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On the feasibility of incorporating the mass concept into conformally invariant action principles Drew, Mark Samuel 1976

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ON THE FEASIBILITY OF INCORPORATING THE MASS CONCEPT INTO CONFORMALLY INVARIANT ACTION PRINCIPLES by MARK SAMUEL DREW B.A.Sc, University of Toronto, 1970 M.Sc,  University of Toronto, 1971  A THESIS SUBMITTED III PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the Department of Physics  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1975  In p r e s e n t i n g an the  thesis  in partial  f u l f i l m e n t of the  advanced degree a t the  University  of B r i t i s h Columbia, I agree  Library  this  s h a l l make i t f r e e l y  a v a i l a b l e f o r r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r extensive for by  s c h o l a r l y p u r p o s e s may his representatives.  be  g r a n t e d by  thesis for financial  written  permission.  gain  s h a l l not  D e p a r t m e n t o f Physics The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  October 1 ,  1975  the  Head o f my  Columbia  be  that  thesis  Department  copying or  for  study.  copying of t h i s  I t i s understood that  of t h i s  requirements  or  publication  allowed without  my  ii Abstract Following an examination of the properties of the conformal group i n 4-space, a review i s made of the procedure by which conformally covariant massless f i e l d equations are w r i t t e n i n manifestly covariant form.  By  w r i t i n g the Minkowski coordinates i n terms of coordinates on the n u l l hyperquadric of a 6-dimensional f l a t space with two timelike d i r e c t i o n s , the  a c t i o n of the group i s l i n e a r i z e d and f i e l d equations are w r i t t e n i n  r o t a t i o n a l l y covariant form i n 6-dimensional space.  I t i s then shown that  extending the 6-coordinates o f f the n u l l surface generalizes space to a 5-dimensional space.  Minkowski  Such a g e n e r a l i z a t i o n necessitates  employing a method o f descent to 4-dimensional space from s i x dimensions which d i f f e r s from the usual procedure, and allows one to encompass massive f i e l d theories i n the manifest formalism.  I t i s demonstrated that these  massive f i e l d s can be understood as manifestations i n Minkowski space of massless f i e l d s i n 5-dimsnsional space.  For the case of spinors, the f i e l d  equation can accomodate p r e c i s e l y two species of p a r t i c l e having two d i f f e r e n t masses.  An a c t i o n p r i n c i p l e i s developed i n the 6-space, and a  method of f i e l d quantization i s devised. s p e c i a l cases of spin-0,  spin-i,  As examples of the method, the  and spin-1 f i e l d s are examined i n d e t a i l ,  and minimal coupling of the spinor f i e l d equation i s c a r r i e d out.  The  formalism presented i n t h i s i n v e s t i g a t i o n provides a means by which one can apprehend a massive compensating f i e l d w i t h i n the confines of a gauge i n v a r i a n t theory.  The interactions which are obtained i n Minkowski  space  include not only the usual couplings with massive vector or pseudovector f i e l d s , but as w e l l the pseudoscalar coupling occurs automatically w i t h i n t h i s gauge i n v a r i a n t formulation.  iii  Contents  Abstract  i i  L i s t of Figures  vi  Acknowledgments  v i i  Introduction  1  PART I . CONFORMALLY COVARIANT FIELDS IN MINKOWSKI SPACE 1. The Conformal Group i n Minkowski Space 1-1 . D e f i n i t i o n of conformal transformations  10  1 -2. I n f i n i t e s i m a l conformal transformations  13  1-3» F i n i t e conformal transformations  14  1-4. Isomorphism with rotations i n 6-dimensional space  16  2 . Conformally Covariant F i e l d s 2 - 1 • D e f i n i t i o n of generators  19  2-2.  Representations of d i f f e r e n t i a l operators  20  2-3.  Algebra of generators  21  2-4.  Induced representations  22  2-5.  F i e l d variations  24  2 - 6 . D i l a t i o n invariance  26  2- 7 . Two-particle momentum space representation; experimental consequences  32  3. Conformally Covariant F i e l d Equations 3- 1. Massless Klein-Gordon equation  35  3 - 2 . The Weyl equation  37  3-3*  38  The Maxwell equations  3-4. Weak f i e l d approximation of E i n s t e i n equations 3-5 • Independence of the generators  ,  4 . Attempts at Including Mass 4-1.  Symmetry breaking  4 - 2 . Conformally  43  ••  44  covariant mass  4 - 3 , Interpretation of d i l a t i o n s  48  4- 4 . Other approaches  49  PART I I . FIELDS IN HIGHER-DIMENSIONAL SPACES 5 . Generalization of Minkowskian Spacetime 5 - 1 • Introduction  50  5-2. Descent from 6-dimensional space to Minkowskian spacetime  ..  50  5-3« Necessity of the choice of descent operator  59  5-4. Eigenfunctions of X  f o r the case L=0  61  5-5. Eigenfunctions of Ji f o r the case L#)  63  5-6. The p o s s i b i l i t y of including mass: eigenfunctions of -^(5)  •  65  5-7. Dimension of the conformal representation  67  5-8. Vector f i e l d s i n 6-dimensional space  68  5- 9 . Spinor f i e l d s i n 6-dimensional space  73  6. Two-Dimensional Conformal Group 6- 1. Four-dimensional analogue of manifestly covariant formalism  80  6-2. Four-rotation covariant spinor equation  81  6-3. Infinite-dimensional representation  84  6-4. Solutions of the spinor equation  87  6-5. Spinor solutions i n "Minkowski" space  89  6-6. Introduction of the mass concept f o r eigenfunctions of  92  PART I H . ACTION PRINCIPLES Synopsis of Part I I I  100  ?• R o t a t i o n a l l y Invariant Action P r i n c i p l e s 7-1 • Conservation laws on hyperboloids  10**  7-2. Alternative Euler-Lagrange equations  108  7-3• Angular momentum tensor i n 6-dimensional space  109  7-4. Modified Schwinger a c t i o n p r i n c i p l e  111  7- 5* Canonical (anti-)commutation r e l a t i o n s  114  8. Special Cases 8-1 . Scalar f i e l d  118  8-2. Green's function f o r scalar f i e l d equation  ••  128  8-3. Spinor f i e l d  132  8-4. Green's f u n c t i o n f o r spinor f i e l d equation  143  8-5« Minimal coupling; gauge invariance with massive vector boson  146  9* Concluding Remarks  154  Bibliography Literature cited  157  A d d i t i o n a l l i t e r a t u r e consulted  163  Appendix 1  170  Appendix 2  172  Appendix 3  •  1?4  Appendix 4 Appendix 5  176 •  181  vi L i s t of Figures  Figure 1. Example of an angle preserving conformal mapping.  12  Figure 2 . Examples of motions induced on the n u l l surface L=0 i n a dimensional space as a r e s u l t of conformal transformations 1-dimensional space.  3-  in 82  Acknowledgments  The author i s indebted to Professor F.A. and c r i t i c i s m .  Kaempffer f o r advice  The f i n a n c i a l assistance of the National Research  Council of Canada i n the form of a Postgraduate Scholarship i s g r a t e f u l l y acknowledged.  1 Introduction Within a few years a f t e r i t was established that Maxwell's equations are  covariant under the Poincare group of transformations, Cunningham  (1909) and Bateman (1909) discovered that the symmetry group f o r the vacuum f i e l d equations could be extended to include the f u l l conformal group of transformations on Minkowski space.  A f t e r another twelve years had  passed, Bessel-Hagen (1921) derived the a d d i t i o n a l f i v e conservation laws that hold whenever Poincare covariant f i e l d equations are also covariant under the wider conformal group.  However, i t was not u n t i l a  pioneering paper by Dirac (1936) that there appeared a modern r e v i v a l of i n t e r e s t i n the conformal group among p h y s i c i s t s .  In that work, Dirac  showed how to write conformally covariant f i e l d equations i n manifestly covariant form by e x p l o i t i n g , i n e f f e c t , the isomorphism between the group of conformal transformations i n 4-dimensional f l a t spacetime and the 15-parameter group of rotations i n a 6-dimensional space with four spacel i k e and two timelike dimensions, an isomorphism known already to L i e  (1893) and appreciated by P a u l i (1921) and Weyl (1923). Nevertheless, the  study of conformal symmetry remained outside the mainstream of  p h y s i c a l f i e l d theory, despite a plea by Schouten (1949), u n t i l recently, when i n t e r e s t i n possible wider physical implications of t h i s symmetry was rekindled, t e n t a t i v e l y a t f i r s t , as i n the proposals of Yang (1947) and F i n k e l s t e i n (1955)• but then with increasingly f i r m commitment, notably i n the work of Murai (1953, 1958), Ingraham (i960), Wess (i960), Hepner (1962), Kastrup (1962,1966), Wyler (1968), Mack and Salam (I969). Since McLennan (1956) and Bludman (1957) established the exact conformal symmetry of a l l massless f i e l d s , i t has come to be a widely held view (e.g.  Gell-Mann I969) that exact conformal symmetry i s , i n f a c t , r e s t r i c t e d  2 to massless f i e l d s .  This b e l i e f has directed, accordingly, a s u b s t a n t i a l  e f f o r t into exploring how conformal symmetry i s supposedly "broken" by the a c t u a l world of p a r t i c l e s (see e.g. the review a r t i c l e by Carruthers 1971)* The present work i s aimed at enlarging the a p p l i c a b i l i t y of conformally i n v a r i a n t action p r i n c i p l e s to the d e s c r i p t i o n of massive f i e l d s .  A possible  avenue to t h i s end i s the systematic study of r o t a t i o n a l l y i n v a r i a n t a c t i o n functions i n a f l a t 6-dimensional space with four spacelike and two timelike dimensions.  Although the exploration of t h i s avenue involves enlarging the  arena of physics to the e n t i r e 6-dimensional space with a l l s i x coordinates accorded f u l l independent status, such a r a d i c a l approach i s not attempted c a v a l i e r l y , but i s based on the evidence that the study of f i e l d theories i n higher-dimensional spaces can be a u s e f u l t o o l i n the treatment of s c a t t e r i n g processes. For example, a n a l y t i c a l expressions f o r Feynman d i a grams i n a 5-dimensional formalism have been developed i n recent work by Adler (1972, 1973)• and using t h i s method the c a l c u l a t i o n of vacuum-polariz a t i o n diagrams i n the 5-dimensional formulation can be put i n a manifestly i n f r a r e d - f i n i t e form. The question of how to incorporate mass into conformally covariant f i e l d theories has p e r i o d i c a l l y attracted some attention, and the concept of a "conformally covariant mass" was introduced i n an e f f o r t to render equations with mass formally covariant under the d i l a t i o n s and under the s p e c i a l conformal transformations which extend the Poincare group to non-linear transformations.  Such research was summarized by Fulton e t . a l . (1962a), and more  r e c e n t l y t h i s theme has been taken up by Barut and Haugen (1972, 1973)• who t r y to b u i l d a covariant character into the mass by employing the 6-dimens i o n a l formalism to write a Dirac-type spinor free f i e l d equation i n 6-space. However, d i f f i c u l t i e s a r i s e with both of these related approaches, and some of these problems w i l l be pointed out i n t h i s report, A more i l l u m i n a t i n g approach i s that favoured by Murai (1958), whose work  has been expanded upon by both Ingraham (1960,1971) and C a s t e l l (I966).  3  These authors look upon the mass as a property of solutions of f i e l d equations i n 6-space, which when projected into a s u i t a b l y chosen 4-space can be interpreted as solutions of the usual f i e l d equations i n Minkowski space. The present work comprises an i n v e s t i g a t i o n into and extension of t h i s approach, i n order to develop a procedure by which the mass concept can be incorporated into conformally covariant f i e l d equations by an examination of action p r i n c i p l e s i n 6-dimensional space. As a preliminary step, the conformal group of transformations on Minkows k i space i s introduced i n Section 1, and the various possible i n t e r p r e t a tions of these transformations are pointed out. In Section 2, the method of induced representations used by Mack and Salam (1969) to construct quantum f i e l d theory representations of the conformal group i s reviewed, some l a t e r developments by Callan e t . a l . (1970) are given, and the concept of the scale dimension and i t s connection with the conserved d i l a t i o n current i s examined.  