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

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SPIN-TWO FIELDS AND GENERAL 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 in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh 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 The University of Br i t i sh Columbia 6224 Agricul tura l Road Vancouver, Canada 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 nonlin-ear 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( a V c K\ c \ i ) ) , where Kab is an arbitrary two-form field. In 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, in general, the conventions of Wald (1984) [73] are followed. Abstract index notation is used throughout. 1 In general, small Greek letters a , • • •, fi, v, • • •, etc., are tensor coordinate indices, taking on the values 0,1,2,3; small Roman 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 Roman letters A , B,C, • • •, etc., are spinor abstract indices. Capi tol Roman letters are also used as labels in tensor equations. The metric gab has Lorentzian signature (-+++) except for spinor equations, where the signature is (H ). Round brackets around indices (•••) indicate that the indices are symmetrized, T(ab) ~ \{Tab + Tba) , (0.1) T^abc) — 3? (T a bc ~\~ Tbca ~\~ Tcab ~f" Tbac ~t~ Tcba ~\~ Tacb) • . . lIt is common when working with tensors to use coordinate index notation, i.e., to choose a coordi-nate 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 TM", 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 va, a rank-two tensor as Tab, a rank (^) tensor as Taia2"'a,li)1i)2...ilm, etc.. The abstract indices 'a', V, etc., are not coordinate indices, i.e., va 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. i l l Square brackets around indices [•••] indicate that the indices are anti-symmetrized, T[ab] = \{Tab - Tba) , (0.2) ^[abc] = 3~i{Tabc Tbac ~\~ Tbca Tcba ~\~ Tcab Tacb) . Vert ical 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(a\[bKc]\d) = \{TabKcd — TacKbd + TdbKca ~ TdcKba) • (0.3) The symbol da is used to denote partial derivative: The L e v i - C i v i t a antisymmetric epsilon tensor, eabcd, is taken such that eoi23 = +1 • (0.5) The Riemann curvature tensor, Rabcd, is defined by ( V 0 V 6 - V 6 V A H = +Rabcdwd , (0.6) for arbitrary wc. We use the Einstein summation convention, i.e., summation over re-peated indices is understood. 2 0.2 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] 2Einstein "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]. iv Table of Contents Abstract ii Conventions and Abbreviations iii 0.1 Conventions i i i 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 10 2 The 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 2.4 The usual symmetric rank-two tensor potential for a spin-two field . . . . 24 2.5 The usual Lagrangian formulation for the spin-two equations 28 2.6 Summary 32 3 Spin-two Theories and General Relativity 34 3.1 The standard argument 36 3.2 The gauge invariance approach 44 3.2.1 The "spin-limitation principle" 45 3.2.2 The "extended Gupta program" 48 3.2.3 Normal spin-two gauge invariance 50 3.3 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 4 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 5 Conformal Invariance - Spin-two Coupled with Spin-zero 85 5.1 Conformally invariant linear equations 86 5.2 Spin-two coupled wi th spin-zero - conformal K le in Gordon equation . . . 87 5.3 Nonlinear extension of spin-two coupled wi th spin-zero 91 5.4 Solving for the gauge invariance 93 vi 5.4.1 The general equations 94 5.4.2 The linearized equations 96 5.4.3 The general solutions 104 5.4.4 Uniqueness of the solutions 108 6 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 Outl ine of the calculations 126 6.4 Conclusions 139 Appendices 141 A Representations of the Poincare Group 141 A . l The Lorentz transformations 142 A . 2 The Poincare group invariants 143 A . 3 Irreducible representations of the homogeneous Lorentz group 146 A . 4 Projection operators 147 A.4.1 Vector field 147 A.4.2 Rank-two tensor field 148 B A Brief Introduction to Spinors 150 B . l A geometric picture of spinors 150 B . 2 The curvature spinors 160 C Conformal Transformations 164 C. l Tensor formulation 164 C.2 Spinor formulation 165 v i i Bibliography 167 vin List of Figures B . l (a) A spacetime diagram wi th 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 153 B.2 The geometric concept of a spin-vector as a null flag, (a) The flagpole is defined by the null vector ka, and the flag plane is defined by the half-plane generated by la + aka. (b) The hypersurface t = 1. Under the transformation KA -> eieKA wi th 6 - T T / 2 , la -* -la. W i t h 9 = T T , la -+ la but KA —> — K A . Every nul l flag defines two spin-vectors, KA and — K A . . . 159 ix Acknowledgement First and foremost I would like to thank my thesis advisor, Professor W . G . Unruh. Not only did he suggest many of the problems in 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 wi th my work. Throughout my time as a graduate student, I have have enjoyed discussions wi th A . Borde, S. Fort in , S. Habib, M . Holzer, R. Laflamme, D . Morgan, A . Roberge, J . Thornburg , °and D . Voll ick. I would like to thank my family and friends, M . Holzer in particular, for providing moral support. M y parents deserve special thanks for their understanding and encour-agement. F ina l ly 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 Prus-sian Academy 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, re-sulted in beautiful relations connecting gravitational phenomena wi th 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. The 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]. The precise form of this interaction between matter and spacetime is given by Einstein's equations. Part of the beauty of general relativity lies in its geometric interpretation. Since its formulation in 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 with relativistic field theories, such as electro-magnetism, physicists have sought to explain gravity in terms of a field propagating in a flat background spacetime. In particular, it has been argued, and widely accepted, that the theory of a spin-two field in 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 1This 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 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 in 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 equa-tions 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 writ ing 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 Einstein equations are identical with the usual formulation of the spin-two equations. Thus general relativity can be regarded as the theory of an interacting spin-two 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 in terms of a symmetric rank-two tensor and the relation of these equations to the linearized Einstein 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. Introduction 3 field theory of a spin-two field are outlined. One of the clearest and most general proce-dures is given by W a l d [74], which is reviewed in section 3.3 in some detail: In [74], Wald found that theories which are reducible to the linearized Einstein equations need not nec-essarily 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 imi t i ng the number of derivatives in the Lagrangian to two, then, uniquely gives Einstein's equations. In our analysis, we use the consistency criteria established by Wald in [74]. First 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 divergence-and curl-free vector field. As in 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 Aab. We compare the types of theories that result from a consistent nonlinear generalization of the vector theory in each case. Al though 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 in three dimensions. Chapter 1. Introduction 4 In Chapters 5 and 6 we consider the spin-two problem: 2 We investigate the general-ization of the classical equations of motion for a non-interacting massless spin-two field propagating in 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 with a differentiable metric, gab, of Lorentzian signature. The matter fields of spacetime are described by a stress-energy-momentum tensor, Tab- The Einstein field equations relate the curvature of spacetime, through the Einstein tensor Gabi to the matter content of spacetime, according to Gab — 8irTab , (1.1) where the Einstein tensor Gab = Rab — \§abR, Rabcd is the Riemann curvature tensor, Rab = gbdRabcd is the Ricc i tensor, and R = gabRab is the Ricc 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, gab, and 2The results of the calculations presented in Chapters 4, 5 and 6 have previously appeared as research articles [43], [44]. Chapter 1. Introduction 5 quantities derivable from it are the only spacetime quantities that . . . appear in the equa-tions . . . " ([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 struc-ture, 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, Gao = Rab — \gabR — 8irTab, E q . ( l . l ) , gives gabGab = —R = 87rT, S O that (1.1) can be wri t ten 3 Rab = 87r(Tab-lgabT) . (1.2) In the absence of matter fields, the Einstein equations (1.1) reduce to Gab = 0 . (1.3) Alternatively, from (1.2) the Einstein equations in vacuum (aka. the vacuum Einstein equations) can be written Rab = 0. (1.4) Note that the Einstein tensor, Gab, satisfies the ((twice) contracted) Bianchi identities 4 V a G a 6 = 0 . (1.5) The trace-free part of the Riemann curvature tensor, Rabcd, is the Weyl conformal tensor, Cabcd, which, in four dimensions, is given by Cabcd — Rabcd + 2g[b\[cRd]\a] + \Rda\c9d\b • (1-6) 3This is the form that Einstein originally presented them in [53]. 4It 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 Tah is conserved, i.e., that VaTab = 0 ([53] p. 256). Chapter 1. Introduction 6 Almost simultaneously wi th Einstein's formulation of general relativity, Hilbert show-ed that the equations (1.1) can be derived from the action [53] SEH = J d4x (CG + a M C M ) , (1.7) where the gravitational Lagrangian density CG is given by C G = V ^ R , (1.8) and CM is the matter part of the action. aM is a constant and g is the determinant of the metric, gab. The action (1.7) is called the Einstein-Hilbert action. Variat ion of SEH with respect to the matter fields yields the matter equations of motion. Variat ion of SEH with respect to gah gives the Einstein field equations C O £EHab = = V=9 (Gab - 8nTab) = 0 . (1.9) In this formulation of general relativity, the stress-energy tensor is defined by T a f e E E - f i - L ^ . (1.10) 8ir yj-g 6gab v ' In [53], Pais points out that the two formulations (1.1) and (1.7) of general relativity are not quite equivalent: In writ ing down the equations (1.1), Einstein did not specify the structure of the stress-energy tensor beyond its conservation and its transformation properties. Hilbert , on the other hand, gave a definite form to the stress-energy tensor of matter, namely (1.10). In a letter to Weyl (November 1916), Einstein writes "Hilbert 's Ansatz for matter seems childish to me." (pp. 257 - 258 [53]). Nevertheless, in general, i n curved spacetime, the stress-energy tensor Tab 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 in terms of a metric, this means that Chapter 1. Introduction 7 the metric, i.e., spacetime, is almost flat. In particular, in the weak field approximation to general relativity, it is assumed that one can write the metric as gab = Vab + lab , where 7 ^ is a 'smal l ' deviation from the flat metric, -qao. B y 'small ' is meant that there exists a coordinate system where the components of 7 ^ , 7 a u / , are much less than unity, i.e., |7M I /| <C 1, (e.g., in our solar system, |7M J /| ~ |$| < MQ/RQ ~ 1 0 - 6 ) . Substituting the metric (1.11) into the vacuum Einstein equations (1.3) and keeping only terms that are linear in 7 ^ , give the linearized (vacuum) Einstein equations 5  G(i)*b = R(i)ab _ lv*bR(i) = Q ^ = <9c<9(y)c - | N 7 A " - ldadbj - l v a b ( d c d d l c d - n 7 ) = 0 , (1.12) where the superscript (n) refers to the power of the field(s) appearing in the quantity. 6 A t this order, indices are raised and lowered by the flat metric, r]ab. Note, however, that gab _ ^ab ^ab The linearized Einstein equations (1.12), G^ab = 0, can be derived from the gravita-tional part of the second-order Einstein-Hilbert act ion 7 4 2 ) = - ( V = ^ ) ( 2 ) , = - \ l a b ^ l a h + \ l c h d a d c l a h - \ j d c d d l c d + | 7 ° 7 , (1-13) where 7 = j a a = rjabjab and • = dada. Variat ion of SQ* = / d 4 x , S(l)ab = 6 S $ = G ( L ) A B = Q hab 5The 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. 6In order to obtain (1.12) it is helpful to note that in terms of the Christoffel symbols rc ai, = \gcii.dagbd + dbgad - ddgab) the curvature tensor is written R a b c d = 2d[bTda]c + 2r%[ard6]e = R d c b a . 7The minus sign appears (cf. Eq.(1.8)), since variation will be taken with respect to jab (and not yab, where gab = rjab - jab). Chapter 1. Introduction 8 gives the linearized Einstein equations in vacuum. From (1.4), 0 = R^ = dcddlcd - D 7 , so that Eq.(1.12), G ( 1 ) a 6 = 0, can be written RMab = ddd{ajb)d - | D 7 a 6 - \dadbj = 0 . (1.15) The quantities £ G 1 * a b (defined in Eq.(1.14)) satisfy the (contracted) linearized Bianchi identities daSg)ab = 0 , (1.16) so that the equations of linearized general relativity (1-12) - (1.15) are invariant under the transformation lab —» lab + <9(aA6) , = lab + ^C\c7]ab , (1-17) where C\c denotes the Lie derivative wi th respect to the arbitrary (but small) vector field A a . Note that this gauge invariance is simply linearized coordinate invariance in the linearized theory: Consider the infinitesimal coordinate transformation X" -> x'" = x» - \?(xv) , (1.18) where £ M is small, i.e., of the same order as 7a&. Under the infinitesimal coordinate transformation (1.18), the metric, gab, transforms according to ^ M - < , ( ^ ) = ~ f ^ • (1-19) Taking the linear part of (1.19) gives 7 ^ = 7 ^ + 5 i^C + , (1-20) which is just (1.17) wi th ( a = A a , i.e., the gauge invariance (1.17) of the linearized E i n -stein equations (1.12) is simply invariance under infinitesimal coordinate transformations. Chapter 1. Introduction 9 They are, however, not coordinate invariant. Under general coordinate transformations, the metric rjao in the action is transformed. It is standard to simplify the linearized Einstein equations (1.12), G^ab = 0, by making the algebraic change of dynamic variables 8 lab ~* lab = lab ~ \r}abl • (1.21) Then (1.12) becomes G ( l H = d c<9 ( a76 ) c - \ufh - \r]ahdcddicd = 0 . (1.22) Further, since in vacuum 0 = R{1) = dcddjcd + f 0 7 , Eq.(1.15), R^ab = 0, becomes R(1)ah = ddd{af)d - | D 7 a 6 + \ v a b a l = 0 • (1.23) In terms of 7 c d , the gauge invariance (1.17) of the linearized theory is lab -> lab + <9(A) - lVabdcXC . (1.24) Eq.(1.12), Ga]j = 0, can be simplified further by taking the arbitrary field A A to be such that 9 D A 0 = -2dbiab. Then dalab = 0 and (1.22) (or equivalently, Eq.(1.23)) becomes Q 7 a 5 = 0 • (1.25) This choice of A A does not completely fix the gauge: A A is invariant under A A —• A A + Xa where aXa = 0 . This residual gauge freedom jab —> jab -f d'aXb) ~ \^abdcXc c a n D e u s e ( i to set 7 = 0 in regions where Tab = 0 ([73] pp. 80, 186). Thus, the linearized Einstein equations in vacuum, G(1)ab = 0, after a change of dynamic variable and gauge fixing, can be written • 7 B t = 0 , <9a7a6 = 0 , 7 = 0 . (1.26) 8jab is called the trace reverse of jab since j = —j. This gives the inverse transformation jai, = lab ~ \rjabl-9This condition can always be satisfied to first-order in ~yai, ([53] p. 280). Chapter 1. Introduction 10 These are precisely the equations that were written by Fierz [32], and Fierz and Paul i [33], in 1939, for the equations of motion for a spin-two field, where the spin-two field is represented by jab. 1.3 Summary This completes our brief introduction to general relativity. The Einstein equations are given by E q . ( l . l ) , Gab = 8irTab, which are derivable from the Einstein-Hilbert action (1.7). The quantity, Gab, which appears on the L H S of the Einstein equations, is called the E i n -stein tensor and, as a consequence of the Bianchi identities, is identically divergenceless, \7aGab = 0. The linearized approximation to gravity was obtained by perturbing the metric about flat spacetime, gab = rjab + 7a6, E q . ( l . l l ) . The linearized theory is invariant under the transformation jab —> jab + d(a\b), Eq.(1.17). F ix ing A a , one can write the linearized Einstein equations in the form given in Eq.(1.26). Since this corresponds wi th 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 non-linear 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. Par-ticular attention is paid to the potentials that can be used to describe the spin-s fields, and various possibilities are mentioned. The specific examples of spin-one and spin-two are investigated in more detail. In section 2.4, in order to clarify the relation between the linearized Einstein equations and the equations of motion for a spin-two field, the usual formulation of the spin-two equations in terms of a symmetric rank-two tensor potential is derived. The chapter ends wi th a discussion of the usual Lagrangian formulation of the spin-two equations as originally derived by Fierz and Pauli in 1939 [33], and their argument is reviewed. For the general discussion of the spin-5 equations the spinor formalism is used. The primary reason for this is to make contact wi th much of the literature on massless spin-5 fields. Also, 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 in the spinor for-mal ism rather than in the tensor formalism [57] (see Appendix B for more on spinors). M u c h of the material on spinors that is included in this chapter can be found in Penrose and Rindler (1984) [57], (1986) [58]. See also Wald (1984) [73] as well as other references given in Appendix A and Appendix B . 11 Chapter 2. The Spins Equations 12 2.1 Spin and representations of the Poincare group The laws of physics in Minkowski spacetime are believed to be invariant under transla-tions, rotations and boosts of spacetime, i.e., under the proper Lorentz transformations ( L T ) . 1 For a physical theory that is defined on Minkowski spacetime this means that these spacetime transformations induce transformations of the physical states of the the-ory [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. The 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", m2 and S2, which can be interpreted physically as the squared mass and the squared angular momentum about the centre of mass [73]. In any irreducible representation, these invari-ants 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. Bargmann 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 writ ing the spin-5 equations. M u c h of the literature uses the spinor form of the equations given in the next section. 1See Appendix A for definitions, more detail, and references. Chapter 2. The Spins Equations 13 2.2 The spin-s equations A pure zero-rest-mass field of spin-s in flat spacetime can be represented by a totally symmetric spinor f ield 2 $AXA2...AVL wi th n = 2s indices that, for s > 0, is a solution to ([25], [4], [59], [56]) dMA'^AlA2...An^0. (2.1) These are the free field equations for a massless s p i n - | field in flat spacetime. 3 Equiva-lently, Eq.(2.1) can be wri t ten 4 dB<B$AxA2...An = dB'A\(f>BA2...An • (2.2) Note that the s p i n - | equations (2.1) imply that the field <j>A1A2,...,An is also a solution to the wave equation 5 Vcf>AlA2...An = 0 , (2.3) where • = 3AA'QAA'• There are three additional features of the s p i n - | equations (2.1) 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 2The simplest kind of spinor KA , a spin-vector, is an element of a two-dimensional complex vector space called spin-space. The complex conjugate of KA, KA is denoted KA . Indices can be raised or lowered by the antisymmetric spinor epsilon (.AB- A spin transformation AAB, i-e., nA —* AAgKB, is a unimodular two-dimensional complex matrix. More complicated spinors may be built up from a spin-space 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 Infeld-van der Waerden symbols <raAA (which are usually taken to be proportional to the Pauli matrices). For example VaTb = <raAA abBB1^'AA'TBB . For a more detailed discussion, see Appendix B. 3See 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. 4To see this, write Eq.(2.1) as eALBVA^<t>A1A2...An = 0 and take into account the antisymmetry of the spinor epsilon eAlB. 5To see this, note that dc'A^BC is antisymmetric in AB, since flat space derivative operators com-mute: dC'AdBC' = dc'BdAC> = -dc>BdAc' = dc>[AdB]C'• Then by (B.18), dc-AdBc' = \eAB^- Thus, acting on (2.1) with dA'lB gives dA'lBdAlA'^AiA2...An - \&AiD^.^-.i, = \^<t>BA^...An = 0. Chapter 2. The Spins Equations 14 non-flat spacetimes [73], [57], (iii) there are many possible ways to describe a spin-5 field in 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 - ^29ab , (2-4) (j)-^4> = nw<j>, ' (2.5) 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-{gabi <f>), and T = !F(gab, (f>). Another way of stating this is to say that there 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. The s p i n - | equations wi l l be conformally invariant, then, if V ^ < ^ 2 . . . A n = 0 <- VA^4>AlA2...An = 0 , (2.6) where <f>A1A2...An — Qw<f>A1A2...An f ° r some real w. One can show that the statement (2.6) is true, by expressing V A l j 4 ' i4 > A l A 2 . . . A n in terms of (j>AlA2,„An and S7BB'- Referring to Appendix B on spinors and Appendix C on conformal transformations, the spinor equivalent to the metric conformal transformation (2.4) is given by Eq.(C.6) , eAB —> eAB = fl-xeAB, so that vMA'4MA2...An = v ^ i (nw<f>AlA2...An) = lMB^B'VBB,{SlW<t>MA2...An) = e A l S e ^ B ' n - 2 ( O w V B B ^ l A 2 . . . ^ + V B S ^ ^ M 2 . . . A n ) , (2.7) where the derivative operator only acts on the quantity immediately following unless brackets indicate otherwise. The relation between two derivative operators is given by Chapter 2. The Spins Equations 15 E q . ( C . l l ) , VAA'TB = VAA>TB ~ ^A'BTA, where from (C.9), TAA> = ^ 1 V y t i 4 / 0 . Thus (2.7) becomes V A ^ M A 2 . . . A N = ^BtA^'Vl-2[nW{VBB4MA2...AN-^B'AABA2...AN - ^B'A2<t>ALBAZ...AN B'An4>A1A2...AN-LB) + ionw-1S7BB,ttcj)A1A2...AN\ • (2.8) Note that al l but the first two and the last terms are zero since they are symmetric in A : and B but contracted wi th the antisymmetric eAlB. Substituting in (C.9), £l~1'VBB>Q = TBB'I Eq.(2.8) becomes y A L A ' ^ A L A 2 . . . A N = ft-2 {VA^cj>MA2...AN + TA^ci>BA2...AN + wTA'^B<}>BA2...AN) • (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 • (2.10) This shows that the massless spin -5 equations, \7AlA'i(f>A1A2...AN = 0, with the symmetric spinor (f>A1A2...AN a field of weight w = — 1, . 4>AIA2...AN = ft~1<f>A1A2...AN , (2-11) are conformally invariant. 6 Note that the wave equation (2.3) satisfied by the spin-5 field is also conformally invariant. 7 6Here, we only consider conformal transformations between flat metrics so that V A A ' T B B ' — rAB>rBAI, [58] p. 124. 7In [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>Ai—Ai that satisfies (2.1), is conformally invariant. For example, Bracken and Jessup show that in order for the wave equation for some field TAiA2 j-0 conformally invariant, then T A I A 2 must be a spin-one field, i.e., it must satisfy Maxwell's equations. Chapter 2. The Spins Equations 16 If the Einstein equations in vacuum hold, i.e., if R A B = 0, the spinor equivalent of the Bianchi identities, V[ a i2&c]de = 0, is given by Eq.(B.43), ^ A A ' ty ABCD = 0, (cf. Appendix B ) . These correspond to the equations of motion for a spin-two field, (2.1), so that the Wey l conformal spinor, tyABCD-, represents a spin-two field. However, the equations VAA'tyABCD — 0 are not conformally invariant since tyABCD is a spinor of conformal weight zero, i.e., tyABCD = ty ABCD (see Appendix 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 Relat ivist ic equations can usually be generalized to curved spacetimes by replacing the flat metric, r]ab, by the curved metric, gao, and by replacing the flat derivative operator, <9a, by the derivative operator, V a , associated wi th gab, [73]. However, this procedure of "naturally generalizing" the flat s p i n - | (2.1) does not in 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 4>Al...A„ to the Weyl curvature spinor tyABCD- To see how the algebraic constraints arise, act on the "curved" s p i n - | equations \7AlA'i (f>A1...AN = 0 w i t h V A 2 A i , 0 = V ^ ( V ^ < / > A l . . . A J = V ^ V ^ K ^ . . . ^ . (2.12) From (B.32), (VAA'^BB'-^BB'^AA')^C = (XABCD£A>B< + §A'B'CD£AB)UD- Contracting wi th eA'B' gives V A ' ( A ^ ' B ) B W = XABCD^D- Then, for s > 1, (2.12) becomes V ^ V ^ K ^ , . . ^ = X A ^ B AABA2...An + x A ^ B AAMBM..^ + X A ^ B A J M A . B M . . ^ + • • • + X A ^ B A j M . . . A n ^ B - (2.13) bmce XA(BC)A = Q5 the first t w o terms on the R H S are zero. Therefore 0 = XAlA2BA3<j>A1A2BAi...An H h XALA2BAn^Ai-.An-iB Chapter 2. The Spins Equations 17 = fiA1A2B(AI...ANXALA2BA3) • (2-14) Since only the symmetric part of XABCD appears in (2.14), XABCD c a n D e replaced by the Wey l curvature spinor ty ABCD = X(ABCD)- This gives the following condition for <t)A1...AN-0 = <f>AlAiB{M..jJ*A*)AlMB • ( 2-!5) This 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 if 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 Einstein equations. The Bianchi identity for the vacuum Einstein equations is S7AA'tyABCD = 0, Eq.(B.43) . Since tyABCD is completely symmetric, this means that it represents a spin-two field. Taking tyABCD for <I>ABCD in (2.15), the consistency condition (2.15) becomes tyABC(DtyE)ABC — 0, which is automatically satisfied. 8 Thus the vacuum 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, in 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, in order to be able to formulate the equations in terms of an action principle. Since the solution 4>AXA2-AN °f the s p i n - | equations represents a pure s p i n - | field, it is a 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. 