r  0 1 (<7i  )  i y = (—c^rT') ?  (<7"3  RR  )  =  V2  V2  1 0 1  0  0  -1  = (cr 0 • (B.6) 3  rr  In writing the association (B.5), often the Infeld-van der Waerden symbols a ' a  dropped and one can write symbolically  k  AA  are  7  4  >  (B.7)  Ki K  A n additional geometric structure is needed to represent the phase of K . However, first A  a few more concepts are needed. Different choices for the Minkowski basis vectors e (a null tetrad, for example), could lead to different choices for <x . Indeed, Penrose and Rindler (1984) [57] identify the indices a = AA', b = BB', • • • , etc., so that they write k — K ~K , for example. 6  a  M  M  7  a  A  A  rr  Appendix B. A Brief Introduction to Spinors  155  A spin transformation A B of the spin vector K , K A  A  A  —* A K A  is a unimodular  B  B  two-dimensional complex m a t r i x . This complex linear transformation of K induces a A  real linear transformation of (ft* ) = (t,x,y,z)  which preserves the line element  1  f) ik k , a  h  a  namely, (cf. (B.4)):  K ~K ' A  -+ K ^  A  4  '  =  A K A  (A K )  B  A  B  =  =  ,  B  B  A (K K ')A ' ,, a  B  B  A  b  B  A (k a ')A ' ,, A  a  BB  B  (B.8)  A  A  B  i.e.,  t  +z  x — iy  i+z  x + iy  t+z  x + iy '  y  ^ x — iy  t —z j  i— z  A  x A- iy  y x — iy  t  —  A* ,  (B.9)  z  where A* is the conjugate transpose of A . In fact, a spin transformation induces a unique proper L T A % = A  A B  A > on any M i n k o w s k i vector (not just the null ones). Conversely, A  8  B  to each proper L T there corresponds two spin transformations, namely A  A B  and — A  A B  (cf. E q . ( B . 8 ) ) , i.e, the group of spin transformations is a double valued representation of the proper Lorentz group. T h i s correspondence can be used locally i n curved spacetimes. However, there are certain topological restrictions on the curved spacetime manifold AA, (e.g., time and space orientability) i n order for spinors to exist on AA. (see Penrose and R i n d l e r [57], and W a l d [73]). Higher order spinors may be defined by requiring that their transformation properties under spin transformations are the same as a product of spin vectors w i t h the same indices. T h e connection between general tensors i n a M i n k o w s k i vector space and spinors is made using the Infeld-van der Waerden symbols. For example a real type ( | ) tensor 8  Since det A  A B  -  ^e e AB  A  B  c  C  conjugate using (B.14) and identifying A  Multiplying this by its complex  = 1, £AB = A A ZCD—A'  A AD  D  A  a 6  A ^ A  B  b  ' gives g  ab  =  A A t,g . c  a  d  cd  Appendix B. A Brief Introduction to Spinors  156  is related to the H e r m i t i a n type (|,|,) spinor  t d ab  c  _ AA'  <3a  BB'  c  ab  t  For a vector t , this gives a 't a  AA  '  A  cd — T  T-AA'BB' T  cd ^  B  AA'BB'  CCDD'  by  ' C C D D '  B  (TX \C\\  CCDD'  ,  (O.IU)  (r)-\-[\ l-D- -- -J  •  J  1  = T '. In the special case where t is a null vector,  a  AA  a  can always be written T ' = T T  AA  .ab  <?b & CC& DD't  i  T'  d  A T  AA  A  a  (cf. (B.7)). The set of spin-vectors at a point  A>  form a two-dimensional vector space called spin-space. A basis for spin-space, denoted (er" ) = (o ,i ), A  4  T = 0,1 is such that OAI  A  = 1, and is called a spin-frame.  A  Note  that since primed and unprimed indices are distinguishable, their relative order is not i m p o r t a n t , i.e., T  =  AA>  T'. A  A  T h e antisymmetric spinor epsilon CAB can be defined by e  AB  =6  AB  =e  [AB]  ,  t t  =t  AB  AG  Note that from E q . ( B . 