The momentum space representations  of the conformal generators are discussed, and a review i s made of recent work by Chan and Jones (1974a,b) which uses these representations to show how conformal symmetry places severe r e s t r i c t i o n s on s c a t t e r i n g amplitudes between h e l i c i t y eigenstates. In Section 3» i t i s demonstrated e x p l i c i t l y that the free f i e l d equations associated with massless p a r t i c l e s of s p i n 0 , i , and 1 are indeed conformally covariant.  For a massless spin-2 p a r t i c l e , the  difficulties  associated with the construction of a conformally covariant f i e l d equation are  explored by considering the covariance behaviour of the weak f i e l d  approximation of the E i n s t e i n equations.  As w e l l , Bracken's recent  work on the non-independence of conformal generators a t the s i n g l e p a r t i c l e l e v e l i s considered (Bracken 1973)•  At t h i s point begins the  o r i g i n a l contribution contained i n t h i s report.  A demonstration i s  given that Bracken's r e s u l t s are, i n f a c t , r e s t r i c t e d to the realm of  s i n g l e - p a r t i c l e quantum mechanics, and cannot be extended to quantum f i e l d theory.  This r e s u l t has the significance that the generators  of conformal transformations i n quantum f i e l d theory are a l l independent, i n c o n t r a d i s t i n c t i o n to the s i n g l e - p a r t i c l e case. In section 4, various attempts at including mass i n conformally covariant equations without departing from 4-dimensional Minkowski space are examined, and i n p a r t i c u l a r the idea of the "conformally covariant mass" m i s treated.  This concept i s introduced when d i l a t i o n s are  interpreted  as uniform changes i n units of length (Barut and Haugen 1972), and i t i s shown here that such an i n t e r p r e t a t i o n  cannot be maintained i n  Lagrangian f i e l d theory, and that a conserved d i l a t i o n current cannot be constructed f o r non-vanishing mass even i f conformal transformation properties are ascribed to m. Mack and Salam (1969) have shown how to develop manifestly conformally covariant f i e l d equations f o r massless p a r t i c l e s by considering r o t a t i o n a l l y covariant equations on the n u l l hyperquadric i n 6-dimensional space.  The extension of the 6-dimensional formalism o f f the n u l l hyper-  quadric i n 6-dimensional space i s presented i n Section 5*  In t h i s S e c t i o n ,  i t i s shown that one can s t i l l obtain conformally covariant f i e l d s from the 6-dimensional formalism, but that the conformal group must be extended to a subgroup of the conformal transformations i n 5-dimensional space to achieve a consistent formulation.  It  i s here that this approach departs  from that of Murai, Ingraham, and C a s t e l l , i n that requiring f i e l d s to transform according to the representation of a conformal group autom a t i c a l l y brings into play the spin matrices associated with rotations i n a 7-dimensional space.  This i s due to the f a c t that the 5 - d i n » n -  s i o n a l conformal group i s isomorphic to rotations i n a 7-dimensional space.  Use of these spin matrices leads to a new coordinate-dependent  5 "descant operator" i n 6-dimensional space which serves to eliminate e x p l i c i t coordinate dependence i n the f i e l d equations when they are written i n terms of Minkowski space v a r i a b l e s .  Employing  this  o r i g i n a l procedure, i t i s shown how one can meaningfully ascribe a s p e c i f i c scale dimension to a f i e l d which transforms according to a representation of the 5-dim*nsional conformal group, and the e x p l i c i t example of a scalar f i e l d i s used to demonstrate  that by i n s i s t i n g that  physical f i e l d s be eigenfunctions of the new scale dimension operator one can i n p r i n c i p l e incorporate the mass i n t o conformally covariant f i e l d equations.  In e f f e c t , then, t h i s work forms a bridge between the  work of Murai, Ingraham, and C a s t e l l , and that of Wyler (1968), who worked with the 5-dimensional conformal group.  has  This Section i s concluded  by the development of the 6-dimensional formalism necessary f o r treating spinor and vector f i e l d s from t h i s new point of view. Before embarking on the task of t r e a t i n g p a r t i c l e s with non-vanishing spin, i t i s i n s t r u c t i v e to develop a formally simpler model f o r e l u c i d a t i n g the formal structure of the method developed i n Section 5*  To t h i s end,  i n Section 6 the corresponding r o t a t i o n a l l y invariant theory i s developed f o r f i e l d s i n a 4-dimensional space with one time axis and three space axes. In t h i s space solutions of f i e l d equations can be found e x p l i c i t l y i n terms of the well known eigenfunctions of the Casimir operators of the Lorentz group.  The value of such an exercise l i e s i n the f a c t that the Lorentz  group i s isomorphic to the conformal transformation group i n a 2-dimensional Euclidean space.  I t i s found that one can indeed write down a 4-rotation  covariant spinor equation, and solutions f o r t h i s equation are constructed explicitly.  I t i s found that the proposal made i n Section 5 i s borne out,  that by selecting components of the spinor f i e l d belonging to the various values of the new scale dimension, defined i n terms of the 3-dimensional  6 conformal group, one can f i n d solutions which can be ascribed a "mass" i n 2-dimensional space* With these r e s u l t s , i t i s found opportune to obtain i n Section 7 a systematic method f o r the construction of covariant f i e l d equations i n 6-dimensional space, which flow from an action p r i n c i p l e which i s not b u i l t upon the absence of e x p l i c i t coordinate dependence, since a l l that i s required i s r o t a t i o n a l and not t r a n s l a t i o n a l invariance o f such an action i n t e g r a l .  I t i s found that by replacing Gauss' theorem with a  type o f Stokes' theorem, one can indeed f i n d a d i f f e r e n t type of E u l e r Lagrange equation which y i e l d s r o t a t i o n a l l y but not t r a n s l a t i o n a l l y covari a n t f i e l d equations, and which r e s u l t s from v a r i a t i o n o f the action integral.  By considering the v a r i a t i o n of the action under i n f i n i t e s i m a l  rotations i n 6-dimensional space, an angular momentum tensor i s constructed which gives r i s e t o f i f t e e n d i s t i n c t conservation laws.  When v a r i a t i o n s  of the boundary are taken into account, these conservation laws can be recovered d i r e c t l y from the action i n t e g r a l by means of a Schwinger action p r i n c i p l e f o r quantum f i e l d s , with the modification that the Lagrangian density i s allowed t o depend e x p l i c i t l y on the coordinates themselves. F i n a l l y , since the quantum f i e l d theory generator of an a r b i t r a r y f i e l d transformation i s thus determined, covariant  (anti-)commutation  r e l a t i o n s r e s u l t as a self-consistency condition, and by choosing the time as a quantization d i r e c t i o n one arrives at equal-time commutation relations• Applications of these ideas to s p e c i f i c f i e l d s are treated i n Section 8. F i r s t l y , the simplest case o f the scalar f i e l d i s studied, and i t i s found that one can extract from the 6-dimensional formalism s c a l a r or pseudoscalar f i e l d s which s a t i s f y i n Minkowski space the usual K l e i n Gordon equation with mass.  By a s l i g h t change of f i e l d v a r i a b l e , i t i s  7 also possible to demonstrate that the f i e l d equation s a t i s f i e d by the scalar f i e l d can be written as that s a t i s f i e d by a massless f i e l d  in  5-dimensional space, and so the conservation laws i n Minkowski space which arise from invariance of the action i n t e g r a l can be written simply as the vanishing of the divergences of the four types of current associated with the conformal group i n 5-dimensional space.  The conserved quantities  are found to be the generators of the unitary transformations which represent the transformation group i n the space of quantized s c a l a r f i e l d s , and the derivative of the f i e l d with respect to the time obeys an equaltime commutation r e l a t i o n with the f i e l d  itself.  Next, r e s u l t s by Mack and Todorov (1973) f o r the Green's function f o r the scalar f i e l d are generalized from the s p e c i a l case of the n u l l hyperquadric in 6-dimensional space to the general case.  It  i s found  that the structure of a Green's function i n a space of any dimensionality f o r a r o t a t i o n a l l y covariant f i e l d equation i s dependent on the value of a Casimir operator f o r the group, and that t h i s value can be s p e c i f i e d uniquely by requiring that the Green's function have a simple form  0  The a d d i t i o n a l case of 5-dimensional Euclidean space i s examined and the r e s u l t s obtained i n t h i s case are i n agreement with those of Adler (1972). The spinor free f i e l d equation i n 6-dimensional space i s  deter-  mined by considering the r e s t r i c t i o n s imposed on possible candidates f o r the Lagrangian density by the form of the action p r i n c i p l e of Section 7. The r e s u l t i n g equation i s compared with a d i f f e r e n t advanced by Barut and Haugen (1973)• and i t  spinor equation  i s shown that t h i s other  equation cannot be derived from a hermitian Lagrangian density.  The  o r i g i n a l spinor equation i s then used to show how one can develop f i e l d equations f o r massive fermions using the r o t a t i o n a l l y invariant formalism, and that i n p r i n c i p l e such an equation can accommodate p r e c i s e l y two  8 specie3 of p a r t i c l e having two d i f f e r e n t masses. Constructing the  angular  momentum tensor i n 6-dimensional space, f i f t e e n conservation laws r e s u l t which can again be written i n terms of the conserved currents associated with the conformal symmetry of massless p a r t i c l e s i n 5-dimensional space. Choosing the time axis as the d i r e c t i o n of quantization, one f i n d s that the spinor f i e l d obeys equal-time anticommutation r e l a t i o n s with the adjoint spinor f i e l d . The spinor equation i s then i t e r a t e d to the form of the s c a l a r wave equation, and using t h i s i t e r a t i o n a Green's function f o r the spinor f i e l d equation i s obtained.  This Green's function can then be used to  great advantage, since the d i f f i c u l t i e s associated with the two  different  spinor propagators suggested by Adler (1972), f o r the s p e c i a l case of the n u l l hyperquadric,  are overcome by the use of the propagator developed here.  F i n a l l y , the concept of the compensating f i e l d i s used to introduce the vector or pseudovector f i e l d .  By considering the second-order Casimir  operator i n the vector representation, a covariant vector f i e l d i s obtained.  equation  I t i s found that the gauge group associated with t h i s f i e l d  equation i s that proposed by Stueckelberg  (1938) i n an ad hoc fashion  i n order to render the equation f o r a massive vector boson formally gauge invariant by the introduction of an a d d i t i o n a l s c a l a r f i e l d .  