8To see that ^ ABCD^EABC + *ABCE*DABC = 0, note that the second term can be written •*ABCE-*DABC = -^ABCE^ADBC = ^ 'CB*ABD° = - * A B C A B C D • 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 AI...A„ such that dAtA''iJ>Ai"-A'*Al...An = 0, (2.16) and such that the field (f>Al...An defined by <j>A1...An=dAlA[...dAkA^---A'*Al...An , (2.17) is completely symmetric and a solution to the s p i n - | equations (2.1). Thus ij)A'i-A'kAl„,An is a potential for the s p i n - | field (f>Al...An- To see that <f)Al...An so defined is com-pletely symmetric, rewrite (2.16) as eA''B' eAlB dBBitpA^^A'k Al...An = 0. This implies that dB'BipA'1"'A'kAl...An - dBiAli>A^-A'kB„,An = 0, so that dA'kAkil>A'1"'A'kAi...An is symmetric in Ak and Ai and hence all unprimed indices. Since flat space derivatives commute, repetition of this argument shows that field <j>Al...An defined by (2.17) is indeed completely symmetric. To see that it satisfies the s p i n - | equations, note first that (2.16) shows that if;A'i-~A'kAl___An satisfies the wave equation ^tpA'1'"A'kAl...An — 0. Act ing on (2.17) with dAlA* then gives dA^<f>Al...An = d A ^ d A l B l . . . d A k A ^ B , A ' ^ A ' k A l . . . A n = \ d M A l 2 . . . d A k A , n ^ - - - K A ^ A n = 0. Thus, the field <j>Al...An defined by (2.17) with • ^ A ' 1 ' " A ' k A l . . . A n given by (2.16) is a spin-f field. The definition (2.17) is not effected by certain gauge transformations. This is because the potential ipAl"'Ak Al...An describes a pure spin field in terms of a mixture of spins, i.e., transformations of ipA'i"'A'kAl...An which effect the lower spin parts may not effect (2.17). For example, for s > 1, the definition <j>Ai...An = QAlA\i>A'xA2...An is unchanged under the transformation ij}A'^ A2...An ~^ "fyA'xA2...An + dA<1 A2^Az...Ani 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 in terms of the potential ipA>B by dBB'ipA'B = 0, Chapter 2. The Spins Equations 19 where <J>AB — QAA1^'B- This is not effected by the transformation ipA'B ~^ ^A'B + dA'B^, where DA = 0. A spin-two field (J>ABCD can be described in terms of potentials that satisfy (2.16) and (2.17) in three ways: (i) by ij)A'BCD such that dAA' ipA'BCD = 0, where <J>ABCD — ^AA'i>A BCD- These definitions are unchanged under ij)A'BCD —> ^A'BCD + d A ' D \ B c , where d B B ' k B c = 0, (ii) by ^A'B'CD such that d A A ' ^ A ' B ' C D = 0, where 4>ABCD = dAA>^BB'i>A'B'CD- The gauge invariance is ipA'B'cD —> i>A'B'cD + d A ' c A A ' D , where d D D ' K B ' D = 0, (iii) and by ^ A ' B ' C ' D such that dAA'^A'B'c'D = 0, where 4>ABCD = dAA'dBB'dcC^A'B'C'D with the gauge invariance ipA'B'c'u —> ipA'B'c'D + a % A s ' c ' , where dBB>AB'C' = 0. 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 ipA'i---A'kAl...Ani defined by (2.16) and (2.17) [58]. For example, let ipAiA2---A'n be a completely symmetric spinor such that dA*A'^AlA'-A'n) = 0 . (2.18) Then 4>A!...A„ defined by <f>Al...An = d{AA* - -.dAnKi>A1)A>...A>N , (2.19) is a spin-1 field. The symmetrization in (2.18) provides a less stringent condition on the potential tpA1A2'"A'n than its unsymmetrized counterpart (this will become clear below.) 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 ipA1A'2"'A'n —> ipA1A'2'"A'n + dAl(A'2hA'3'"A'n* for arbitrary symmetric A.A3~-A'i. To see that $AX...AN a s defined in (2.19) satisfies the spin- | equations consider dMA'^AL...AN = \dA^ [dAA'2 - - • dAnA'^AlA'2...A>N + dAA'2 - - • dAIA'^A2A>...A'N + ••• + dAA'2 - - - OA^/'^A^A'..^ + dAlA'2... dAn„A'ni>AnAi2...A>n] , Chapter 2. The Spins Equations 20 ( - i ) n - 1 a ^ . . . d A n K d A ^ A l A ' ^ - \ u e ^ . . . d A < - ^ A 2 A , . . . K _ ^ lndAnA'idA< ...dAn_2A'^An_lA^Ali...K - \ u d A ^ ... dAn_A'^An^A,3...Aln\ . (2.20) Consider now just the last term: - \udM*>. ..dAn_A'^AnA'^A,...A,n = -\dAA>. . . d ^ n s j y ^ ^ , . (2.21)' Substituting § E < ^ = dAlA'ndAnA>n into (2.21) and relabeling gives -cV> • • • d ^ - i d j ^ d * ^ ^ ^ . ^ = {-\)n-ldA2K .. . d A n K d A ^ A A ' ^ - A ' ^ . (2.22) Repeating for similar terms in (2.20), Eq.(2.20) becomes dMA'^Al...An = ( - l ) " " 1 ^ . . . d A n A , n d A ^ A / ^ - A ^ , (2.23) which equals zero from (2.18). Thus the field ipAlA'f"A'n is a potential for a s p i n - | field 4>A1...AN-Moreover, Eq.(2.18) is conformally invariant: ^MA[^AK...K) = ^ B B ^ W ^ A A ' 2 . . . A ' N ) + ^ B B , ^ - " ^ ] , = r r - V i B { W T B { A H A A ' 2 - K ) + V B [ A H A A * ~ K ) - ? A I { A ' ^ B A > - K ) - f T B c ^ ^ ^ 1 0 ' 1 ^ - ^ + • • • + T B C ^ ^ A ^ ^ ) 0 ' \ , (2.24) where al l but the first three terms in the last equality are zero. Taking w = — 1 the first and th i rd term cancel showing that the equations (2.18) are conformally invariant. 2.3 Tensor equivalent of the spin-one and spin-two equations Having established some general properties of the spin-s equations in 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 Equations 21 2.3.1 Spin-one To find the tensor equivalent to the symmetric spinor §AB-, referring to ( B . l l ) and the discussion preceding (B.24), mult iply § A B by 1A'B> (to obtain the paired (tensor) index combinations AA', BB') and add to <J>AB"^A>B> its complex conjugate. This gives the real spinor FABA'B' = 4>AB~eA'B< + <j>A'B'EAB <"» Fab • (2.25) Referring to (B.22), Fab is an antisymmetric tensor and FABC'C' = 2<J>AB- TO find the tensor expression for the spin-one equations of motion, dAA' 4>AB = 0, take the divergence of FABA'B' and its dual, *FABA'B> = ^FABB'A', dAA'FAA>BB> = dAB,<i>AB + dBA'$A,BI <- daFab, (2.26) b^'*FAA.BB- = -idAB><l>AB + idBA'4*B> d[aFbc]. (2.27) Setting (2.27) to zero gives dAB'4'AB = 9BA'<f>A'B'- Taking in addition (2.26) set to zero gives dAB'(j)AB = 0. Thus one sees that the tensor equivalent to the spin-one equations are Maxwell ' s equations dAA' (j>AB = 0 ^ { daFab = 0 • . (2.28) d[aFbc] = 0 From (2.18), a spin-one field can also be described in terms of the potential ij)AA', where dA^A'ipAB'* = 0, and <J>AB — ^B\B^)A)B>- TO obtain the tensor equivalent to the potential IJ)AA>, one need only add to I\)AA' its complex conjugate ipAAi AAA< = ^AA> + ^AA' <"»• Aa , (2.29) which is the familiar electromagnetic vector potential. Substituting <f>AB = dB'\B^A)B' into (2.25) (and making use of dA^A'ij)AB'* = 0) then gives the usual Fab ~ 2d[bAa]. The gauge invariance ipAA' —» 4>AA> + dAA'h corresponds to Aa —> Aa + da\. The potential Chapter 2. The Spins Equations 22 I\>AA> that is defined by (2.16), (/>AB = dB>B^PAB', and (2.17), dAA'ipAB' = 0, for a spin-one field, can now be shown to be less general than the corresponding potentials with symmetrizations: Lowering the B' index of dAA'ipAB' = 0, gives dAA'ipAA' = 0, which, referring to (2.29), corresponds to daAa = 0, so that the unsymrnetrized potential (2.16) corresponds to electromagnetism in the Lorenz gauge. Hence, also DA = 0. 2.3.2 Spin-two Similarly, the tensor equivalent to the symmetric spinor 4>ABCD, defined by (2.16) and (2.17) wi th n = 4, can be found by mult iplying 4>ABCD by IA'B'^C'D' and then adding this to its complex conjugate: KAA'BB'CC'DD' = ^ABCD^A'B'^C'D' + <f>A'B'C'D'EABCCD Kabcd • (2.30) From Appendix B , this is precisely the spinor equivalent to a rank-four tensor Kabcd, which has the symmetries K(ab)cd = Kabl<cd) = K[abc]d = K a b a d = 0, where KABCDA'A'CC' = ^(J>ABCD- Taking the divergence of (2.30) gives dAA KAA'BB'CC'DD1 = QAB'<f>ABCD^CD' + &BA ^A'B'C'D^CD ^ d a K a b c d . (2.31) Equat ing this to zero (considering parts symmetric and anti-symmetric in CD) give the spin-two equations of motion dAB'<i>ABCD = 0 <-> d a K a b c d = 0. (2.32) Note that, unlike in the spin-one case, there is only one tensor equation corresponding to the spin-two equations. This is because d[aI(hc}de = 0 daKabcd = 0. To see this, first note that d[aKbc\de = 0 is equivalent to d a * K a b c d = 0, where *Kabcd = \ e a b e l K e j c d . h 9 9To see that d[aKbc]de = 0 *-> da*Kabcd = 0, consider the following: d[aKbc]de = 6{a69b6^djKghde = - ^ a b c m e f 9 h m d f K g h d e = - \ e a b c m d j { \ e ^ m K g h d e ) = - \ e a b c m d s * I < i m d e . Conversely df*Rfmde = d s { \ c ^ h m K g h d e ) = ±et>hmduKgh]de, [57]. Chapter 2. The Spins Equations 23 Then o\aKbc]de * KAA'BB'CC'DD' = — ^ B'^ABCD^C'D' + <f>A'B'CD'€CD • (2.33) Equating this to zero is equivalent to equating dAA'KAA'BB'CC'DD' to zero. Thus a spin-two field can be represented by a rank-four tensor Kabcd which has the symmetries K(ab)cd = Kab{cd) = K[abc]d = K a b a d = 0, and which satisfies daKabcd = 0, or equivalently, d[aKbc]de — 0- For example, the Weyl conformal tensor, Cabcd, can represent a spin-two field. If the vacuum Einstein equations hold, Rab — 0, the Riemann curvature tensor, Rabcd, also represents a spin-two field. To obtain the tensor equivalent to the potential i)AA'BB'i where dcc' ifiA'B' CD = 0 and 4>ABCD = dAA'dBB'tyA'B'CD-, from Eqs.(2.16) and (2.17), add to ipAA'BB' its complex conjugate h-AA'BB' = IpAA'BB' + ty AA'BB' ^ ^Ib •> (2.34) where h^b is a symmetric traceless rank two tensor. 1 0 Contracting dcc' tyA'B' CD — 0 with IA'C'ZB'D'I gives dcc'tyccDD' = 0. This corresponds to daha"b = 0. Also, since ^i^AA'BB' — 0, D^a6 = 0- Thus, the tensor equivalent to the spin-two equations where the spin-two field is represented by the potential tyAA'BB> according to Eqs.(2.16) and (2.17) dAA <PABCD = 0 d c c ' V B ' c D = o d«hTab = o , , } { . (2-35) <j>ABCD = QAA'QBB4A B CD J { ° K B = 0 where ty>AA'BB' + AA<BB> ^ab- This is the form of the equations of motion for a spin-two field that were written by Fierz in 1939 [32]. They are precisely the gauge fixed 1 0It is clear that the tensor tab corresponding to a spinor TAA'BB1 that is symmetric T(AB)(A'B') must be symmetric and traceless. Conversely, to show that the spinor TAA>BBJ that corresponds to the symmetric and traceless tensor tab must be symmetric, note that: tab = tba *~* TAA'BB' = TBB'AA'- Contracting the spinor part with eAB gives eAB TAAIBB' — —£ABTAB<BA' so that eABTAA>BB' is antisymmetric in A'B'. However, in order that gabtab = 0 <-> eABeA B TAAIBBI = 0, eABTAA'BB' must be zero. This implies that TAA'BB' is symmetric in AB [57]. Chapter 2. The Spins Equations 24 equations for linearized gravity (1.26). From 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 Kaoca> in terms of h^b, from (2.34) note that (J>ABCD = dAA'dBB>tyA'B'cD + dAA'dBB'tpAB CD = dAA'dBB'hA'B'CD, since dcc"if>ABC'D> = 0. Now consider the spinor KAA'BB'CC'DD' defined by ^AA'BB'CC'DD' = ~[pAA'^DD'^CC BB< ~ ^BB'^DD'^CCAA' + dBB'dcC'hDD'AA' ~ QAA'^CC'^-DDCBB') <-> ~4<9[a<9|[d/l^|b] , (2.36) which, due to its symmetries, can be written as, (cf. Eqs.(B.22) and (B.30)), KAA'BB'CC'DD' = XABCD^A'B'^C'D' + X.A'B'C'D'EAB^CD , (2.37) where XABCD — \KABCDAA'CC''• Contracting (2.37) with eA'B'ec'D', gives o A'ct B'r XABCD — 0(A OB hCD)A'B> , = 9{AA 9BB ty>CD)A'B' + d(AA 0BB tycD)A'B' = &ABCD • (2.38) Comparing this wi th Eq.(2.30) (putting 4>ABCD = dAA'dBB'tyA'B CD in (2.30)), shows that KAA'BB'CC'DD' is simply KAA'BB'CC'DD'- Therefore, from (2.36), Kabcd = -4<9 [ a d | [ d / ^ w . (2.39) Thus, the potential ipAA'BB' defined according to Eqs.(2.16) and (2.17), corresponds to the tensor potential h^b related to the spin-two field K a b a i according to (2.39). The transformation TAA'BB1 —• TAA'BB' + QAA1 ^-BB1 , where dAA' KAB1 — 0, corresponds to hlb -> h%b + d(a\h), where D\b = 0 and dAXA = 0. 2.4 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. From (2.32) a tensor Kabcd which has the symmetries K(ab)cd = Kab(cd) = K[abc]d — Chapter 2. The Spins Equations 25 K°bcd = 0, and which satisfies d[aKbc]de = 0> c a n be taken to represent a spin-two field. We have already seen in section 2.3.2 that one way of choosing potentials showed that Kabcd could be writ ten in terms of a traceless symmetric tensor h^b as Kabcd = ~^d[ad\[dh^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 Kabcd which has the symmetries K ^ c d = Kab(cd) — K[abc}d = 0, and which satisfies d[aKbc]de — 0, consider each of the symmetries of Kabcd in turn: K(ab)cd = 0 and d[eKab]cd — 0, imply that there exists a rank-three tensor Tabc such that Kabcd = d[aTb]cd , (2.40) since d[edaTb]cd = 0. Note that Eq.(2.40) is invariant under the transformation Tabc —> T a 6 c + daVbc where vab is an arbitrary rank-two tensor. The symmetries Kab(cd) — 0 and K abc]d — 0, give the following two equations, respectively, that the potential Tabc must satisfy dialed) = 0 , (2.41) d[arbc]d = 0 . (2.42) Wr i t ing T a ; , c in terms of its symmetric and its antisymmetric parts, Tabc = ^ (5 a 6 C + Aabc), where Sabc = T( a j ) c , and Aabc = T[ a&] c, and substituting this into (2.42), gives 0 = d[aTbc]d = \(d[aSbc]d + d[aAbc]d). Since d[aSbc]d is zero, this gives d[aAbc]d = 0 , (2.43) so that one can introduce the potential E]a(, defined by Aabc = d[aEb]c , (2.44) since d[adb]^cd = 0. Note that 3ab is invariant under the transformation Eab -> S a ! ) + da\b , (2.45) Chapter 2. The Spins Equations 26 where A a is an arbitrary one-form field. Returning to Eq.(2.41), (2.41) implies that one can introduce the symmetric rank-two tensor flao by T a (6c ) = daClbc , (2.46) since d[adb]flcd = 0. Expanding out Eq.(2.46) -danbc = -\(Tabc + Tacb) , (2.47) wri t ing cyclical permutations d c n a b = l ( r c a b + r c b a ) , (2.48) and adding the preceding three equations, yields db^ca + dcflba — daflbc = T( 6 c ) a + T[ c a]j — T[ba]c • (2.49) Wri t ing Tabc in terms of Sabc and Aabc gives Sabc = dbflca + dcflba — daVlbc + Aabc + A a c j , . (2.50) Combining results, from (2.40) Kabcd = \{daTbcd — dbTacd) , — \{daSbcd — dbSacd + daAbcd — dbAacd) . (2.51) Substituting i n (2.50) for Sabc gives Kabcd = 4 (dadcflbd + dbddftac — daddflbc — dbdcClad + daAbcd + dbAadc + daAdbc + daAdcb - dbAdac - dbAdca) • (2.52) Chapter 2. The Spins Equations 27 Wri t ing this in terms of ^ a o v ia Eq.(2.44), gives Kabcd = d[ad\[cQ,b]\d\ - l(d[ad\[cEb]\d] + d[ad\[cZd]\b]) . (2.53) Since "Eab only appears in the form 'EL(ab), o n e c a n define the symmetric rank-two tensor field jab by jac = -2{flac - E ( a c ) ) . (2.54) F rom Eq.(2.45), (2.54) is invariant under Jab -> Jab + 9(ah) • (2.55) This gives Kabcd = 2d[adp7c]|(,] , (2.56) (cf. (2.39)). Note that the R H S of (2.56) is just the linearized Riemann tensor, -fi^ &L' f ° r the metric gab = i]ab + jab- However, in order to represent a spin-two field, Kabcd must also be traceless, Kabad — 0. Setting the trace of (2.56) to zero gives K c a c b = ddd[ajb)d - \^lab - \dadbj = 0 , (2.57) where j — j a a = r]abJab- These are the equations that are usually taken for a spin-two field. Eq.(2.57) is just the linearized vacuum Einstein equations, Ra]j = 0, Eq.(1.23). A t this point, as discussed in 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)) dajah = 0 , a j a b = 0 . (2.58) In summary, in trying to find a potential for a tensor Kabcd, where K(ab)cd = Kab(cd) = K[abc]d = Kcbcd = 0, we were led quite naturally to the potential jab related to Kabcd by Eq.(2.56). The equations of motion for the spin-two field, then, are the linearized Einstein equations (2.57). Let us investigate this result further: As previously mentioned, the Weyl Chapter 2. The Spins Equations 28 conformal tensor, Cabcd, can be taken to represent a spin-two field. It has the symmetries C(ab)cd — Cab{cd) = C[abc]d = C°bcd = 0. Then the spin-two equation corresponding to V A j 4 ' 4 > A B C D — 0, is WaCabcd = 0. The Riemann curvature tensor, Rabcd, can also be taken to represent a spin-two field provided that the vacuum Einstein equations hold. Rabcd has the symmetries R(ab)cd = Rab(cd) — R[abc]d = 0, and V[aRbc]de = 0 is identically satisfied. The spin-two equations are then taken to be the vacuum Einstein equations, R°bcd — 0. Let us compare these two formulations: The Weyl tensor and the Riemann tensor are related by Cabcd = Rabcd + 2g[b\[cRd]\a] + \Rga[c9d}b • (2.59) Taking the divergence of (2.59) and using the Bianchi identities (1.5), gives VdCabcd = | V t 6 (Ra]c - \ga]cR) , (2.60) Setting this equal to zero, gives the spin-two equations VaCabcd — 0. However, in terms of Rabcd, the spin-two equations are Rab = 0. Whi le Rab = 0 implies VaCabcd — 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 The usual Lagrangian formulation for the spin-two equations There is one remaining hurdle to overcome in order to complete our discussion of spin-two fields, namely, a Lagrangian formulation of the equations needs to be found. 1 1 Fierz and Paul i (1939) [33], were probably the first to try to find a Lagrangian formulation for the 1 1 "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. Chapter 2. The Spins Equations 29 (massive) spin-two equations of motion, which they took to be, as written by Fierz in 1939 [32], ° 7 a r6 = ^ 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 both of these equations (2.61) from a variational pr inciple , 1 2 Fierz and Paul i introduced an auxiliary scalar field $ (see also Chapter 5 ) . A general ansatz was then made for the Lagrangian that was a functional of both jjb and The derived field equa-tions were required to give (2.61) as well as $ = 0. These requirements give conditions on the unknown numerical coefficients of terms appearing in the general Lagrangian den-sity. The zero-rest-mass spin-two equations were then obtained by setting K = 0. This argument led Fierz and Paul i (though not uniquely) to the second-order Einstein-Hilbert action by identifying 7 = $ . The starting point for many discussions of interacting spin-two theories is the La -grangian found by Fierz and Paul i , (e.g., [67], [28], [29], [20]). For this reason, it is useful to outline their calculation. They took the general second-order Lagrangian (in our notation) C%% = K 2 ( C 6 7 j 6 7 T a 6 + c 2 $ 2 ) - C 5 7 j 6 n 7 r a 6 - C l 7 T c ^ a 4 7 j 6 - C 4 $ a c ^ 7 j d - c 3 $ D $ , (2.62) where jjb = jab — | ^ a b 7 for some arbitrary symmetric rank-two tensor field jab. (In [33], Fierz and Paul i began by setting c 4 = C5 = CQ — 1. Of course, one of the parameters can be fixed without loss of generality.) Variat ion of Spp = / d4xCpp wi th respect to 7 ^ and 1 2Fierz 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. Chapter 2. The Spins Equations 30 gives the equations of motion u lab v Jed v Jab v Jcd = 2c6n2iTab - 2c1dcd{aiTh)c - 2 c 5 D 7 T a 6 - c4dadb<S> - f ^ ( i C a ^ d 7 c T , + i c 4 n $ ) = 0 , rfrl = ^ = 2 c 2 « : 2 $ - 2 c 3 D $ - c4dadblTab = 0 . (2.64) To find conditions on the values of the Cj ' s , Fierz and Pauli noted that the double di-vergence of (2.63) taken together with (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. The divergence of (2.63) is daSFl)Pab = 2 c 6 K 2 d a l T a b - ( C l + 2c 5 )c9 c D7 T 6 c - \ C l d b d c d d l T c d - \c4Udb® . (2.65) Note that if C\ = —2c$, the second term vanishes and (2.65) becomes da£&ab = 2 c 6 K 2 d a 7 T a b - db ( \ C l d c d d l T c d + | c 4 n $ ) , . (2.66) which implies that daSFx)Pah = dcjTcd = 0 if $ = 0 and dcddjTcd = 0. Taking the divergence of (2.65) gives dadb£F1)Pab = ( 2 C 6 « 2 - ( C l + 2 C 5 ) n - i C l n ) r - f c 4 n n $ , . (2.67) where T = dcdd-fTcd. 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. This determinant A is det A = 4 [C2C6K4 - K2 (c 2 (c 5 + f c i ) + c 3 c 6 ) • + (c 3 (c 5 + | C l ) - ^ c 2 ) • • ] . (2.68) In the massive case, K / 0 so that det A ^ 0 if c 2 (c 5 + f c i ) + c 3 c 6 = 0 , c 3 (c 5 + | C l ) - ^ c 2 = 0 . (2.69) Chapter 2. The Spins Equations 31 If c i = — 2c 5 , Eq.(2.69) reduces to C l c 2 = - 4 c 3 c 6 , c i c 3 = ^c 4 . (2.70) For AC 7^  0, if (2.69) is satisfied, then (2.63) and (2.64) imply (2.61) and $ = 0. Fierz and Paul i chose the values c 4 = c 5 = c 6 = 1, c\ = —2c5, c 3 = —| , and c 2 = —| . However, note that other choices are possible. The 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) imply (2.61) and $ = 0. However, the massless equations are invariant under lib - lib + 0 ( A ) - lVabdcXC , (2.71) $ $ + dcXc , (2.72) where X0 is an arbitrary one-form field. Then one can choose the gauge such that daiarh = 0, Eq.(2.61) and $ = 0. Identifying $ = 7, (2.73) 7tT& = lab - \Vabl , (2.74) the equations (2.63) and (2.64) (with K = 0) become j r W = _ j R ( i ) = 0 ; (2.75) the linearized Einstein equations in the absence of matter. The Lagrangian (2.62) be-comes CfP = -A(^-gR)(2) . (2.76) (Variation of (2.76) wi th 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 Equations 32 In summary, Fierz and Paul i began with a general ansatz for a second-order La -grangian i n terms of the fields -jjb and $ , Eq.(2.62). The coefficients were fixed by demanding that the equations of motion must give $ = 0 and (2.61). In the massless l imi t , however, the gauge invariance of the theory must be exploited in order that the equations of motion give $ = 0 and (2.61). In this manner, then, Fierz and Paul 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 hab starting from the Lagrangian density CF = 42) + habTab , (2.77) where Cp* is a general ansatz for a second-order Lagrangian in hab, and Tab is the stress-energy tensor. Variat ion of CF gives the equations of motion £ F = ^ L = £{FAH + Tab = 0 . (2.78) ohab Then the condition that the stress-energy tensor, Tab, is conserved, i.e., that daTab = 0 gives that Sp* is the second-order Einstein-Hilbert action (1.13), i.e., Cp* = Cp*H, and that Sp*ab is the linearized Einstein equations, (1.12), i.e., Sp*ab = EQ*0,1'. 2.6 Summary The important points to remember from this chapter are the following: A spin-two field is represented by a completely symmetric spinor <J)ABCD that satisfies the spin-two equations dAA'' 4>ABCD = 0. In terms of tensors, a spin-two field can be represented by the rank-four tensor Kabcd, where K(ab)cd = Kab(cd) = K[abc]d = Kchcd = 0 and VaKabcd -0. Whi 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, jab. 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. In the next chapter, some of these arguments are discussed. Chapter 3 Spin-two Theories and General Relativity General relativity is a geometric theory of gravitation in which the structure of spacetime and the matter content of spacetime are inorexably intertwined. The spacetime itself is viewed as a "dynamic participant in physics" - the curvature of spacetime "tells matter how to move" and the matter content of spacetime "tells spacetime how to curve." The geometric nature of general relativity (aka. geometrodynamics) contrasts sharply wi th other field theories which describe physical forces such as electricity and magnetism in terms of dynamical fields on a "God-given" flat background spacetime. The 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 in the description of nature led physicists to try to interpret field theories such as electromagnetism from a geometrical point of view [82]. More predominantly, on the other hand, the success of field theories led physicists to try 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. The cla im 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 (Gupta [40], Feynman [31]). Then 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 (Kraichnan [46], Feynman [31], Thir r ing [67], Wyss [83], and Deser [22]). These methods were in 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 Wald [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 l imit of a quantum theory of gravity, (e.g., Boulware and Deser [9]). Almost al l of these arguments concluded the uniqueness of Einstein's equations. W a l d [74] however, found another possibility - in 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 spin-two field is generally covariant are reviewed. In section 3.1, first the standard argument originally proposed by Gupta [40] and Feynman for nonlinearity of the spin-two equa-tions is given. Then it is shown how various extensions of this argument (Feynman [31], Thi r r ing [67], Wyss [83], and Deser [22]), have been used to arrive at the Einstein equa-tions. 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 in 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 Wald [74] are discussed in some detail. Since in 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 in section 3.3. Through-out, 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 wi th Maxwell 's theory of electromagnetism. Whi le 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 with 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]. From a field theoretic point of view, then, the differences between Maxwell 's equations and Einstein's equations would be related to this difference in spin. Gupta [39]-[41] used this difference in spin in 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 al l matter. This 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 Chapel H i l l Conference in 1956. Feynman's point of view was somewhat different (cf. [31]). He argued that gravitational forces result from the vir tual exchange of a spin-two particle called the graviton (see also [77], [9]).) The "standard" argument ([47], [39]-[41], [67], [31], [83], [22], [28]) that any consistent theory of a spin-two field must be nonlinear (in contrast with theories of a spin-one field which have a consistent linear formulation) is based on a Lagrangian formulation together wi th the four principle assumptions: (1) the spin-two field is represented by a symmetric Chapter 3. Spin-two Theories and General Relativity 37 rank-two tensor field, (2) the spin-two field obeys the linearized Einstein's equations (referred to as the equations of motion for a spin-two field in 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, Tab, (4) the stress-energy tensor either (i) is required to be derivable from the action by variation wi th respect to the flat metric nao (Deser [21]), or (ii) is defined canonically (Gupta [40]), or (iii) is specified precisely (Thirr ing [67] and Feynman [31] take particle matter, Wyss [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. The argument for nonlinearity is as follows: The 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, in order to derive these extended equations from an action, higher-order terms must be added to the action. However, these higher-order terms in turn give rise to new terms in the stress-energy tensor. In order to get these new terms from an action principle, the action must in turn be further extended. However then new terms arise in 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 in 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 in 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 G \ Eq.(1.13), and the linearized Einstein equations, SQ = = 0, Eq.(1.14), (assumptions (1) and (2)). According to M T W [47] p. 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 CG by demanding that it give rise to divergenceless equations of motion (see Chapter 2). Others, (e.g., Thir r ing, [67], Fang and Fronsdal [28]), chose CG since it is the Lagrangian determined by Fierz and Paul i (2^  [33] (the first to write down CA , [47], see Chapter 2). However, as pointed out in Chapter 2, the Lagrangian found by Fierz and Paul i is not unique (nor do they claim it to be). Indeed, it was remarked in Refs. [28] and [67], that any equations related to SQ *AB by a transformation of the form jab —* lab + c>labl, are equivalent, where c is a real number. Thus, it was argued, that one could, without loss of generality, formulate the spin-two equations in terms of a rank-two symmetric tensor field, jab, that satisfies the (2) linearized Einstein equations, £ G * a b , where £ G * a b = Next it is asserted that to the spin-two equations, £ G * a b = 0, must be added a symmetric source term, T^ab, for the spin-two field, jab, (where for the moment, the superscript (n) indicates the powers of jab appearing in the term). The source term, T(° ' a f > , is constructed from the other fields with which the spin-two field interacts. The Chapter 3. Spin-two Theories and General Relativity 39 equations for the spin-two field are then £$ah = 8wT^ab, (3.1) which are derivable from the modified Lagrangian density + OLMCM , (3.2) by variation wi th respect to 7^ , where CM = C M * — ^ iabT^0)ab and O J M = 167T. However, since the linearized Einstein equations are divergenceless, da£G*ab = 0, the R H S of (3.1) must also be divergenceless, daT^0)ab = 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. This is most clearly illustrated by choosing a specific matter model, say scalar matter as described by the Klein-Gordon equation. In the noninteracting theory, Q^rp(o)at - g p r 0 p 0 r t i o n a l to the matter equations of motion. Hence, the conservation of T'( 0) a b follows from these equations. However, one finds that in 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 in the stress-energy tensor. This problem could then be resolved by having on the R H S of Eq.