4 ) (with K  AA>  9ab  kk a  b  c  = K K '), A  = k k = 2 det a  a  =  =S  B  A  AA  C  B&  J j •  ( ) R12  one gets  K ' =  ^AB~tA'B'K 'K  (era) = ( _?i  ,  B  AA  2(K V ' 00  1  - /c 'r ') , 01  10  ,  (B.13)  from which we identify gab  <-> tABtA'B'  (B- ) 1 4  •  Indeed, CAB is used to raise and lower spinor indices according to the following sign conventions: = K ,  Ke A  B  AB  e K AB  B  =K .  (B.15)  A  (Note that K TA = — K,AT and K K,A = 0.) One can also check that A  Cabcd  A  KAA'BB'CC'DD'  A  = i{^ACKBD^A'D'€-B'C'  ~ tADtBCtA'C'ZB'D')  •  (B.16)  Appendix B. A Brief Introduction to Spinors  157  T h e two-dimensionality of spin-space gives the important identity = 0.  CA[BCCD]  (B.l7)  Contracting ( B . l 7 ) w i t h an antisymmetric spinor T  = T^ ^ gives the following very  AB  AB  useful relation AB  =  T  [AB]  T  y B  =  T  D  D  (  B  1  8  )  This gives a useful technique to find spinor equivalents of various tensorial quantities and visa versa. For example, let us find a spinor expression for the spinor w i t h a real rank-two antisymmetric tensor F FAA'BB'  =  FAA'BB'  associated  = F[ y T h e strategy is to first decompose  ab  ab  into symmetric and antisymmetric pieces  —FBB'AA'  Fb  <->•  FABA'B'  —Fba  <->  —FBAB'A  a  =  —  FAB(A'B')  =  1  + F B[A'B']  +  —FBA(B'A>)  (B.19)  ,  A  FBA[A'B'\  (B.20)  •  E q u a t i n g terms antisymmetric and symmetric i n the indices AB from (B.19) and (B.20) and using (B.18), one finds  FAB(A'B')  FAB[A'B']  =  —FBA(B'A')  —  FBA[A'B']  Substituting (B.21) into (B.19) w i t h  — F[AB](A'B')  —  §AB  F( B)[A'B<]  —  = \(-ABFC '{A>B>) =  A  ,  C  \(-A<B>F(AB)C'  (B.21)  •  C  one gets that the spinor equivalent  \F( B)C' ' C  A  to a real antisymmetric rank-two tensor is  Fb a  =  *-» FAA'BB'  F[ab]  =  4>ABtA'B'  + 4>A'B' AB e  (B.22)  •  U s i n g (B.16) and (B.22) one obtains a very simple spinor operation for the process of tensor dualization (which holds for F  * F  a b  =  ^abcdF  cd  not necessarily real)  ab  *-+ *F BA'B> A  =  iF BB'A< A  =  ~iF AA'B> B  •  (B.23)  Appendix B. A Brief Introduction to Spinors  158  R e t u r n i n g now to the geometric argument, recapping, associated w i t h any non-zero spin-vector n  is a unique future-pointing null vector k defined by (B.7), k  A  called the  a  flagpole.  a  <->  K K ', A  A  However, this correspondence does not distinguish between spin-  vectors differing only by phase. one tries to construct from K  In order to obtain a more precise geometric picture another Minkowski tensor.  A  Referring to ( B . l l ) , what  is required is a combination of K ^ ' S which has spinor indices appearing i n the paired combination AA', B B ' , • • •, etc., and which is H e r m i t i a n . We have already looked into the simplest such combination, namely K K . A  f» i.e., K K A  e''  A B  «_>K  One of the next simplest possibilities is  A>  K  A B  +  T  ^ - A ' - B '  ^  {  A  are invariant under K the pair (K ,T ) A  —» — K .  A  ab  ah  A  constitute a spin-frame, i.e., K\T  A  A  A  - KT.  B  finds that p  B  ab  A  Substituting this expression for e  AB  where k <-> K K ' A  A  A  up to an  into the R H S of (B.24) one  ab  =  k  t  ^g_ j  ^  ab _ b rk  25  and l is defined by a  l  A  This defines r  can be written as p  Since r  = 1.  