From the  analysis of the vector f i e l d , i t i s found that t h i s extra f i e l d component a r i s e s i n a natural way i n the r o t a t i o n a l l y i n v a r i a n t theory, and that the properties o f the Stueckelberg formalism are reproduced automatically. However, i t i s found that as w e l l as recovering the vector ( ^ S^B. U J ) or pseudovector ( f ^ tf^B ^ )  coupling from the minimally  coupled  spinor f i e l d equation i n 6-dimensional space, one a l s o obtains the a d d i t i o n a l pseudoscalar  type meson-nucleon coupling (  *f 40.  9 Since only rotations and not translations are considered as elements of the group of transformations in 6-dimensional space, the treatment of the scattering theory of massive particles from the viewpoint set out here presents several attractive advantages.  The f i e l d eouations and  Green's functions found in this work can be used to develop covariant Feynman rules after the fashion of Adler (1972), so that the study of the behaviour of scattering amplitudes under conformal transformations can be made less complicated due to the fact that the non-linearity associated with the special conformal transformation subgroup i s removed by going to the manifestly covariant formalism. The details of this perturbational approach are not developed here, but this report serves the purpose of laying the foundation necessary before such a task can be embarked upon.  10  PART I . CONFORMALLY COVARIANT FIELDS IN MINKOWSKI SPACE 1 . The Conformal Group i n Minkowski Space 1 -1 . D e f i n i t i o n of conformal transformations Conformal transformations are mappings of the set of coordinates y  of a  point i n Minkowski space into another set y'k, (1.1)  y* *  y'  k  = y' (y) k  .  where i n a l l instances r a i s i n g and lowering of indices i s accomplished means of a metric  (1.2)  y  =  k  by  S j describing a f l a t space, k  S yJ . y » = kj  ^jV't,  k  £j k  = diag(-1 ,-1 ,-1 ,+1 )  .  These mappings are defined by the requirement that the l i n e element (1.3)  ds (y) = 2  S dyJdyk Jk  at the point y i s transported by (1.1) i n t o another l i n e element (1.4)  ds (y«) = 2  SdyOdy«k kj  which d i f f e r s from the o r i g i n a l l i n e element by not more than a spacetime dependent (1.5)  factor, ds (y«) = e 2  2 3  ^)  ds (y) 2  .  Since the transformation (1.1) y i e l d s the r e l a t i o n  (1.6)  ,  d y ' = Oy' /*yJ)dy.5 k  k  i t follows that the requirement (1.5)  (1.7)  i s met provided  SjktV'VWOy^Ay") = «  $mn •  2  Transformations s a t i s f y i n g t h i s constraint are c a l l e d "conformal" because the  cosine of an i n f i n i t e s i m a l "angle"  (1.8)  cos 9  1 2  » (dyi)  k  (dy ) 2  k  (dyj  2  dy )-* 2  2  is l e f t invariant. The constraint (1.7) so,  uniquely determines the set of mappings (1.1).  Even  i f i t i s kept i n mind that the same metric i s used f o r both sets of co-  ordinates y^ and y'k there are s t i l l several possible physical  assumptions  11 that one can make which give rise to the same definition ( 1 . 5 ) .  Each of  these assumptions w i l l lead to the same condition ( 1 . 7 ) . F i r s t l y , one can assume that (1.1) represents a displacement of a l l the points of Minkowski space, as opposed to a change of coordinate axes. situation is illustrated in Figure 1.  This  This approach i s not the same as  viewing (1.1) as transformations of the coordinate system, with the metric transforming as a tensor, since in such a situation the metric changes by the multiplicative factor e ~  2<r  ( y ) for transformations satisfying ( 1 . 7 ) , so  that the line element i s l e f t invariant.  The mathematical advantage of (1.5)  i s that the unnecessary use of curvilinear coordinate systems i s avoided. Secondly, one may again define conformal transformations by ( 1 . 5 ) , but assume that  Sj^ transforms as a tensor density of weight  of the coordinates (Fulton e t . a l . 1962a).  under changes  Since the determinant of the  metric tensor transforms as a scalar density of weight 2, the Jacobian of the transformation (1.1) i s hy/dy* | =e"^ (y). r  This means that  Sj i s l e f t k  unchanged, but the line element i s no longer invariant under coordinate transformations.  Alternatively, this type of transformation may also be  looked upon as a coordinate transformation which i s accompanied by a variation of the metric alone that serves to map the f l a t 4-space with curvilinear coordinate system y'^ onto a f l a t 4-space with the Minkowski metric (Wess 1971)o Finally, (1.5) can be regarded as a change of units, since i f one has  (1.9a)  dyj » d y ' J = e ( y ) dyj  (1.9b)  ds »  ,  ?  2  d s ' = G'jkdy'Jdy' 2  11  Sj • G'j k  = e  2  ?  (y)  k  G^y^y*  then in order that d s transform under a change of units as the square of 2  a length, (1.10)  ds » 2  ds'  2  = e  2 ?  ( y ) ds  2  ,  one must have  (1.11) It  ^jk *  G'  jk  S S  j k  .  should be kept in mind, however, that since $ has the units of momentum  12  2±  >  Figure 1. Example of an angle preserving conformal mapping. ds j 1  and  ds*2  The vectors  d i f f e r from ds^ and ds^ i n length only by the common  scale factor exp(<r).  The angle formed between d s / j and d s ^ i  same as the angle formed between ds_j and dsp.  s  ^  e  1  3  multiplied by length, then changing the units of time and length by the same f a c t o r necessarily e n t a i l s a change i n the numerical value assigned to For any of these i n t e r p r e t a t i o n s , (1.5) constitutes a g e n e r a l i z a t i o n of the Poincare transformations, which correspond to the choice  S^y)^.  Unless  stated otherwise, the f i r s t physical meaning of the conformal transformations, as displacements of a l l the points of Minkowski space, w i l l be used i n the following.  Arguments i n favour of t h i s choice w i l l be presented i n Section 4.  1-2. I n f i n i t e s i m a l conformal transformations To determine those conformal transformations that are continuously connected with the i d e n t i t y , i t i s s u f f i c i e n t to consider i n f i n i t e s i m a l transformations (1.12a)  y'  (1.12b)  e2 ( y ) = 1+2 S*(y)  k  = y+  Sy (y)  k  k  .  ?  The constraint (1.?) reads then*  &jk( « J  (1.13)  + B  SyJ.mX S V  $y .n) = (1+2?) Smn  .  k  so that (1.14)  Vn.m  + $y »n = * z  Taking the trace,  (1.15)  S  m  ? = i $y»  n  ,  K  =  one obtains  k  S  m  n  By antisymmetrizing i n m and (1.17)  ^  §y .ink + $ y . k n )  U  Sy .k " Wk'm^'n  =  S  m  £( 8 y . m  k  mn 'k ?  •  k,  and operating on t h i s r e s u l t with (1.18)  •  v i s found to be  and operating on (1.14) with (1.16)  mn  ^'k  mn  " $kn  ^'n  bj,  - S y k . ) . j = I mn^'lk " Skn^'jm m  one can f i n d a condition on  n  •  cr by antisymmetrization i n n and j ,  "For p a r t i a l d i f f e r e n t i a t i o n s any of the notations ^ X=X,i ^XAy w i l l be used when convenient. The product x y =x^y4-x*y. denoted by xy, with x x denoted by x . Also, ]/L and c are set equal to unity unless stated otherwise. =  k  k  i s  k  2  k  c  k  k  14  0 9)  ^mn ^ j k " ^kn - j m "  J  Sjra » k n  ?  ?  +  $jk » m n ?  =  *  0  By specifying various i n d i c e s , one can e s t a b l i s h that t h i s equation requires (1.20)  ^.mn =  •  0  Thus & can be w r i t t e n i n the form (1.21)  «=• = v +2/? y  where  are f i v e i n f i n i t e s i m a l parameters.  (1 .22)  k  *y , + \y , n  m  m  k  = 2(<r +2 f y ) k  These d i f f e r e n t i a l equations f o r more constants of i n t e g r a t i o n (1.23)  Sy  k  .  k  n  may be integrated to y i e l d , with t e n  <x and fc^ =- c J , k  k  = * + f jyJ+(cr+2 ^ k  Substituting i n t o (1.14) y i e l d s  k  )y - f yj k  y  j  k  .  k  y j  The i n f i n i t e s i m a l transformations are, then, (1.24a)  TRANSLATIONS:  S y • <*  (1.24b)  HOMOGENEOUS LORENTZ TRANSFORMATIONS:  (1,24c)  DILATIONS:  (1.24d)  SPECIAL CONFORMAL TRANSFORMATIONS:  k  k  ^y  k  e^yj  =  (gj*s_£kj)  S y = <*-y k  k  S y = 2y ( p y ) - ^ y k  k  .  2  The number of parameters associated with each of these types of transformat i o n i s 4,6,1,4 r e s p e c t i v e l y , and each type forms a subgroup w i t h i n the 15 parameter group of v a r i a t i o n s ( 1 . 2 3 ) .  1-3.  F i n i t e conformal transformations  The 15 parameter group of f i n i t e conformal transformations that are continuously connected with the i d e n t i t y consists of (1.25a)  TRANSLATIONS: « k = k + k  (1.25b)  HOMOGENEOUS LORENTZ TRANSFORMATIONS:  (1.25c)  DILATIONS:  (1.25d)  SPECIAL CONFORMAL TRANSFORMATIONS: y ' = ( y - b y ) / ( 1 -2by+b y2)  y  y  , k  y  = sy  a  y  , k  = R jyJ k  k  k  k  k  2  The i n f i n i t e s i m a l transformations (1.24) are recovered by w r i t i n g (1.26)  a =ct k  k  ; RJj = %^+  ^  ; s=1 + <r ; b = p k  k  .  2  The transformations inverse to (1.25) are formed using the parameters - a , -R^j» -s, and -b^.  For example, the inverse of (1.25d) i s  k  (1.27)  y ^ y ^ + b V ) [l+2(by')+b (y') ]" 2  2  2  1  where, i n keeping with (1.2), the inner products i n (1.27) involving the t r a n s formed coordinates y' are formed by using the Minkowski metric S j .  The proof  k  of (1.27) follows from the i d e n t i t i e s  (1.28a) (1.28b)  [l+2(by')+b (y') ]2  2  1  = 1-2(by)+b y2 2  y« +b (y«) = y * [^(by^y ]" k  k  2  2  .  1  The group of s p e c i a l conformal transformations can r e a d i l y be shown to c o r respond to spacetime dependent d i l a t i o n s , since from (1,25d) one finds that the square of the i n t e r v a l between two points y and x i s transformed i n t o  (1.29)  (y -x») =^ (yO-xO)(y' -x» )= ,  k  2  ilzlil  k  k  .  [l -2(by)+b y ] [1 -2(bxj+^x ] 2  2  2  S i m i l a r l y , the i n f i n i t e s i m a l i n t e r v a l ds formed from the difference dy i n the components of the two points y and y+dy, becomes  (1.30)  ds'^Sijdy^dy'J=ds  2  [l-2(by)+b y ]" 2  2  2  where dy'* is the difference in the components of the two points y  1  and y'+dy',  into which y and y+dy are mapped by (1.25d). One can now investigate how t r a j e c t o r i e s are transformed under s p e c i a l conformal transformations ( c . f . Rosen 1968).  I f the motion is given along a  curve with coordinates y«5, parametrized by the arc length s , then the tangent vector to the curve is u«5=dy«Vds. The mapping (1.25d) induces a mapping on the tangent vector, and at the point y' of the new curve, parametrized by s ' , the transformed tangent vector is  (1.31)  u'J=dy»J/ds'=tey' /dy )u (ds/ds')= J  k  j [l-2(by)-t-b y ]+bJ [4(uy)(by)-2(uy)-2y (bu)] +yJ [2(bu)-2b (uy)l 1-2(by)+b y 2  U  k  2  2  2  where use has been made of (1.30).  2  Factoring the quantity u^ out of u=UjjV,  the tangent vector can be written compactly by defining a quantity  (1.32)  u' =u V j  4  j  .  2  such that  16  From (1.32) one obtains (1.33)  1-v' =(1-v?)/(V ) . 2  2  |f  Hence the transformation (1.25d) preserves the property that the tra,)ectory has a speed l e s s than, equal t o , or greater than the speed o f l i g h t , even though from (1.29) one can see t h a t the t i m e l i k e or spacelike character of the i n t e r v a l between two p o i n t s i s not n e c e s s a r i l y preserved, under a f i n i t e s p e c i a l conformal transformation. (y-x)  2  This i s possible because i f the s i g n o f  changes, the s p e c i a l conformal transformation must become s i n g u l a r at  a point y  0  on the t r a j e c t o r y between y and x; points on e i t h e r side o f y  are transformed t o i n f i n i t y i n opposite d i r e c t i o n s , and the transformed  0  tra-  j e c t o r y remains always t i m e l i k e ( l i g h t l i k e , s p a c e l i k e ) , i f the o r i g i n a l trajectory i s timelike ( l i g h t l i k e , spacelike), 1-4,  Isomorphism w i t h r o t a t i o n s i n 6-dimensional  space  The conformal transformations are isomorphic t o the r o t a t i o n s i n a 6-dimen s i o n a l space w i t h metric S^g=diag(-1,-1,-1,+1,-1,+1). follows.  (1.3*0  T  This can be seen as  Introduce new, diraensionless, coordinates rfay*  where H i s i n v a r i a n t under t r a n s l a t i o n s and homogeneous L o r e n t z transformations, but transforms under d i l a t i o n s as  (1.35) and under s p e c i a l conformal transformations as  (1.36)  K'  =[l-2(by)+b y ]* 2  2  .  I f one introduces f u r t h e r the redundant v a r i a b l e  (1.37) then one has the c o n s t r a i n t  (1.38)  \  0  .  In terms o f the coordinates >^ = (T» 1 » 1 )»  (1.39)  x ^ - l ? >=-(V-f) 3  t h i s can be w r i t t e n  w h e r e  so t h a t ^5=-i(H+^);  V=i(*-A)  SABW  (1.40)  *0 .  7  A  Now the transformations of the  are l i n e a r , and the transformation laws  corresponding to (1.25) ar© (1.41a) (1.41b)  ^' =rf+a x; ^ R j^;  (1.41c)  1» =*f;  (1.41d)  ^i =^.b A;  k  *»=*;  k  , k =  k  k  2  X'=s>  A ^ S - V ;  k  x'=2(avj)+a H+X *'=A  K'=-2(bv()+X+b A; 2  k  A'= X  f o r t r a n s l a t i o n s , homogeneous Lorentz transformations, d i l a t i o n s , and s p e c i a l conformal transformations, r e s p e c t i v e l y . These transformations leave the quadratic form (1.40) i n v a r i a n t , and they are thus properly characterized as rotations i n the space spanned by the coordinates v r \  For i n f i n i t e s i m a l  transformations one can write (1.41)  (1.42a)^ =€ vv -(°< -p ) l ( +(S ^ (1.42b) k M«VrV»l +< l k  k  J  k  k y 5+  0<k  k  6  j  k  5  (1.42c)  W -( 'k*fk>vlk+o',l5 6  0  or, written more  (1.43)  n 6  compactly,  K EV A=  B  with (1.44)  E V =-E ; a  b  b a  E ^V=-E a 6  E =/^ =.E ; a 5  a  5 a  6 a ;  E  5 6  =o^E  6 5  .  Equation(1.43) i s obviously the 6-dimensional analogue of the r o t a t i o n s (1.24b). Since the transformations (1.41) are orthogonal, they leave invariant the metric i n 6-dimensional space,  (1.45)  l' =W . B  A  so that i f one defines the covariant components of y' as  (1.46)  y jn j/ ,  ,  H,  =  S  y j A  l V ,  ^  =  k  / K » = g  j  k  y '  k  ,  then the Minkowski metric i s also l e f t i n v a r i a n t .  (1.47)  d S ^ n "l =d\ dn.k+d('|6+'|5) 2  d  A  A  d(^ -^ )  then with the r e s t r i c t i o n (1.40) one finds (1.48)  dS =v£ d y dy =v£ d s . 2  k  2  k  6  5  Defining the l i n e element  2  Since dS ds  2  1 8  i s invariant under the rotations (1.41), t h i s r e l a t i o n shows that  transforms under conformal transformations as H  2  that ds  2  i s m u l t i p l i e d by s  transformations  2  .  From (1.41), i t follows  under d i l a t i o n s , and under s p e c i a l  conformal  i t s transformation law i s given by (1.30).  Among the transformations that cannot evolve continuously from the i d e n t i t y the "transformation by r e c i p r o c a l r a d i i " (or "reciprocation") defined as (1.49a)  y'^y^y  2  i s of p a r t i c u l a r i n t e r e s t . (1.49b)  y'Sr  Since i t has the consequence  - 2  i t amounts, according t o (1.30), t o an interchange o f H a n d \, (1.49c)  K'=-X;  X'=-K  6  k so that the coordinates ^ and ^  remain unchanged, and only the coordinate  i s inverted,  (1.49d)  ^.5=.^.  ^ =*i; k  n  ,6 6 ^ = n  k  ( A l t e r n a t i v e l y , the transformation (1.49) can be represented by  V  =-^ ; K  Any s p e c i a l conformal transformation can be decomposed i n t o a r e c i p r o cation, a t r a n s l a t i o n , and a r e c i p r o c a t i o n .  Indeed, i f one c a r r i e s out on  the reciprocated coordinates  (1.50)  y" =-y /y k  k  k  a translation y (1 .51)  2  k  k  -* y +a  one obtains  y" =-(y +a )/[y +2(ay) a ] k  k  k  2  2  +  k /2  k  so that upon repeated r e c i p r o c a t i o n y  (1.52)  «^ -y /y  one has  y ^ K ^ - a V ) / [l-2(ay) aY] +  which i s i d e n t i c a l with the s p e c i a l conformal transformation (1.25d) i f a =b . k  k  19  2. Conformally Covariant F i e l d s 2-1. D e f i n i t i o n of generators A field  tXy) i s said to transform according to a representation S(c)  of the transformation (2.1)  y-» y'=T(c)y  characterized by the parameter c i f (2.1) implies (2.2)  t(y)-»  f(y')=s(c) Y(y)  and (2.3a)  T(  C 1  ) T(c )=T(c ) 2  3  implies (2.3b)  S(c ) S(c )=S(c ) . t  2  3  T of t h i s operation by  Defining the generator (2.4)  S(c)=e  i c P  ,  one has f o r i n f i n i t e s i m a l transformations (2.5)  S(c)=I+icr  .  Introducing the operator d(c) by (2.6)  d(c)Y(y)=Y(T- y) 1  and defining the generator g by (2.7)  d(c)=e  i c g  ,  then, f o r i n f i n i t e s i m a l transformations, one has (2.8)  d(c)=1+icg .  The entire v a r i a t i o n of ^ (2.9)  can therefore be represented by an operator D(c),  T(y)=D(c)Y(y)=d(c) s(c)Y(y) .  corresponding to a generator g, (2.10)  D(c)=e * . ic  Then f o r i n f i n i t e s i m a l transformations, (2.11)  D(c)=I+icg=(1+icg)(i+icr)=I+ic(g+r)  and hence (2.12)  i=g+r .  ,  Thus the v a r i a t i o n of the f i e l d may be w r i t t e n as (2.13)  S '=y«(y).r(y)=icIr(y)=ic(g r)H (y) ;  T  +  .  The quantum f i e l d theory u n i t a r y operator U(c) corresponding to D(c) i s defined such that applied to f i e l d operators H^(y)t (2.14)  u- (c) Y ( y ) U(C)=D(C) V(y) 1  .  Then defining the generator G by (2.15)  U(c)=e Cic  §  and considering i n f i n i t e s i m a l transformations, one finds from (2.14) that (2.16)  [y<y).G]*5Y(y> .  2-2. Representations of d i f f e r e n t i a l generators To obtain representations f o r the operators d(c) associated with the conformal group, one defines operators p^, mj^s-m^j, (p , k j such that (2.17a)  d(a )=e ( P)  (2.17b)  d(wJ )=e-( / )  (2.17c)  d(s)=e (  (2.1?d)  d(b^)=e ( ) .  k  i  a  i  k  i  i  l n  s  2  w J k m  Jk  >  bk  Using the i n f i n i t e s i m a l parameters (1.26), these operators become (2.18a)  d(o<. )=1+ioc p k  k  k  (2.18b)  d(^ )=1 -(i/2)6*M  (2.18c)  d(«r)=1+i«r- 9  (2.18d)  d(jJ )«1+i^kj .  k  j k  J  Carrying out the i n f i n i t e s i m a l transformations T'^y i n the argument of the r i g h t hand side of ( 2 . 6 ) , one finds (2.19a)  d(* )Y(y)=[l-*\]Y(y)  (2.19b)  d(gJ ) H ( y ) = [ i H €  (2.19c)  d(<r)H (y)=[l-o'y b ] f(y)  (2.19d)  d ( f ) V ( y ) = [ l - ( 2 y y i - y ^ ) ] *f(y) .  k  k  J  ;  J k  ( y \ - y o ) ] H'(y) j  k  j  v  k  k  J  v  J  r  k  ; j  2  k  Comparing (2.19) with the d e f i n i t i o n s (2.18), one finds the representation of the d i f f e r e n t i a l generators  (2.20a)  pjj = i\  (2.20b)  m  (2.20c)  j k  l  = i(yji -y ^) k  V =  (2.20d)  C  k  i y \  kj = i C Z y ^ j j - y ^ ) = ^ j f - i y 2  2  ^ •  Th© f i r s t two generators are recognized as the usual differential forms of the momentum and angular momentum, while the generator of dilations i s called the " v i r i a l " .  Since the special conformal transformations w i l l be shown to  form a subgroup which i s isomorphic to the translation group i n 4-space, the generator kj may be called the "complementum" (Kaempffer 1971). Indeed, under the reciprocation ( 1 . 4 9 ) , the momentum pj is transformed into the differential generator ky  To see t h i s , consider the inverse of  the reciprocation, (2.21)  y =-y' /(y') k  k  2  .  Under reciprocation the momentum becomes (2.22)  2-3.  P  j  * p«j = iW  = iOy^y'^/ay  J  1 1  = i(2y y a -y ^) = kj n  2  j  n  e  Algebra of generators The  algebra of the self-adjoint generators G=(P ,Mj ,<|>,K^) defined by ( 2 . 1 5 ) . k  k  can be inferred from the commutation relations of the differential generators ( 2 . 2 0 ) , with the result ; [ M , P ] = i ( S P ^ P ) ; [<M ]=-iP ; [Kj.Kj-0 :  [Pj.Pj-0 (2.23)  [ j. K  p  ] - ^jk* =  k  j k  [Mjk]  2 i  =0  i  + M  n  jk  n k  )  1  r  n j  [V'^^  k  k  ( $  kmVV ^kn jm-SjmMkn) ;  h.«J=i(^ VW m  k  M  M  M-o »  frj-fl-aj  From this algebra, i t can be seen that the group of special conformal transformations has the structure of an abelian translation group in four dimensions. In keeping with section 1-4, i t should be possible to identify this algebra with that satisfied by the generators of (pseudo-)rotations in a f l a t 6-dimensional space in which an invariant interval i s defined by using the metric tensor S.^^diagt-I ,-1 i-1 ,+1,-1 i+1 ).  Introducing the 15 operators  J A R = - J R  (2.24)  Jmn =  ; J  6  = $ ;  5  *  =  <  ;  J  6m = ^ V m )  •  K  the commutation r e l a t i o n s (2.23) take then the standard form f o r  (2.25)  [ KL. MN] J  J  ^LM KN  =  J  ^KN^LM" ^ L N K > T ^ K I V L N ^  +  J  From the commutation r e l a t i o n [ $ » P ] - i P =  k  (2.26)  [({j.P ] =-2ip2  or  2  [A.P ]^? 2  2  k  follows by  with /\ =i(j)  This implies that the only discrete eigenvalue of P continuous spectrum of P  rotations  •  iteration  . i s zero, and that the  2  cannot s t a r t at any f i n i t e value (Wess 1 60). Q  2  Thus, exact d i l a t i o n a l symmetry means that the mass spectrum i s either zero or continuous.  Therefore one may work either with f i e l d representations of  the conformal group with a continuous mass spectrum (Ruhl 1973)  • or with  representations associated with zero mass.  2-4.  Induced representations  To f i n d the generators G defined i n (2.15), one must f i n d operators s a t i s f y i n g the commutation r e l a t i o n s (2.23) and having the a d d i t i o n a l properties (2.16), e x p l i c i t l y  (2.27a)  [Y(y).P ] = P Y ( y )  (2.27b)  [Wy).*^] = ^ V ( y )  (2.27c)  [f(y).$] = y ^ y )  (2.27d)  [^y),^] =  k  k  v  k  k^(y)  where now the generators g = ( p » k  m  j « f% k^), defined as i n (2.12), k  contain both the external and the i n t r i n s i c attributes generate the respective conformal transformations.  of the f i e l d  that  These generators can be  found by the method of induced representations, using a representation of the " l i t t l e group" which leaves the point y=0 f i x e d to determine of the conformal group. sional  The l i t t l e group i s generated by the f i n i t e - d i m e n -  ( " i n t r i n s i c " ) attributes  generators m j . If, k j . k  representations  of the f i e l d  <r^,-±J  t  H^, contributing to the  These matrices are defined by employing (2.27) at y=0.  (2.28b)  [?((>).M ] =  (2.28c)  [H>(0),4)] =  (2.28d)  [W0).K.j] *  The  ^  k  <T. f(0)  jk  k  -i£ Y(0) K. <f(0)  .  are the spin matrices, and &  and the " i n t r i n s i c complementum".  and  Hj are c a l l e d the "scale dimension"  They are defined s i m i l a r l y i n any represen-  t a t i o n ; e.g. f o r (2.20) one has (2.28c •) so that  [Vj.p] = ( - i ) ( - ^ j ) ^=-1  f o r the d i f f e r e n t i a t i o n operator.  In (2.28b c,d), one assumes t  that momentum cannot e x i s t i n i n t r i n s i c form, so that the basis i n the index space of *f can always be chosen such that  (2.29)  VK, = P L v  k  -iV Si k  .  Now the entire generator g can be found by using the t r a n s l a t i o n property (2.30)  Y(y) = e ^ ^ ) Y ( 0 ) e - ^ P y )  and the i d e n t i t y (2.31)  fr(y>.G]  -e^Py)  [^(O).G(-y)] e-i(Py)  where G(-y) i s defined by (2.32)  G(-y) = -X*y)Q,H*y)  = G-i[(Py),G]-i[(Py),[(Py),G]]+....  B  On account of the commutation r e l a t i o n s ( 2 . 2 3 ) , one f i n d s (2.33a)  P (-y) = P  (2.33b)  M (-y) = M - i [ P ^ M ^ ] = M  (2.33c)  <|)(-y) = H f P ^ . f l  (2.33d)  Kj(-y) = K . - i ^ . K ^ - i [ P y . [ P ^ . K ^ ^ j ^ ^ y ^ ^ y ^ P y - y ^ j  k  Jk  k  j k  = *+y"P  j k +  (y . P ^ j )  n  n  n  so that, using the d e f i n i t i o n s (2.28)  [f(0).P ]  (2.34a)  [T(0),P (-y)] =  (2.34b)  [Y(0).M. (-yj] =  (2.34c)  [H»(0).$(-y)] = - i i  (2.34d)  fV(0),K (-y)]  k  k  k  k  ^ ^ ( 0 ) + [ (°).(y/ -y j)] f  p  k  ^(0)+  k  [r(0),y P ] k  k  = (Kj-ai/yj^^^O)*  [^(0)  ,(2y . y ^ - y ^ ) ]  Upon s u b s t i t u t i o n , from (2.31) one obtains ( c . f . Mack and Salam  19^9)  .  24 (2.35a)  p  = 1\  (2.35b)  m. =  (2.35c)  y = -ii+iy\ = -ii+ <f  (2.35d)  ^  k  = p  k  <yi( vy V ^ jk =  yj  k  - H 2i£y r  +m  k  . 2yV. i(2 /a -y >.) = 2  +  n +  y  k  V^V^'V** *  By s e t t i n g y=0 i n these expressions, i t can be seen that the generators <Tj , - i i , K j also s a t i s f y the defining algebra ( 2 . 2 3 ) .  of the l i t t l e group 2-5.  k  F i e l d variations  The v a r i a t i o n ( 2 . 