(3.1) the total stress-energy tensor of matter including the spin-two field. This is assumption (3). Before proceeding, note that there are alternative methods to obtain a divergence-less 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]. As another example, in [21], Deser and Lau-rent were interested in 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. The nonlocal projection operator Pao = r]ai> — n~1dadb was used to project out the divergenceless part of in [31])). The quantity that appears on the R H S of (3.1) is then the divergence-free tensor Jab = (PacPbd+pPabVcd + qPabPcd)Tcd. The constants p, q were taken to be p = q = - 1 to agree wi th Newtonian gravity. The 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, in this example, the price is nonlocality or auxiliary fields. Returning to the principal discussion, since the various arguments differ slightly de-pending 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 wi th respect to the flat metric, rjab, [21], so that (the argument using the canonical definition of the stress-energy defined by Tcb = — r]ab£M, where 4> represents the fields, is identical [40]). Therefore, consistency the stress-energy tensor (which was taken for the specific example of particle matter (as (3.3) where (3.6) Chapter 3. Spin-two Theories and General Relativity 41 Then variation of (3.6) wi th respect to jab gives (3.4), (the superscript (2) on T^ab means here that T^ab is composed of two powers of the fields in general, i.e., T^ab can also 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), in [28], Fang and Fronsdal remark that in the 1962 thesis of E . R . Huggins (California Institute of Technology) that Huggins "attempted to construct [the third-order term in the action C^1) by requiring • • • that ^ 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 in advance. The 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 main discussion of the standard argument, variation of (3.6) with respect to r}ao now gives an additional term in the stress-energy tensor, namely j i(0 )a6 _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 CM\ so that variation of the augmented action with respect to i]ab gives (3.7). A n d so on. This reasoning leads to the conclusion that "any self-consistent theory of an interacting massless, spin-two field in flat spacetime is as nonlinear as Einstein's equations" [28]. Accepting 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 in particular, the arguments of Deser [22], and of Feynman [31]. In [22], (see also [9]), Deser uses the above reasoning in the absence of matter, (i.e., rp(o)ab _ to argue that any consistent theory of a spin-two field must be Einstein's equations. In particular, consistency is defined by the conservation of Tab, and Tab is defined by the variation of the action with respect to r]ab, i.e., consistency is realized by Chapter 3. Spin-two Theories and General Relativity 42 the condition [9] SCdq^Tlcd) _ aMTab__ <XMSC{lab,Vcd) ^fy Sjab 2 2 6r)ab For £ ( 2 ' Deser takes the linearized Palat ini action CP\ (The Palat ini Lagrangian Cp for the full Einstein equations takes the metric density, ^/—~ggab, and the connection coefficients, Tabc, as independent variables (see M T W [47] p. 491 f.). Variat ion of Sp with respect to Tabc gives the usual expression for the Christoffel symbols in terms of the metric. Variat ion of Sp wi th respect to \/~-ggab gives Rab — 0. Deser obtains the linearized form by substituting \f—ggab = rjab-\-hab, where hab is a tensor density, into Cp and keeping terms of second-order in hab and Tabc.) Combining the equations that Deser obtains from variation of Cp* wi th respect to hah and Tabci gives R^ — 0. Adjoining the tensor rab = then gives Rab — —KTAB, where K is some constant, which Deser asserts can be derived from the full Palat ini 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^* = / druaub and Tab = uaub, where ua is the particle four-velocity and r is the proper time. The matter part of the action can then be written SM = j dr{Vab + lab)Tab , (3.9) and the total action is given by Eq.(3.2), SG +O:MSM- The spin-two equations are given by Eq.(3.1), £Q*AB = STrT^ab. 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 Tab which follows from (3.1) says that a particle moves on a straight line in contradiction with the equations of motion derived from SM- Therefore, (3.9) needs to be modified by adding terms involving higher powers of the fields to 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, ~fab, and that, therefore, the matter only couples to the spin-two field in the form stated in Eq.(3.9), (rjab + jab)Tab. This gives h i m the complete gauge invariance of the theory. He is then able to determine the form of £ ( 3 ' . Constructing invariants of the infinitesimal transformations, he finds that the complete action is the Einstein-Hilbert action, SEH, evaluated wi th respect to the metric gab — Vab + lab- Wald [74] has shown that the (implicit) assumption that the complete matter Lagrangian is given by C M * + C M * — gabTab, 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. Then, l imi t ing the number of derivatives in the action to two, one obtains the Einstein-Hilbert action. Wald points out that the assumed linear coupling of matter to the spin-two 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 Feynman obtains the result that the equations are generally covariant wi th respect to the combination gab = i]ab + 7a6 but not even for a metric related to gab by a change of variables such as jab —• Tab + ^cdJcaJbd- In other words, it is certainly possible to relax many of the assumptions that are made in [31]. We briefly mention the work of Weinberg [77], and Boulware and Deser [9], who ap-proach 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. They assume the Lorentz invariance of the S-matrix to-gether wi th (1) and (2) above and that the forces are transmitted by the vir tual exchange Chapter 3. Spin-two Theories and General Relativity 44 of gravitons. Fang and Fronsdal [28] and Wald [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 in [9]. For example, Wald points out that the "low-energy l imi t" of a theory of quantum gravity is obtained by l imit ing the number of derivatives appearing in 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 in itself. We prefer to deal wi th the classical problem directly, by methods that avoid quantum extensions that are irrelevant to i t" [28]. 3.2 The 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 in 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, did not depend on the particular form taken for the stress-energy tensor. A n alternative approach, in 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 with matter was also considered [74]). These works concentrated on the gauge invariance of the theory. Whi le 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. Again 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 inves-tigated 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 ten-sor field 7a(, that excluded spin-one. The equations were to be derivable from an action principle and to be compatible with the condition where a is an arbitrary number and m is the mass of the field jao. They call (3.10) the spin-limitation principle. 1 A key assumption that Ogievetsky and Polubarinov make is 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, 3 the infinitesimal variations 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, yab, (16 components) contains a spin-two field (5 com-ponents), 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). 2They 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 - 1 . 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. 3 In 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. m2(dajab + a d b l ) = 0 , (3.10) Chapter 3. Spin-two Theories and General Relativity 46 This argument leads Ogievetsky and Polubarinov to an equation that is very similar to the integrability condition derived by Wald in [74]. (However, the decision to restrict the dimension of the coupling constants l imits 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 imitat ion of the dimension of the coupling constants, the general form for the variations is restricted to zeroth- and first-order quantities. Substituting in these general expressions they find, up to a change of variables, that interacting theories compatible wi th the spin l imitat ion condition are generally covariant. Ogievetsky and Polubarinov also include interactions with 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] with that of [28], and [74], it is helpful to outline the calculation of Ogievetsky and Polubarinov in [51] in our notation. They took the general Lagrangian C 0 p = C0p{jab, ddab, dddcjab, (j>, da<f>, • • •), where '• • •' represents other fields which we shall not consider. Wri t ing the Lagrangian as COP = C0p - lm2(jabjah + 0^ 77) , (3.11) the equations of motion can be written cab COP — f - m 2 ( 7 a f e + m / a 6 7 ) = 0 (3.12) where *a6 OP — hab 84> (3.13) daSaOP = da£$P - m2(dalab + a# 67) = 0 • (3.14) Chapter 3. Spin-two Theories and General Relativity 47 Therefore, if da8op is identically zero, or zero as a consequence of the equations of motion, then (3.10) follows. da£op = 0 if the following generalized divergence identity holds da [Bahcd£^P + DahF0p) = C \ d Z 0 d P + AbTOP - m2{dalab + adbj) , (3.15) where B a b c d is a function of j a b , rjab, and tabcd; C a c d is a function of dcjab: r)ab, and tabcd] Dab is a function of <f>, r)ab, and eabcd; and Ab is a function of da<j>, rjab, and eabcd. B(o)abcd = sa[c8bd) + pr]abr)cd, C ^ a c d = 0, D^ab = 0, and A^b = 0. In the case of zero mass the last term in (3.15) vanishes. Equivalently, COP must be invariant under the following infinitesimal gauge transformations Sxalcd = BabcddaXb + CacdXa , (3.16) 8XJ = DabdaXb + AbXb . (3.17) Restricting the dimension of the coupling constants, the most general expression for the tensor fields are4 B a b c d = 8\c8bd) + pr,abVcd + J2btB\1)ahcd , i=0 C\d = J2 c£\1)ecd , Dab = d3D^)ab , A" = a2A^b , (3.18) where the precise form for the Bfbcd, 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 -fab, i.e., there is no coupling between different fields, (cf. Eqs.(5.30) and (5.31)). The condition that (3.16) form a group gives - 8^a89a)jab = 8Xajab , (3.19) 4 In terms of the parameters used in [51], 26i = b, 2bj = d, 263 = /, 64 = e, 65 = c, bg = a,60 = —g, 2ci = b', c 2 = c', c 3 = e', c 4 = a', 2c5 = / ' , 2c6 = d', c 0 = -g'. Chapter 3. Spin-two Theories and General Relativity 48 where \a is some function of 9a and tpa, (cf. Eqs.(5.36) - (5.38)). Substitution of the assumed form for the exact gauge invariance, 8\ajab = ^lab + £J^ 7a&, given by (3.16) and (3.18), and assuming that Xa = xi°\ gives a system of 17 equations for the un-known coefficients. 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 wi th no interactions or, (2) a theory with the gauge invariance 8\cJab = \C\cgab . (3.20) The condition that the infinitesimal transformations (3.17) form a group is given by (3.19), where now Vga is a vector defined by (3.17). Ogievetsky and Polubarinov find 8X4 = \CX^ . (3.21) 3.2.2 The "extended Gupta program" In [28], Fang and Fronsdal are interested in extending the standard argument proposed by Gupta [40] for the nonlinearity of any spin-two theory (cf. section 3.1). El iminat ing the matter model, they seek a nonlinear generalization of the linearized Einstein equa-tions in terms of a formal power series in the coupling constant K , which they call the "Gupta program". The (consistency) condition that the equations of motion linear in K are divergenceless allows them to find the term in the Lagrangian that is linear in K (restricting the number of derivatives). Assuming that the complete theory must be in-variant under a Lie algebra of infinitesimal transformations, Fang and Fronsdal are able to show, using deformation theory, that the determined linear gauge invariance leads to the Einstein equations. In more detail, the complete Lagrangian density is written oo UF = 42) + E * n ~ 2 £ F F ( n ) , (3.22) n= 3 Chapter 3. Spin-two Theories and General Relativity 49 where the CpF(n) a r e assumed to be polynomials in j a 0 and its first-order derivatives, and CFF{3) has no constant term and no term linear in jab. The equations of motion are then cab — 0L>FF _ c(l)ab n-l cab (o oq\ CFF — — CG -t- 2_j K CFF(n) J [o.ZO) °lab n-2 where the SFF^ are polynomials in jab and its first and second-order derivatives, and < ? F F ( I ) has no constant term. To order K, the fact that the linearized Einstein equations are divergenceless implies that da£pbF(2) = 0 identically or as a consequence of the equations of motion. Thus one can write da£pbp(2) = Cbcd£c^cd f ° r some Cbcd- Fang and Fronsdal prefer to write this schematically as 4^ = 0 _> da€{^Fab = 0. (3.24) In order to determine £ j?F (3 ) they assume that it is a homogeneous polynomial of order three, i.e., that £ F F ( 3 ) = £^ 3 '• Writing out the most general expression for C^3\ they find, up to a change of variables, that either £( 3) = 0 or that £ ( 3 ' = £Q\ (note that they have restricted the number of derivatives appearing in C^3*). Comparing their results with the work of Feynman [31] (and Wyss [83]) who used the condition Cbcd = T^bcd (obtained for particle matter) to find that £( 3) = CQ\ Fang and Fronsdal also adopt Cbcd = 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]". The paper concludes with 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 Einstein 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 in al l of the preceding arguments is to l imit the number of derivatives appearing in the Lagrangian to two. It is the relaxation of this condition that leads to the counter-example to the "folk theorem" found in [74]. Wald '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 in 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, in a perturbation expansion, each linear solution is tangent (in the space of functions) to an exact solution. This implies that if the original linear equations satisfy a linear identity, (e.g., the linearized Bianchi identity) then the complete nonlinear equations wi l l be consistent if they satisfy a (to be determined) generalized identity. This identity, or equivalently, infinitesimal gauge invariance of the action, wi 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). The resulting equation is Chapter 3. Spin-two Theories and General Relativity 51 referred to as the fundamental integrability condition in [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. Wald 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 in lowest order ja\, couples to the stress-energy tensor Tab, where Tah is determined from the action by variation with respect to the metric gab — r]ab + 7a&, he conjectures that this would eliminate the possibility of finding theories having normal spin-two gauge invariance. However, he is only able to show this in certain cases. As "perhaps the simplest" example of a theory with normal spin-two gauge invariance, Wald suggests Cw = CG* + (R^*)n, where n > 2. Of course, if the further decision is made to l imi t the number of derivatives in Cw to two, then it is no longer possible to find nonlinear spin-two theories wi th 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." 5 However, the point of Wald's result is that alternatives to generally covariant spin-two theories are indeed possible. They may be ruled out by further restrictive assumptions, as Wald points out, such as specifying how 7^ couples to matter, or the decision to not consider-higher derivative theories. Moreover it is important to be aware that alternatives exist 5It should also be mentioned that in [19], the term (R^)n = £ j n i in Cw is reinterpreted as jabTab, where Tab = = • P a i ( i j W ) n - 1 = (r]aba - 5 a 5 6 ) ( i2W) n - 1 , (OPab 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. Chapter 3. Spin-two Theories and General Relativity 52 and to know when they can be used. In any case, in Chapter 6, we exhibit nonlinear spin-two 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 in this thesis, we employ the consistency criteria of Wald [74] to investigate a number of questions regarding nonlinear theories, in this section, these arguments of are discussed in detail. We 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, in 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. This is a problem that Wald investigated in [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, wi th particular emphasis on the basic criterion for consistency that is presented in [74]. 3.3.1 Consistent nonlinear extension of Maxwell's equations The equations of motion for a massless spin-one field in Minkowski spacetime are given by Maxwell ' s equations, (cf. Chapter 2), daFab = 0 , (3.25) d[aFbc] = 0 , (3.26) where Fab = —Fba. Eq.(3.26) implies that there exists a one-form field Aa such that Fab = 28[aAb] , (3.27) Chapter 3. Spin-two Theories and General Relativity 53 since d[adbAc] = 0. Note that this specification for Fab, (3.27), is unchanged under the transformation Aa -> Aa + daX , (3.28) where A is an arbitrary scalar. With the choice of potential Aa, (3.26) is an identity while (3.25) are the equations of motion which can be derived from the action «S|^ S{?M = - \ Jd4xFabFab, (3.29) by variation with respect to Aa, 6S{2) jr(i)a = _ A o . = 2dbd[bAa] = dbFba = 0 , (3.30) oAa where, the numerical superscript (n) indicates the power of the field(s) (in this case Ab) appearing in the term. Note that (corresponding to the gauge invariance (3.28)), satisfies the identity da^1)a = 0 . (3.31) In order to determine the types of consistent nonlinear extensions of the linear theory (3.27) - (3.31), consider the action SAA that is obtained by adding terms of third-order or higher in Aa to C^l • There are no restrictions on the types of terms added to C^l except that the number of derivatives at each order are to be bounded by a (fixed) integer N.6 Variation of SAA with respect to Aa gives the complete nonlinear equations of motion, r- = -r^r = F{1)a + f{2)a + ^ ( 3 ) a + • • • = 0 , (3.32) SAa y J which we want to be local partial differential equations involving a bounded number of derivatives. denotes the quadratic part of TaJF'3'0 the cubic part, etc.. Given any Lagrangian density, it is always possible (theoretically) to write down the corresponding 6The analysis of Fang and Fronsdal [28] was limited to polynomial Lagrangians, and the analysis of Ogievetsky and Polubarinov [51] was limited in order. Chapter 3. Spin-two Theories and General Relativity 54 equations of motion, Ta = 0. However, since the linear term, J-^a, in the equations of motion is identically divergenceless, in general, as discussed in 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), Fa — 0, perturbatively, i.e., try to find a one-parameter family of exact solutions Aa(e) to Ta = 0 such that 0, (3.33) 4,(0) = 0. (3.34) For example, take (3.35) where A^* = 0, A^* is the linearized solution, A^* is the second-order perturbation, etc.. The first-order equation in this perturbative expansion is ^[A0l)] = 0 . (3.36) Solution of this linear equation (3.36) gives the linear perturbation A^\ which should be a reasonable first approximation to Ab. Solving the second-order equation f ( " ' r f ) l + ^ rfU<1,] = oI (3.37) for Ab should give a better approximation to Ab, where J:^a[_, _] is the quadratic part of Ta'. However, recall that the first-order quantity is identically divergenceless, Chapter 3. Spin-two Theories and General Relativity 55 daF^a = 0, Eq.(3.31). Taking the divergence of (3.37) and using (3.31) then gives «9 a^ a[4 1 ) 54 1 ) ]=0 - (3-38) 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 in per-turbation theory, it gives additional conditions that the (already solved for) linear per-turbation must satisfy. This is the consistency problem. Only the solutions of the linear equations (3.36) that are also solutions to the second-order equation (3.38) are allowed. However, in general, not all solutions (if any) of (3.36) wi l l also be solutions to (3.38) - the two equations are, in general, incompatible [74]. Similar problems occur at higher orders. For, example, taking the divergence of the third-order equation ^a[A{b3)] + J*Va[A?\ 42)] + J*2>[AC2\ A(b1]] + ^ a [ A ^ \ A?\ Ad1]] = 0 , (3.39) gives dajM'iA1), 42)] + d ^ [ A ? \ A^} + d ^ l A ^ , ^ , Ad1]] = 0 . (3.40) Eq.(3.40), a third-order equation, gives further conditions on the first- and second-order solutions A^b \ A^2*. In general, higher-order equations in the perturbation expansion 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 in a nonlinear extension of J-~^a, consistency problems may arise since is divergenceless, i.e., since the linear quantity J7^ obeys the linear divergence identity (3.31). The strategy employed in [74] to avoid these pathologies is to find a generalization of the divergence identity (3.31) such that if the quantities J-a satisfy 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 wi th the lower-order perturbation equations - if one generalizes the equations of motion, one should also generalize the associated identities and corresponding infinitesimal gauge invariances. Wald shows that it is sufficient that T^l)a satisfies an identity of the form daTa = CaTa + B\daTh + Da\dadhr + • • • , (3.41) where the sum on the R H S is finite, to ensure consistency in the perturbation solutions. The tensors Ca, Bab, and Dabc are locally constructed from Ab and its derivatives, rjab and (in four dimensions) eabcd; at zeroth-order C^ = 0, B^ab = 0, D^0)abc = 0, (so the first-order part of (3.41) is just (3.31)). For example, to see that Eq.(3.41) together with 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): daT{2)a = Cal*F^a + B ^ \ d a ^ ) b + D^abcdadbF{x)c + • • • . ' (3.42) Evaluating (3.42) at the linearized solution A^b* gives daJ^a[A{hx\ Ai1*] = C l 1 ^ ] &1)a[AP] + BW\[AW] d^iAi1*} + D^abc[Ad1]] dadb^c[A{b1]] + • • • = 0 , (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) imply 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^b* and A^b2\ aa^2'a[4i2),41)] + da^a[A^\A{2)] = C^IA,1*} F^lAi2*} + ClVf] ^ (1)a[4a)] + B^MA,1*} da^b[A^} + B^\[Ad2)] daT^b[A^] + t)^abc[A^} dadbT^c[Ad2)] + b^abc[Ae2)] dadbT^[Ad1]] + ••• (3.44) = CW[AP] ^ (1)a[42)] + BM'b[AP] daT^b[A^] + b^abc[A[l)\ dadb^[Ad2)] , 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 A[^\ 5a^3)-[41),41),41)] = CW[AN ^ > [ A [ h l \ A ^ } + CW[A?\AU] ^ [ A ^ ] + B^KiA^} d a ^ b [ A ^ \ A ^ ] + BW\[A$\AP\ da^h[A^] + D ^ M ? } dad^\A$\AW) + bW*c[A£\AP] d a d ^ i A i 1 * } + ••• = CP[Aliy\ &2)a[Aliy,AW] + B ^ A ^ ] d a T ^ b [ A { h x \ A ^ \ + b^abc[A(b1]} d a d t ^ i A ^ , A ? ] + ••• . (3.45) Using the second-order equation (3.37), (3.44) becomes d a ^ a [ A \ 2 \ A ^ ] + daJ*2*[AWAl2)] = -Cf )[4 1 )] ^(2)B[41)>41)] -B^MA^] d*?W*[A£\ A?] - D ^ M P ] dadbT^a[A[l\ A?] • (3.46) Add ing (3.45) and (3.46) gives (3.40). Similarly, at higher orders, the divergence of the nth-order perturbation equation wi l l be satisfied as a consequence of the lower-order equations together wi th (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 in the previous sections. Again , 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 nonlin-ear equations of motion, Ta = 0, are local partial differential equations that involve a bounded number of derivatives. This requirement can be achieved by l imit ing the num-ber of derivatives appearing in 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 in BAB and only Chapter 3. Spin-two Theories and General Relativity 58 one derivative in Ca. W i t h this restriction, an equal number of derivatives occur on each side of (3.41). Otherwise, including say the term Dahcdad0J-c on the R H S of (3.41), in a perturbation expansion, a single derivative of J7^0 would be related to a double deriva-tive of J r ( n _ 1 ) c which would in turn be related to a double derivative of ^ ( n _ 2 ) c 5 and so on, i.e., "successive J7^0 would contain a larger and larger total number of derivatives", (unless the higher derivative terms conspire to cancel or vanish). Wald points out that there may be other ways of ensuring that the equations of motion, Ta = 0, are local partial differential equations involving a bounded number of derivatives. Indeed he refers to this assumption as "probably the most significant loophole" in his analysis. However, his procedure is st i l l at least as general as al l of the previously mentioned procedures and is manifestly more general than most. 7 The restricted form of the identity (3.41) is then daTa = CaTa + B\daTh . (3.47) In order to ensure that a generalization of Maxwell 's equations is consistent, then, it is sufficient that the quantities Ta satisfy an identity of the form (3.47). It remains now to solve for the unknown tensor fields Ca and Ba0. It is convenient to rewrite (3.47) as follows 8 8a (B\Fb) = CaTa , (3.48) where B^ab = Sab, and Ca°* = 0, but are otherwise the same as their hatted counterparts ( l imit ing 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 = jdAx\[da(BabFh)-CaFa 7For example, in [51], the identity corresponding to (3.47) is restricted to the form daTa = cTaAa. 8Where we have made slight changes from [74] to be consistent with later calculations. Chapter 3. Spin-two Theories and General Relativity 59 JdAx [BabdaX + CbX] Th , (d4x [Babdax + cbx] s4f (3.49) Eq.(3.49) is the statement that the action SAa is invariant under the infinitesimal variation This infinitesimal gauge symmetry wi l l arise from an exact gauge symmetry if the com-mutator 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, Wald gives the following argument: Let AA be the manifold of field configurations, AA — {Aa}- Then the action S is a scalar function on A4, S : AA —> 3£, and the infinitesimal variation 8\Ab, (3.50), defines a vector field on AA. Let Wp be the subspace of the tangent space Vp at p e AA that is generated by the collection of vector fields at p that satisfy (3.49), i.e., Wv is the vector space composed of the infinitesimal gauge symmetries of S. Let W denote the set of Wp, W = {Wp\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 in Af coincides with W. These submanifolds are called integrable submanifolds. 