A  SO that  = 1 can be written equivalently as  A  A  = KT  defined by (B.24)  TO picture p , introduce the spin-vector T  A  additive complex multiple of K . T h e relation K T  a  )  B  a  AB  M  m u l t i p l i e d by t ' ', (to give the correct index combination for a tensor), added  B  to its complex conjugate. Note that both k defined by (B.7) and p  e  B  KT' A  a  is not uniquely determined, p  ab  A  + TK' A  .  (B.26)  is invariant under the transformation  l -> l + ak a  A  a  a  ,  (B.27)  for a real. T h e set of / ' s (B.27) describes a two-dimensional half-plane, called the flag a  plane, perpendicular to the light cone along k  a  (see Figure 2). T h e flag plane together  Appendix B. A Brief Introduction to Spinors  159  Figure B . 2 : T h e geometric concept of a spin-vector as a n u l l flag, (a) T h e flagpole is defined by the n u l l vector k , and the flag plane is defined by the half-plane generated by l + ctk . (b) T h e hypersurface t = 1. Under the transformation K — » e K w i t h 0 = T T / 2 , l -> - l . W i t h 9 = TT, l - • / but K A - » • E v e r y n u l l flag defines two a  a  A  a  a  spin-vectors, K  a  A  and  a  —/c" . 4  a  l9  A  Appendix B.  A Brief Introduction to Spinors  160  w i t h the flagpole give the geometric picture of a spin-vector as a null flag. However, even this does not suffice to distinguish entirely between distinct spin-vectors: Note that under the transformation K  A  (referring to (B.26)) l  a  —> e K l9  (and therefore T  A  — > e~ T )  A  T9  — > — l , and w i t h 9 — ir, that l a  rotates the null flag back to itself, but n  a  goes to — K .  A  A  A  w i t h 9 — 7r/2, that  —> l , i.e., a rotation by 2-7T a  This is the most complete  geometric interpretation that can be obtained in V [57]: to each Minkowski null flag there corresponds two spin-vectors. A spin-vector field K (X) A  is the assignment of a spin-vector to each point of the  spacetime manifold. T h e field K (dropping from now on the explicit x dependence) thus A  defines a null flag at each point of Ad (which is a structure i n the tangent space (the M i n k o w s k i vector space) at each point). A spinor covariant derivative operator  \7 AA *-> V  9  1  a  is a map which acts on spinor  fields of type (™ ^) and sends them to spinor (fields) of type ( j ^ /^i), {K AA }B  1  i.e., VAA '• {K } —> 1  B  T h i s derivative operator is real, linear, satisfies the Leibnitz (product) rule and  is such that VAA'f  = SAA'J <-* d f,  where 8AA' <->• d  a  a  is the ordinary (flat) derivative  operator and / is an arbitrary scalar field. In addition, we take VAA' to be torsian free, i.e., ( V ^ . V B B / - VBB'VAA')!  B.2  = 0 <-> 2d d f [a  = 0.  b]  The curvature spinors  T h e R i e m a n n curvature tensor  10  R bc  d  a  is defined from the noncommutativity of two  derivative operators acting on an arbitrary one form field w , 2V[ V ]W d  Similarly, one can define the s p i n o r  11  XAA'BB>C  D  A  b  c  =  R w. d  abc  d  from the noncommutativity of two spinor  That the spinor derivative operator has two indices is related to the correspondence between spinors and actual directions in spacetime. Note that R bc has the same sign with respect to metrics of signature (+ ) and (-+++) since the sign of the covariant derivative is not effected by a sign change of the metric. "Notation of [73]. 