1 3 ) of a f i e l d  Y due to the most general i n f i n i t e s i m a l  conformal transformation may now be written = (i^ ? -(i/2)€J Sj +i<r7+ip^kj)  (2.36)  k  k  k  k  with the generators g as given i n ( 2 . 3 5 ) •  The c l a s s i c a l transformation law  ( 2 . 2 ) takes the form  r ( y ' ) = [i-(i/2)e o- +(<r+2pJyj^  (2.37)  .  jk  Jk  A f i e l d i s said to transform as a s c a l a r , vector, tensor, spinor under homogeneous Lorentz transformations i f the spin operators are represented by (2.38a)  <Tj  (2.38b)  (<Tj )  (2.38c)  (  = 0  k  k  (2.38d)  m  = i(S j S - & j £ ) B  k p  =S ( m  rs  r  P  k  ^ r vector ^  «>)V^kO"  ( T j = (i/4) [^j.yj k  where the ^  Y  B  r  0-j r k  f o r scalar  f o r spinor  s  r a  f o r tensor  y »  Y  are the 4x4 Dirac matrices f o r 4-component spinors Y .  k  Y s a t i s f y i n g Euler-Lagrange equations a r i s i n g from  Consider now f i e l d s a Lagrangian density  £(Y,^ Y). k  Defining the conjugate momentum T* as the  fourth component of the vector (2.39)  ¥  k  =  >2/> Y k  .  the f i e l d equations can be w r i t t e n (2.40)  ^  = u / ^ r  .  Then i f the canonical equal-time commutation (anticoramutation) r e l a t i o n s (2.41)  | / ( x . t ) , f(y..t)l. = ; i  5(x-yJ  are s a t i s f i e d , f o r boson (fermion) f i e l d s , the quantum f i e l d theory generators  25 G, defined i n (2.16), take the form (2.42a)  P =  (2.42b)  M  (2.42c)  () =  (2.42d)  t \ y ) djr  k  K  k  = j_  j k  y  k  Z  o \ y ) dy_  = f  j  d  S. \y)  t  y  =  t  K \ y ) dy.  with  (2.43a)  t " = TI^Y -J  (2.43b)  S  (2.43c)  D  (2.43d)  k  j k  n  « - -iir  X  ^  +7^-7^."  = -TTlS' + y \  y  n  S/^+if ,  = ( e y ^ K ^ t ^ i r ^ J y j ^ ^  n K  n  n k  j  fields Here, i t i s understood that a l l independently varied f i e l d s are to be summed over, and that the Ji and H . f o r  are the matrices given by ( 2 . 2 8 c , d ) .  3  [ c . f . Mack and Salam (1969), who r e s t r i c t t h e i r attention to f i e l d s with vanishing  K - i n the above c u r r e n t s . ]  Even though the operators (2.42) generate the f i e l d v a r i a t i o n s (2.36) whether or not the v a r i a t i o n of the Lagrangian density under conformal transformations i s a divergence, only when the divergences of the currents (2.43) vanish are the operators (2.42) independent of the time t .  Only  when t h i s i s the case do the generators (2.42) s a t i s f y the d e f i n i n g algebra (2.23) of the conformal group. The currents (2.43) can be written i n a more transparent form whenever  kn one can f i n d a second rank symmetric tensor u (2.44) If  TT i^ n  -iir (r Y = * k  n  k  u k n k  such that  '  t h i s condition i s s a t i s f i e d , upon subtraction of a term whose divergence  vanishes i d e n t i c a l l y the d i l a t i o n current (2.43c) can bo w r i t t e n i n the form (Callan e t . a l . 1970)  26 (2.45)  D" = y* e  n k  where the energy-momentum tensor G . d i f f e r s from t . J J k  k  by a term with  zero divergence, and s a t i s f i e s  e. . =0  ;  n  (2.46a)  n  0  =  j n  & n  ,  .  and also s a t i s f i e s (2.46b)  e  = 0  n n  provided the d i l a t i o n current (2.45) i s divergenceless.  When the condition  (2.44) holds, ©-. takes the form u k  (2.47)  e  = TJ + i x ^ , ^  Jk  k  where T^ i s the symmetrized energy-momentum tensor (Belinfante 1939) k  (2.48)  T J = T J = t0k+(i/2)J (--nO y-km k  k  m  v+ir  m ^jkvjj^ ^ m y j  and (2.49)  X^ "" = - $ ^ J k $ 1  r a  +  J nk rak ^ u  n  .  u  Callan e t . a l . (1970) have also shown that i f one assumes a l l f i e l d s to have  .=0 the condition (2.44) i s necessary as w e l l as s u f f i c i e n t ,  and the complementum current can be w r i t t e n  (2.50)  K.  n  =  (2y y -y S )e k  j  2  k  ;j  n k  .  Therefore, the above procedure f o r f i n d i n g a traceless energy-momentum tensor can c e r t a i n l y be c a r r i e d out f o r massless f i e l d s with  K .=0 and 3  spin 0, j , or 1, which w i l l be shown to have divergenceless d i l a t i o n currents f o r suitable scale dimension.  2-6.  D i l a t i o n invariance  Conservation laws f o r the currents (2.43) arise from considering the transformation behaviour of the action i n t e g r a l  (2.51)  I = J/(V,> y)  dV  k  under i n f i n i t e s i m a l conformal transformations.  The derivation of the  conservation laws f o r momentum and angular momentum i s w e l l known ( e . g . Schweber 1961), and these conservation laws take the form (2.52)  t  n k  ,  = 0  n  where t^" and S ^  n k  ;  S.  n k  .  n  = 0  are given by (2.43).  S i m i l a r l y , one can derive the conservation law f o r the d i l a t i o n current by making use of the transformation law of a f i e l d under d i l a t i o n s .  For  i n f i n i t e s i m a l d i l a t i o n s s=1+**,  S T * = ry*  (2.53)  the transformation law (2.37) i s , with the matrix £ defined as i n (2.28c),  (2.54)  vy.(y.) = ( 1 c r i ) S » ( y ) +  ,  so that the v a r i a t i o n o f the f i e l d under pure d i l a t i o n s characterized by the i n f i n i t e s i m a l parameter  (2.55)  ^=<r(^-y\)Y  o~* i s  .  This terminology amounts to defining the scale dimension of the c o o r d i n ates y (2.56)  k  as  =+1, on account of the transformation law dy»  k  = s d y = (1 + <r) dy k  14  ,  and enables one to determine the behaviour under d i l a t i o n s of other geometrical objects i n terms of that basic scale dimension. Now, i f the i n t e g r a l (2.51) i s to be invariant under d i l a t i o n s , the v a r i a t i o n of X. must be a divergence,  (2.57) i.e.  (2.58)  IX = -} (<ry .O = <r ( - 4 - ^ ) / k  k  (U/o<r) = (-4-^)71  .  Comparison with (2.55) shows that i f one can meaningfully ascribe a dimension Jl^ to the Lagrangian density X at a l l , then d i l a t i o n a l invariance, f o r the case of constant a* , i s guaranteed i f the dimension of (In any event, X  /  i s X^ =-4.  i s necessarily a c-number since X i s a s c a l a r . )  28 When this condition i s obeyed, one can obtain an expression f o r the conserved current due to d i l a t i o n a l invariance i n the usual fashion by considering cr to be coordinate dependent, so that the d i l a t i o n has the e f f e c t s (2.59)  Y ^ r ^ ?  (2.60)  -N Y* n  Y=U-y\)V  »ith  \ r +O trO P+<r* 9 c  ,  v  a  B  .  Then  UX/^cr) =  (2.61)  O W ) K \ ?  and  =v a-y\)f n  |S*/>G <r)] =-rr 9 n  (2.62)  B  where TT" i s defined as i n (2,39).  D i f f e r e n t i a t i o n of (2.62) gives  ^nf^AOn*')] = O n ^  (2.63)  +  A i ?  •  On account of the f i e l d equations (2.40) one can write (2.63), using  \D*Aa crJl  (2.64)  (2.61),  =  n  Now, i f one i n s i s t s on eq. (2.58) as the expression of d i l a t i o n a l invariance, then eq. (2.64) has the form of a conservation law (2.65)  D , n  n  - 0  f o r the canonical d i l a t i o n current vector  (2.66) where t  D* = n k  - T V+y  k  (v\f  -£ X ) = -TTW  W  N  K  i s the canonical energy-momentum tensor (2.43a).  The f i r s t term  on the r i g h t hand side of (2.66) may be termed, i n analogy to the corresponding term i n (2.43b), the " i n t r i n s i c d i l a t i o n current" vector.  In terms of the  symmetrized tensor (2.48) the vector (2.66) may be written (2.6?)  D ^ - ^ i t ^V-< / >^m<- k^ i  =-TTW + y  k  T  2  n  +  T  i  k  l  M?+  ^  <r  n k  1'-Tr  ^ > -(i/2n [y (-F a> K  m  k  The l a s t terra i n t h i s equation i s the divergence  k  n r a  ntf  * "  r a- m  + 1  Y) =  k  n  k  n T  .  of a tensor antisymmetric  i n (n,m) and can therefore be discarded without a f f e c t i n g the conservation law (2.65) which remains v a l i d i f one defines the d i l a t i o n current as (2.68)  D' = y \ - ^ i f + i i r < r n  n  k  » f .  29 Aiming at the form (2.45) f o r the d i l a t i o n current, introduce now the tensor  through (2.47) and write  (2.69)  V*=^ -hfaiy\»S ^ n  +iifV  m  n k  f.  The second term on the r i g h t hand side of (2.69) can be dropped without a f f e c t i n g the conservation of the r e d e f i n e d d i l a t i o n current  (2.70)  D* = y ^ ^ U / ^ + X ^ - i r ^ Y + i ^ < r 7 n  k  because a simple c a l c u l a t i o n shows that  (2.70  (y x k  n k  >).  n >  ^(x  n k  ^x J-). k  n j m  4[yk(g3n  Since one has a l s o , by d e f i n i t i o n (2.49), the  (2.72)  X / ^ X / * * )  U k  '"-s''%J)].  n J m  =o  .  identity  = > u*n k  the p o s s i b i l i t y of writing the d i l a t i o n current i n the form (2.45) i s c o n t i n gent upon the p o s s i b i l i t y of w r i t i n g , as follows from inspection of (2.70),  eq.(2.44) (2.73) If  ir ^H -iTr o- H = n  ;  k  n  ,  k  >  ukn k  •  t h i s condition i s s a t i s f i e d the d i l a t i o n current can be written i n the  form (2.45), where &  n k  i s constructed after the p r e s c r i p t i o n (2.47) with  (2.49).  The conservation of (2.70) implies that &  (2.7*)  e * =T ^ * - ^ -T ^ - u ^ =0 .  n k  is traceless,  k  In other words, whenever the condition (2.73) i s s a t i s f i e d , the trace of the symmetrized tensor (2.48) i s n e c e s s a r i l y the divergence  (2.75)  T  k k  =  >O n  ukn k  > = ^(^^•i^^H')  o r , equivalently, the trace of the canonical energy-momentum tensor i s the divergence of the i n t r i n s i c d i l a t i o n current vector,  (2.76)  t  k k  = ^ (TT ^r ) n  n  .  Nowhere i n the foregoing have the elements of the scale dimension matrix been s p e c i f i e d , except i n the statement (2.57) that the v a r i a t i o n of M* must be such that ;£ has the scale dimension -^=-4.  P a r t i c u l a r matrices Jl arise from  ,  30  consideration of the momentum Tr^(y) canonically conjugate to  M'(y),  from consideration of a p a r t i c u l a r dynamics f o r the f i e l d M^. siderations, one i s led to matrices  i.e.  From these con-  which are multiples of the u n i t matrix,  with the m u l t i p l i e r c a l l e d the "canonical" scale dimension, also denoted byJ?. As an example, consider the neutral scalar massless f i e l d N-, (2.77) 2=H H'. S'. . 7  n  described  by  n  n  n  The requirement of d i l a t i o n a l invariance of the action i s s a t i s f i e d i f one ascribes to the f i e l d  the canonical scale dimension  =-1•  To see  how  t h i s value derives from the canonical momentum d e n s i t i e s (2.78)  TfJ 2 ^ / M V j Y )  =  > H> d  .  assume that the canonical equal-time commutation r e l a t i o n s are obeyed  [H'W.^YCy)]^^  (2.79) so that the (2.80)  = i S(x-jr)  operator  fot)  =  ^{-M^f +  r,^Y, -iy^, k  v m  f\  i n  } dx  s a t i s f i e s the commutation r e l a t i o n (2.81) where J (2.82) Now  [M'(y).ftt)]  i(y\->W(y)  =  i s as yet unspecified.  [Ay),*(t)]  S i m i l a r l y , one arrives at (Cheung  =1(^+3^)^(7)  .  i f (j) i s independent of time, then taking d/dt of (2.81) one  (2.83)  [Ay).*]  =  1971)  obtains  i(y 3 -^+1)TT (y) k  4  k  so that comparing (2.83) and (2.82), one f i n d s that the canonical value i s necessary i n order that the canonical momentum i s ~R^ YIn =K  t h i s requirement i s simply a statement of the covariance  Jl=J\  fact,  of the commutation  r e l a t i o n s (2.79) under d i l a t i o n s , since (2.84)  S(sx-sjr) = s "  3  $(x-y_)  so that, f o r the example above, (2.85)  [*'(«), T T ' ^ s y ) ] ^ ^  implies X =-1 .  = a *" 2  1  [T(X),  Ay)]  S i m i l a r l y , the canonical scale dimensions of noninteracting  31 s p i n - j and spin-1 massless f i e l d s are _3/2 ascertained the value of H  t  e . g . f o r the Lagrangian density  u  k n  =u  n k  Once one has  the traceless energy-momentum tensor 9 ^ can be k  constructed by employing the condition  (2.86)  and -1 r e p e c t i v e l y .  (2.77)•  (2.73)  to f i n d the tensor u ^ ; k  one can use  = -IS^f* .  That d e f i n i t e values of the scale dimension must be assigned to f i e l d s H\  i n order that the equal-time commutation r e l a t i o n s be preserved  under d i l a t i o n s , does not exclude the p o s s i b i l i t y of d i f f e r e n t a s s i g n ments of the scale dimension when these commutation r e l a t i o n s are no longer necessarily obeyed, and when i n t e r a c t i o n terms are added t o Lagrangian densities describing free f i e l d s (Wilson  1969).  