9 If the submanifolds can be found, then one can obtain an action S wi th 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 9 I t is helpful to recal l the definit ion of integral curves, which are a special case of integral submanifolds: Let v be a smooth vector field on the mani fo ld M, (i.e., Wp = v\p and W = v). T h e n there exists a unique f ami ly of curves M < M f i l l ing M such that for each point p e M, the tangent to the curve passing th rough p is v\p> (i.e., the tangent space of each point 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, and the integral submanifolds , which are curves, can always be found. If d i m W 7 > 1, then the submanifolds can be found i f and only i f (by Frobenius 's theorem [73]) the commutator of two elements of W lies i n W. Otherwise, i t is possible that the ' W - p l a n e s ' twist a round ' " and cannot be smoothly 'added up ' so that a smooth scalar funct ion cannot be defined. 8xAb = BabdaX + CbX . (3.50) 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 X 1 (3.51) where V#, V ^ , and Vx represent infinitesimal gauge directions, given by (3.50), in the manifold of field configurations. Substituting (3.50) into (3.51) gives Se (Babdaty + Cbty) - <v {B\da0 + Cb9) = B\daX + cbX, (3.52) where Sg denotes the linear change in the quantity induced by the variation (3.50). Since Bab involves no derivatives of Aa, and Cb, at most one derivative of Aa, Eq.(3.52) can be writ ten dBa dC dC -—^8eAcdaty + -^-8eActy + b da8eActy oAc dAc odaAc -{9 <- ty) = B\daX + CbX , (3.53) which, using (3.50), can be further specified: b (Bdcdd9 + Cc9) daty + ~ - (Bdcdd9 + Cc9) ty dAr + dCb ddaA, •da (Bdcdd9 + Cc9) ty - (6> <- V) = B\daX + CbX • (3.54) Eq.(3.54) is an integrability condition for the unknowns Bab, Cb, and x- In order to solve (3.54) then, one must find a scalar field x, which depends on 0 and ty (and Aa, i]ab, etc.,), and expressions for the tensor fields Bab, Cb, (which are independent of 9 and ty). This is simplified somewhat by observing that, by definition, there is some freedom in specifying Chapter 3. Spin-two Theories and General Relativity 61 the fields Bab and Cb: First note that since the field A in Eq.(3.50) is arbitrary, (3.50) is unchanged by the rescaling A - / A , (3.55) where / is an arbitrary function that is locally constructed from Ab , rjab, and (in four dimensions) eabcd, wi th / ^ ° ' = 1. Correspondingly, the tensor fields Bab, Cb transform as follows: r>a D ' a r r>a JJ b —• ti 6 — JfJ b , Cb - C'b = Babdaf + fCb. (3.56) Also , expressions for Bab or Cb, that are related by a change of variables are considered to be equivalent. Under an algebraic change of dynamic variables, Ab -» Ab(Ab,r]cd,eabcd) , (3.57) the quantity Ta transforms according to ^ r = ^  = ^ £ £ = **££. (3.58) 8Aa 8Ab dAa dAa Substituting this into (3.48), one finds that the tensor fields Bab, Cb, transform under a change of variables (3.57) according to dAc B\ -+ B\ = Bab dAb ' — 8 A Cc -> Cc = C b l r f . (3.59) oAb Thus one seeks expressions for Bab and Cc up to a rescaling (3.56) and a change of 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 in a power series Chapter 3. Spin-two Theories and General Relativity 62 i n Ab, B \ = E B ( n ) a » > Cc = E Cin) , X = E X W , (3-60) ra n n and to first try to solve the zeroth-order part of the integrability condition (3.54) (0 *-» = dbX{0) • (3.61) The most general linear expressions for the tensor fields (restricting the number of deriva-tives) are given by B^ab = 0 , = ClAc , (3.62) where c\ is an arbitrary constant. Substituting these first-order expressions into the zeroth-order equation (3.61), gives d ( W - edcty) = dbX(0) • (3.63) Taking the curl of (3.63), gives C i = 0. Thus, the first-order expression for the infinites-imal gauge invariance (3.50) is 8^Aa = 0. Indeed, it can be shown by induction [74], that the general solution of the complete integrability condition (3.54) is Bab = 8ab, Cc = 0, so that a nonlinear generalization of Maxwell 's equations has normal spin-one gauge invariance, 8xAa = dax-3.3.2 Summary of Wald's consistency criteria In summary, Wald'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 in Aa to the second-order action SJfjh, Eq.(3.29). Al though one can, in principle, obtain some sort of equations of motion from such an extended action, in general, since the linear quantity is divergenceless, consistency problems may arise in a perturbation solution. Such problems can be avoided, however, Chapter 3. Spin-two Theories and General Relativity 63 if the generalized quantity Ta satisfies a generalization of the linear divergence identity. This generalized identity can be interpreted in terms of invariance of the full action Se,Mi under infinitesimal variations of the field, 8\bAa. These infinitesimal variations w i l l correspond to an exact gauge symmetry if the space of variations is involute. This 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. This consistency criteria has been applied to a number of problems. In [74], in addition to considering extensions of Maxwell 's equations, Wald investigates extensions of a collec-tion of spin-one fields as well as extensions of the linearized Einstein equations. In the for-mer case, the generalization of a collection of k spin-one fields, {Aa1,..., Aak} = {A/}, obeying the linear equations J~^a = 2dbd^bAa\ = 0, was found to have Yang-Mil l s infinitesimal gauge invariance, S^Aa*1 = Da\^ = 0 a A M + c'lv^AaKAy, where — —c^irv, and c^^c"^ = 0, (i.e., & V 1 ^ defines a Lie algebra on the (internal) vector space { A M } ) . In the second case, Wald finds that a nonlinear extension of the l in-earized Einstein equations, G^ab = 0, can have normal spin-two gauge invariance or general covariance, (see Chapter 6). In [18], Cutler and Wald find that the nonlin-ear generalization of a collection of spin-two fields, {jab^}, obeying G^ab = 0, has the infinitesimal gauge symmetry, 8\b^ab^ — V ( a A 6 ) / i = d(ahf ~ ^cab llvKv, where rc0&"„ = ^a"a0(g-1)cdal/(2diajb)/ - dd-fab'3), g^bv = 7]ab8^„ + a^^jabP, and = a"(l^ ), a^u[-Ka0p]\ = 0, (i.e., defines an associative commutative algebra). 1 0 In [75], these infinitesimal gauge invariances are interpreted precisely in terms of algebra-valued tensor fields and diffeomorphisms on algebra manifolds. 1 0In [60], Reuter constructs a six-dimensional gravity model which has this new gauge symmetry. Chapter 3. Spin-two Theories and General Relativity 64 3.3.3 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 consis-tent (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 ex-plore 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. 1 1 Application of this equation to a candidate infinitesimal symmetry 1 1 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 that fc|{<£} is simply the trivial symmetry <f) —• <f>. (In other words, K, is 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, jao, coupled to a spin-| field, (j>A, is, in particular, considered). The equation for the induced variation of the field equations by an arbitrary infinites-imal local symmetry 8<f>A, (where (f>A represents generic fields, and the index A, represents generic indices), is derived in [1] as follows, (which is essentially identical (with minor modifications in notation) to what appears in [1]): Let 1 2 <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 (b)> $<t>A = HA[4>B\ (ii) (f)A = •§p<f>^1\a=o=[3 is an arbitrary field variation of compact support. 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 of the the h o m o m o r p h i s m from Q to <?|{0}-) Consider the symmetries g e Q that are related up to an element k e fC - these are the (left) cosets, g + IC = {g + k\k e K-}, oiQ which pa r t i t ion Q. T h i s set of equivalence classes is called the quotient set and is denoted QjK, = {g + K\g e Q}, which is i somorphic to N o w , take a representative f rom each element oiQ/K.. T h i s collection need not have L i e algebra structure. However, i t gives rise to the gauge symmetries of the theory. For example, the commuta tor of any two such representatives may have terms propor t iona l to the equations of mot ion . Unless evaluated on the so lu t ion space {(j)}, the commutator may fai l to lie i n the collection of representatives oiQ/K,. In other words, the commuta tor of two inf ini tes imal local symmetries of the Lagrang ian that " implement the gauge invariance" may not also be one. However, i t is not clear that the class of symmetries (3.50) w o u l d consti tute such a restricted representation of the inf ini tesimal symmetries. In any case, rather than take the cond i t ion that the space of variat ions be involute, i n [1], an alternative equation, namely (3.68), for the induced var ia t ion of the equations is used. 1 2 I n this discussion, the letters a and'/?, are used as parameters, and the symbol ' 0 ' ( 'nought ') is used as a l abe l . T h i s deviates f rom the index convention used elsewhere i n this thesis. Chapter 3. Spin-two Theories and General Relativity 66 since for fixed ft, 8(j)A^ is a local symmetry of the action S. Now consider the following: a = 0 8=0 da I d r s s i r A d f / ' d d y r a J A U^°A a = 0 / 3 = 0 - 1 d{3 Q = 0 8=0 da d/3 a,f3 (8£)Aj>A+[£A d£ dftdaj d{8fy-a = 0 8=0 a = 0 8=0 j + e d(3 A8{8fydf/ 8f/ dp a = 0 8 = 0 , {S£)A + S B S - ^ 8 9 A (3.65) (3.66) (3.67) where (8E)A — ^^-\a=Q-p. Since <f)A is assumed to be an arbitrary field variation of compact support, this series of equations shows that 0 = (8S)A + £ B8(8<j>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)^, in more detail. In particular, in going from (3.65) to (3.66) (where both the a and f3 dependence in 8(j>A^ are retained) d IQ,(3 8{8f/) d£_ dp Q = 0 8=0 a = 0 8=0 (3.69) use of the condition (ib) was made. 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)AP = S^AI^B'^} o n fl i s o n l y through the intermediate cf)^, (as stated in condition (ib) above, however, it does not seem necessary to assume that there is no explicit dependence of 8§aA on a). It is not immediately obvious that this condition is compatible with the Chapter 3. Spin-two Theories and General Relativity 67 condition that <f>^ is an arbitrary field variation, (condition (ii) above). Whi le 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>A^ depends explicitly on 4>s^i and hence, so must <j>jf, i.e., <f>A 3[a, (3, <t>c'^\i s o w e assume this as well. Consider the following smooth two-parameter family of field configurations: + <*PXA[<I>B] + <XPXA<1>B13], (3.70) where nA, £A, and XA, a r e arbitrary functions of (f>%'B; and rjA , £A, and XA, are arbitrary functions of 4>s- Then k°>>0 (3.71) 6f/ da ' which, for constant f3, f30, is required to be a one-parameter family of local symmetries of the Lagrangian, 6<f>A8. Also , from (3.70), df/ d(3 a = 0 B=0 = &[*B] + U[<l>aB a = 0 6=0 (3.72) which is required to be an arbitrary field variation of compact support, (j>A. From (3.71), d dfx a,B^ d/3 V da (SVA[f/}df/' V 6f/ da a = 0 6=0 A[?B d4>\ B S=0 ba/ da a = 0 B=0 + XA a=0=B + XAW/] a=0=P (3.73) In going from (3.70) to (3.73), one sees that in order that (3.69) holds, it is necessary that the last three terms in (3.70), EAI^B'13], XAI^B], and XAWB13]-) are not included in the expression (3.70) for <f>A0, (otherwise, there is explicit dependence of [3 in (3.71) and 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>A3, (3.70) must be restricted to the form f / = + *VAIM + ^ A [ f / \ + p&fo] . (3.74) (Note that if the term rjA[(j)^] was not included, the condition (3.69) would be identically zero, (which is the reason why 8<f>A3 is written vs ft<f>A)-) The question is whether, wi th these restrictions, that <f>A is s t i l l an arbitrary variation. From (3.74), (3.72) reduces to, a = . 0 /3=0 = &[M • . (3-75) d/3 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)A8 does not depend explicitly on (3. Consequently, in 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 in some calculations, the condition (3.68) enables us to eliminate or put further constraints on some of the parameters appearing in the expression for the first-order gauge invariance, parameters which we were not able to include in 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. On the other hand, it is also possible to have equivalent descriptions of a theory in terms of different potentials. For example, a tensor and its dual can give equivalent descriptions of a theory yet be different types of tensors. When 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 in 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 in three dimensions. 4.1 Divergence- and curl-free vector field The theory of a divergence- and curl-free vector field can be described in terms of a scalar field, or, in terms of an antisymmetric tensor field, Aab. 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 be-haviour [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 Va in four space-time dimensions that satisfies the equations d[aVb] = 0 , daVa = 0 , (4.1) where [•••] denotes antisymmetrization. There are four ways of introducing potentials to describe this theory: Defining Va = da(f>, one can write the theory in terms of the scalar field (j). Then d[aVb] = 0, and variation of the action S?) = -yd*xda<l>da</>, (4.2) with respect to <f>, gives the equations of motion 1 6S(2) EE —i- = • = daVa = 0 , (4.3) where the numerical superscript denotes the power of the field appearing in the expres-sion. (Note that Eq.(4.3), JF ( 1 ) = daVa = 0, implies that Va = dbAab locally, where Aab = _Aba^ Alternatively, by defining Va = dbAab, one can write the vector theory (4.1) in terms of the rank-two antisymmetric tensor field Aab. Then, daVa = 0, and variation of the action S{Z = -\$dAx dcAac8bAab , (4.4) with respect to Aab, gives the equations of motion £c(2) 4V = j~bf = d[adcAb]c = d[aVb] = 0 . (4.5) throughout, boundary terms are ignored. Chapter 4. Potentials and Nonlinear Generalization 71 (Note that Eq.(4.5), Aa^ = d[aVb] = 0, implies that Vb = db<f> locally, for some scalar field <j).) These equations are unchanged by the transformation Aab _^ Aab + g^labc} ^ ^ where A a 6 c is an arbitrary rank-three tensor field. In four dimensions the gauge transfor-mation (4.6) can be written Aab Aab + ^bcdQ^ ^ ^ where A c is an arbitrary one-form field, and eabcd is the Lev i -Civ i t a totally antisymmetric tensor. Also , Aa\* satisfies the linear identity V f i l ^ O . ^ . (4.8) Equivalently, the vector theory (4.1) can be described in terms of the duals of (f> and Aab, *<f> and *Aab. 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) in terms of a scalar field, the quantity Eq.(4.3), does not satisfy some tensor identity. Hence, there are no constraints, of the k ind discussed in section 3.3, on the types of terms that can be added to the Lagrangian. This means that any non-pathological addition to £ ^ wi l l result in 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, wi l l give consistent equations. For example, the cubic potential v(<f>) = ij>dcm, (4.io) is one possibility. Chapter 4. Potentials and Nonlinear Generalization 72 O n the other hand, in the formulation of the vector theory (4.1) in terms of Aab, the quantity A^b satisfies some tensor identity, namely the identity (4.8). This imposes restrictions on the types of terms that can be added to the Lagrangian. In particular, to assure consistency in 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) SXaAcd = D[cd]ahda\b + Ea[cd]Xa , (4.11) where the unknown tensor fields Dabcd and Eabc are locally constructed from the fields Aab, eabcd and rjab (the Minkowski metric), with D^abcd = eabcd, E^abc = 0. The requirement that the space of variations be involute, then gives an equation that can be solved for these unknown tensor fields. One solution for Dabcd and Eabc gives, for the most general gauge invariance of the extended theory, SXaAcd = ecdabda\b . (4.12) In theories wi th this type gauge symmetry, Acd can only appear in the action in the form dcAac. 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{x1]aAcd = J 1 ( 4 a e a ^ c U e + A 6 a e c d a e a 6 A e ) + d 7 A a f e e c d a 6 a e A e + {dx - 2d7){Aabteba[cde\d] + AaeecdbedbXa) Chapter 4. Potentials and Nonlinear Generalization 73 + e1(d[cAabed^abeXe + dbAbaeacdeXe) + e7da Abeebecd Xa + (ei - 2e7)(daAbeea^cXd^ + daAbeebcdaXe) , (4.13) where d\, d7, &\ and e7 are arbitrary constants. (Eq.(4.12) is just the special case d\ = d7 = ei = e7 = 0.) Higher-order calculations may put further conditions on these parameters, however, in light of the complexity of (4.13), it is not clear how one could obtain a solution to al l orders other than in the special case d\ = d7 = e\ — e7 — 0, (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 in the two cases: SJf* can be extended arbitrarily whereas there are fairly strict conditions on possible additions to SA^ . 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 in three dimensions. The equations of motion for a massless spin-one field in Minkowski spacetime are given by Maxwell 's equations, (see Chapters 2 and 3), 3aFab = 0, d[aFbc] = 0, (4.14) where Fab = —Fba. One usually formulates this theory in terms of the vector potential Aa by defining Fab — 2d[aAby Then d[aFbc] = 0 and variation of the action Sl% = -iJ<rixFttbF*b , (4.15) with respect to Aa, gives the equations of motion 8S(2) jrU)» = = 2dbd[bAa^ = dbFba = 0 , (4.16) oAa Chapter 4. Potentials and Nonlinear Generalization 74 (which imply that Fab = dcGabc locally, where Gabc = &abc]). These equations are invariant under the transformation Aa -» Aa + daX , (4.17) where A is an arbitrary scalar. Also, J7^0, satisfies the identity daF{1)a = 0 . (4.18) Alternatively, Maxwell 's equations can be formulated in terms of the completely anti-symmetric rank-three tensor field Gabc, by defining Fab = dcGabc. Then daFab = 0 and variation of the action (4.15), wi th respect to Gabc, gives the equations of motion 6S{2) = = \^ddGah¥ = \d[cFah] = 0 , (4.19) (which imply that Fao = d[aA^ locally). These equations are invariant under the trans-formation gabc _^ gabc + Q^abcd] ^ where A\abaIi is an arbitrary rank-four tensor. Correspondingly, JF^ ' satisfies the identity 0[e*£]] = 0 • (4.21) In four dimensions these two descriptions of Maxwell 's equations are identical, since Qabc = eabcd^ Qabc a n d ^ are duals of each other). However, in three dimensions, Qabc _ ea6c,05 for s o m e a r b i t r a r y scalar field ty, and A[abcd] = 0. It is not difficult to see, then, that in the formulation of (4.14) in terms if), (i.e., Gabc), the linear equations (4.19) no longer have a gauge invariance and J7^ does not satisfy some tensor identity. There are therefore no restrictions on possible additions to SG . O n the other hand, in the description in terms of the vector-potential Ab, the identity ( 2) (4.18) constrains the possible extensions of CAa to theories that are invariant under the Chapter 4. Potentials and Nonlinear Generalization 75 infinitesimal transformation (cf. (3.50)) 6xAb = B\daX + CbX , (4.22) where Bab and Ca are locally constructed from the fields Ab, r]ab, and eabcd (in four di-mensions), wi th BWba = Sba and C f ) = 0, (cf. section 3.3 and Ref.[74]). In [74], (cf. section 3.3), it was found that the most general solution for Eq.(4.22) in 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 gen-eral gauge invariance resulting from a consistent nonlinear generalization of Maxwell 's equations in three dimensions can be written 6xAa = daX + kXeabcdcAb + 2k2XAbd[aAb] + ••• , (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, SA* and SQ* , of the theory of a spin-one field in three dimensions, need not necessarily give rise to equivalent types of nonlinear theories. This 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 in-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 in three dimensions (formulated in terms of the vector potential Aa) are outlined. Briefly, we adopt the criteria of [74] in order to ensure that the nonlinear theory SAAB arising Chapter 4. Potentials and Nonlinear Generalization 76 f r o m the gene ra l i za t ion of SA*, (4.4), is consistent . T h i s enta i ls , as discussed i n d e t a i l i n sec t ion 3.3, f i nd ing a genera l i za t ion of the l inear i den t i t y (4.8), r e in te rpre t ing th is i n (2) t e rms of an i n f i n i t e s i m a l gauge invar iance of SA^: and requ i r ing that the space of these va r i a t ions is i n v o l u t e . T h u s one arr ives at an in t eg rab i l i t y c o n d i t i o n for a n u m b e r of u n k n o w n tensor f ields. It is expedient to e x p a n d these unknowns i n a power series i n te rms of the f ie ld Aao and first o b t a i n an express ion for the l inea r i zed gauge invar iance of SA^b w h i c h can be s imp l i f i ed b y equivalence under a change of var iables a n d the inherent a rb i t ra r iness i n the de f in i t ion of the u n k n o w n tensor fields. In the case of the vec tor the-o ry (4.4) , th i s leads to the express ion (4.13) for the in f in i t e s ima l l inear gauge invar iance . H o w e v e r , i t is unc lea r h o w to f ind a so lu t ion to a l l orders. In the case of e lec t romag-n e t i s m , th is leads to the express ion (4.23) for the second-order gauge invar iance . I n this case, we are fur ther able to m a k e use of the iden t i t y (3.68) to show that (4.23) reduces to n o r m a l spin-one gauge invar iance (4.17). 4.3.1 Vector theory T h e l inea r vec to r theory (4.1) can be genera l ized by add ing powers of the f ie ld Aab of order three a n d higher to the second-order ac t ion SA*b, Eq . (4 .4 ) . T h i s gives rise to the non l inea r a c t i o n SAab a n d the genera l ized equat ions of m o t i o n Aab, B y the a rguments of [74] (see sect ion 3.3), these general iza t ions of SA* w i l l be consis tent i f the quan t i t i e s Aab satisfy a genera l i za t ion of the l inear i d e n t i t y 5[c«4-o6] = 0> E q . ( 4 . 8 ) , w h i c h is of the f o r m da (DcdabAab) = EbcdAab , (4.25) where D ^ a b c d = tabcd, E ^ a b c = 0. T h u s , the in f in i t e s ima l gauge invar iance of the non-l inea r t heo ry is t a k e n to be SXaAcd = D ^ a b d a \ b + Ea^Xa, Eq . (4 .11) . T h i s i n f i n i t e s i m a l Chapter 4. Potentials and Nonlinear Generalization 77 gauge invariance w i l l arise from an exact gauge symmetry if for al l one-form fields A a , 7r a there exists a one-form field Xa such that [VxM = VXc , (4.26) where V\a, VVb, and VXc represent infinitesimal gauge directions, given by Eq.(4.11), in 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 8Akl Dklab(daXbdc7rd - 8airb8cXd) + Efkl(Xf8cnd - wf8cXd) + ( * A Q £ c t e / 1 K - ( < ^ a £ c [ e / ] ) A c = D^cd8cXd + Ed^Xd , (4.27) where 6\a etc., denotes the change in the quantity induced by the infinitesimal variation 8XaAab, Eq.(4.11). Before attempting to solve (4.27) for D 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 ^ j~)[ab]ce jd Ec[ab] _^ rjlab^eQ^c^ + E<i[ab]jc^ ? where fab is locally constructed from Aab, r)ab, and eabcd, with f ^ a b = 8ab, and that (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 ^ j-yefcd jj<c[ab] _^ £cef dAah dA*f ' 8Aab 8Aef (4.30) Firs t , consider the zeroth-order part of (4.27), which involves zeroth- and first-order quantities, eklab8c7rd8aXb"AU + ekla\c8d8aX{ ^ 8Akl 88dAkl Chapter 4. Potentials and Nonlinear Generalization 78 - ( A C <-> Tc) = eefabdaXd0) . (4.31) The most general expressions for the first-order tensor fields are, (limiting the number of derivatives), D(l)[ab]cd = dieedf[bva]cAef + faabed^ + ^ab^cd + ^ [ y ] * + ^ [ a ^ c + d7eahef AefVcd + d8eecf[aAefVb]d + dgeabecAde = £ d i Z > | 1 ) [ a 6 ] c ' ' , E(i)f[ab] = e i t ^ d a ] A c d + e2ecabfddAdc + e48fAab + e5d[aAb]f + e6dcAc[aVb^ + e7ecdabdfAcd + e8edabcdcAfd + e 9 e d e c ^ f ddAce = £ e ^ f ) / M , (4-32) i = l where the d{ and the ej are arbitrary constants. (The terms d3€ecd[bAa]e a n d e3€cfd[bdcAa]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^ab is f(1)ab = hAab + f2eabcdAcd . (4.33) Substituting (4.33) into the first-order part of the transformations (4.28) shows that i = 9 D(l)[ab]cd _^ D>{l)[ab]cd _ ^ dlDi:X)[ab]cd + f1Dgl*[ah]cd - 2 f2D^a^Cd i=l t=9 — 4:f2D^a^Cd EE ^  d'.£)(1)^cd i=l E(l)f[ab] E'(l)f[ab] _ ^ e i E J 1 ) f [ a b ] ~ / i E ^ 1 ) / [ a 6 ] - 2f 2 E{ 1 ) f [ a b ] t'=l -4/241)/ta6]==Ee;.41)/[o6], (4.34) i=l i.e., fi effects e8 and d9, and / 2 effects e4, e6, c?4, and <i5. However, the most general second-order change of variables, A^2)e* = a3eabc^Ae\Abc, does not effect any of the terms in the expressions (4.32). 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(d2 - d1)d[eiradf]Xa + 2{dx -d2 + d8 + d'9)dKJ a\\ fl - 2(d8 + 4 )d a 7 r [ e d | a | A / ] + 2{d2 + d'9 - 2d7)daTTadle\K + d'Atejahdcir cda\b + d'5e ab*dcK f]da\b + d6e ab^ d e]ndaX + 2{e1 + e'8- 2e7)Tradad[e\f] - 2(e'8 + e9)^de]daXa - 2{e'8 + e9)^en\^ + e'4eefab7r cdcdaXb + e5e abcKcd f]daXb - (A a ^ ;r a) = eefabdaX{b0) • (4.35) Note that the parameters e2 and e'6 do not appear in (4.35). In order to find conditions on the parameters di and e;, one can take the divergence of (4.35) so that the R H S disappears. Then, choosing various cases for the fields A a and 7ra, gives conditions for the dl and e;. This task is somewhat simplified by combining terms on the L H S into divergenceless groups, which in addition to vanishing when acting on (4.35) with de, give possibilities for the form of x i ° ' - Combining techniques (see Chapter 5 and Ref. [43] for more details about this approach), we find di = d2 , d8 = d1- 2d7 , e 9 = e x - 2e 7 , \d's = d'4 = e 4 , de = e 5 = 0, d'9 + d2 — 2d7 = e'8 + et — 2e7 , xi0) = [(di + 4 - 2d7)\deabcddairb + d'4(irddd\c - rddcXd)} - (ira ~ Xa) + kc , (4.36) where d[akb] = 0. In the special case where A a and ira are constant one-forms, the 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 ; e i = 0 or e'6 = 0 . (4.37) F rom 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 gWAab = D W W c d d c X d + £(i)/MA/ = dx{eednbda]XdAef + eabedAcedcXd) + d'4{Aabdc\c + \A^dc\b^) + d7eabefAefdc\c + (di - 2d 7)e e c /t°a cA 6U e / + (4 + e i - 2e 7 - dx + 2d 7 )e a 6 e c A'' e 0 c A ( J + d f e ^ ^ U r f A / + ecabJddAdcXf) + d'4dfAabXf + e'6dcAc[aXb] + e7ecdabdfAcdXf,+ e8edabcdcAfdXf + {ex - 2e7)edec[aXb]ddAce , (4.38) where the fx and f2 have as yet not been specified. It is not clear how best, or if there is indeed a best way to choose fx and f2. The complexity of (4.38) seems to defy determination of the gauge invariance to al l orders, except in the special case where al l 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 in (4.38). However, it is reasonably tricky and involved to check this. (We have checked in the special case where A a is a constant field, so that only the e; terms remain in (3.68). Beginning wi th no conditions on the e;, (3.68) confirms that eg = ex — 2e 7 and ex = e2.) 