9  10  d  a  Appendix B. A Brief Introduction to Spinors  161  derivative operators acting on an arbitrary spinor field u>c  (yAA'^BB  — VBB JAA>)UC  1  =  ,X  D  B y considering action of ( V A A ' V B B — V B B ' V A A ) ° 1  1  n  a  r  e  (B.28)  •  XAA'BB'C UD  l spinor u>cc, one finds that  a  the spinor equivalent to the R i e m a n n tensor can be written Rab/  = XAA'BB'C £C  *-> RAA'BB'CC'  DD  T h e symmetries of R b a  d  c  D  + XAA'BB'C'  D  e  D  c  •  D  (B.29)  allow one to further simplify this expression.  Alternatively, one can find the spinor equivalent exploiting the symmetries R( ) ab  Rab(cd) =  R[abc}d — 0 directly. a  RAA'BB'CC'  1 2  a  = XABC CA'B'£C'  DD  =  Following precisely the same procedure used to derive  Eq.(B.22) one finds that R( b)cd = R b{cd) - 0 i m p l y Rab/  cd  D  + $ABC'  D  ~(-A'B'(-C + C C . ,  D  (B.30)  D  where the curvature spinors XABCD and QABC'D' are defined by  In addition, since R ^ABCD'  XABCD  =  \R(AB)A  $ABC'D'  =  \R(AB)A'  a b c d  = R  A  A  {CD)C  = X^ B)(CD) ,  C  A  (which follows from R( )  c d a b  ab  = R (cd)  cd  ab  is real and XABCD = XCDAB (which implies XA(CD)  A  (B.29) shows that XAA'BB'CD = XABCD^A'B'  (VAA'^BB  1  - VBB'V'AA')WC  + $A'B'CD£AB  D  ab  abc  d  SO that  •  D  a  = R[abd\d — 0)  — 0). Comparison w i t h  = (XABC ^A'B> + $A'B'C £AB)WD  A l s o , m a k i n g use of the equivalence R[ bc]d = 0 <-> *R b (cf. footnote 9 on page 22), R[ ]  (B.31)  C°' (CD') = $(AB)(C"D') •  (B.32)  = 0, where *R bcd — \ ab e  c  a  ei  Refcd,  = 0 can be shown to be equivalent to the condition  that A = A , where A = \X 12  Notation of [57].  A A  B  B  •  (B.33)  Appendix B. A Brief Introduction to Spinors  162  W r i t i n g XABCD i n terms of symmetric spinors, XABCD = ^ABCD + A(6 C^BD + ^AD^BC) ,  (B.34)  A  the spinor equivalent to a real tensor R bcd having the symmetries R( b)cd = Rab(cd) — a  a  R[abc]d = 0 can be written r> d ^ D it & <-» KAA'BB'CC a  DD'  _  iTr  *ABC  —  c  DtA'B'tC  D'  ,  >R  + ® ABC  £>'-  ^A'B'^C  + A(e^cejB + tA t c)l 'B'^c ' D  D  D  A  B  T h e spinor $ABCD'  £>  + ex. .  (B.35)  is called the R i c c i spinor and the spinor tyABCD = X(ABCD) is called  the W e y l (conformal) spinor. Contracting (B.35) over b <-> BB'  and d <-> D D ' one finds  that Rac  P y l A ' C C " = 6Ae 4c£yl'C'  —  j  2$^C7^'C •  (B.36)  T h e traceless part of Rabcd-, Cabcd-, the W e y l curvature tensor, is Cabcd <-> CAA'BB'CC'DD'  = ^ABCD^A'B'^CD' + ^A'S'C'd'eASeCD  •  (B.37)  A l s o , from (B.36), one finds that R  <-* 24A ,  Rab ~ \Rgab Gab  (B.38)  *-*  -2<S>ABA>B> ,  <->  -6Ae  A S  (B.39)  e ^ B / - 2$ABA'B'  (B.40)  •  T h e E i n s t e i n equations i n vacuum are then  Rab = 0 <-» <  tyABA'B  1  — 0  (B.41) A = 0  Recalling that V[ Rbc] a  = 0 <->• V * i ? 6 c d = 0, (cf. footnote 9 on page 22), one finds that a  de  a  the B i a n c h i identity can be written V i? [ a  6 c ] d e  = 0  VV*ABCB =  VB^VCOA'B'  -  2e V ,A B{c  D)B  .  (B.42)  Appendix B. A Brief Introduction to Spinors  163  In vacuum this becomes V '*ABCD  = 0 ,  AA  (B.43)  which are the zero-rest-mass equations for a spin-two particle. However, (B.43) is not conformally invariant: T h e spin-^ equations, ^ ' <pA ...A AlA 1  1  with  <t>Ai....An  tyABCD,  a  n  —  0>  a  r  e  conformally invariant  symmetric spinor of weight w — —1, cf. Eq.(2.11). T h e W e y l spinor  on the other hand, has conformal weight w = 0, cf. Eq.(C.14).  Appendix C  Conformal Transformations  Consider a manifold AA w i t h a Lorentzian metric g .  