However,  whenever the defining commutation r e l a t i o n s of the conformal group  (2.23)  are s a t i s f i e d , the scale dimension Jl must necessarily be a multiple of the u n i t matrix ( i . e . each component of the f i e l d must have the same value of i.) f o r f i e l d s which transform according to an i r r e d u c i b l e of the Lorentz group.  representation  This follows, as a consequence of Schur's lemma,  from the commutation r e l a t i o n (2.87)  f<rjk.i] =  0  which must be s a t i s f i e d by the generators of the l i t t l e group at y=0. Since using Schur's lemma i s contingent on the  generating an i r r e d u c i b l e  set of matrices, Z must act on y as a c-number (and reducible representations of the Lorentz group can be completely decomposed, so that i n each i r r e d u c i b l e subspace I i s a multiple of the u n i t m a t r i x ) . Therefore, i t i s important that conformally covariant f i e l d s be assigned a d e f i n i t e scale dimension Ji i f they are to transform according to an i r r e d u c i b l e representation of the Lorentz group. F i n a l l y , t h i s argument can be c a r r i e d further to show that such f i e l d s cannot possess an i n t r i n s i c compleraentum K'j. considering the commutation r e l a t i o n  This r e s u l t i s a r r i v e d at by  32 (2.88)  [ .,i] K  = Kj  which states that f o r a general representation of the conformal group K j must be a singular matrix which can be put i n t r i a n g u l a r form with i n the diagonal (Mack and Salam 1969).  zeroes  Hence f o r an i r r e d u c i b l e repre-  sentation of the Lorentz group, with J- represented by a c-number, v4 .  3 must vanish i d e n t i c a l l y . 2-7. Two-particle momentum space representation; experimental consequences Since the v i r i a l operator (j) does not commute with the momentum operators, ,p.l=-iP., then eigenstates of momentum P.?k) cannot be eigenstates of <{). Therefore one cannot meaningfully define the " v i r i a l of one-photon momentum eigenstates", say.  This s i t u a t i o n i s comparable to trying to define the  i n t r i n s i c p a r i t y of a right-hand c i r c u l a r l y polarized photon, which i s impossible since the p a r i t y operator changes the p o l a r i z a t i o n .  To construct  v i r i a l eigenstates, then, one must form superpositions of momentum eigenstates. As w e l l , states with non-vanishing  expectation values of the energy must  be formed from v i r i a l eigenstates by smearing i n the eigenvalue virial.  of the  For t h i s reason, consideration of the v i r i a l eigenstates does not  appear to be the most d i r e c t means to understanding  the p h y s i c a l meaning of  the extra conservation laws associated with conformal symmetry. Nevertheless, t h i s does not mean that these conservation laws are devoid of p h y s i c a l consequence.  In f a c t , one can determine d i r e c t l y the r e s t r i c -  tions placed on 2 - p a r t i c l e scattering amplitudes by conformal symmetry, by looking at the action of products of 2 - p a r t i c l e operators on amplitudes. Recently, Chan and Jones (1974a,b) have demonstrated that, i n scattering involving four p a r t i c l e s of a r b i t r a r y spin, exact conformal invariance places  See, however, Kastrup (1965,1966a), who has investigated the formalism necessary to accomodate "eigenstates" of the v i r i a l with non-real eigenvalues — v i z . a p a r t i c u l a r i n d e f i n i t e metric i n H i l b e r t space.  33  severe r e s t r i c t i o n s on scattering amplitudes between h e l i c i t y eigenstates. F i r s t l y , the momentum space equivalents of the conformal generators are written down f o r f i e l d s with  K .=0  ( 2 . 3 5 )  and spin s related to the canonical  scale dimension X  by s = - ^ - 1  ( i . e . s c a l a r s , spinors, and second-rank tensors,  but not v e c t o r s ) .  Writing the massless quantum f i e l d  as the Fouxder  M'(y)  transform of h e l i c i t y \ =-s eigenfunctions w*(p_) m u l t i p l i e d by a n n i h i l a t i o n and creation operators a * ( £ ) and b \ p _ ) , one can f i n d the action of the +  generators on these operators.  This determines another s p i n - s representa-  t i o n g" of the conformal group such that f o r any of the a n n i h i l a t i o n and creation operators ( 2 . 8 9 ) .  [a.G] = f a  ;  [p,a ] = g * * +  From these operators, i t conformal operators.  Let two p a r t i c l e s with momenta p_^ and p_g s c a t t e r i n t o  the process are massless.  a  -  s  a  a n (  * suppose that a l l the p a r t i c i p a n t s  in  Assuming that G167=0, then  <1jvj.' |4 .' ) -  where now g ^ * ^ ^  .  i s straightforward to f i n d the 2 - p a r t i c l e  two p a r t i c l e s with momenta 2.y2^*  (2.90a)  +  El  fe  ( S j ^ E ^ a J V f c ) "  2 - p a r t i c l e representation of the conformal group.  From  t h i s equation, one can see that  (2.90b)  <?J+gJ)  = (g +g ) S  where S. i s the S-matrix.  3  4  S  x  minus the u n i t matrix contains a 4-dimensional  d e l t a function which ensures momentum conservation, and commuting the 2 p a r t i c l e operators through t h i s d e l t a function one obtains t h e i r action on the T-matrix T^.  As shown by Gross and Wess  ( 1 9 7 0 ) ,  a l l the operators  remain unchanged, except f o r the v i r i a l operator <f , to which an extra must be added.  Then the condition  ( 2 . 9 0 b )  places f i f t e e n  4i  differential  constraints on T . , and by going to the centre of momentum frame and choosing  34 the z - a x i s perpendicular to the scattering plane, Chan and Jones have shown that these constraints amount to three possible cases.  (i) ( X + X - X - X 4 ) = 0 ; 1  2  3  y i e l d i n g T x =0;  (iii)  ( i i ) ( > + X - > 3 - X ) / 0 and A ^ X g (or \ f \^) 1  ( X,+Xg-X  subject to a d i f f e r e n t i a l  These are:  2  4  - X ^)#> and X = X = - \ y - X ^ , with ? t  £  x  constraint but not necessarily zero.  In summary, then, an experimental consequence of conformal invariance the exact symmetry case i s h e l i c i t y conservation, i n the case constraint.  A^+A  = 2  ).^= X £ - A 3=- A 4. when the amplitude obeys a =  A^, except differential  in  t  35 3. Conformally Covariant F i e l d Equations One can demonstrate e x p l i c i t l y that i n order f o r free f i e l d equations i n Minkowski space to be covariant under the s p e c i a l conformal t r a n s f o r mation subgroup, i t i s necessary that boson f i e l d s possess a scale dimension ^=-1, while fermion f i e l d s must have i = - 3 / 2 .  In t h i s s e c t i o n , the c a l c u -  l a t i o n i s c a r r i e d out f o r the cases of s c a l a r , spinor, and vector massless f i e l d s , - and d i f f i c u l t i e s encountered with spin-2 f i e l d s are pointed o u t . F i n a l l y , i t i s shown that while the conserved quantities associated with d i l a t i o n s and s p e c i a l conformal transformations can be written i n terms of the momentum and the angular momentum i n quantum mechanics, i n quantum f i e l d theory a l l four types of conserved quantities associated with the conformal group are independent. 3-1 • Massless Klein-Gordon equation A massless scalar f i e l d  (3.D  ^jS'(y) =0  Y i s governed by the Klein-Gordon equation  ,  and transforms under i n f i n i t e s i m a l conformal transformations as (3.2)  W(y) = ( W ^ - U ^ )  6 J m + i c r ^ - i ^k-j) t ( y ) k  j k  .  Equation (3.1) i s covariant under conformal transformations provided the field  (3.3)  S " ( y ) , defined by H"(y) = Y ( y ) + SY(y)  i s also a s o l u t i o n of the same equation, i . e .  (3.4)  ^  j f «(y) = 0  .  I f t h i s i s the case, then one also has that ( y = > / ^ y m  (3.5)  g  , m n  , m  )  Vm y r ( y ' ) = o n  with (3.6)  2 ' ™ s ( > , y ' / o y J ) ( ^ y ' / ^ y ) g^ m  n  k  k  so that interpreting the conformal transformations (1.24) either as mappings between d i f f e r e n t  points or as changes of the coordinate system one i s  l e d to the same r e s u l t . This can be shown f o r i n f i n i t e s i m a l s p e c i a l conformal transformations,  (3.7)  * y  k  = ZykpJyj-pV .  by c a l c u l a t i n g  (3.8)  ( > y  ,  n  W ) =\  m  f 2 %  m  . f o  k  + 2 y » } . - 2 f y .  .  Then, to f i r s t order i n j5 , one has  (3.9)  g*  =\  m  so that It  m  0 ^ 7 ^  .  (3.4) implies (3.5).  i s evident that (3.1)  i s covariant under pure d i l a t i o n s , as i s also  the case f o r translations and Lorentz transformations. conformal transformations, (3.4) D'Alembertian Y^}j with k  (3.10)  n  is satisfied i f  vanishes,  For s p e c i a l  the commutator of the  i.e.  [VVj.^i] = • 0  Substituting f o r k^ from  (2.35). with  a c-number, the l e f t hand side  1  of t h i s equation reads, f o r general s p i n ,  (3.11 ) For is  [V}j.k ] = - 4 C M  ff- J^44±y ^ij  n  a massless s c a l a r f i e l d , ^ j j ^ O ,  o  n  e  sees immediately that  s a t i s f i e d only f o r solutions of (3.1)  the scale dimension with J£=-1.  (3.12)  ^  j  V  f  '  (  .  n  n  (3.4)  that are eigenfunctions of  In t h i s case one has  y  ) ={i-f (2y y ^ -y ) +6y )]^^r(y) . n  2  k  n  k  n  n  which shows that V^jH^v) transforms as a s c a l a r f i e l d with scale dimension -3» • The divergenceless p r o b a b i l i t y current density associated with (3.1)  (3.13) If  j  k  - iY*> T*-it} V* k  the i n t e g r a l of j  k v  is  .  over y_ i s to be a number, and hence d i l a t i o n a l l y  4  invariant,  then j  must have a scale dimension - 3 , which i s indeed guaran-  teed by assigning the canonical value i=-1  to the f i e l d  Y,  37 3-2. The Weyl equation The 4-component spinor representation i s given by (2.38d), (3.1*)  <r  - (i/4)  jk  [Yj.Yj .  Neutrino f i e l d s are governed by the Weyl equation  -iA/ = 0  (3J5) and,  using the defining r e l a t i o n of the C l i f f o r d algebra s a t i s f i e d by the  the Dirac matrices Y«5  (3.16)  = 2SJ  fr .Y ] j  k  k  .  the i t e r a t e d form of the Weyl equation i s  (3.17)  ^Yj? = 0 .  The set of matrices generated by the  <5"j  k  i s a d i r e c t sum of two i r r e d u c i b l e  representations of the Lorentz group which are related by the p a r i t y operator. Since the generator of d i l a t i o n s commutes with the p a r i t y and Salam 1969), J. acts on ^  operator,(Mack  as a c-nuraber i n each i r r e d u c i b l e subspace,  and the i n f i n i t e s i m a l generator of s p e c i a l conformal transformations f o r neutrino f i e l d s i s given by  (3.18)  kj = i ( 2 y ^ - y ^ _ - 2 i y ) + 2 y ( r ' 2  k  k  j  j  j k  .  The commutation r e l a t i o n (3.11) can be conveniently employed to determine the value of ^ n e c e s s a r y to ensure that whenever M'(y)  is a solution.  [VV ] k  (3.19)  n  YHy) is  a  s o l u t i o n of (3.1?)  Substituting (3.1*0 into (3.11). one obtains  " -^+3/2)1^+2^^^^  so t h a t , on the space of solutions of the Weyl equation, i n order f o r 1^ to be a symmetry operator one must have i=-3/2. that  tf^jf  In t h i s case one finds  transforms as a spinor with scale dimension - 5 / 2 .  The divergenceless p r o b a b i l i t y current density associated with (3.15) i s  38 (3.20)  j  k  =9 Y V  with  k  Thus the scale dimension of i n t e g r a l of j  y  =  .  must indeed be ^=-3/2 i n order that the  have zero scale dimension.  3-3. The Maxwell equations The 4-component vector representation i s given by (2.38b).  A non-  i n t e r a c t i n g massless vector f i e l d A i s governed by the Maxwell equations  (S >jV-A )A k  (3.21)  N  n  N  = 0 .  so t h a t , i n t h i s case, covariance of the f i e l d equations i s guaranteed provided  (3.22)  [(^i ^ ).(KV)%]»{(^ij)(^)VC^„)^)V r  n  -(k )P (^^K(k )P (i J )} = o n  r  s  r  n  s  *  when acting on a s o l u t i o n of (3.21).  Substituting the e x p l i c i t form of  the spin matrices, one obtains  (3.23)  [(  V ^ V ^ V ^ s ]  S  -*i(1+4)$  21(1+*)^ p s  +  21(1+4)^,-  \-M*iy UP ^ - W k  .  