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 in the example of three-dimensional electromagnetism, we do not pursue the effects of (3.68) on (4.38) in a more general scenario. Choosing (somewhat arbitrarily) e'6 = 0 from Eq.(4.37), using f2 to set d& — 0, and fx to set d'g + dx — 2d7 = 0, Eq.(4.38), the most general solution to the zeroth-order part of the integrability condition (4.27), can be written as Eq.(4.13) with xi°* — kc- Again , taking the special case dx = d7 = ex = e7 = 0, gives the gauge invariance (4.12). Chapter 4. Potentials and Nonlinear Generalization 81 4.3.2 Three-dimensional electromagnetism The formulation of this problem is given in [74], and discussed in detail in section 3.3. Summarizing, the consistency arguments of [74] require that generalizations of Maxwell 's equations have the infinitesimal gauge symmetry S\Ab = Babda\ + C(,A, Eq.(4.22). The 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 Ba0, Cb, transform according to Eq.(3.56). Under a change of variables Ab —> Ab(Ab,r)cd,eabcd), Eq.(3.57), Bab, Cb, transform according to Eq.(3.59). (Note that A^ = ri\Ac, and A^ = 0.) In four dimensions, the most general expressions for the first-order tensor fields are given by Eq.(3.62): CJ.1* = C\AC, and B^ab — 0. In three dimensions, however-, it is possible to write the more general expressions B^ba = WAceabc, C^ = ClAa + c2eabcdcAb . (4.39) Substituting (4.39) into the zeroth-order part of the integrability condition (3.54), (which is Eq.(3.61)), gives 2b1eabcdbtydce +-Cl(il>da6 - 0daty) = daX{0) • (4.40) Taking the curl of this equation shows that ci = h = 0 , (4.41) and that x^°* 1S a constant, which, checking at second order in the case of constant ip, 9, is zero. However, c2 is s t i l l completely arbitrary. Because this problem is not too complicated, we can go to the next order. The most general second-order tensors are BW\ = b2AbAa + b3SbaAcAc, Ci2) = c3AcdaAc + c4AadbAb+ c5AbdbAa, (4.42) Chapter 4. Potentials and Nonlinear Generalization 82 and the most general expressions for / ( 2 ) and Aa are f{2) = fxAaAa , (4.43) Aa3) = n2AaAbAb . (4.44) Under the second-order rescaling, (4.43), one gets B W \ _ B'Wba = b2AbAa + (b3 + f1)8baAcAc, C[2) -> C f ) = (c3 + 2 / 1 ) A c 5 a A c + c 4 y l o a 6 A 6 + c 5 A 6 5 6 A 0 , (4.45) and under the second-order change of variables, (4.44), one gets Bi(2)ba ^7(2)^ = ( & 2 + 2 n 2 ) A 6 A a + (6 3 + n 2 ) O c A c , (4.46) where b3 = 63 + /1. Substituting (4.42) into the first-order part of the integrability condition (3.54), gives [(62 - 2b'3)dbtyda0Ab + (4 + c5)tydadb6Ab + (4 - c22)tydb0daAb + c4tyda0dbAb + (c 5 + c 2 2 ) ^ 5 6 ^ 6 A a + c4tyO0Aa} - (0 <-• tf>) = 8aX(1) , (4.47) where b2 = b2 + 2n2, b3 = b'3 + n 2 , and c'3 = c3 + 2 / i . B y inspection, must be of the form Ab(0dbty — tydb8). Adding and subtracting (for example) (b2 - 2b'3)[tyda8b0Ab + tydb9daAb] - (0 <-» ty) , (4.48) to (4.47), gives "(62 - 2b'3)da(Ab0dbty) + (4 + c5 + b2- 2b'3)tydadb0Ab + c4tyda0dbAb + (c3 - c2% - 2b'3)tydb08aAb + (c 5 + c22)tydb0dbAa + c4tyU8Aa • -(9~ty) = daX[l) , (4.49) Chapter 4. Potentials and Nonlinear Generalization 83 Taking the curl of (4.49) one finds 4 + c 5 + b2 - 2b'3 = 0 , c 5 + c 2 2 = 0 , c 4 = 0 , (4.50) and = ( 6 2 - 2b'3)Ab(0dbty - tydb9). From Eq.(4.43), we choose fx = -63 - " 2 to set 6'3 = 0, and from (4.44), we choose n2 = -\b2 to set b2. (Then also dbx^ = 0.) W i t h these choices, the second-order expressions (4.42) for the tensor fields reduce to 5 ( 2 ) 6 a = 0 , Ci2) = 2c 22A cd[aAc] . (4.51) U p to second order, then, this gives the infinitesimal gauge invariance 6\Aa = daX + k\eabcd cA b + 2k 2xA bd[aAb], Eq.(4.23), where k = c2, as a solution to the zeroth- and first-order parts of the integrability condition (3.54) with dax^ — 0. However, in 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 Ta', (4.16), by an arbitrary infinitesimal local variation of the Lagrangian (4.15), is 0 = S\J~ a + J~b —r—8\Ab , oAa + ( 4 ' 5 2 ) The zeroth-order part of (4.52) is simply Sx°*J r(1* a — 0, which, by definition, is true. The first-order part of (4.52) is 0 = 40)^(2)a + ^ 1 ) a - 8C ^ A - ^ g U A ^ . (4.53) Substituting pWa, from Eq.(4.16), and S^Aa, from Eq.(4.23), into (4.53), and taking A constant so that the quantity ,7-*(2)a drops out, gives 0 = kX(e cd^ ad b]dbdbAc-e bacdcd dd[dAb]) , = kXDFcbeabc . (4.54) 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 theo-ries 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. However, this assumption is more restrictive than necessary. In particular, it implies that the gauge invariance of the linear theory is lab - » 7a6 + d(a\0). It was noted in Chapter 2, on the other hand, that the massless spin-5 equations, (2.1), can be regarded as being conformally invariant. The 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 or-der to incorporate the possibility of theories with conformal invariance, a more general formulation of the spin-two equations is necessary. One method of describing the spin-two equations in terms of a symmetric rank-two tensor, jab, such that the equations are 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 Paul 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 con-sider 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. Conformal Invariance - Spin-two Coupled with Spin-zero 86 5.1 Conformally invariant linear equations The massless spin-two equations, VAA'(J>ABCD = 0, Eq.(2.1), with 4>ABCD a field of confor-mal 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 - ttrjab , (5.1) where f l is an arbitrary real scalar field (cf. Appendix C ) . One way of finding such a theory is by brute force: Write out the most general second-order action in ^ a b (assuming second-order derivatives), £ M G I = a l 7 a b a 7 a b + a 2 l c b d a d c l a b + a3ldc8dicd + a 4 7 n 7 , (5.2) where the a, are arbitrary constants, and derive the equations of motion, c 0 ( 2 ) = ~ ^ — = 2 a 2 d c c/( a7 6 ) c + 2 a 1 D 7 o 6 + a3da0bi + nab(a3dcddlcd + 2 a 4 D 7 ) = 0 . (5.3) OJab 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 + ±a3)dadbiab + (2ax + a 3 + 8 a 4 ) n 7 = 0 . (5.4) This gives the conditions that 2a2 + 4et3 = 0 and 2ax + a3 + 8a 4 = 0. For the sake of preciseness, take the one free parameter to be, say, a2, and fix a x = — 4 - Then 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 3 = - | a 2 , a 4 = ! ( a 2 - | ) . (5.5) In Chapter 6 we consider the case a 2 = | , and find interacting spin-two theories that are conformally invariant. 1 However, in this type of formulation of the equations, the "normal" spin-two gauge invariance, 8~fab = d(aipb), °f the equations is lost. 2 One way to formulate a conformally invariant spin-two theory in terms of 7A&, but to maintain the "normal" spin-two gauge invariance, i.e., to find a linear theory that has the local symmetry hab = d(a\b) - tt'qab , (5.6) is to introduce, along with jab, an auxiliary field, the simplest one being a scalar field, <f>. In the next section, we consider the linearized massless conformal Klein-Gordon equation in curved spacetime, as a candidate linear theory. 5.2 Sp in- two c o u p l e d w i t h sp in-zero - c o n f o r m a l K l e i n G o r d o n e q u a t i o n First let us consider the (full nonlinear) conformally invariant theory described by the action St = - | J d4xy/^(gabWa^Vb^ + eft$ 2) , (5.7) 1 I n [26] (see also [5]) the Lagrangian that corresponds to a 2 = | is written down. 2 F r o m (5.3), the requirement that d£$£b = 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). Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 88 and the equations of motion resulting from variation of (5.7),3 / = ^ ( V a V a $ - Cm) = 0 , (5.8) \y/=g{Z&Gab + eab) = o, (5.9) where, Qab = ^ _ l ) ^ ^ y c $ V c $ + (1 - 2 £ ) V a $ V 6 $ + 2 £ # a b $ V c V c $ - 2 £ $ V a V 6 $ . (5.10) Taking £ = | , Eqs.(5.7) - (5.9) are conformally invariant [54].4 Under a conformal rescaling of the metric, gab —*• u>2gab, the equations transform according to T% —• Tt, — u>~3J-£, and 8£b —»• Sf' = u>~6££b, and the action is equivalent to §t up to a boundary term, which we ignore (since throughout we assume locality). The quantities Tt, and ££ b satisfy the identities V a £ f - l g a b V a $ T t = 0 , (5.11) l<f>Tt-gab€f = 0 , (5.12) (where we are taking £ = | ) . Correspondingly, the equations of motion are invariant under the transformations gab -> gab + V(aAb) + u2gab, $ -» $ + l A ^ + w 1 * , (5.13) 3 T h e equations (5.8) are sometimes referred to as the massless conformally coupled K l e i n - G o r d o n equations i n curved 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 a 3 a $ = 0. T h e usual prescr ipt ion for generalizing equations to curved spacetimes is s imp ly to make the " m i n i m a l subst i tu t ions" , rjab —> gab, and da —• V a , [73]. T h i s yields V a V a $ = 0. However, V a Va<J> — £.R<I> = 0, where the constant £ is arbitrary, is another possibi l i ty consistent w i t h the " m i n i m a l subst i tu t ions" prescr ip t ion [73]. T a k i n g £ = | makes the equations conformally invar iant . 4 P a r k e r [54] has a neat argument to show that the equations (5.9) are conformally invariant : One can work out that Gab = Gab + 6 6 a & / w 2 , where Gab = Gab(gab) - Gab(u>2gab), (see, e.g., [73]). T a k i n g LO = $ gives Gah{§2gab) = Gab(gcd) + 60 o j (ffcd, $ ) / $ 2 . One then also has Gab(&gab) =jSab(gab) + 6@ab(®2,gab)/&- Since $2gab = &gab, ( taking J > = w ~ ^ $ ) , equating Gab(^2gab) = Gab($2gab) gives Gab(gcd) + 6@ab(gcd,®)/$2 - Gab(gab) + d>Qab(^2,gab)/^2> which is precisely the statement that the equations (5.9) are conformally invar iant . ogab Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 89 where A a is an arbitrary one-form field. Let us now linearize this theory to obtain a linear theory of a spin-two field, 7^, coupled to a spin-zero field, (f>, that is conformally invariant. The second-order part of the action (5.7) is second-order action S f = -ljd4x[dacf)da^ + ( ( ^ { 1 ) R ^ + R^+2RWcf>)] , (5.14) = - \ Jd4x \da<t>dj + ([w + \ i ) ( d c d d l c d - dcdcl) + Ybdlad[a7c]*}} , (i.e., substitute ga°b = rjab, g$ — -)abi = 1, and $( x) = <j> into (5.7) and retain terms that are second order in the fields). Variat ion of (5.14) gives the equations of motion s s P 77(1) 6<f> SS. = dada<f> - £ ( d c d d l c d - d a d a l ) = o , (2) ?(l)ab _ _ 1 £ [ G ( 1 ) a 6 + 2{r)abdcdc(j) - dadh<i>)\ = 0 , (5.15) where G^ab is the linearized Einstein tensor. The quantities J7^ and £ ^ a b satisfy the linear counterparts to (5.11) and (5.12), (I)a6 = 0 , \ff] ' Vabe^ = 0 (5.16) (5.17) (recall that £ = | ) . Thus, the linear gauge invariance of the theory (5.14) is lab -> lab + <9(aA6) - VlTjab (5.18) Alternatively, we could have obtained the linear theory (5.14) - (5.18) by brute force: Wri te out the most general second-order action in 7^ and <f>. ' M G 2 = «i7a6 Vlab + a 2 l c h d a d c l a b + a 3 7 d c d d l c d + a 4 7 D 7 + a5<f)dadbiab + ae^O-f + a74>U(j> , (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, *A2) ^ = slab J J C ( 2 ) 8<f> c(l)ab _ voMG2 c(l)ab . o a o i j , a & n J . n cWn. = — = tM'Gi + a50 a <p + a6ri •<£ = 0 , (5.20) 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^MG2 = U ' Kf MG2 VabOMG = U , (5.21) where k is an arbitrary constant. This gives a2 = —2a,i, « 3 = 2aj, a 4 = — a x , fcas = —4oi, &a 6 = 4ai, fc2a7 = —6ai. For preciseness, fix a x = — ^ . These conditions give Scc[2) = -\j d4x [C(y/=^R)W - l(RilU - ^ 0 ° ^ ] , (5.22) together with the equations of motion 8(f) r n ( 2 ) r(l)a6 _ °^>CC _ i > G^ab-\(r}abdcdc<f>-&l^'<f>) (where £ = | ) . The quantities Tcxc and £cc, a 6 satisfy identities p, p(l)a6 oacc = 0 and the linear gauge invariance of the theory is (5.23) (5.24) (5.25) lab -> lab + 0(a\b) - Q,T)AB , d> —> ^ — &f2. (5.26) With A; = — | this gives precisely (5.14) - (5.18). Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 91 5.3 Nonlinear extension of spin-two coupled with spin-zero Thus we have a linear theory of a spin-two field, ^ a b , coupled to a spin-zero field, <f>, that 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, £cdc = 0 and Tec = 0, Pcd _ 8 S c c _ _ SScc _ n 9_. by adding higher powers of the fields, 7aj, and <f>, to the action Scc. The linearized form of the equations (5.27) are given by Eqs.(5.23), £<cCcd = 0 and Tec = 0- Following precisely the same reasoning as in section 3.3, consistency of the nonlinear equations is ensured if the exact quantities £ c 1 c c d and TQC satisfy the two generalized identities da(Bahcd£cdc + DabTcc) = Cbcd£cdc+AbTCc , (5.28) Hab£cbc + GTCC = 0 . (5.29) These identities are imposed in 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 = Babcdda\b + C a c d \ a + nHcd, (5.30) 6<f> = DabdaXb + AbXb + VLG , (5.31) where B^)abcd = 6a{c6bd), = -h0riab, D^ab = 0, G^ = g0, C^\d = 0, = 0. To obtain the linear theory (5.15) - (5.18), h0/g0 — 2, (e.g., g0 = |, hQ = 1), and to obtain the linear theory (5.22) - (5.23), h0/g0 — —1/k, (e.g., g0 = —fc, ho = 1). Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 92 The tensor fields Babcd, Hab, G, D a b , Cacd and A 6 , are locally constructed from 7^ , rjcd, (f), and (in four dimensions) eabcd- Cacd and Ab can include single derivatives of lab and 4>. (Note that the second term in (5.31) can be absorbed into the third term by setting f l -AbXb/G + 0 . Then (5.31) becomes S</> = D a b d a \ b + f l G and (5.30) becomes 6fcd = BabcddaXb + ( C a c d — AaHcd/G)\a + CtHcd. Also, note the symmetries B a b c d = B a b ( c d ) , Hab = H { a b ) , and C \ d = C \ c d 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 = V ( c A r f ) - gcdh0n , (5.32) 6<f> = lgabXbVa4> + g0Sl , (5.33) where V a is the derivative operator associated with the metric gab = nab + qjab (cf. Eqs.(5.89) and (5.90)). Although 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 Sgab = V ( a A 6 ) , (5.34) = -\ga%Va4> + g& , (5.35) where, gab — e9° gab, <j> = —(j), Xb = Xbeg° , and V a is the derivative operator associated wi th the metric gab. In other words, the gauge invariance (5.32) and (5.33) is simply general covariance. In 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 in terms of a symmetric rank-two tensor field, j a b , wi th "normal" spin-two gauge invariance, coupled with 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 wi l l drop two of the principal assumptions made in the analysis of this chapter: we consider, on the one hand, formulating the spin-two theory in terms of a symmetric rank-two tensor field, 7a&, which does not have "normal" spin-two gauge invariance, and, on the other hand, formulating the equations in terms of a traceless, rank-two tensor field, -fjb- I*1 both instances we find theories which are not generally covariant. We are also able to find theories that are conformally invariant. 5.4 Solving for the gauge invariance According to the consistency arguments of section 3.3, the nonlinear extensions of SGC (cf. Eqs.(5.22) - (5.26)), See-, w i l l be consistent if See is invariant under the infinites-imal 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, 0a and ipb, Co a n < i ^> © a n d T , pairs of one-forms and functions, \c and IT, ra and A , <j>b and T, respectively, such that [Vea,V^} = VXc + Vn, (5.36) [Vc.,Va] = VTc + VA, (5.37) [Ve,Vr] = V+e + Vr, (5.38) where Vga, etc., are vector fields on the manifold of field configurations defined by Eq.(5.30) and (5.31). 5 5 The calculation is considerably more involved than in [74] since it involves the coupling between two fields. Although both jai, 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 hat, and (j) fields is investigated. Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 9 4 5.4.1 The general equations Rewri t ing ( 5 . 3 6 ) - ( 5 . 3 8 ) in terms of the tensor fields Babcd, Hab, G, Dab, Ab, and Cacd, via ( 5 . 3 0 ) and ( 5 . 3 1 ) , we find first, from ( 5 . 3 8 ) , that cj>c = 0 , and F = 0 , since [VQ, VT] = 0 . Next , E q . ( 5 . 3 6 ) , i n component form, can be written + dBabcd dChcd dChcd datyb (B9hefdgeh + c 9 e f e g ) + -^datyb (Defdeef + Aee tyhdg (Babefdaeb + c \ s o a ) + jfijWe {D9hdgeh + A9eg) + ef dde<j> BCb tyh (BabefdJb + C\sBa) + -Qf-tyb (D9hdg6h + A99g) - {oc <- tyc) = B a b c d d a X b + c \ d X a + n# c d ( 5 . 3 9 ) and, BT)ab BDab 8atyb (Befcdde0f + Chcd9h) + ~^datyb (Dcddc6d + AC8C) d% cd + tyb86cAb] - (0C <-> tyc) = DabdaXb + AcXc + EG . Final ly , E q . ( 5 . 3 7 ) , in component form, becomes {B^cddeQ + C<cd(j) + ^  (D<ddc(d + A%c) fdBcdabrj 8Bab I —: iief + -dcC cd \ 9% ef d<f> G and, BC BC ^—(Bcdab8c(d + Cdab(d) + -7r7(Dabda(b + Aa(a) - (b6QAb = D abBan + Aara + GA ( 5 . 4 0 ) - CdSaCdab = Bcdab8CTD + CDABTD + AHab , ( 5 . 4 1 ) ( 5 . 4 2 ) died d<p The notation 8^b is used to denote the linear change in the quantity induced by the variations 8^ab = Babcddatpb + Cacd?pa and 8^b<f> = Dabdaipb -f Aatya, °TpbV " ') = —hb'Jab + ™ o^Ociab H ^7~d4>b(P + a a / 8^bOc<p , ab d d c i a b d<f> ddc4> ( 5 . 4 3 ) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 95 and SQ is used to denote the linear change in the quantity induced by the variations Afi7a6 = &>Hab and Sa<f> — O G , djab ddcjab d<t> ddccf> 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 Babcd, Hab, G, Dab, Ab, Cacd, and Xa, IT, r 0 , and A. This is tractable order by order in -fab and <f>. Note first that there is a certain arbitrariness in the definition of the tensor fields, which can be seen by considering that, since ipb and f l in (5.30) and (5.31) are arbitrary, the equations (5.30) and (5.31) are unchanged by the transformations ipa —> fbetya arid f2 —> eO, where fab and e are locally constructed from jab, <j> and t]ab wi th f^°*ab = Sab and e^0) = 1. Correspondingly, the tensor fields are defined up to the transformations nab . nlab nae rb n cd —> rs Cd = rs cdj e , . file rjab a re , s~tb fe U cd —> U cd — rS cdOai 6 + ^  cd) 6 5 ^a6 ^ j-ylab £)ae Jb A* _> A'e = Abfeb + Dabdafeb, Hab —> H'ab — eHab , G - • G' = eG. (5.45) Also note that, under a change of variables <j> -»• ^(^,7 0 i ) , 7a6 -» 7o6(^7cd), (5-46) the tensor fields transform according to Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 96 O-jcd 0(f> Ab _ ^ = C ^ - f A ^ r r T r _ r r 7^crf . p 7^crf Ofab 0(p G V = H * 1 2 - + G § . (5.47) This 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 The linearized equations The next step is to expand the tensor fields in a power series in the fields 70& and <j), (e.g., Babcd = 5TJn B^n*abcd, etc.) and to substitute the zeroth- and first-order expression for the tensor fields into the zeroth-order part of equations (5.39) - (5.42). The most general linear expressions for the tensor fields are B ^ a b c d = bo7e{ced)abe + Wr)ablcd + b2jabrjcd + b3lVabricd + bAl8\c8bd) + b58b[clad) + b 6 8\ c l b d ) + b7^abr,cd + b8<f>8\c8bd) = £ biB?)abcd , j=0 C { l ) e c d = cQda-fb{ced)abe + cxde^cd + c2d{c-yd)e + c38e{cdd)-y + c4dbjb{c8ed) 8 + c5dejr]cd + c 6d&7 eVd + c7de4>r]cd + c88e{cdd)<f> = ] T cxC\l)ecd , i=0 3 (l)ab D(i)ab = d l r , a b 7 + d2rb + d3Vabcf> = J2diDl1)ab , t = l A ( 1 ) 6 = a1db-f + a2db<f>-ra3da7ab = J2^A (1)6 i 5 t'=l Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 97 H{1)ab = hir]abj + h2fab + h3riab<f> = E > i=l G ( 1 ) = gi-y + 92</> = J 2 9 i G ( i 1 ) • (5.48) To see how the terms in (5.48) are effected by the transformations (5.45), write the most general expressions for f^be and 32 / ( 1 ) 6 e = / l 7 ^ e + /276e + / 3 ^ e = E / ^ ( 1 , e . t=l 2 e(1) = eii + e2<l> = YleiEV . (5.49) j=i Substituting (5.48) and (5.49) into the linear part of (5.45), gives at first-order 5/(1)abe/ = E bM1)a\d + hB^abeS + f2BPabef + f3B^abef = £ KBl1)abcd , i=0 i=0 8 C ' W / = E c.-C?^ + /aC7i1)6e/ + /2<7<1)6e/ + / 3 G < 1 ) b e / EE £ cJCp* , i=0 t=0 j=i t=i - E ^ + f f o e i G ^ + ^ G ^ E i E ^ . (5-50) i=l i-l and D'Wab = D^ab, A / ( 1 ) b = A * 1 ) 6 , i.e., modifies the terms 5l1)a6e/ and G | 1 ) 6 e / , (i.e., / i :ffects b4 and C 3 ) and can therefore be used, for example, to eliminate one of these terms; F\e affects the terms B<Q*ab ef and C ^ ^ e / ? (i- e-5 h affects 66, c 2 ) ; i * 1 ^ affects the terms BQ1*abej and G ^ 6 ^ , (i.e., f3 affects 68, c 8 ) ; E^ affects the terms and G P , (i.e., e i affects / i i , 51); and affects the terms H^}d and G ^ , (i.e., e 2 affects / i 3 , #2)- However, at this point, it is not clear how best to exploit these parameters. We could simply state that, wi th the benefit of hindsight, we choose certain values for / 1 ? / 2 , / 3 , e i , and e 2 However, rather than l imit ing our choices at this point, we wi l l simply suppose that the values of /1, f2, f3, e i , and e 2 are fixed, without stating explicitly their 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), b4, c 3 , 6 6, c 2 , b$, c 8 , hi, h3, gi, and g2, by a prime: b\, c'3, b'6, c'2, b'8, c'8, h[, h'3, g'i, and g'2. To see how the first-order expressions (5.48) are effected under a change of variables according to (5.47), note that the most general change of variables (up to second order) can be writ ten 7a6 = mi7ab + rn2r]ab<j> + m3yr)ab + m4^ab<f) + ra5rca677 + ™el<xC1bc + m7j"fab + msj4>r]ab + m9(f>(j>T}AB + mwr]abfcd-<(cd , <f> = ni<f> + n2j + n 3 # + " 4 7 7 + n 5<^7 + " 6 7 c r f 7 c d • (5.51) Substituting (5.48) and (5.51) into the zeroth-order part of (5.47), gives B { 0 ) a b e f = mi8\e8bs) + m3r)efrb , D{0)a» = n2fb , Hi°d = (-mih0 + g0rn2 - kh0m3)incd = h0r]cd , G(0) = g0rii - 4 / i 0 n 2 = g0 , (5.52) and, C^°*bef = 0, A'°) 6 = 0. The form for the zeroth-order gauge invariance (5.18) gives mi = 1, m2 - 0, m3 = 0, ni = 1, n2 = 0. A t this point we take mi — 1, m3 = 0, and n2 = 0 but leave m 2 and n\ unspecified. W i t h these choices, substituting (5.48) and (5.51) into the first-order part of (5.47), one finds that the expressions for the first-order tensor fields are effected by a change of variables as follows: B W a b e f = J2b'M1)a\d + ^ M1)abcd + (2mw + m 2 d 2 ) B ^ b c d + (2m 5 + m2di) B ^ a b c d + m7B^abcd + m 6 2# ) a b c d + m6B^abcd + (m8 + m2d3)B^abcd + m4B^abcd = E M ^ c - , D W a b = rn J2 diD\1)ab + 2n4D[1)ab + 2n6D{21)ab + n5D{31)ab = ]T d'iD?)ab , »'=! i=l Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 99 ef - 2-sCi°i cd + m 2 a 1 ^ 5 e / + m 2 G 3 C 6 e / + m2d2L7 ef — 2_^C^i cd ' i=0 i=0 i=l t=l 11$ = J2KHS + (m2g[+g0ms-ho(8m5 + m7 + 2m10))H[l)d i=l + {gom4 - h0(im7 + 2m 6 )) i J ^ c d + {2gom9 - h0(m4 + 4m 8 ) + m2g'2) H^]] 3 = ^ h'iHicd , i = i 2 = n i E ^ + {g0n5 - h0{Sn4 - 2n 6 )) + ( 2 5 o n 3 - 4/i 0n 5) G 2 X ) = E ^ G f , (5.53) « = i and, C^bef — C^bef, A^h = A ^ ) 6 , i.e., m 4 affects the terms with coefficients b'8, / J 2 , and / i 3 ; ms affects 63 and h[; m6 affects b5, b'6, (in particular the combinations, 65 = \(b5-\-b'6)), and h2; m7 affects fei, b'4, (in particular, the combination, b'x = |(fri + &4) ), ^ i , and / i 2 ; ^ 8 affects 67, /I'J and / i 3 ; m 9 affects / i 3 ; and m 1 0 affects b2 and Z^; 723 affects g'2\ n4 affects d\ and g[; n 5 affects c/3, ^ and g2; n 6 affects d 2 and g[. Whi le we can unambiguously choose m9 = — (m2g'2 — /to(4m 8 + m 4 ) Jrh'3)/2g0 so that h'3 = 0, and n 3 = — (ni<72 — 4:hQn5)/2g0, it is not immediately clear what is the best choice for the remaining parameters. However, we view them to be fixed at this point. Quantities effected by a second-order change of variables are denoted by an overline. Final ly , note that the R H S of the zeroth-order part of (5.39) is d(cxd°* — rjcdh0I[(0\ and the R H S of the zeroth-order part of (5.40) is g0U{0). Substituting 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^cdab dC^dab — -dctydde0f + —-tpd cde9f Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 100 h0 / < 9 D ( 1 ) c d dA^d ^ + Vab— —z dcipdde9j + — ipddcde6f 90 V dlef OdcJef , - (ec <- tyc) = a(BXg> . (5.54) Similarly, the R H S of the zeroth-order part of (5.41) is <9(aT^0' — T]abhoA^°\ and the R H S 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) \ — 1- n^hn—; dn%> i -u dB^dah _ dB^dab H riefh0— #0 w- h rjab—  h rjcdh0- „ c*7cd C7 e / d</> g0 \ dfab d^cd -9o-Q£)(l)ab 8<f> - Q ^ K b - V*%C*S§]AM = d(aTV> . (5.55) 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). This gives, [\boHdabe (dlelec)datyb + dc)ele[datyb) + (h - b'4)d{ced)datya +§(&5 - b'6)d{c0adlaltyd) + \codad{cem)ahetye + \c'2tyedcddee + ( C l + \c'2)i>edsd(ced) + (4 + \c4)ty[cdd)deee + \c4ty{cn0d) + (C5 + | C 6 + fo{ai + | a 3 ) ) Vcdtyededa9a + § (c 6 + | a 3 ) W ^ 6 ] - ( ^ ^ V c ) = a(cxg), (5.56) and, [(ax + | a 3 ) ^ e 5 e 5 a r + \a3tyenee] - (6C <-> 0C) = g0Iii0) . (5.57) (Note that b2, b3, 2&i = bx + 6 ' 4 , 265 = b5 + 6 ' 6 , 6 7 , 6 ' 8 , J 1 ? J 2 , d 3 , do not appear in (5.56). This means that there exists a change of variables such that these quantities can be set to zero (cf. (5.99) - (5.110)). Also, c 7 , c'8, a 2 , h'i, h2, h'3y g'x, and g'2 do not appear.) 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\ + h0(h + b2 + 46 3) - g0b7 + I (g\ + hQ{Adx + d2) - g0d3)\ Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 101 + ttd{c(d) (h2 + h0(4b~U + b5 + b'e) - gob's) + ({cdd)n (h0(c'2 + 4c 3 + c 4) - g0c'8) + r]cd(eden [h0(Cl + 4c 5 + ce) - g0c7 + ^  (ft0(4ai + a 3) - g0a2)] = d^rff , (5.58) and, (g\ + K{4dx + d2)- gods) nda(a + (h0(Aai + a3) - g0a2) (ed eft = c?oA ( 0 ) . (5.59) (Note that the parameters h'3 and g'2 do not appear in the integrability condition (5.58), 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 ^ e that, by inspection, xd°^ m u s t be a linear combination of the terms tydda0a - eddar, tyadaed - eadatyd, tyaddea - eaddr , edabe (eedatyb - tyedaeh) . (5.60) Therefore, adding the term (which is zero) . (4 + \cA)d{ced)d atya - (c 3 + \c4)d[ced)datya + ( C l + \c'2)d[cead\a\tyd) -{Cx + §4 W ^ i - A + \bQdad{ce[b\ed) abe^ e - \b0dad{ce\b\ed) abe^ e, (5.61) to the L H S of (5.56), and rearranging, gives d(c (\c2rdd)Ba + (Cl + \c'2)Vd\a\6d) + (4 + \cA)tyd)daBa + \bQed)abe8edatyb) +12b0deeiced)abedatyb + (l\ - Eu + 4 + l2ci)d(cBd)datya + ( | (6 5 - b'e) + Cl + \c2)d{ce adHtyd) + \{K + co)dad{c0wed) abetye + \c^(cn0d) + (c 5 + \ce + |(«i + I « 3 ) ) r)cdtyededa0a + \ (c 6 + | a 3 ) r,cd</>eD0e] - (*c ~ </0 = 5 ( c xi) ] • (5.62) F rom (5.62), one can identify xd 0) = [(4 +  12c4)tyddaea + (ci + |4)^°aa^ + |40 o^° + i 6 o e / 6 e ^ 0 a ^ ] - (0c *-> 0C) • (5.63) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 102 Then (5.62) reduces to }2bodee{ced)ahedatyb + (b\ - E>4 + c'3 + lc4)d(ced)datya + i(6o + c0)dad(cem) ahetye +(|(6 5 - h) + cx + \c'2)d(cead]a]tyd) + \c4ty[cmd) + \ (<* +1«3) r / C ( ^ e ° 0 e + (c s + i c 6 + fQ{ai + |a 3 )) W e «9 e <9 a 0 a ] - (6»c ^  Vc) = 0 . (5.64) It is now relatively straightforward to see that the coefficients of each of these terms must be zero. For example, take ipa to be a constant one-form. Then (5.64) becomes i(bo + c0)dad{celb]ed)abetye + \cAty{cned) + (c 5 + |c6 + | ( a x + |a 3 )) Vcdtyededa6a + \ (c 6 + | a 3 ) / / c ^ e ° ^ e = 0 . (5.65) Choosing 0 a such that n# a = 0 and da0a = 0, reduces (5.65) to \ (b0+c0)dad(c8^ed)abeipe = 0. This implies that bo + c 0 = 0. Returning to (5.65) (with b0 + c 0 = 0), taking = 0 but da0a 0, shows that we must have c 5 + \c% + ^"-(ai + \a3). Now only the second and the fourth terms remain in (5.65). Taking ?