A conformal transformation is a  ao  rescaling of the metric 9ab -> dab =  tt 9ab  (C.l)  2  where fl is an arbitrary positive, real scalar field. Such a transformation can be regarded as a "shrinking or stretching" of the manifold. Since a vector is spacelike, timelike or null w i t h respect to both g  ab  and g , manifolds that are related by a conformal transformation ao  have the same causal structure. In this appendix, we give a few of the relations used i n the m a i n text.  C.l  Tensor formulation  F r o m ( C . l ) we have that the inverse metric g of  g  ab  of g  aa  is related to the inverse metric g  ab  by 9  ab  so that and V  ab  g bg  bc  a  a  = g bg  bc  a  =  $a c  -» 9  ab  ,  =  (C.2)  One (uniquely) defines the two derivative operators  associated w i t h the metrics g b and g a  V  a 5 6 c  ab  V  a  respectively by Vg  = 0 ,  1  a  bc  = 0 .  (C.3)  Recall that derivative operators are linear, satisfy the Leibnitz rule and their action on scalar fields is equivalent to the ordinary derivative operator. In addition, we assume that they are torsian free. 1  164  Appendix C. Conformal Transformations  To relate the two derivative operators V  165  and V , note that the action of ( V — V ) on  a  a  a  an arbitrary vector field t defines the (^)-type tensor h  (v B y evaluating V g a  cd  b  .  c  (CA)  is found to be given by Q  c  ab  + ^bdad - Vdgab).  \g yagbd  ab  a  — 0 using (C.4) and (C.3), Q  ab  Q ° by  2  - V ) t = Qjt  B  a  c  ab  =  Substituting i n ( C . l ) and (C.2) then gives  Q  = 2V  c  ah  ( a  In nS  c b)  - g g V  In ft .  cd  ab  d  (C.5)  Equations ( C . l ) - (C.5) can be used to (arduously) calculate the transformation properties of other tensors and equations. metric g  ab  For example, the curvature tensor R d abc  for the  can be written i n terms of "untilded" quantities by substituting ( C . l ) - (C.5)  into V[ Vj,]to a  ( = \Ra. w ).  In particular, one can show that the W e y l tensor  d  c  bc  d  is conformally invariant, i.e., C  d abc  =C  abc  d  .  C  d abc  Similar equations can be straightforwardly  determined i n the spinor formalism which, i n particular, gives a much simpler demonstration of the conformal invariance of the spin-^ equations.  C.2  Spinor formulation  In terms of spinors, associated w i t h the conformal transformations ( C . l ) and (C.2), (since g  CAB^A'B'-,  ab  Eq.(B.14)) are the spinor epsilon transformations  tB A  1  A B  = VtAB ,  e  so that 8A = SA (also ZA'B' = ft^U'B', ^ B  'VAA' <-» V  a  B  a n d VAA'  <-> V  a  AA  ~ V >)K  C  AA  Notation of [57]. Q b corresponds to C b of [73]. c  a  - e  =  A B  =  ft ^' '). -1  5  ft~V  B  ,  (C.6)  T h e derivative operators  defined by VAA'^BC = 0 and VAA'CBC — 0 are related by  (V > 2  B  A B  c  a  = ®AA'B K C  B  (C.7)  Appendix C. Conformal Transformations  T h e action on the field r  for some (io' ) spinor field ®AA'B • C  ;  VAA>)K T  166  = 0 to be (V A< - ^ AA*)T = -®AA>B T -  B  ( B y evaluating [V A> - ^  C  B  A  B  can be found from ( V ^ ' —  B  C  where T '  <-> t for some arbitrary vector field t , one finds that Q  QAA'C' €C  •) T h e torsian-free condition can be used to show that  b  BB  B  B  c  b  @AA'B  = i^-AA'^B  C  + TA'B^A  0  AA')T ' BB  A  ao  <-> QAA'C ^C' ' B  ,  0  B  +  (C-8)  for real TLAA' and TA'B [57]. Evaluating VAA'^BC using (C.6) - (G.8) gives 1AA> = n- v <n  ( c . 9 )  l  AA  and ILAA' = 0. T h e n (C.7) becomes ( V ^ i - VAA,)K  =T,6 K  C  a b  c a  ,  B  (CIO)  and also  - V A')T  C^AA'  A  = -TA'BTA  B  These expressions agree w i t h (C.5) on tensors T '  (C.ll)  •  t . 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