s  and comparison with (3.21) shows that J. must be -1 f o r k  r  to be a symmetry  operator on the space of solutions of the Maxwell equations.  In that case,  the l e f t hand side of (3.21) transforms as a vector with scale dimension - 3 . 3-4. Weak f i e l d approximation of E i n s t e i n equations In the weak f i e l d approximation, the g r a v i t a t i o n a l f i e l d may be described by a second-rank symmetric tensor f i e l d h ^ i n Minkowski space, which i n k  the free f i e l d case s a t i s f i e s the equation ( P i r a n i 1964)  (3.24)  [\ f\\ n  Sfs-Aafr^-^nS^^V^i^rsl  =  0  •  39 It  i s of i n t e r e s t to a s c e r t a i n whether e q . ( 3 . 2 4 ) , as i t  stands, i s conformally  covariant i n the sense outlined i n the preceding sections; >'  the metric I  throughout and assuming that h  i.e.  using  transforms with a s p e c i f i c  r s  scale dimension ^ , h ' ( y ) = <r«-y\)h (y)  h (y) *  (3.25)  r s  rs  r s  under i n f i n i t e s i m a l d i l a t i o n s . be not covariant ( c . f .  .  In f a c t , the f i e l d equations turn out to  Deser 1 9 7 3 ) .  To study the behaviour of (3*24) under s p e c i a l conformal transformations, one must define the generator kp, with four i n d i c e s , such that the d i f f e r e n t i a l part m u l t i p l i e s the f i e l d h^  u  with the product of Kronecker deltas  Si-t& u» i « « r  S  e  (kp)Vu = i( ypy Vy ^p- ^p)s tS +2y (cr  (3.26)  with  2  m  2  r  2  s  m  u  ff'  given by ( 2 . 3 8 c ) .  p n i  )  r s t u  Covariance of (3.24) i s guaranteed only i f  t h i s generator commutes with the d i f f e r e n t i a l  operator of the f i e l d equation.  The r e s u l t which emerges i s a c t u a l l y of the form  (3.27a)  [ t t ^ S V v ^ ^ ^  even when acting on f i e l d s h  * 0  which s a t i s f y the f i e l d equation.  t u  The actual c a l c u l a t i o n of t h i s commutator i s straightforward but tedious, and has been relegated to Appendix 1.  Substituting the tensor form of the  spin matrices, the f i n a l r e s u l t f o r the l e f t hand side of ( 3 . 2 7 a ) i s  (3.27b)  Wy ^ > S sV)„> S t->u> 5 t^ j' S y2i{2(i^)S SVS Stu}p n  p  J  k  n  k  k  ;i  J  <  t  k  tu  -4<^V^ V  5  s  P  P  t^-  -2i{(i-i)s sJ ^r -3s s )j J  pu  t  tu  u  pt  k  *  Jk  t  +  40 When operating on a symmetric tensor h^ =h ^, t h i s expression becomes u  M y  (3.27c)  p  ^ \ S V ^ ^  ^i$ V MS J  u  p  u p  h). +2ii$ ^ u  p  +2i(2+j?)(h AhJ , ) k  k  p  where h ^ 3  1 1  .  p  While the f i r s t term i n this expression shows that the  e n t i r e f i e l d equation transforms i n part as a second-rank symmetric tensor with scale dimension c a n c e l , f o r any choice of 1,  as expected, the remaining terms do not For t h i s reason, the equation i s not covariant,  and hence the space of solutions of the f i e l d equation does not form a representation space of the conformal group. 3-5» Independence of the generators The quantum f i e l d theoretic version of the four types of conservation laws associated with conformal transformations i s given i n section 2-5.  There, i t was assumed that each type of generator conserved  i n time i s independent of a l l the others, so t h a t , f o r example, no generator can be written i n terms of a l l the other generators.  On the o n e - p a r t i c l e  l e v e l , however, i t has recently been shown by Bracken  (1973) that f o r a  wide c l a s s of representations of the Lorentz group, f o r f i e l d s  which  s a t i s f y a massless f i e l d equation of the type (3.28)  c- b T k  Jk  V,Y = 0 ,  the generators y> and k j , when acting on Y , can be written i n terms of p"k and m j . k  metric tensors.  Here, A  i s f f o r spinors and 1 f o r second-rank antisym-  For example, f o r the spinor representation, given by (2.38d),  (3.28) takes the form (3.29)  (i/2) Y j J ^ f  0 ,  41 and  has the form  (3.30) where  ^ ^=  "6 T£  ^ i  e  5  i J k l  2^ ^  i , and  ^.123+- ^ *  s i o n s , with  (3.31)  = d/4) *  =+  ^  i  ¥  the permutation symbol i n four dimen-  s  i t e r a t e d version of (3«30) reads  e  (y^f) = -i(3+2S  y  .  mJ )Y . k  jk  For the scalar representation, although there i s no r e l a t i o n s h i p of the form (3.30), there i s an equation analogous to (3.31).  -l(2+m  (3.32)  jk  m^)^ .  which holds i f one assumes  (3.33a)  y = i(y\+1)  (3.33b)  »  j k  = m. = i ( k  y  j  Vy V k  and  (3.34)  V ^ C y ) = 0 .  One can also f i n d an expression f o r the generators k j , by obtaining k^ from the  relation  (3.35)  M  = ( n + 2  +  X  2  - 1 )Y  f o r general X , where n=(m^ . m ^ i ^ - j ) . Although these relationships show that i t i s possible to express the generators of d i l a t i o n s and s p e c i a l conformal transformations i n terms of the generators of translations and Lorentz transformations at the l e v e l of s i n g l e - p a r t i c l e quantum mechanics, t h i s i s not the case i n quantum f i e l d theory.  To see t h i s , one can attempt to construct an operator out of the  operators M j  k  such that $  2  i s given i n terms of M^jJM^ as i n (3.32).  Such an  i d e n t i f i c a t i o n holds only when the operators act on one-particle Fock s t a t e s , just as the operator P^P^ i s zero f o r massless f i e l d s only when acting on one-particle s t a t e s .  42 To see e x p l i c i t l y how such e q u a l i t i e s break down at the many-particle l e v e l , consider a complete set of s t a t e s , made up of combinations of the Fock states which are eigenstates of the momentum operators, labeled by the set of r e a l quantum numbers (j,ra;j ,m'\oL ). 1  They s a t i f y the set of  equations (Yao 196?) (3.36a)  {j -j(j+1)}|j,m;j\m';o<>  (3.36b  {J -J'(  2  =0  j',m';^>  )}  , 2  =0  (3.36c)  (J^-m) |j,m; j ' . m ' ;<*>  (3.36d)  (J« -m»)|j,m;j«,m« oc>  (3.36e)  (i<|>-°0|j,m;j\ni ;*>  3  ;  ,  =0 a 0 =0  where, defining Ij=(M23» 31 « 1 2 ) and N=(Mji|i .M^.M^-j), J M  M  a  and J '  &  (a=1 ,2,3)  are  given by (3.37a)  J  (3.37b)  J '  = £(L +iN )  a  a  a  a  = |(L -iN ) a  a  2  2  .  1  The f i v e operators J , J ' . J ^ . J ' ^ . I * can be simultaneously diagonalized i n any representation of the conformal group. When acting on t h i s b a s i s , the pertinent operators have the properties (3.38a) (3.38b)  ( +<* )jj,m;j«,m';c</ = 0 2  2  Y  { ^ k M ^ - ^ j + I J + j U j ' + O j l j . m ^ ' . m ' ; ^ =o  .  Comparing e q s . ( 3 . 3 8 ) with the i d e n t i f i c a t i o n (3«32), one can see that the relation (3.39a)  (j) = - i ( 2 + M . M ) 2  jk  k  holds only when (J) and M j  k  act on the set of states  o<. = 1+2[j(j+1)+j'(j'+1)]  (3.39b)  2  j j,ra; j ' ,m'\<*C] having  .  j u s t as the Klein-Gordon equation has the quantum f i e l d theory equivalent  P ^  2 only when P  acts on the set of s i n g l e - p a r t i c l e momentum eigenstates.  Hence  f o r a general l i n e a r combination of the basis s t a t e s , the r e l a t i o n (3.39a) does not h o l d .  2  43 4. Attempts at Including Mass 4 - 1 . Symmetry breaking As a f i r s t step i n considering the p o s s i b i l i t y of including the mass concept i n conformally covariant f i e l d equations, one can determine  the  e f f e c t s of mass terms on the conservation laws established i n section 2-5. By straightforward  application of the d e f i n i t i o n s of the currents  and K*^  k  to Lagrangians with mass terms, one finds that the conservation laws are broken by the presence of mass, so that the v i r i a l and complementum are no longer independent of time. In general, suppose the Lagrangian density £. i n the action i n t e g r a l (2.50) i s not d i l a t i o n a l l y i n v a r i a n t ,  but so that instead of (2.57) i t s  due to d i l a t i o n s (2.53). with constant parameter (4.1)  <r,  variation  i s of the form  -cr^ (y X)-crF k  k  where F i s some f u n c t i o n .  If  that i s the case then the derivation of the  conservation law (2.65) i s modified, i n the l a s t step from eq.(2.64), by substitution of  (4.2)  Otfr <r)  = -\(y /)-F k  instead of (2.58), leading to the equation <*.3)  = F  f o r the divergence of the d i l a t i o n current vector.  Thus the function F  can be said to break the conservation law (2.65). As an example consider the scalar f i e l d  Y described by the Lagrangian  density (4.4) If  X = Hi^J  the scale dimension of  Vf  -m Y ) . 2  2  Y i s taken to be -^=-1, and m i s considered to be a  parameter invariant under d i l a t i o n s , then the v a r i a t i o n of jC due to the d i l a t i o n (2.53) can be computed, using the transformation law (2.55)  44  (4.5)  V * t+<K-i-y\)Y  with the r e s u l t , f o r the case of constant G ', -  (*.6)  %£ = - c y y ^ ) - ( r m V k  2  2  .  Thus, the parameter m i s seen to break the d i l a t i o n a l invariance i n that i t gives r i s e to a function F as i n (4.1),  (4.7)  F = m ^ 2  2  .  Now the d i l a t i o n current (2.45 ), which reads i n t h i s case  (4.8)  is  D* =  y {(2/3)\Y ^ T - O / S H ^ n j n  conserved only i f m=0.  and  +(S73)(g VVM a Y)+ n  k  n  k  Equation (4,3) can be worked out i n terms of ^  t  i s seen to hold as an i d e n t i t y on account of t h e f i e l d equation  (*.10)  V ^ - h ^ Y  =0  that flows from the action p r i n c i p l e with (4,4) as Lagrangian d e n s i t y . 4-2.  Conformally covariant mass  I f one regards the conformal transformations (1.25c,d) as (spacetime dependent) changes of u n i t s , then, as was pointed out i n s e c t i o n 1-1, i t is  necessary that the numerical value of $ must vary, i n a spacetime depen-  dent f a s h i o n .  Then requiring that at each point i n Minkowski space a new  unit of mass be chosen, such that $ has the new numerical value 1, means that the unit of mass must be transformed oppositely to that of lengthy i n order, that e . g . the speed of l i g h t c written i n terms of the Compton wavelength ^ , c $ / ( X n O . be l e f t unchanged. Here one invokes the assumption that the numer=  i c a l value of the mass m i s unaffected by changes of the length u n i t , so that the change of the mass unit i s a separate transformation. Thus, accompanying every unit transformation by a corresponding mass unit transformation which holds the numerical value of )ji at unity amounts to the assumption of an i n f i n i t e s i m a l v a r i a t i o n ( c . f . Fulton e t . a l . 1962a)  45 Sra = < f b m in  (4.11a) ' x  and (4.11b)  oBs2pJyjj^m  with - ^ - 1 1 f o r d i l a t i o n s and s p e c i a l conformal transformations r e s p e c t i v e l y , =  m  for  a mass m which i s "conformally covariant" i n t h i s sense. For a s c a l a r p a r t i c l e with mass , s a t i s f y i n g the equation  (4.12)  (Yhj+m )^)  = 0 , '  2  the transformation property o f Y under d i l a t i o n s i s assumed to be (*.13) Employing (4.14)  Y ' ( y ' ) = <riY(y) + Y(y)  .  the d e f i n i t i o n ( 4 . 1 1 a ) , one finds that  [<V>y )My' )+a ] P(y») = ,d  ,2  v  j  = (1-2o-+/ r)> ^Y(y)+(1+24^+^Om f(y) j  =  2  <  = V-ZS+JcX^+m^VM  =0 ,  where the second e q u a l i t y r e s u l t s from s e t t i n g ^^=-1 •  Then one can say  that i n the new system o f u n i t s , the f i e l d equation (4.12) i s again s a t i s f i e d . I t i s i n the sense o f eq.(4.14) that the anomalous transformation laws (4,11) are u s u a l l y employed (e.g. Hamilton  1972).  However, i n Lagrangian f i e l d theory, one must use the complete transformat i o n property of the f i e l d ,  (M5)  Y»(y> - Y(y) +  o - ( i - y \ ) Y(y)  .  Using t h i s transformation law f o r the f i e l d , but continuing to use the v a r i a t i o n (4.11a) f o r the mass, one finds that i t i s not p o s s i b l e to r e t a i n covariance o f the Klein-Gordon equation (4.12) under d i l a t i o n s , as long as the mass i s non-vanishing, since (4.16)  y ^ Y ' t y ^ Y ' t y )  = (i+icr-2<r-2^y ^^f(y)+ k  (1+^+24or--(r'y^ )m f(y) = - C y \ ^ W 2  +  k  «m o-y\Y(y) . 2  7  )  =  46 These calculations have been based on the behaviour of eq.