/>a#a = 0 but 9a otherwise arbitrary, gives c 4 = 0, and so, finally, c 6 + ^ a 3 = 0. Eq.(5.62) has been reduced to [\bodee{ced)ahedatyb + (by ~ VA + c'3)d{c8d)datya + {\{h - b'e) + ej + ! c 2 ) 0 ( c 0 a 0 | o | ^ ) ] - (0C ~ ^ c ) = 0 . (5.66) Taking, in (5.66), da9b — Aab — A[ab] to be a kil l ing vector field and datyb = Sab = S(ab) to be symmetric, give (|(6s — b'e) + C\ + | c 2 ) = 0. This leaves the first two terms remain in (5.66). Taking datpb = Aab = A[ab] and 0 a arbitrary, gives bo = 0. Finally, we have that (b\ - b'4 + c'3) = 0. Alternatively, one can get conditions on the coefficients in (5.56) by acting on (5.56) (or (5.62)) wi th dfdg and antisymmetrizing over / and c, g and d. This eliminates al l the symmetrized derivative terms, in 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 9a and tpa give conditions on the remaining terms. This gives 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 3 + | c 4 = 0 , i ( 6 5 - h) + C l + |4 = 0 , (5.67) goc5 + hai = 0 , g0c6 + h0a3 = 0 , c0 = cA = b0 = 0 . (5.68) Similarly, from (5.58), one finds that ^ ( 0 ) = (M4 + 4c 3 + c 4) - g0c8) ^ , (5.69) and h'i + h0(bi + h + 463) - goh + | (g\ + /*o( 4^ + 2cl2) - g0d3) = 0 , (5.70) h0(ci + 4c5 + ce) - g0c7 + ^ (/*0(4ai + a3) - g0a2) = 0 , (5.71) h2 + h0(4V4 + b5 + b'e) - gob's - h0(c'24c'3 + c4) + g0c8 = 0 . (5.72) In order to simplify the conditions (5.67) - (5.72), it is now opportune to take advan-tage of the transformations (5.45) and (5.47) to fix the values of some of the parameters appearing in the equations (5.49) and (5.51). Note that there are a number of equivalent ways of choosing these parameters. We choose fi = - c 3 -> c 3 = 0 , f3 = - c 8 -> 4 = 0 , ei = (-#1 - gons + h0(8n4 + 2n6))/go -> g\ = 0 , m4 - -b'8 -> b'8 = 0 , m 5 = - | ( 6 3 + m2c?i) —> 63 = 0 , m6 = -b5 - » 6 5 = 0 , 6'6 = (4 - 65) , m 7 = -2(&1 + ^ 4) ^ 4 = ' m § = _ ( & 7 + m24) -> h = 0 , m 9 = - ( / i 3 - / t 0 ( " i 4 + 4m 8) + m2g'2)/2g0 -> A' 3 = 0 , Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 104 n 3 = -(g'2 ~ 4:h0n5)/2g0 -* g2 = 0 , raio = ~\{b2 + " ^ 2 ^ 2 ) — • &2 = 0 , n 4 = - » J i = 0 , n5 = -dz -> J 3 = 0 , n6 — -\d2 —> cl 2 = 0 . (5.73) (Note that e2 is redundant since e2 affects only g2 and h3, while mg affects only h'3, and n 3 affects only g'2.) Combining results, we have b0 = b\ = b2 = 63 = b'4 = b7 = b's = c 0 = c' 3 = c 4 = c ' 8 = 0 , J i = d2 = d3 = / ? i = h'3 = g\ = g>2 = 0 , (5.74) and h2 + h0(b'6 - c2) = 0 , - | 6 ' 6 + c x + |c ' 2 = 0 , <70c5 + h0ax = 0 , g0ce + h0a3 = 0 , g0c7 + h0(a2 - C l ) = 0 , (5.75) where we have not yet specified the parameter f2 from (5.49). 5.4.3 The general solutions One obvious choice for f2 is to take f2 — b5 — b6 so that b'6 = b5 and then, from Eq.(5.73), b'6 = 0. (Equivalently, one can choose f2 = —c2 so that c'2 = 0. Then b'6 / 0.) This leaves the four arbitrary parameters c'2, a x , a 2 , and a 3 , from (5.75), in terms of which the remaining non-zero parameters, C5, c 6 , c 7 , h2l can be written c i = - | c ' 2 , ft2 = c'2h0 , goes = -ho^ , g0c6 = - / t 0 a 3 , 5oC7 = -h0(a2 + | c ' 2 ) . (5.76) Substituting (5.74) and (5.76) into the first-order expressions for the tensor fields, gives C{1)\d = \c'2 {2d{cld)e - delcd) - r / c , | ( a i a e 7 + a3dbYb + (a 2 + § 4 W ) , A ( 1 ) 6 = a i 5 6 7 + a3da7ab + a2dbcf> , = c'2ho7ab , (5.77) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 105 and B~'{1)abcd = 0, D{1)ab = 0, = 0. Defining JW" = - i (o 1 a f c 7 + a3a.7B6 + a 3 ^ ) - ^ c 3 a V , (5-78) 9ab ~ Vab ~ C 2 7 ab , (5.79) the most general first-order solution (5.77) to the integrability conditions (5.39) - (5.42) can be writ ten c ^ \ d = - r w e c , + V c d l 0 , A™ = -goJW - |c'25V , H$ = -gUho , (5.80) together wi th , X(d0) = \c'Madd6a-6Ma) , n ( 0 ) = j - o l ( a i + la3)rdedaO a + la3tyen6e]-(0c~iPc) , r 6 ( 0 ) = h0c'2tt(b , A* 0* = j;(h0{U1 + a3)-g0a2)Cden, (5.81) where T^ecd is the linearized Christoffel symbol for the metric (5.79). These equations give the first-order gauge invariance S(1hcd = -T{1)ecdXe + r / c ^ o J ( 1 ) e A e + c'2h0lcdn , 8W<f, = -gQjMh\b-\c'2\hdbct>, (5.82) i.e., SW-fcd = ( V ( c A d ) + gcdh0XeJe - gASl)™, and 6^<f> = (-\d 2gabV a(j>\b - g0JbXb + Qgo)^. 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 h0 and g0): Babcd = Sa{cS%Dab = 0,G = go, and C\d = -Y\d + g'cdh0Je , Ab = -g0Jb - \c2gabV acf> , Hab = -gabh0 , (5.83) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 106 together wi th , Xd n n A = \c2 ty°vA - 9avdVa) + c 2 r > d j c e c - edjctyc) , (fll + \az) i/>eVeVa0a + \azgahrnea] - {6C <- 0C) hoC2tt(b , j- (^ 0(4a! + a 3 ) - g0a2) ( e V e f i , (5.84) where V a is the derivative operator associated with the metric (5.79), gab — ^ab ~ c 2 7 a f > , and J e is arbitrary. Then the most general gauge invariance of the complete theory is ^ 7 c d = V( c A r f ) + gcdh0Ja\a — tthogcd , H = -\c2gab\bVa<f> + g0n-g0JbXb. (5.85) (5.86) Let us rewrite this by making the following substitutions: let q = —c2 (so that (5.79) becomes gab = rjab + ? 7 o &), mult iply (5.85) by q, let 0 —• tt/q + JdXd, and let Xd —» Xd/q. Then (5.86) can be writ ten {*9cd = V ( c A d ) - gcdh0n = ^ a b A b V ^ + ^ f t (5.87) (5.88) where, from (5.52), / i 0 = ho(l — gQm2), g0 = <7oni) and in order to obtain the linear equations (5.23), h0/g0 = ho/go = —1/qk (where this latter equations allows us to absorb q in to 6 go)- Thus, to obtain the linear gauge invariance (5.26), we take m2 = 0 and n i = 1 so that = ho and g0 = go, and thus also h0/go = ho/go = — 1/?^ which gives the original linear equations. W i t h these considerations, Eqs.(5.87) and (5.88) can 6Indeed, we could have begun by including q explicitly in the equations (cf. the constant c in [43]): in S(?Q, ^cc a n < ^ ^cc°''> k v setting yai, —> qyab- Then variation of S^QQ with respect to qyab gives Sccah• ^ n e identities, (5.28) and (5.29), would than be modified by replacing £c1Q,b by q£c^ab• The infinitesimal gauge symmetries, (5.30) and (5.31), would be the same, however instead of ho/go = —1/fc we would have gotten ho/go — —l/qk. Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 107 be writ ten Sged = V ( c A t i ) - gcdh0tt , (5.89) 6<f> = lgabVa<j>\b + g0n. (5.90) However, there exists a change of variables that shows that the gauge invariance (5.89) is simply general covariance, namely, ftp l gab -»• gab = e*> gab, (5.91) tf, j> = -(f>. (5.92) From (5.89), one gets ^ 6 = e » " I V ( a A b ) + h-gabVd<f>Xd) , (5.93) and from (C.4) and (C.5) Sgab = (V(aA6) + ^ A ( 6 V a ) ^ , (5.94) where V a is the derivative operator associated with the metric (5.91). Then, defining \ b = A 6 e ^ , (5.89) and (5.90) become Sgab = V ( a A 6 ) , (5.95) H = -\gah\VA4> + g0h. (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, in 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 al l orders. Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 108 5.4.4 Uniqueness of the solutions Let us remark on the uniqueness of the solutions (5.95) and (5.96). First , we find con-ditions on the possible changes of variables, icd and (j), such that one is able to set ab B c d = Sa(c5bd), Hab = -{h0rjah + q>yab), D = 0, and G = g0: (i) F rom Eq.(5.47), there exists a change of variables such that the tensor Babcd can be set to Sa(cSbd), i.e., B a b c d = Satc6bd), if there exists a 7 c d such that died _ g-lab _ j^-labjjef ^7 led dlab " c d e f ~ d<f> • Substituting this into the condition d2%d &%d (5.97) dfabdjed dicdd^a and taking the zeroth-order part, gives = 0 (5.98) V dicf where, from (5.51), cd dfab + # 7 cd d<f> (0) ' Q£){l)ab dE-Wef d% ef dfab = 0 , $7 cd d<t> (0) m2r]cd (5.99) (5.100) Eq.(5.99) is the zeroth-order condition that there exists a change of variables jcd such that B a b c d = 6a(c6bd). (ii) Similarly, there exists a change of variables such that Hab = —(h0rjab + ?7a&), if there exists a icd such that died = _G-^loT1ab + q j a b ) _ Q-^HJ1^ d(j> Substituting (5.101) and (5.97) into Plef d2%d d-Jab (5.101) d<f>d~fcd dfcdd(f> = 0 (5.102) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 109 and taking the zeroth-order part gives ( d B ^ a b e f h 0 d B ^ c d e f \ d<f> Vef r. go Olab 1 dG™ +~^f~Ji— go Olab + rn2rief 'dD^ab hodD^cd' d<f> Vcd go dfab (h0 - h0) go dfab go —6 a(eS /) = 0 . (5.103) This is the zeroth-order condition that there exists a change of variables jcd such that Babcd = 6a(c8bd) and Hab = -(h0r]ab + qiab)-(iii) F rom Eq.(5.47), there exists a change of variables such that Dab = 0, if there exists a <b such that dlcd~ B a b D d<t> Substituting (5.104) into 82(j> d2 Ofabd-Jcd dfcddfab and taking the zeroth-order part, gives the condition = 0 , d<f> (o) where, from (5.51), (0) = 0 dl d<f> (iv) Final ly , there exists a change of variables, cf>, such that G = go, if — — go^r — Lr flabT. • d(j> d"1ab Substituting (5.108) and (5.104) into d2<f> 0 , djcdd<j> d<j>d~fcd and taking the zeroth-order part, gives godGW fdD^ab h 0 d D ^ + ni I — — Vcd-9o dja V 9<f> go d<yab (5.104) (5.105) (5.106) (5.107) (5.108) (5.109) (5.110) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 110 Eq . (5 .110) is the zeroth-order c o n d i t i o n tha t there exists a change of var iables <f> such tha t Dab = 0 a n d G — go-N o w , let us consider the spec ia l case of (5.83) where c 2 = 0, w h i c h is (of course) also a s o l u t i o n of the zeroth-order i n t eg rab i l i t y condi t ions (5.54) and (5.55). T h e first-order tensor fields can then be w r i t t e n C ^ e c d = T]cdh0J{1)e, A^b = -g0J{1)b, = 0, B^abcd = 0, D^ab = 0, a n d = 0, so tha t , 8^lcd = r]cdh0J{l)e\e, a n d 6^<f> = —g0J^bXb- T h e n a comple t e so lu t ion to the f u l l i n t eg rab i l i t y cond i t ions , (5.39) - (5.42), is Babcd = 8a(cfibd) ? Cecd = rjcdh0Je , Hab = —hoVab , Dab = 0 , G = g0, Ab = -g0Jb, d{cXd) = 0 , d(CTD) = 0 , (5.111) a n d , LT, A are g iven by (5.81). T h i s gives the gauge invar iance for the comple te theory hcd = d{cXd) + r]cdho\eJe - rjcdh0$l , S(f> = gQtt - goJbh . (5.112) O n e c a n show by i n d u c t i o n tha t (5.111) is the un ique so lu t ion (up to a change of var iables (5.47), and the resca l ing f reedom (5.45)), cor responding to the first-order cond i t ions c 2 = 0, (5.75), a n d (5.74) [73], [43]. W e have shown that Eq. (5 .111) is t rue for n = 1, (i .e . , for B^abcd, e tc . , w i t h n = 1). N o w assume that (5.111) is t rue up to a r b i t r a r y n. T h e n the n th -o rde r i n t e g r a b i l i t y cond i t ions , us ing the i n d u c t i v e hypothes is , are the same as the zero th-order i n t e g r a b i l i t y cond i t ions , (5.54) and (5.55), w i t h the (l) r ep laced by (n+i) (however , w i t h o u t yet m a k i n g use of the two freedoms (5.47) and (5.45)). R e a r r a n g i n g , these equat ions are •/dB^abcd g J B("+l )e / e \ h() /dD(n+l)ab QD(n+l)ef A ^7e/ dlab J ^ go \ Olef • dfab + [ t y a S ^ C ^ a c d - (0e <- tyc)} = d{eX^ , (5.113) datybde0j Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 111 and, 'QB^)abcd h0dB^+1^bcd\ h0 fdD(n+Vab /i 0 9 D ( n + 1 ) a t Vef ~ + Vcd— ~xl 'ef 3 ' 'lcd 'ef a 0(p go 0~fef J g0 V o<P 90 O^ef A v c d ^ - - i ^ S ^ l (-g0ndc(d) - < d ^ C ^ d a b = dlcTtf , (5.114) go <?7ab 90 Olab J ' where C^n+^acd = C^n+1^acd + h0r]cdA^+l^/go. In order to show that Eqs.(5.113) and (5.114) imply that (5.111) holds for (n +1), first let tya and ( a be constant one-form fields so that the first terms in both (5.113) and (5.114) vanish. Then, operating on (5.113) and (5.114) with dgdh, and antisymmetrizing over g and c, h and / , the RHS's of (5.113) and (5.114) are eliminated, and these equations can be written C ^ t A ^ ^ i c ] = 0 , (5.115) 4 % % C ( n + 1 ) ° r f | | c ] = 0 . (5.116) Eq.(5.115), is the statement that the quantity <9[9<9|[/lC,(ra+1)aa']|c] is invariant under the transformation jab — > ^ab + d(a\b). In order for this to hold, fab can only appear in the expression d[gd\[hC(n+^a^ in the form d[ad\[cjb\]d]. However, while d[gd\[hC(n+1)ad]\c] is an (n + l)th-order quantity and thus contains (n + 1) 7a6's> it is assumed to involve at most three derivatives. Consequently, the sum of all terms with more than one ~fab must cancel. Considering terms with only one jab, for n > 1, it can be seen that (5.115) can 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[gd^h,C^n+1">a^ = 0, which implies that (j(n+i)d^b m u s t kg a symmetrized derivative, i.e., that there must exist a tensor g^n+1^ed such that C ^ d a b = d(Jn+1^ed) . (5.117) Taking g^n^ed = — / ' n + 1 ) e d , one sees that the C^n+1^dab can be set to zero by the rescaling freedom (5.45). Substituting C(n+^dab = 0 into (5.113) and (5.114), operating on the Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 112 resulting equations wi th dgdh, antisymmetrizing over g and c, h and / , and taking special cases for the fields tya, 6a, (a, J), (cf. [73]) gives the equations /dB(n+^abcd h0 dB(n+Vabcd\ h0 fdD^n+^ab h0 dD(n+Vab\ V d<f> V e f g0 d*fef J Vcdg0 \ d<f> V e f g0 d 7 e / ) ho d G ^ 1 8 H ^ —oVcd^ ~ = 0 • (5-119) % Olab g0 Ofab Comparing these equations with (5.99) - (5.110), one sees that, from (5.99), Eq.(5.118) is the condition that indicates that the tensor field Babcd can be set to 8a'c8bd) up to order n by a change of variables (assuming the inductive hypothesis), provided that rn 2 = ho/go, i.e., that there exists a change of variables such that Babcd = <5a(c ;6d), Having made this choice, referring to (5.103), Eq.(5.118) becomes the nth-order condition that Dab can be set to zero by a change of variables. Similarly, Eq.(5.119) is the nth-order condition (5.99), wi th q = 0, that there exists a change of variables that allows one to set both Babcd = 6a(c5bd) and Hah to —horjab-, (again, provided that m 2 = h0/go which, from (5.52) gives ho = 0). Final ly , wi th these choices, Eq.(5.119) becomes the condition that G can be set to g0 (together with Dab = 0). Thus, if m 2 = ho/go (and consequently h0 = 0), then (5.111) is the unique (up to a change of variables and the rescaling freedom) solution to the integrability conditions (5.39) - (5.42) with the first-order conditions c 2 = 0, (5.75), and (5.74). Taking m 2 = h0/go gives h0 = 0, and Eq.(5.111) becomes Babcd = 8 \ c 8 b d ) , C\d = 0 , Hab = 0, D a b = : 0 , G = g0 , Ab = -g0Jb , (5.120) which gives 8^cd — d(c\d), and 8<j> — g0(p, — JbXb). Lett ing S) —> 0 + JbXb, we get 8^cd = d(cXd) , 6<f> = g0Sl • (5.121) Chapter 5. Conformal Invariance - Spin-two Coupled with Spin-zero 113 This argument can immediately be extended to the case where c 2 7^  0, [74], [43]: Define AB^ah, AHab\ etc., to be the difference between the nth-order parts of two arbitrary solutions to the integrability conditions (5.39) - (5.42) having the same first-order solutions (5.75) and (5.74), i.e., AB^ab = 0, AH$ = 0. By a repetition of the preceding inductive proof, one can set AB^)ab = 0, AH$ = 0, AD^ab = 0, AG ( n ) = 0, AC^%d = AC^ecd + h0gcdAA^e/g0 = 0, V ( c A X i ) _ 1 ) = 0, and V ( cArj ) n-1 ) = 0. Thus the solution (5.95) and (5.96) is unique (up to a change of variables and the rescaling freedom). C h a p t e r 6 N o n l i n e a r , N o n c o v a r i a n t , Sp in-two T h e o r i e s 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 in part upon this belief ([10], [23], [38]). Recently, however, Wald [74] found another possibility, namely, theories which are reducible to the linearized Einstein equations (1-12) could simply have "normal" spin-two gauge invari-ance. However, if the spin-two field interacts with matter, a "quite natural assumption" about the coupling of 7a& to matter seems to eliminate this possibility. 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 in the "natural" manner specified in [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, 7a&, is chosen as the potential. In the thi rd case, however, a traceless symmetric rank-two tensor is used as the potential. We find types of theories which, when interpreted in terms of a metric gab, are invariant under the infinitesimal gauge transformation yab —> ~fab + V( a V c A"| c |b), for any two-form field Kab. We also find classes of theories that are conformally invariant. Thus, one sees that 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, VAA'<J>ABCD = 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 rank-two 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 in addition to, being divergenceless and/or traceless. In Chapter 5, we wrote down the most general second-order Lagrangian in 7a&, £ M G I = a i 7 a 6 a 7 a 6 - ) - a2Ybdadc-jab + a3-ydcdd"/cd + a4-yD^, Eq.(5.2), and the correspond-ing equations of motion, SM^Q = 0, Eq.(5.3). The condition that the quantity , be (identically) divergenceless, gives the linearized Einstein equations. The requirement that SM^Q be traceless, gives the following restrictions on the at- parameters: 2a2 + 4 a 3 = 0, and 2ax + 0 3 + 8a 4 = 0, (5.5). Now, consider the requirement that £M^QB satisfies (iden-tically) b^cdh£iMGi = ®- This condition gives the relation a2 = —2ax. W i t h this choice, the equations of motion that are obtained for 7^, are invariant under the transformation lab —*• lab + d(adcK\c\b)- After an appropriate change of dynamic variable and choice of gauge, the equations of motion can be written n~yab = fj, dAJAB = 0, Eq.(1.25), the equations of motion for a spin-two field. This is the starting point for the alternative formulations of the spin-two equations studied in this chapter. Following, are the La -grangians considered, together wi th the corresponding equations of motion, identities, and symmetries: (o) Firs t , by way of comparison, the usual formulation of the spin-two equations is out-lined. The Lagrangian density is given by the gravitational part of the second-order Einstein-Hilbert action Eq.(1.13), 4? = - h - * D 7 ° 6 + bcbdaddab - \idcddlcd + i 7 n 7 , (6.1) variation of which gives the linearized Einstein equations (1.12) co(2) g(l)ab = = Qcd(a7b)c _ l^ab _ iQagb^ _ ^ab^gd^ _ ^ = Q ? Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 117 where £ ^ a b obeys the linearized contracted Bianchi identities Eq.(1.16), dae§)ab = o. (6.3) Correspondingly, the equations of linearized gravity are invariant under the transforma-tion (cf. E q . (1.17)) lab * lab + d[a\b) • (6.4) (i) As a first alternative to the usual formulation (o), we consider the Lagrangian obtained from , Eq.(5.2), by fixing the proportionality between and a2 to be a2 = — 2a\, = - \ k i l a b ^ a b + \ h l c h d a d c l a b + k 2 l d c d d l c d + fc37°7 , (6-5) where k\, k2, and k3 are arbitrary constants, (where we have set a\ = —\k\, (kx ^ 0), a3 — k2, a4 — k3). The subscript K is simply a label which identifies the quantities associated wi th the Lagrangian (6.5), (l< refers to the constants "&") . Variat ion of C$ gives the equations £ o(2) e[i)ab = _K_ = k i d c d ( a 7 b ) c _ l k l a ^ b + k2da&i + r,ab{k2dcddlcd + 2 ^ Q 7 ) - 0 , (6.6) 0"fab where £ ^ a b satisfies the identity d[cdb£(K)a]b = o, (6.7) (2) and, therefore, CK is invariant under the local infinitesimal transformation lab lab + d{adcK\c\b) , (6.8) where Kab 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 8iab = d(a(b), where ( b , rather than being completely arbitrary, is divergenceless, da(a — 0. In other words, (b = daKab, for arbitrary Kab — —Kba). Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 118 (ii) As a second alternative formulation, we consider the special case of CK where the potential 7 a & appears only in the form 1 iab — \r]abl, 42) = -\l*^ab + \lchdadclah - \idcddlcd + £ 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 Cc' gives the equations j,c(2) 4 1 H = Y^  = DCD{A^ B)C ~ - - b^^-rcd - | ° 7 ) , (6-10) "lab where £ c 1 ^ a b satisfies the two identities d[edbSg)a]b = 0 , Vab£C - 0 , and, consequently, C$ is invariant under the transformations lab ->• 7a6 + d(adCK\c\b) + On a 6 , where Vt is an arbitrary scalar field. (iii) As a final example, we consider formulating the theory (1.26) in terms of the sym-metric traceless potential 7 ^ , and take the Lagrangian 42) = -Hb*iTab + \iTcbdadcil. (6.14) 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 ijb is traceless (r stands 1 T h a t is, combine the demand that the quanti ty (5.3), be traceless, which gives the condi t ion (5.5) on the parameters a,, (i.e, 2 a 2 4- 4 a 3 = 0, and 2a i + a 3 + 8 a 4 = 0), w i t h the demand that d^db^MGhL = ^ ' w h i c l i further specifies that a 2 = — 2a\. T a k i n g ai = — 1 (for preciseness), gives (6.9). Note that this is precisely the Lagrang ian that F ie rz and P a u l i found [33] w i t h the arbi t rary scalar field replaced by 7. (6.11) (6.12) (6.13) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 119 for traceless). (This Lagrangian can be obtained from Cc , by taking the fundamental dynamic variable 7JJ,, to be the traceless combination iao — ^//ai7- Alternatively, C^ can be obtained from C$ by setting the field 7^ in Eq.(1.13) to be traceless.) Varying C?\ subject to the constraint that 7^ remains traceless, gives the equations /;c(2) = 41)a6 - i ^ ! ? = 0 , (6-15) where R^ab and Rj^ are the linearized Ricc i tensor and linearized curvature scalar for 7^, respectively. The quantity £ ^ a b , satisfies the same identities as £ c ^ a b , d[cdb£{T1]a]b = 0, Tjab£^)ab = 0, (6.16) but the equations are invariant only under l l - lab + d{adcKm . (6.17) This completes our listing of the alternative linear formulations of the spin-two equa-tions. Note that al l of these linear theories are fully Poincare invariant since taking Kab to be an arbitrary linear or quadratic function of the coordinates gives the full Poincare group. 6.2 Nonlinear, noncovariant spin-two theories Next, 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 in 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 in a perturbation Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 120 expansion. The condition that the space of variations be involute, gives an integrability condition which we try to solve order by order in the fields to find the infinitesimal gauge invariance. Having determined the first-order gauge invariance of the complete theory, because higher-order calculations are extremely messy, we then try to find a solution, in special cases, to all orders by inspection. We verify the validity of our choice by direct substitution. Following, are the results of the calculations. The details are outlined in the next section. (o) In [74], a consistent nonlinear generalization of the the standard formulation of the spin-two equations, CQ\ CG, is shown to have the infinitesimal gauge symmetry 2 6x«7cd = B\cdda\b + CbcdXb , (6.18) where B^abcd — ^(JJctyb, a n d C^°] = 0. The condition that the space of variations be involute, gives an integrability condition, the first-order solution of which can be wr i t ten 3 S^lab = Cl ( A a 5 c 7 a 6 + 2 7 c ( a ^ ) A C ) , (6.19) where ci is an arbitrary constant. Wri t ing (6.19) in terms of the metric gab = rjab + 2ci~fab, one gets S^hab = \ [Cxagab}{1) , (6.20) where | [C\agab]^ = ci (A a d c 7 ab + 2,yc(adb)\c) is the linearized form of C\agab, the Lie derivative of gab wi th respect to the vector field A a . Note that in the expression C\agab, indices are raised and lowered wi th the metric gab so that within the square brackets indices are raised and lowered by gab. However, once C\agab is linearized, i.e., [C\agab]^\ 2 W h e r e the fo rm of (6.18) differs s l ight ly from the corresponding equation i n [74]. 3 E q . ( 6 . 1 9 ) is equivalent to the corresponding expression i n [74] up to a change of variables and a rescaling A c —+ fcb\b. Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 121 indices are raised and lowered by the flat metric 7]ab. U p to first order, then, the infinites-imal gauge invariance of a consistent nonlinear extension of linearized gravitation can be writ ten &\«lab = d(a\b) + c i ( A c d c 7 o 6 -I- 2 7 c ( a d 6 ) A c ) = \C\agab , = 12(Xcdcgab + 2gcibda)\c). (6.21) Moreover, (6.21) is a complete solution to the integrability condition (6.49), 5\a/yab = \£>\°-gab = V ( a A b ) , (6.22) 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-jab = d(a\b), is also a possibility for the gauge symmetry of the complete Lagrangian CG-Recal l that a theory is said to be generally covariant if "the metric, gab, and quantities derivable from it are the only spacetime quantities that can appear in the equations of physics" [73]. In the context of a theory of a spin-two field, jab, in flat spacetime, the 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]ab, [74]. In [74], Wald has shown that any theory wi th 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 CK , CK, are ensured to be consistent if they have the infinitesimal gauge symmetry feac7e/ = d b d d K a c D a c b d ( e f ) + d b K a c B a c \ e f ) + K a e C a \ e f ) , (6.23) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 122 where D^acbdef = TjadSce8bf, B^acbef = 0, and C^acef = 0. We find that the solution of the zeroth-order part of the fundamental integrability condition (Eq.(6.66) below) gives, for the first-order part of Eq.(6.23), +dbrbd(eKj)a + d [ e V ) a d m K a m + \ d a l e j d m K m a + d m d [ e l m a K j ) a ) + b 1 2 d a i d { e K ] a l f ) + c 7 d m d u l K \ m ] e ) + bnVefda7dcKca , • (6.24) where b, bx2, &n, and c7 are arbitrary constants. Interpreting this in terms of the metric Eq.(6.24) can be written xi1) 9ab - Vab - bjab , (1) (6.25) ^(a^CK\c\b)\ + (b12 ~ §&)3 c 70(a#|C|&) + (c 7 - \b)dcd^K^b) + b11r]abdd-idcKcd , (6.26) where [ V ( a V c / v | c | 6 ) ] ( 1 ) = [gcd V ( a V { d [ K m } ( 1 ) = -b (7abd{ed]alKf)b + d[elabd\a\Kf)b + ^ \ e d s ) K b a + d{edb7af)Kab + dblabd{eKs)a + d ( e l f ) a d m K a m + \ d a l e f d m K m a + d m d ( e l m a K f ) a + ± d a i d { e K W ) + \dmdulK\m]e)) , (6.27) and V a is the derivative operator associated with the metric gab. As noted after Eq.(6.20), wi th in the square brackets indices are raised and lowered with the metric gab. However, once V ( a V c A ' | c | ( ) ) is linearized one gets V r „ V c A i \c\b) (i) so that in Eq.(6.26) indices are raised and lowered by the flat metric rjab. Using the equation (3.68), in the case where k2 = —\ki, (6.26) simplifies to C :7«6 = [ V ( a V C A | c | 6 ) ] 1 + hlVabd^FKai (6.28) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 123 and in the case where k2 ^ — \k\, (6.26) simplifies to V ( a V c / v , c|6) (1) (c - \b) [d(a (dcjKm) + a^d^d'Ka] , (6.29) where a = (\k\ + /C2)/(^1 + 4fc2). In the special case (c =) 612 = c7 — \b, (which is the case when k2 = ~\k\), and b\\ = 0, we are able to show that &Kaclab = V ( a V C A | c | f e ) (6.30) is a solution to al l orders. Of course, if we set b — 0, we get the t r iv ia l solution (!>A'QC7a& — d { a d c K m , Eq.(6.8). (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 Ha°b = nab. In this example, we find that the solution of the zeroth-order part of the fundamental integrability condition can be written til™ d1) h (rbd{edlalKf)b + d{elahd\a\Ks)b + d^^d^Kba + d{edbrf)Kab +dblabd[eKf)a + d ( e l f ) a d m K a m + \ d a l e } d m K m a + dmd{elmaKj)a) + Vef (c6dcdbrcKab - b9dcrbdaKcb - bwdbrbdcKca ~ bnda^dcKac) , hnjef . (6.32) In the special case where c 6 = bg — bw = bn = 0, we find the complete solution hab = V ( a V c / \ | c | 6 ) + flgab , where V a is the derivative operator associated with the metric (6.33) gab = Vab + h~fab , (6.34) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 124 and gabdcgab = 0, i.e., gab is a constant-determinant metric. The condition that 7 a j only appears in the combination •jab — \rjab-f at linear order holds to all orders in this special case. Also, h = 0 gives 8^ab = d(adcK\c\b) + £lr]ab, Eq.(6.13), as a general solution. (iii) In this last example, because 7J(, is traceless, extensions of eft are l imited to theories wi th the infinitesimal gauge invariance <^ac7eT/ = Pg\f(dbddKacDaMgh + dbKacBac\h^ (6.35) ftryj = P 9 \ ^ H g h , (6.36) where P9kef = 6°{Jhf) - (6.37) projects out the trace. We find that the solution of the zeroth-order part of the funda-mental integrability condition can be written C/7ar6 = d(7J{adcddKb)d + jTciadb)ddKdc + dc^bddKdc) , (6.38) WlL = 0 . (6.39) This infinitesimal gauge invariance can be interpreted in terms of a constant-determinant metric in the following way: Let gab be a symmetric field defined in terms of the traceless field 7J6 by 9*b = e ^ T \ a C r,M e^ T ) d b ) , (6.40) where the indices have been raised by the flat space metric r)ab, and / is an arbitrary constant. Since rjah^b = 0, det <7ab is a constant. Then, we can see from the zeroth-order gauge transformation (6.17), that = Jd(adcKm . (6.41) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 125 Also , from the first-order infinitesimal gauge transformation (6.39), we have = fd [jJ(adcddKb)d + ~fTc{adb)ddKdc + d^TabddKdc] + \f [iTc(adh)ddKcd - ^ d ^ K ^ ] . (6.42) Taking d = \ f, this becomes [SKacgab}{1) = f2{iT\adb)ddKdc + ldc7lbddKdc) , (6.43) which is just [SKacgab]{1) = / [ f c ( . V b ) V ^ ] ( 1 \ = M ? ] , (6-44) where £ a = | V c A c a , and denotes the Lie derivative with respect to the vector field c«. Although we can write the solution Eq.(6.39) succinctly in terms of gab, because of the highly nonlinear relationship between gab and 7j6 , it is difficult to find a complete solution, 6j^b, 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 = Jd4xR(gab) , (6.45) where R(gab) is the curvature scalar for the constant-determinant metric gab. The equa-tions of motion for (6.45) are Rab - \R gab = 0 , (6.46) and the gauge invariance of the theory is ^Kacgab = C^gab , = gc(aVb)VdKdc , (6.47) T c T 7 (a7b)c Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 126 where Rab is the R icc i tensor for the metric gao. Despite the fact that we cannot write 8^ i n closed form, we can infer that generalization of CT can lead to classes of consistent theories of a spin-two field with the gauge invariance (6.47) where gao is related to 7j6 by Eq.(6.40). The theory (6.45) was first considered by Einstein in 1919 [27]. It is particularly interesting since any solution of Eq.(6.46) is a solution of Einstein's equations with cos-mological 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. Again , for comparison, we include the example of linearized general relativity, a calculation already carried out in Ref. [74]. Due to the complexity of the remaining examples, only the principal equations are mentioned. The method of solution is sketched in more detail in Chapter 5. (o) The requirement adopted for consistency of the theory given by Eqs.(6.1) - (6.4), is that extensions of CQ\ CG, yield equations of motion 8Gd, which satisfy the generalized divergence identity da (B\CD£GD) = Cbcd£cGd . (6.48) In other words, the action SG has the infinitesimal invariance 8\a^cd — BabcddaXh + CbcdXb', (6.18), where Babcd and Cbcd are functions of yab, Vab, and eabcd- The tensor Cbcd can include derivatives of 7^ . Then, for al l vector fields 0a and tpb, in order that the space of variations be involute, one must exhibit a vector field x°, such that [Vga, V^b] = Vxc, where V$a, etc., are vector fields on the manifold of field configurations defined by Eq.(6.18). Rewri t ing this commutator in terms of the tensor fields Babcd and Cbcd v ia Eq.(6.18) gives Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 127 the integrability condition ^^d^b [B\efdg6h + cgef6a} +^Va, (B\efdjb + c a e f e a ) *l>h (BabefdJb + Caef9a) - (6C <- Vc) = BabcddaXb + CacdXa • (6.49) Before proceeding to solve (6.49) order by order in the unknown fields Babcd, Cbcd, and X a , note that the infinitesimal gauge invariance is unchanged by Babcd —• B'abcd = Baecdfbe, and Cecd -> Cecd = Babcddafeb + Cbcdfeb, where f(°)b = 6ab, (cf. the first two of (5.45)). In addition, the tensor fields transform under a change of variables according to Babef — • B bef = Babcd — , Cbef —> Cbef = Cbef —— , (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 = C7 = 0: B{x)abcd = b0je(c£d)bea + hSabycd + b2iab7]cd + b3j6abricd + b4y6a\crjd)b + b5rjb(cYd) + M a(c7d)6 , Cell = coda-yb(c£d)eab + CXd^cd + C2d(cfd)e + C3rje(cdd)-y + C4dbfb ( cT/ d) e + c5dejr]cd + cedbjbericd . (6.51) The first-order part of the rescaling freedom fbe, fb^e = fifSbe -f / 2 7 6 s , is such that /1 affects 64 and c 3, and f2 affects 66 and c2. The most general change of variables up to second order is lab - Vllab + P2Vabl + P 3 7 ( a ° 7 b ) c + PlVabU + Psllab + PeVabl^Jcd • (6.52) This gives at zeroth order B^abef = Pi + P2^abi]ef-I and cfjj = 0, so that the zeroth-order gauge invariance is S^lab = A) + p 25 cA 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 p2 in the equations for the moment, rather than setting pi = 1, and p2 = 0. A t first order, we find that p 3 affects the term wi th coefficients 65 and b'6 (in particular the combination, (65 + b'6)); p 4 affects 63; p 5 affects 61 and b'4 (in particular, the combination, (61 + 64)); and pe affects b2. A t this point we can take p 4 = — |(pi&3 + P2(W + 46 3 + b'4)), so that b3 — 0, and p6 = -\(p\b2 + p2(Ab2 + 65 + b'6)), so that b2 = 0. Substituting the expressions (6.51) into the zeroth-order part of the integrability condition (6.49), gives [\b0 (de6iced)abedai>b + d(ce]e]ed)abedai;b) + (b\ - b',)d{ced)da^a +l(h-b'6)d{ceadla^d) + \codad{centd)abe^e + \c'2i>edcddee +(Cl + \c'2)^eded(ced) + (c'3 + \cA)^(cdd)de9e + \cA^(caed) + ( c 5 + lc6)VcdiPededJa + lc6VcdiPeaee] - {0e <- Vc) = d{cx$ • (6.54) Proceeding as in Chapter 5 (or as in [74]), one finds that b\ - b\ + c'3 + \cA = 0 , (6.55) 5 ( 6 5 - 6 7 6 ) + c 1 + | c 2 = 0 , (6.56) c 0 = c 4 = c 5 = c 6 = 6 0 = 0 , (6.57) X d0 ) = [c'3^ddja + (Cl + \c'2)rdaed + \c'^addea\ - [0C <- </>c) . (6.58) Taking fi = —c 3 sets c'3 = 0. The change of variables t e r m p 5 can now be used unambigu-ously: takeps = - | p i ( 6 i + 6 4). Then bi-*b\ = | p i (61 — 6 )^ and 6 4 -> b'4 = - | p i ( & i - & 4 ) , i.e., b'4 = —bi. Then (6.55) becomes 61 — 6'4 = 2\ = 0, so that bi = 6'4 = 0. There are st i l l the two freedoms p3, f2, as well as Eq.(6.56). We mention two ways of choosing the remaining parameters: (1) Taking f2 — b5 — b6 gives that b'6 = 65. Then taking p3 = —b5 gives 65 = 0 and b'& = 0. The only two remaining parameters are c\ and c'2 which are related by Eq.(6.56) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 129 d + |c ' 2 = 0. Then B'^abcd = 0, and C'^d = c1(de'ylcd — 2d(c-fd)e). The first-order solution to the zeroth-order integrability condition (6.49) is then wri t ten 4 6{£fab = c1(delab-2d{alb)e)XE, (6.59) xi0) = Cl(eedar-^daBe). (6.60) The R H S of (6.59) is the linearized Christoffel symbol —T^eab for the metric gab = rjab + 2ci7 at so that (6.59) can be written S^^ab = [V^A;,)]^ 1 ' where V Q is the derivative operator associated wi th the metric gab. Wald shows that the unique (up to a change of variables and the rescaling freedom) complete solution to the integrability (6.49), with the first-order part (6.59), is #Aa7ab = V ( a A b ) , (6.61) W i t h Xa = C l ( 0 e V A 0 E - V>eVa0e). (2) Another way of choosing the final two freedoms f2 and p3, is to take f2 — —c2, so that c'2 = 0, and p% = —65, so that 65 = 0. The only two nonvanishing parameters are then related by ci - | 6 ' 6 = 0, so that B'^)abcd = 2c16a^d)b, and C'^d = cxdefcd. This gives the following first-order solution to the integrability condition (6.49), 4-7ai = cx{\adclab + 2lc{adb)\c) = \\C^gabf) , (6.62) x(°)° = c 1 { r d e e a - e e d e r ) = c1[c^eaf) , (6.63) where C^a denotes the Lie derivative with respect to the metric gab — nab + 2ci^ab. This is a complete solution of (6.49), h*fab = \£\«gab = V ( a A 6 ) , X a = CiC^9a . (6.64) 4Note that p2 is still arbitrary. Alternatively, we could have started with the term &oi?7cd6ai in B(°>bcd, so that 6^jed = d{c\d) + b0ir)cdda\a. Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 130 Consider now the interaction of CG wi th matter fields, 4>, (where ^ is a generic symbol for any matter field), so that the complete Lagrangian can be written, CA = CG — OLCM-I where £M is the matter Lagrangian, and a is a constant. Assuming that the matter equations of motion hold, i.e., that ^j^- = 0, according to the results given above, CA w i l l either be generally covariant, or invariant under normal spin-two gauge transformations. In [74], examples of Lagrangians, CA, having normal spin-two gauge invariance are constructed. However, it is also pointed out that, assuming that jab couples directly to the stress-energy tensor Tab at lowest order, i.e., CM^ = J^abTab, and assuming that Tab is defined according to some prescription, say Tab is obtained by variation of CM wi th respect to the flat metric, this possibility seems to be eliminated. This is illustrated using the example of a Klein-Gordon scalar field, S$ = f da<j>da<f). The problems encountered are precisely those discussed in Chapter 3 (in particular Ref. [83]). (i) We l imit the possible generalizations of £ $ (cf. Eqs.(6.5) - (6.8)) to theories with the infinitesimal gauge invariance 6Kac^ef = d b d d K a c P a c b d ( e / ) + dbKacBacb(ef) + KacCac(e}), Eq.(6.23). Then, for al l two-form fields Kab, and L a b , we must exhibit a two-form field Mab, such that [VK^VLJ = VMA» , (6.65) where Vf t a i , etc., are vector fields on the manifold of field configurations defined by Eq.(6.23). Rewri t ing (6.65) in terms of the tensor fields D a b c d e f , B a b c e f , and Cabef, v ia Eq.(6.23) gives the integrability condition dDacbd(ef) n m n g h d P m n 3 h { e f ) b d a U kl S L> kl dbddLacdgdhK„ H [dbddLac(dpKmnBmnpki + KmnCmnki) — {Kac <-> Lac)] Olkl + \dbLac8KabBacb(ef) + LacSKacCac(ef) - (K ac «-> Lac)] = d b d d M a c P a c b d { e f ) + d b M a c B a c b [ e f ) + M a c C a c ( e f ) . (6.66) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 131 In order to solve (6.66) for Dabcdef, Babcef, Cahef, and Mac, note that the arbitrariness in the definition of the tensor fields, (the freedom to rescale Kab — » fabcdKcd)i is mn I ac 5 r)[mn](bd) r\[ac](bd) fu (ef) ^  r> (ef)Ja B[mn]b(ef) - facmnB^b{ef) + 2ddfac mnD [a^ bd\ef), (6.67) C H ( e / ) - * dac\ef)facmn + dbfac mnB^b{ef) + dbddfac mnD^bd\ef) , where fj$mn = 8am8cn. In addition, the tensor fields transform under a change of vari-ables according to D M ( H ) W ) - D^bd\gh)^-, B^\ef) - B^\gh)^-, (6.68) (-/) - ^ ( , ^ -First we consider the zeroth-order part of the integrability condition (6.66), which involves at most first-order quantities. Wri t ing out the most general linear tensor fields £ > ( 1 ) [ a c ] ( M ) ( e / ) = d ^ d ( e V b  ^ f) + rl 2c^ ( /7 6 ) [ c<5 a ]e) + 4 7 [ c ( e * a l f)VM + drf 6*> f ) BWM\ef) = WdHfV c]b + b2db1[c(e8  ^ f) + b3d[cr\eSbf) + b^\e8 chbf) + bsdk^ak6c\e6bf) + bedkyk(e6^f)V^b + b7d{e^e s ) ^ b + hd{eibH*f) + b9Vefd^b + bwriefdK7KVB + b n V e f d ^ b + b12d[c76a\e8bf) + b13d{el8^f)^b, CM[ca\ef) = c^^f) + c2$\eaY ]f) + c3d{ed^ a]f) + cA8 (^ed^dk1})k + c56 {^edf)dk7 a]k + ceriefdkd [a7 c]k + c7d(ed[cy6a]s) , (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 \d2dhdmKmadad{eL})b + \d2dadmKmbdad(eLJ)b + d';d{edmKlmaldadnLf)n + §(&4 + h)daLbUde)dbdmKma + (&x - lb7)dad(edmK]m]f)dnLna - \{b2 + h)daLimde)dadmKmb + | ( c 2 + c5)d(fDdmKlmlaLe)a + lhodmKmad{eLf)a + \b7dedsdmLmadnKna + lhd{edbdmK]m]adJ)Lba + \c3ded}dadmKmbLab + 1-d3(nLa(edadmKf)m + nLa{edf)dmKam) - \(b2 + b4)dbdadmKm[edbLf)a + lb6dmKm{eadnLHf) + i(c2 + c4)adadmKm(eLf)a + \c1{dadmKm{eL})a + d(edmKlmlaLf)a) - \b9VefdbdadmKmcdbLac - $b10riefndmKmc&'Lbc - lc6r,efDdadmKmcLac - (Kab ~ Lab) = d(edaM$f) , (6.70) where d'[ = \(d\ + d4) and d4 = \(d\ — d4). (Note that d", d5, d6, bn, bi2, b13 and c7 do not appear.) To get conditions on the terms appearing in (6.70), we note that the R H S of (6.70), d{edaM$f), can be eliminated in two ways, (1) by taking the trace of (6.70), and (2) by acting on (6.70) with dgdh and antisymmetrizing over g and e, h and / . Addit ional ly , the L H S can be grouped into traceless pieces and pieces that are symmetrized derivatives. This 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, d3 — b6 = ci = b2 = b4 = 0 , b = b3 = c3 = - & 5 = -b7 = -d2 = b8 , C EE C 2 = -C4 , c 5 = -(6 + c), bt = (d- \b) , (6.71) b9 = b10 = ce = 0 . (6.72) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 133 In the special case where Kao and Lao are constant two-forms, the first-order part of the integrability condition (6.66) (which wi l l only involve first-order quantities in this special case) gives the added condition that c = 0. Further simplification can be obtained using the two freedoms (6.67) and (6.68). The most general second-order change of variables is Tab = n3l(aclb)c + n4rjab-fj + n577a& + nerjabj0 -ycd . (6.73) We find that n3 can be used to eliminate d'{; n5, d6; and n^, d5. The most general linear expression for fabcd is If fi = — fi = f[, then f[ affects d2, d4, b3, bb, 67, 6 8 , c 3 and c 5 . f3 affects c/ 6 , bi2, 613 and c 7 . We use the two freedoms in (6.74) to set d = 6 i 3 = 0. This gives the linear gauge invariance (6.24) (or (6.26)), SKaJab = [ V ( a V ^ | c | 6 ) ] ( 1 ) + ( f e 1 2 - | f e ) 5 c 7 % / ^ ) + ( c 7 - \b)dcd(alKm + bnVab&i&Kd , (6.75) which is a solution to the zeroth-order part of the integrability condition (6.66), where Ma°j is given by M $ = b [Kb[adf]dmLj + Kb[adbdmL]mlJ] + \dmKm[jdmLa]m\ - (Lac <- Kac) . (6.76) In the special case when by2 — C7 = | 6 , and 6 n = 0, a solution to the complete integrabil-i ty condition (6.66) having the first-order part given by Eq.(6.24) is 8fcac7ab = V( a V c /f| C|6), Eq.(6.30), where V a is the derivative operator associated with the metric gab = rf^ — b^ab, Eq.(6.25), and Maf is given by Maf = b [Kb[aVf]VmLmb + Kb[aVbVmL[mlf] + i V m / , m [ / V M L Q ] M - {Lac <-* Kac) • (6.77) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 134 This 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\2 and cj. The equation (3.68) gives further restrictions on the constants 6 1 2 , bn, and c7, namely, c = b12 = c7 , (6.78) bn(h + 4fc2) = (c - (\kx + k2) . (6.79) Thus, the four arbitrary parameters b, 6 1 2 , bn, and c7, are reduced to the two arbitrary parameters b and c. In the case where k2 = — \k\, bn remains arbitrary, but c = This gives the infinitesimal symmetry S^lc7ab = [ V ( a V c i f | c | 6 ) ] ( 1 ) + bnr}abddfdcKcd, Eq.(6.28). In the case where k2 ^ —\k\, bn = a(c — \b), where a — (\k\ + k2)/(ki + 4& 2 ) . This gives 6£ljab = [ V ( a V c A | c | 6 ) ] ( 1 ) + (c - i6)[a(a(ac7^|c|6)) + a^d^Ka], Eq.(6.29). When considering additionally the interaction of with matter, direct extensions of the arguments for nonlinearity of the spin-two equations discussed in Chapter 3, seem to eliminate the possibility of a theory with the tr ivial gauge symmetry Sx^-Jab = d(adcK\c\b), (6.8). We also remark that the criteria (2.70) that Fierz and Paul i gave for second-order (2\ Lagrangians in fab, applied to CK , would give the following restriction on the arbitrary ki coefficients k2 h = \h + \k2 + f y 1 , (6.80) fci so that one parameter would st i l l be completely free. (In cases (ii) and (ii i) , c3 = c 4 = 0 so that (2.70) is automatically satisfied.) (ii) For extensions of £ £ \ (cf. Eqs.(6.9) - (6.13)), we are only interested in theories that have the infinitesimal gauge invariance (6.23) and the conformal invariance (6.31). Then, for al l two-form fields Kab and Lab there must exist a two-form field Mab and a scalar field T such that [VKab,VLab] = VMab + VT. (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 dhddMacDacb\e}) + dhMacBac\el) + MacCac{ef) + rHef . (6.82) Furthermore, for al l two-form fields Nab, and scalar fields fi, there must exist a two-form field Qao, and a scalar field A , such that [VNab,VQ] = VQab + VA. (6.83) Substituting the gauge invariances (6.23) and (6.31) into this equation gives a second integrability condition n,dbddNac OH, — V kl — tiki Olkl acbd (ef) # 7 'ki + tt^(dbNacBacbkl + NacCackl) - dbNac6QBac\ef) - Nac6QCac(ef) dbddQacDacb\e}) + dbQacBac\ef) + QacCac(ef) + AHef . (6.84) The arbitrariness in the tensor fields is given by Eq.(6.67) together wi th Hab —• cHab (6.85) where <r'0' = 1. The first-order expression for a, o^ = 017, affects h2. In addition, under a change of variables the tensor fields transform according to Eq.(6.68) together wi th Olab Hab —• H<. cd cd (6.86) We use n 3 , n4, n5 and n& from Eq.(6.73) to set cl", h2, d6 and d5 to zero respectively. Sub-sti tuting the most general expressions for the linearized tensor fields, Eq.(6.69) together wi th H^Kb = hjab + h2ji]ab , (6.87) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 136 into the zeroth-order part of the first integrability condition, Eq.(6.66) with the R H S given by (6.82), gives the zeroth-order equation (6.70), except that the R H S is now diedaM$}) + T%ef. (6.88) Substituting the first-order expressions (6.69) and (6.87) into the zeroth-order part of the second integrability condition (6.84), gives the additional equation {hi -d4 + di- 4:d6)ad(edcNm + (c 4 + c 5 - c 3 - 4 c 7 ) d ( e d c O i V | c | / ) + (65 - 63 - 64 - 4b12)dcnd(eNW) + (h -be-br- ih3)d{efldaNf)a + (bw ~ W - b9 - Un)rjefdandcNca = d(edaQ{$f) + A < % e / . (6.89) Solving the first zeroth-order integrability condition, (6.70) with the R H S given by (6.88), as in the previous example, we get (6.71). Solving (6.89), we get h = hi , & !3 = -6x2 = -\(b+h) , c7 =-\c+\(b+h) . (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, wi th Ma°j given by (6.76), and T(°) = [\b9dhdadmKmcdaLbc + \bwndmKmcdaLac + \c&OdadmKmcLac\ — (Lac <-» Kac) , Qa0} = 2hVLNa} , A ( 0 ) = (610 - \h - b9 - Abn)dattdcNca . (6.91) In the special case bw = b9 — ce = 6 n = 0, a solution to the integrability conditions (6.66) (with R H S (6.82)) and (6.84), wi th the first-order solution (6.32), is given by Eq.(6.33), Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 137 where Maj is given by Eq.(6.77), and T = Qaf A 0 , 2hftNaf , - | A v a n v c j v c o , (6.92) where V a . is the derivative operator associated with the constant-determinant metric (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 CT , (cf. Eqs.(6.14) - (6.17)), is required to have the gauge invariances (6.36). This yields the fundamental integrability conditions (a Jjacbd a r\mnop \ _ _ l l D m n r P s t _ ± D a c M \ ^ ^ ^ J ^ + dDc cbd ef Hi dbddLac [(dpKmnBmnpst + KmnCmnst) — (Kac <-» Lac)\ + Pefgh (dbLac6KabBac\ej) + Lac6KacCa\ej)) - (Kac <-> Lac) = Pefgh(dbddMacDacbdef + dbMacBacbeI + MacCacef + rHef) , (6.93) and, pef pst V d%i T (dbNacBacbst + NacCacst) H, 3D acbd ef kl djki Pef9h(dbNac6uBacbef + NacSQCacef) = Pefgh{dbddQacDacbdef + dbQaeBacbef + QacCacef + AHef) . (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 Pabcd: one projects out the overall trace, and the other arises from the variation with respect to jjb, i.e., fc) = M fc) = pab hab djab 6jJd cd hld (6.95) Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 138 D{l)[ac}{bd) (ef) B(l)[ca)b (ef) = The most general expressions for the linear tensor fields 5 are the same as (6.69), except that they do not include any of the terms involving rjef (since P e / g/ l7/ e/ = 0) or the trace Of Jab, + f / 4 7 T t C ( e » a I ( 6 ^ ) / ) , hdWe}^ + b2db~fT[C(e6a]f) + b3d^Ta\e6bf) + 6 4 5 [ C 7 T | i > l ( / ^ 1 e ) + hdk<yT[ak6c\e6bf) + b 6 d k 7 ^ c f ) V ^ b + h d { e j T [ c f ) ^ b + b8d(e7Tbwf), c i ^ [ c ( e 7 T a ] / ) + c 2 S [ c { e D ^ f ) + c 3 d { e d ^ f ) + c^c{eda]d%k + c ^ c { e d f ) d k l T a ] k , = hxlTab. (6.96) We have the usual arbitrariness in 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[Jcjb^. Also , a change of variables jjb —> T"J6, induces Q(1)[C (ef) pefkiD[ac)(bd){ef) PefkiB^b(ef) PefkiC[ac\ef) PefkiHef pef r)[ac](bd) ®1kl r^ m.Ti J-s ( (ef) pef p>[ac]b (ef) dj ki pef (~<[ac] -T mn^1 (ef) pef rr H r ab-tlef 7T1 djT Imn dlli hL T mn U p to second order, fab = ™l7a6 + m2{lJaClI)c ~ llW^ab) (6.97) (6.98) 5 We have also considered terms involving the antisymmetric Levi-Civitaepsilon tensor, eabcd, 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. 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, bw and c 6 terms do not appear and included on the L H S are the terms \{bs - 64 - h)dadbdmKmcdbLca + i ( 6 5 - h - h)UdmKmadnLna + | ( c 4 - c 5 - c3)DdadmKmcLca] . (6.99) Substituting the expressions (6.96) into the zeroth-order part of the second integrability condition (6.94), simply gives nhid{edaNW) = d{edaQ\%} , (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) wi th Kf] = 2ddmKm[adnLf]n + k[aJ], Q[°b] = 0 , (6.101) where kab is a two-form field such that $( ed afc| a |/) = 0. 6.4 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 spin-two equations. In particular, we considered the three formulations (i) CK , Eqs.(6.5) - (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) CK , could lead to theories with the infinitesimal gauge invariance 8jab = d(adcK\c\b), Eq.(6.8), or 8jab = V ( a V c A ' | c | i ) ) , ( 2) Eq.(6.30); that generalization of (ii) Cc , could result in theories with the symmetries Chapter 6. Nonlinear, Noncovariant, Spin-two Theories 140 hab = d(adcK\c\b) + flrjab, Eq.(6.13), and 6jab = V ( a V c A " | c | & ) + ftgab, Eq.(6.33); and that generalization of (iii) C?\ could lead to theories having the infinitesimal gauge invariances 6jJb — d(adcK\c\b), Eq.(6.17), and 8gab = C^gab, Eq.(6.47). Moreover, there may be additional classes of allowable theories. We have only succeeded in finding general solutions in certain special cases. The requirement that at linear order the dynamical variable jab (lab) c o u p l 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 VAA'4>ABCD = 0, Eq.(2.1). Other formulations, in 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 by considering interactions with 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 Relat ivist ic field theories are based on invariance properties under the proper Lorentz transformations (LT) . 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 equa-tions. The 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 transfor-mation 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 ILT . 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 in 1939 [80], and the associated differential equations (as well as a summary of [80]) are given in Bargmann 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 142 A . l The Lorentz transformations A relativistic system defined on the spacetime (M,gab) is invariant under translations, 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 to another system of coordinates xm is given by x M x " 1 = A V + A" , (A.l) where A M is a constant four vector and the matrix A is such that 77^ = A 7 r f i A p t / r y ^ p . The set of al l L T form a group called the the Poincare group (PG) or the inhomogeneous Lorentz group. For example, performing a second L T on ( A . l ) , one gets = A ^ x " + ,^1 (A.2) where A 2 , i = A 2 A i and A 2 , i = A 2 A i + A 2 . This means that any representation D(A, A) of the P G must satisfy the group multiplication property / J ( A 2 > A 2 ) D ( A 1 , A 1 ) = J D ( A 2 A 1 , A 2 A 1 + A 2 ) . (A.3) The set of L T wi th A M = 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 ° 0 > 1, i.e., no space or time inversion. Since the symmetries of nature are believed to be the proper L T , i.e., the translations, rotations and boosts of the spacetime, in what follows, only the proper transformations are considered (however, the adjective "proper" w i l l , in general, be dropped). Under a L T , a field \P transforms according to = D ( A , A ) t f . (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 with a unit norm and are determined up to an arbitrary phase (as in 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 2 , \2)U{AU Xi) = ± C / ( A 2 A 1 , A 2 A X + A 2 ) . (A.5) The 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 This infinite dimensional representation is a double-valued representation of the P G , 2 and can be writ ten completely in terms of irreducible pieces. Elements of the represen-tation space are called spinors. See also Wald (1984) [73]. To find the invariants which characterize the irreducible representations, it is convenient to consider the ILT . A.2 The Poincare group invariants A n infinitesimal L T , i.e., a transformation that is infinitesimally close to the identity transformation, is given by -> x"1 = + LO\)X" + e" , (A.6) where the six real parameters = u^„] describe rotations, and the four real param-eters eM describe translations. The unitary matrices corresponding to the infinitesimal transformations (A.6) can be written U{1 + w, e) = 1 + \uahJab - ieaPa + 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,$). 2 More correctly, ISL(2,$) is a representation of the (twofold) covering group of the P G . Appendix A. Representations of the Poincare Group 144 where Jab = and pa are called the generators of the transformations .and correspond to the total angular momentum and the four-momentum of the system. 3 In the case where the field ty in (A.4) is a vector field, the generators of rotations and boosts Jab are the four-by-four matrices (J"13)^ = —2i6a[^St3„}. Then (A.7) reduces to (A.6) . The total angular momentum Jab can always be written in terms of the orbital angular momentum Lab and intrinsic angular momentum Sab. The generators satisfy the following commutation relations [Jab, Jed] ~ ^9[b\\cJd\\a\ , (A-8) be Jed] = 2igc[bPa] , [pa,Pb] = 0 . (A.9) (These commutation relations can be brought into a more familiar form, by identify-. . . . . . —* —* ing K% = J and J1 = 2tljkJ3 where K is the generator of boosts, and J , the an-gular momentum three-vector, is the generator of rotations (i,j,k take on the values 1,2, 3.) Then one may rewrite (A.8) as the three commutation relations [J\ J J ] = ie^kJk, [K\K>] = -ie^kJk, \J\K>\ = ie*kKk.) 3The 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^1^ where = J^eit t J and i,j,k range from 1 to 3. In a relativistic system this relation becomes Jjj" = 28xi,ip"^, the angular momentum of a particle at event B with four-momentum p^ about event A, where 8x^ is the four-vector from event A to B. Then J23, J 3 1 , J21 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°l components (Weinberg (1972)[76]).) For a system described by the (conserved) energy-momentum tensor T^v, the Tl° 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 s d3x (Sx^T"0 - 8x"T'i0) 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^ — —8atlpv-\-8aup11 where 8a11 = a^ — a^ and pM = f (PxT110. One can define the intrinsic angular momentum four-vector 5M by Sn = -^tapltlpa J^1 which is invariant under a change of reference point, where m2 — —p^p11- Note that in the rest frame of the particle S = J and 5° = 0. Also, in any frame 5M has only three independent components since S^p11 = 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 Ll"/ and intrinsic 5"" parts as Jf" = + L»v where S»v = ^paSpea^v 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. Appendix A. Representations of the Poincare Group 145 A group "invariant" is a nonlinear function of the generators which commutes with al l the generators, (i.e., a Casimir operator). Defining the Pauli-Lubanski vector Wa by Wa = -\tahcdpbScd , (A.10) one can show that [Jab, Wc] = 2igc[aWb], and [pa, Wb] = 0. Thus one sees that m 2 = - P a P a , W2 = WaWa , ( A . l l ) commute wi th al 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 2 and W2 characterize an irreducible 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, and representations wi th (ii) zero-rest-mass with integral or half-odd-integral spin: (i) The irreducible representations wi th finite mass are characterized by the values of m 2 > 0 and W2, wi th W2 = m2S2 = m2s(s + 1), where S2 is the squared spin angular momentum and the spin s = 0, | , 1, • - - . The space of states that transform according to this representation can be given by the set of completely symmetric spinors (f>A^-A" that are solutions to the differential equation ( • -\-m2)<f>Al-An = 0, where • = dAA'dAA'. (ii) The irreducible representations with zero-rest-mass and integral or half-odd-integral spin are characterized by the values of m2 — 0 and, since W2 = 0, Wa = spa where the helicity (spin) 4 s — 0, ± | , ± 1 , • • • , is the component of the spin S parallel to the momentum. The space of states that transform according to this representation can be given by the set of completely symmetric spinors (j)Ai-An that are solutions to the differential equation C U J A J ^ 1 " " 4 " = 0. See also Chapter 2. 4More accurately, the magnitude of the helicity is called spin. The two possible values of the helicity correspond to the two states of polarization. Appendix A. Representations of the Poincare Group 146 A.3 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 i f e ± iJ01) = \{Jl ± iK{) . (A.12) the commutation relations (A.8) can now be written as the commutation relations for two independent sets of angular momentum matrices [Sl+, S{] = id\S>l ' SL] = ieijkSk. , [SI, St} = 0. (A.13) The representation of angular momentum matrices J in terms of irreducible matrices denoted (here) by is well known: The standard representation is one in which J 2 = J • J and Jz are diagonal. This infinite dimensional matrix can be written as the direct sum of irreducible submatrices, (i.e., matrices that can not be written in 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 Jz and J2 — j(j + l ) . Two commuting sets of angular momentum matrices, Ji and J 2 , can be added to give the set of "total" angular momentum matrices J = J\ + J2 which can be written in terms of irreducible submatrices J = j f a + J a ) © j(h+32-\) © . . . © j(\h-h\) ( f o r details, see any quantum mechanics text, for example, Cohen-Tannoudji, D i u and Laloe (1977) [17] pp. 644-660 and pp. 1003-1042). 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 in 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 | ) - 5 Thus a 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 Projection 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 Aa = 1Aa + 0Aa = 1PabAh + 0PabAb , (A.14) where the pre-subscript («) indicates the spin content of the quantity, and the s P a b are the vector projection operators that project out the spin s part of the field. The projection operators i P % and oPab are given by r P \ = 6 \ - ^ , (A.15) 0 P \ = - I f . (A.16) The operator \ P a 0 is the transverse projection operator, i.e., pa (iPabAb) = 0, and the operator oPab is the longitudinal projection operator. 5 For example, substituting (Ja<3)^„ — -2z'(5a[ /J<$ /3„] into the expressions for 5± (A.12) gives S± • S± \l = s±(s± -f 1) which implies that (s + , .s_) = ( | , | ) . Appendix A. Representations of the Poincare Group 148 For a vector field, the generators of rotations and boosts J are the four-by-four matrices (Jaf3)^ = —2i8a ^ 8^ ^ . The squared Pauli-Lubanski vector W2 is then (W2yb = -2p2 ^8\ - ^ . (A.17) The action of (A.17) on 1PabAb and 0PabAb is given by {W2)\ ( i P 6 c A c ) = m2s(s + l){1PabAb) = 2m2{1PabAb) , (A.18) (VK 2 )% ( i ^ V H = 0 , (A-19) which gives spin s = 1 for the transverse part of Aa, iPbcAc, and spin s = 0 for the longitudinal part, oPbcAc A.4.2 Rank-two tensor field A rank two tensor hab 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 hab = 2^a6 + \hab + Qhab = ( 2 ^ 0 6 ^ + iPab^ + oPab^hab 1 (A.20) where the projection operators sPabcd project out the spin-s part of a (o) tensor. For a (o) tensor, the generators Jab are given by (Jab)cdef = -2i(r)cya8b}e8j -f r)d[a6b]f8ce), so that the squared Paul i-Lubanski vector is (W2)abCd = -±p28aC8bd + 2p2(VabVCd ~ Sad8bC) + 8p(oJ>%d> - 2(71abpCpd + V^PaPb) • (A.21) Note that the action of (W2)abcd on a tensor field h^bT that is transverse and traceless (pahTJ = 0, nabhTJ = 0) is {W2)abcdhTJ = m2s(s + l)hTabT = -6p2hTJ , (A.22) Appendix A. Representations of the Poincare Group 149 so that the spin of a transverse and traceless (o) tensor is s = 2. The operator which projects out the spin-two part of a (o) tensor is (see [34], [49]) 2PabCd = 6jC6bQ ~ 1 ^ + j>(jlPaPbPCpd + VCdPaPb) ~ ^J{a{CPb)Pd) • (A.23) Then (W2)abcd (2-Pc/^ e/) — 6 m 2 (2Pabcdhcd) and the rank-two tensor 2Pabcdhcd is trans-verse and traceless. The projection operators for the spin-one part are given by i P a b c d = 6a[c8bd] - paPbPcpd + fA*{cP*)Vd) , (A-24) which can be writ ten in terms of three orthogonal parts, Pi cd p cd . p cd . p cd ( \ t ) r \ - ab — \"\ab + \*2ab. + \"Zab i \A.ZO) where iPiabcd = ^ ( ^ ^ P ^ - w y ) , (A.26) lP2ab°d = SaV ~ ^ P b ] p V , (A.27) iPsabcd = $6[alcpb]p4 , (A.28) (the last two operators are zero on a symmetric tensor field). The spin-zero projection operator oPabcd is given by 0PabCd = § (VabVCd + ^PaPbpY ~ ^ ( w V + PaPbV*)) , (A.29) which can be writ ten in terms of two orthogonal parts, dPabCd = oPlabCd + O^afe^ , (A.30) where iPiabcd = ltd ((1 + a)Vab - (1 + ± a ) ± P a P b ) , (A.31) iP2abcd = | ( - a « a 6 r / c d + f,PaPbPcpd - jt(riabPcpd ~ 4avcdPaPb)) , (A.32) where a is an arbitrary real number. Appendix B A Brief Introduction to Spinors Physical quantities are most commonly described by tensor fields and satisfy tensor equations. In a system that is locally like 9ft4 and which has a Lorentzian metr ic 1 (for example, the spacetime in which we live), there is another mathematical formalism avail-able, namely the two-spinor calculus. Spinor calculus encompasses tensor calculus and, i n many ways, is simpler. The concept of a spinor was briefly introduced in 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 in the main text. The approach here follows roughly that of Penrose and Rindler (1984) [57] (to whom the reader should refer for further discussion, details and proofs) and is not intended to be complete. 2 B.l 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. 3 We shall see that associated wi th a spin-vector KA (the simplest type of spinor) is a unique future 1When 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. 2There 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. 3Although 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. 150 Appendix B. A Brief Introduction to Spinors 151 pointing Minkowski null vector 4 ka, which can be interpreted as a "flagpole", together wi th a two-dimensional half-plane orthogonal to ka, which can be interpreted as the "flag plane". The flagpole together wi th the flag plane give the geometric picture of a spin-vector as a "null flag" in a Minkowski vector space. However, under a rotation by 2ir, while the nul l flag rotates into itself, the spin-vector goes to its negative. Thus every Minkowski nul l flag corresponds to two spin-vectors. First , we briefly discuss Minkowski vector spaces. Let V be a Minkowski vector space, i.e., a four dimensional inner product space wi th a metric of signature (+ ). For example, in 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. 5 On a curved spacetime manifold, Minkowski vector spaces are the tangent spaces of points. Denote an element of V, a Minkowski vector, by va. In a basis {e^} = {e 0 a , e i a , e 2 °, esa], va is written in terms of its components = u 1 , u 2 , v3) as va = u M e M ° — v°e0a + v1e1a + v2e2a + v3e3a, where ' V is a label indicating which component of va or which of the four basis vectors {e^ a } , and V is the (abstract) vector index. In the standard representation, each point in 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). The components of va are then the coordinates (t,x,y,z). This is the conventional way of representing Minkowski vectors in terms of Minkowski coordinates. The 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 4The null vectors, i.e., vectors na such that nana = 0, are taken to be future pointing by convention. 5 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£4) with a Lorentzian metric. By spacetime manifold is meant a set of points with the local structure of a Minkowski spacetime. Appendix B. A Brief Introduction to Spinors 152 span V) in terms of complex numbers. Consider the future pointing null vectors originat-ing from an arbitrary origin. This nul l cone can be represented in a coordinate system (t,x,y,z) by any time slice, say t — 1, which is the hypersurface x2 + y2 + z2 = 1 (see Figure 1). This 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 wi th the point / = oo. B y defining the pair of complex numbers (/c 0 ,^ 1 ) (not both zero) by / = £ . (B-2) the point / = oo wi l l correspond to some finite label, say (1,0). This pair ( K ° , K X ) , coordinatizing the future pointing nul l vectors of V, can be regarded as a coordinate representation of the spin-vector KA. The spin-vector, or one-index spinor, KA is written in terms of these components « r , T = 0,1, as ( « r ) = (K°, re1)", where "r" is a coordinate index indicating which component of KA, and UA" is the (abstract) spinor index. From this geometric construction, it is clear that associated with any spin-vector KA is a future pointing nul l direction. Note, however, that since the time slice taken as a representative slice of the nul l cone was arbitrary, this association does not distinguish between KA and XKA, where A is a non-zero complex number. A useful way of writ ing the relationship between the coordinates of Minkowski vectors and the components of spin-vectors can be obtained by inverting the stereographic map-ping ( B . l ) . After a rescaling by A (so that now only nA and eL9KA, for 0 real, correspond to the same nul l vector) one finds y = 77= ( K V - a1*0') , z = -j= («°K° ' - K1*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 154 where ( « r ' ) = K1') denotes the complex conjugate of nA, KA. Rearranging (B.3), one gets 72 t + z x + ly ^ x — iy t — z j I „ . 0 —l' \ AC /C K K K K K K t —0' — 1' (B.4) Equations (B.3) and (B.4) can be written in terms of the quantities oaAA\ called the Infeld-van der Waerden symbols, as follows: k* = a^rr'Ktnr , k — K K ^ (B.5) where (k11) = (t,x,y,z). The components <7Mrr' of uaAA' have been chosen here to be proportional to the Paul i matrices 6 rr'\ 1 / 1 0 (cr r r , J , (<7i ) = o i y V2 0 1 1 0 \/2 0 i - i 0 = (—c^rT') ? (<7"3RR ) V2 1 0 0 - 1 = (cr 3 r r0 • (B.6) In wri t ing the association (B.5), often the Infeld-van der Waerden symbols aaAA' are dropped and one can write symbolical ly 7 k 4 > Ki K (B.7) A n additional geometric structure is needed to represent the phase of KA. However, first a few more concepts are needed. 6Different choices for the Minkowski basis vectors eM a (a null tetrad, for example), could lead to different choices for <xM r r . 7Indeed, Penrose and Rindler (1984) [57] identify the indices a = AA', b = BB', • • • , etc., so that they write ka — KA~KA , for example. Appendix B. A Brief Introduction to Spinors 155 A spin transformation AAB of the spin vector KA, KA —* AABKB is a unimodular two-dimensional complex matrix. This complex linear transformation of KA induces a real linear transformation of (ft*1) = (t,x,y,z) which preserves the line element f)aikakh, namely, (cf. (B.4)): KA~KA' -+ K ^ 4 ' = AABKB (AABKB) , = Aab(KBKB')AA'B,, = AAB(kaaABB')AA'B,, (B.8) i.e., t + z x + iy x — iy t — z j i + z x + iy y ' A ^ x — iy i — z t + z x A- iy A* , (B.9) y x — iy t — z where A* is the conjugate transpose of A . In fact, a spin transformation induces a unique proper L T A % = AAB AA B> on any Minkowski vector (not just the null ones). 8 Conversely, to each proper L T there corresponds two spin transformations, namely AAB and — AAB (cf. Eq. (B.8)) , i.e, the group of spin transformations is a double valued representation of the proper Lorentz group. This correspondence can be used locally in curved spacetimes. However, there are certain topological restrictions on the curved spacetime manifold AA, (e.g., time and space orientability) in order for spinors to exist on AA. (see Penrose and Rindler [57], and Wald [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 with the same indices. The connection between general tensors in a Minkowski vector space and spinors is made using the Infeld-van der Waerden symbols. For example a real type ( |) tensor 8Since det AAB - ^eABe AACABD = 1, £AB = AcAADbZCD- Multiplying this by its complex —A' conjugate using (B.14) and identifying A a 6 A ^ A B ' gives gab = AcaAdt,gcd. Appendix B. A Brief Introduction to Spinors 156 tabcd is related to the Hermit ian type (|,|,) spinor T A A ' B B ' C C D D ' by _ AA' BB' c d .ab AA'BB' (TX \C\\ <3a <?b & CC& DD't cd — T CCDD' , (O.IU) iab T-AA'BB' (r)-\-[\ t cd ^ T CCDD' • l-D-J--1-J For a vector ta, this gives aaAA'ta = TAA'. In the special case where ta is a null vector, TAA' can always be written TAA' = TATA> (cf. (B.7)). The set of spin-vectors at a point form a two-dimensional vector space called spin-space. A basis for spin-space, denoted (er"4) = (oA,iA), T = 0,1 is such that OAIA = 1, and is called a spin-frame. Note that since primed and unprimed indices are distinguishable, their relative order is not important, i.e., TAA> = TA'A. The antisymmetric spinor epsilon CAB can be defined by eAB = 6AB = e[AB] , tABtAG = tBc = SBC , (era) = ( _?i J j • (R12) Note that from Eq.(B.4) (with KAA> = KAKA'), one gets 9abkakb = kaka = 2 det KAA' = 2(K00V1' - / c 0 1 ' r 1 0 ' ) , = ^AB~tA'B'KAA'KB& , (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: KAeAB = KB , eABKB = KA . (B.15) (Note that KATA = — K,ATA and KAK,A = 0.) One can also check that Cabcd KAA'BB'CC'DD' = i{^ACKBD^A'D'€-B'C' ~ tADtBCtA'C'ZB'D') • (B.16) Appendix B. A Brief Introduction to Spinors 157 The two-dimensionality of spin-space gives the important identity CA[BCCD] = 0 . (B . l7 ) Contracting (B . l7 ) wi th an antisymmetric spinor TAB = T^AB^ gives the following very useful relation TAB = T[AB] = 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 FAA'BB' associated wi th a real rank-two antisymmetric tensor Fab = F[aby The strategy is to first decompose FAA'BB' = —FBB'AA' into symmetric and antisymmetric pieces Fab <->• FABA'B' — FAB(A'B') + FAB[A'B'] , (B.19) = —Fba <-> —FBAB'A1 = —FBA(B'A>) + FBA[A'B'\ • (B.20) Equating terms antisymmetric and symmetric in the indices AB from (B.19) and (B.20) and using (B.18), one finds FAB(A'B') = —FBA(B'A') — F[AB](A'B') = \(-ABFCC'{A>B>) , FAB[A'B'] — FBA[A'B'] — F(AB)[A'B<] = \(-A<B>F(AB)C'C • (B.21) Substituting (B.21) into (B.19) with §AB — \F(AB)C'C' one gets that the spinor equivalent to a real antisymmetric rank-two tensor is Fab = F[ab] *-» FAA'BB' = 4>ABtA'B' + 4>A'B'eAB • (B.22) Using (B.16) and (B.22) one obtains a very simple spinor operation for the process of tensor dualization (which holds for Fab not necessarily real) * F a b = ^abcdFcd *-+ *FABA'B> = iFABB'A< = ~iFBAA'B> • (B.23) Appendix B. A Brief Introduction to Spinors 158 Returning now to the geometric argument, recapping, associated wi th any non-zero spin-vector nA is a unique future-pointing null vector ka defined by (B.7), ka <-> KAKA', called the flagpole. However, this correspondence does not distinguish between spin-vectors differing only by phase. In order to obtain a more precise geometric picture one tries to construct from KA another Minkowski tensor. Referring to ( B . l l ) , what is required is a combination of K ^ ' S which has spinor indices appearing in the paired combination AA', B B ' , • • •, etc., and which is Hermitian. We have already looked into the simplest such combination, namely KAKA>. One of the next simplest possibilities is f» «_> KAKBeA'B' + T ^ - A ' - B ' ^ { B M ) i.e., KAKB mult ipl ied by tA'B', (to give the correct index combination for a tensor), added to its complex conjugate. Note that both ka defined by (B.7) and pab defined by (B.24) are invariant under KA —» — KA. TO picture pah, introduce the spin-vector TA SO that the pair (KA,TA) constitute a spin-frame, i.e., K\TA = 1. This defines rA up to an additive complex multiple of KA. The relation KATA = 1 can be written equivalently as eAB = KATB - KBTA. Substituting this expression for eAB into the R H S of (B.24) one finds that pab can be written as pab = katb _ rkb ^ ^g_25j where ka <-> KAKA' and la is defined by la KATA' + TAKA' . (B.26) Since rA is not uniquely determined, pab is invariant under the transformation la -> la + aka , (B.27) for a real. The set of / a 's (B.27) describes a two-dimensional half-plane, called the flag plane, perpendicular to the light cone along ka (see Figure 2). The flag plane together Appendix B. A Brief Introduction to Spinors 159 Figure B .2 : The geometric concept of a spin-vector as a nul l flag, (a) The flagpole is defined by the nul l vector ka, and the flag plane is defined by the half-plane generated by la + ctka. (b) The hypersurface t = 1. Under the transformation KA — » el9KA wi th 0 = T T /2 , la -> - l a . W i t h 9 = TT, la - • / a but K A - » • Every nul l flag defines two spin-vectors, KA and —/c"4. Appendix B. A Brief Introduction to Spinors 160 wi th 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 KA — > el9 KA (and therefore TA — > e~T9TA) wi th 9 — 7r/2, that (referring to (B.26)) la — > — la, and with 9 — ir, that la —> la, i.e., a rotation by 2-7T rotates the nul l flag back to itself, but nA goes to — KA. 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 KA(X) is the assignment of a spin-vector to each point of the spacetime manifold. The field KA (dropping from now on the explicit x dependence) thus defines a null flag at each point of Ad (which is a structure in the tangent space (the Minkowski vector space) at each point). A spinor covariant derivative operator 9 \7 AA1 *-> V a is a map which acts on spinor fields of type (™ ^ ) and sends them to spinor (fields) of type ( j ^ /^i), i.e., VAA1 '• {KB} —> {KBAA1}- This derivative operator is real, linear, satisfies the Leibnitz (product) rule and is such that VAA'f = SAA'J <-* daf, where 8AA' <->• da is the ordinary (flat) derivative operator and / is an arbitrary scalar field. In addition, we take VAA' to be torsian free, i.e., (V^.VBB/ - VBB'VAA')! = 0 <-> 2d[adb]f = 0. B.2 The curvature spinors The Riemann curvature tensor 1 0 Rabcd is defined from the noncommutativity of two derivative operators acting on an arbitrary one form field wd, 2V[AVb]Wc = Rabcdwd. Similarly, one can define the spinor 1 1 XAA'BB>CD from the noncommutativity of two spinor 9That the spinor derivative operator has two indices is related to the correspondence between spinors and actual directions in spacetime. 10Note that Rabcd 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]. Appendix B. A Brief Introduction to Spinors 161 derivative operators acting on an arbitrary spinor field u>c (yAA'^BB1 — VBB,XJAA>)UC = XAA'BB'CDUD • (B.28) B y considering action of (VA A'VB B 1 — VB B'VA A 1 ) ° n a r e a l spinor u>cc, one finds that the spinor equivalent to the Riemann tensor can be written Rab/ *-> RAA'BB'CC'DD = XAA'BB'CD£CD + XAA'BB'C'D ecD • (B.29) The symmetries of Rabcd allow one to further simplify this expression. Alternatively, one can find the spinor equivalent exploiting the symmetries R(ab)cd = Rab(cd) = R[abc}d — 0 directly. Following precisely the same procedure used to derive Eq.(B.22) one finds that R(ab)cd = Rab{cd) - 0 i m p l y 1 2 Rab/ RAA'BB'CC'DD = XABCDCA'B'£C'D + $ABC'D ~(-A'B'(-CD + CC. , (B.30) where the curvature spinors XABCD and QABC'D' are defined by XABCD = \R(AB)AA {CD)CC = X^AB)(CD) , $ABC'D' = \R(AB)A'A C°' (CD') = $(AB)(C"D') • (B . 3 1 ) In addition, since R a b c d = R c d a b (which follows from R(ab)cd = Rab(cd) = R[abd\d — 0) ^ABCD' is real and XABCD = XCDAB (which implies XA(CD)A — 0). Comparison wi th (B.29) shows that XAA'BB'CD = XABCD^A'B' + $A'B'CD£AB SO that (VAA'^BB1 - VBB'V'AA')WC = (XABCD^A'B> + $A'B'CD£AB)WD • (B.32) Also , making use of the equivalence R[abc]d = 0 <-> *Rabbc = 0, where *Rabcd — \eabei Refcd, (cf. footnote 9 on page 22), R[abc]d = 0 can be shown to be equivalent to the condition that A = A , where A = \ X A B A B • (B.33) 12Notation of [57]. Appendix B. A Brief Introduction to Spinors 162 Wri t ing XABCD in terms of symmetric spinors, XABCD = ^ABCD + A(6AC^BD + ^AD^BC) , (B.34) the spinor equivalent to a real tensor Rabcd having the symmetries R(ab)cd = Rab(cd) — R[abc]d = 0 can be written r> d ^ D DD' _ i T r D- - D' , >R £ > ' - £ > i t a & c <-» KAA'BB'CC — *ABC tA'B'tC + ® ABC ^A'B'^C + A(e^cejBD + tADtBc)lA'B'^cD' + ex. . (B.35) The spinor $ABCD' is called the Ricc i spinor and the spinor tyABCD = X(ABCD) is called the Wey l (conformal) spinor. Contracting (B.35) over b <-> BB' and d <-> D D ' one finds that Rac PylA'CC" = 6Aej4c£yl'C' — 2$^C7^'C • (B.36) The traceless part of Rabcd-, Cabcd-, the Weyl curvature tensor, is Cabcd <-> CAA'BB'CC'DD' = ^ABCD^A'B'^CD' + ^A'S'C'd'eASeCD • (B.37) Also, from (B.36), one finds that R <-* 24A , (B.38) Rab ~ \Rgab *-* -2<S>ABA>B> , (B.39) Gab <-> - 6 A e A S e ^ B / - 2$ABA'B' • (B.40) The Einstein equations in vacuum are then tyABA'B1 — 0 (B.41) A = 0 Recall ing that V[aRbc]de = 0 <->• V a * i ? a 6 c d = 0, (cf. footnote 9 on page 22), one finds that Rab = 0 <-» < the Bianchi identity can be written V [ a i ? 6 c ] d e = 0 VV*ABCB = V B ^ V C O A ' B ' - 2eB{cVD)B,A . (B.42) Appendix B. A Brief Introduction to Spinors 163 In vacuum this becomes VAA'*ABCD = 0 , (B.43) which are the zero-rest-mass equations for a spin-two particle. However, (B.43) is not conformally invariant: The spin-^ equations, ^AlA'1<pA1...An — 0> a r e conformally invariant wi th <t>Ai....An a symmetric spinor of weight w — —1, cf. Eq.(2.11). The Weyl spinor tyABCD, on the other hand, has conformal weight w = 0, cf. Eq.(C.14). Appendix C Conformal Transformations Consider a manifold AA wi th a Lorentzian metric gao. A conformal transformation is a rescaling of the metric 9ab -> dab = tt29ab ( C . l ) 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 wi th respect to both gab and gao, manifolds that are related by a conformal transformation have the same causal structure. In this appendix, we give a few of the relations used in the main text. C . l Tensor formulation From ( C . l ) we have that the inverse metric gab of gaa is related to the inverse metric gab of gab by 9ab -» 9ab = , (C.2) so that gabgbc = gabgbc = $ac- One (uniquely) defines the two derivative operators 1 V a and V a associated wi th the metrics gab and gab respectively by V a 5 6 c = 0 , Vagbc = 0 . (C.3) 1 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. 164 Appendix C. Conformal Transformations 165 To relate the two derivative operators V a and V a , note that the action of ( V a — V a ) on an arbitrary vector field th defines the (^)-type tensor 2 Qab° by (v B - V a ) tb = Qjtc . (CA) B y evaluating Vagab — 0 using (C.4) and (C.3), Qabc is found to be given by Qabc = \gcdyagbd + ^bdad - Vdgab). Substituting in ( C . l ) and (C.2) then gives Qahc = 2 V ( a In nSb)c - gabgcdVd In ft . (C.5) Equations ( C . l ) - (C.5) can be used to (arduously) calculate the transformation prop-erties of other tensors and equations. For example, the curvature tensor Rabcd for the metric gab can be writ ten in terms of "untilded" quantities by substituting ( C . l ) - (C.5) into V[ a Vj,]to c (= \Ra.bcdwd). In particular, one can show that the Weyl tensor Cabcd is conformally invariant, i.e., Cabcd = Cabcd. Similar equations can be straightforwardly determined in the spinor formalism which, in particular, gives a much simpler demon-stration of the conformal invariance of the spin-^ equations. C.2 Spinor formulation In terms of spinors, associated with the conformal transformations ( C . l ) and (C.2), (since gab CAB^A'B'-, Eq.(B.14)) are the spinor epsilon transformations tAB 1 A B = VtAB , e A B - e A B = ft~VB , (C.6) so that 8AB = SAB (also ZA'B' = ft^U'B', ^  B = ft-1^'5'). The derivative operators 'VAA' <-» V a and VAA' <-> V a defined by VAA'^BC = 0 and VAA'CBC — 0 are related by (VAA> ~ VAA>)KC = ®AA'BCKB (C.7) 2Notation of [57]. Qabc corresponds to Ccab of [73]. Appendix C. Conformal Transformations 166 for some (io';) spinor field ®AA'BC• The action on the field rB can be found from ( V ^ ' — VAA>)KBTB = 0 to be (VAA< - ^ AA*)TB = -®AA>BCTC- (By evaluating [V AA> - ^ AA')TBB' where TBB' <-> tb for some arbitrary vector field tb , one finds that Qaoc <-> QAA'CB^C'B' + QAA'C'B€CB •) The torsian-free condition can be used to show that @AA'BC = i^-AA'^B0 + TA'B^A0 , (C-8) for real TLAA' and TA'B [57]. Evaluating VAA'^BC using (C.6) - (G.8) gives 1AA> = n-lvAA<n ( c . 9 ) and ILAA' = 0. Then (C.7) becomes ( V^i - VAA,)KC = Ta,b6acKB , (CIO) and also C^AA' - VAA')TB = -TA'BTA • (C.ll) These expressions agree with (C.5) on tensors Taa' ta. 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