(4.12) under d i l a t i o n s , and since the parameter  a" was assumed to be independent of  spacetime, the mass m was assumed to undergo at most a uniform scaling  by the f a c t o r O+^j ')* 0  However, the transformation law (4,11b) of the  mass under s p e c i a l conformal transformations involves multiplying m by a  0 f*Vj4n^* +2  spacetime dependent factor  Indeed, the p o s s i b i l i t y o f a  coordinate dependent d i l a t i o n parameter cr was e s s e n t i a l f o r the d e r i v a t i o n of the conservation law  (2,65),  so that i t might seem reasonable that  by requiring that m too be looked upon as a function of the coordinates, one might r e t a i n the d i l a t i o n a l covariance o f the f i e l d equation even when the mass term i s present.  In t h i s case, the anomalous transformation laws (4.11)  must be replaced by variations which i n analogy to (2.35c,d) are given by  (4.17a)  W = <T ( i ^ - y ^ m  (4.17b)  Sm = H<f  fyyXVfX^ty^j)}"  f o r d i l a t i o n s and s p e c i a l conformal transformations. The transformation law (4.17a) serves the purpose of restoring the d i l a t i o n a l invariance o f the action i n t e g r a l , since i f one chooses scale dimension o f X. i s J.^ =-4,  -4n-1 =  then the  and the v a r i a t i o n of X. i n the case of  constant <T i s a divergence  (4.18)  S t = -<r\(y £) k  s  -«-yy /)-mW-m^ Sm-rmY y ikin 2  k  2  k  .  However, although the action i s i n v a r i a n t , the f i e l d equation i s no longer satisfied i f  Y ( y ) i s replaced by Y ' ( y ) and m by m'=m+Sm given by (4.17a).  In f a c t , f o r the Klein-Gordon equation (4.12) one finds  +(1+icr'+2i^-2<r'y d ) ( m Y ) + m k k  2  +*-y\(* O . 2 v  In a d d i t i o n , the d i l a t i o n current (4.20)  D = - ^ f + y \ n  n  D",  as defined i n  (2.43c)  ,  i s no longer conserved, and f o r the s c a l a r f i e l d one has  (4.21)  D , = m ^+my (^ m)H n  2  k  n  ;2  k  47  .  The reason f o r t h i s i s that i n the derivation of the current (4.20) i t  is  now necessary to include the v a r i a t i o n &m of the mass, so that  (4.22)  O^/W)  =  O^AYr?  +(^A*)m  + * 1$ n  where  (4.23a)  (4.23b)  9=  d - y \ ) Y  m = (4-3^) m .  Combining (4.22) with the r e s u l t  (4.24)  [fct/aftn*-)] = ^ T Y  .  n  with the help of the f i e l d equation (4.12), one obtains  (4.25)  [ } c r ) ]  .„ = C * £ A < r - ) - U ^ m ) m  .  Hence, whenever (4.18) h o l d s , so that  (4.26)  = -\(y .£) k  .  one has  (4.27)  D , = U//>m)(Vy\) » • n  n  The transformation laws (4.17) are seen to elevate the status of the mass to that of an independent f i e l d v a r i a b l e , and indeed unless t h i s step i s taken, the f i n a l term i n (4.22) i s absent.  This omission leads to the  replacement of the r i g h t hand side o f (4.27) by zero, i n disagreement with the e x p l i c i t c a l c u l a t i o n (4.21),  However, the approach leading to (4.22)  necessitates as a concomitant circumstance the existence of an a d d i t i o n a l f i e l d equation f o r the mass, flowing from the action p r i n c i p l e .  For the  Lagrangian corresponding to the scalar f i e l d , this equation i s simply  (4.28)  (SjC/Sm) = U X A m ) = -m Y  2  = 0 .  C l e a r l y , t h i s requires either m=0, thus reverting to the d i l a t i o n invariant massless case, or y =0, which amounts to the empty case of no f i e l d whatsoever.  48 4 - 3 . Interpretation of d i l a t i o n s The various interpretations section 1-1  of conformal transformations outlined i n  can now be evaluated i n the l i g h t of the above considerations.  F i r s t l y , i t i s evident that the i n t e r p r e t a t i o n of d i l a t i o n s as uniform changes of the u n i t of length means that equations with mass terms possess symmetry under changes of u n i t s , i n the sense that the f i e l d equation with Y , n, and y replaced by  v  f',  m , and y* i s again s a t i s f i e d . 1  However,  one cannot e s t a b l i s h a conservation law i n Minkowski space associated with t h i s symmetry*, since  when (4.13) and (4.11a) are obeyed, one has f o r a  s c a l a r f i e l d with Jt=-\  [using (4.18) with ^ = 0 ,  (4.29)  and (4.25) with m=-m]  D , = m Y . n  2  2  n  Also, since the metric  i s l e f t unchanged, under conformal transforma-  tions as discussed here, these transformations are not simply changes of the coordinate system. (c.f.  Because u n i t transformations are ruled out by (4.29)  Coleman and Jackiw 1971, p.555), one may i n t e r p r e t d i l a t i o n s as  mappings from points i n Minkowski space onto d i f f e r e n t points i n spacetime— the f i r s t i n t e r p r e t a t i o n suggested i n section 1-1 — and i n t e r p r e t s p e c i a l conformal transformations as spacetime dependent  dilations.  This i n t e r p r e t a t i o n i s analogous to " d i l a t i o n s " i n n o n - r e l a t i v i s t i c macroscopic physics, i n which the space and time variables are allowed to transform separately.  For example, i n the Kepler problem the o r b i t of  an object i s transformed i n t o another possible o r b i t provided the period i s transformed with the 3/2 power of the parameter specifying the d i l a t i o n  'One can force (4.29) into the form of a conservation law by inventing a vector f i e l d gj s p e c i f i c a l l y f o r t h i s purpose (Hamilton 1973), such that  but t h i s seems somewhat a r t i f i c i a l i n view of the f a c t that according to (4.16) the f i e l d equation s t i l l can not be made covariant under the t o t a l variations of the f i e l d (4.15), oven with the aid of such a device.  49 of the semi-major axis of the o r b i t e l l i p s e . In f a c t i t i s possible to determine even larger groups of transformations of t h i s s o r t , corresponding to n o n - r e l a t i v i s t i c versions of the conformal transformations with space and time transforming d i f f e r e n t l y ,  which are  symmetries of the Schrodinger equation including mass terms (Barut 1973). 4-4. Other approaches The conclusion r e s u l t i n g from (4.28), that mass m breaks the invariance of a massless f i e l d for the case of constant  dilational  <r and a f i e l d  Y  defined as a function of 4-dimensional Minkowski space, could conceivably be avoided by augmenting the Lagrangian density £. by a d d i t i o n a l terms containing m, and possibly the derivatives ^ m , i n expressions other than  2 ? m f  .  However, such a procedure would necessarily be somewhat  arbitrary,  due to the absence of any guiding p h y s i c a l p r i n c i p l e s by which to construct a "mass f i e l d equation" that would avoid the consequences of (4.28). Even remaining within f i e l d theory i n 4-dimensional spacetime there i s  still  the p o s s i b i l i t y of assuming that l o c a l conformal invariance obtains, characterized by parameters that vary from point to point i n a Riemannian 4-space with l o c a l i n e r t i a l coordinates y^«  Then, one could construct  invariant actions provided s u i t a b l y defined compensating f i e l d s , such as the "dilaton f i e l d " (Rosen 1971), were introduced.  However, since there i s ,  at present, i n s u f f i c i e n t experimental evidence f o r the actual existence of such f i e l d s invented f o r t h i s purpose, the treatment of compensating f i e l d s i n 4-dimensional spacetime w i l l not be pursued here. There remains the p o s s i b i l i t y of considering f i e l d s i n higher dimensional spacetimes with projections onto 4-dimensional space s a t i s f y i n g equations resembling the equations governing conformally covariant f i e l d s i n Minkowski space.  The examination of t h i s p o s s i b i l i t y i s the purpose of the remaining  sections i n t h i s  report.  PART II.  50  FIELDS IN HIG HER-DIMENSION AL SPACES  5. Generalization of Minkowskian Spacetime  5-1.  Introduction  D i f f i c u l t i e s associated with attempts at encompassing the mass concept within conformally covariant f i e l d theories were outlined i n the previous sections.  The remainder of t h i s report consists of an examination of  f i e l d theories i n spaces which are generalizations of Minkowskian spacetime, i n an e f f o r t to extend the a p p l i c a b i l i t y of conformal symmetry to the case of massive p a r t i c l e s . There are many possible approaches to t h i s problem (Murai 1958. C a s t e l l  1966, Ingraham 1971, Wyler 1971. Barut and Haugen 1972), but i t seems reasonable here to formulate a theory that extends the theory of conformal symmetry 1  i n 4-dimensional space yet does not a r b i t r a r i l y depart from i t ,  i n the  sense that the f i e l d equations derived i n higher-dimensional spaces should be expressable i n terms of f i e l d s i n Minkowski space which transform according to a transformation law which c l o s e l y resembles that associated with a representation of the conformal group, as given i n (2,36).  This  means, i n p a r t i c u l a r , that when written i n terms of Minkowski space v a r i a b l e s , i n keeping with the discussion of section 2-6 these f i e l d s should possess a d e f i n i t e value f o r the scale dimension, so that equal-time commutation r e l a t i o n s can be imposed on free f i e l d s and a quantization procedure can be c a r r i e d out. 5-2, Descent from 6-dimensional space to Minkowskian spacetime In section 1-4,  i t was shown that conformal transformations i n Minkowski  space can be l i n e a r i z e d by the introduction of the conformal coordinates * ^ , as defined i n (1.34) and (1.39). A  assumed to vanish i d e n t i c a l l y .  There, the variable L=Ylk\k  was  For t h i s case, Mack and Salam (1969) have  51 shown that by defining f i e l d s over the coordinates v^  manifestly conformally  t  covariant f i e l d equations can be formulated i n 6-dimensional space f o r f i e l d s which correspond to massless s c a l a r , spinor, and vector f i e l d s i n 4-dimensional space. Perhaps the most obvious generalization of Minkowskian spacetime, suggested by the conformal coordinates  v \ \ i s obtained by enlarging the arena of  physics to the entire 6-dimensional space with metric S^g=diag(-1,-1,-1 ,+1 ,-1 ,+1 ). This enlargement amounts to introducing the s i x independent variables y , vt , L related to the  by  k  (5Ja)  (5.1b)  y = ^ /(^ -^ ) ; * = ^ - ^ k  k  6  v ^ x y *  5  5  ; L=^  ^ ^ ( L - ^ - ^ y  ;  2  * y +K  \f  A  ) ^ ^  2  ;  (^6)  2  \ =1 ={L+ * 6  2  $  with the understanding that the constraint (1.40) does not h o l d . proposition one conceives a l l physical f i e l d s variables (5.2)  -  ^y )/ " 2  By t h i s  7~ as functions of s i x  «\ , A  M\)  = X«(y .*,L) k  defined i n the entire 6-dimensional space and considered, i n p a r t i c u l a r , as functions of L unless stated otherwise.  Such a function w i l l be said  to be defined "on the hyperquadric" only i f the s p e c i a l case L=0 i s assumed. The d i f f e r e n t i a t i o n operators the d i f f e r e n t i a t i o n operators  (5.3)  7  A  = (oy /M )^ k  A  k  jf^^A^ V  + (°  K  can be expressed i n terms of  »  / H  A  a  s  ) V  +  OL/H )> A  L  .  E x p l i c i t l y one has  (5.3a)  M l / * ) V 4 2 n * = 0 / * ) ^ ,+2Ky^  T  ,  / * )2>J+2xyJ>  2  52 (5.3b)  * K 1U  (5.3o)  ^ =-(1/K)y\+^+2^ =-(1/H  so  )y\-V  5  +  2^ =(i / * )y\-V  - [ ( 1 / * ) - * - * y ]> =-^ 2  l  6  L  5  L  )y a +^ + [ < L / K ) + X k  R  - n y  2  ] ^  6  that  i\ Jf =  V + 2Lb  A  (5.*) The  L  .  d i f f e r e n t i a l generators of the rotations i n 6-dimensional space,  defined as  (5.5)  « »i<TA-WA> ffl  i n analogy to the corresponding operators (2,20b) i n 4-dimensional spacetime, read e x p l i c i t l y (5.5a)  m.  (5.5b)  m  and  k  =  5 k  k  i[(n /x- J V v ^ n V V l 5  ^-ipW^VvVrk***]  (5.5c) (5.5d)  = i<yjo -y ty  k  m  =  6 5  i ( y \ - > O j  are seen not to contain o , ii  (5.6)  m  6 k +  m  5 k  = i^  . i.e.  One has also  = ^fo  \  ^ Jf  v ^ - ^  ^  using the t h i r d r e l a t i o n ( 5 . 1 a )  and,  V V'vV ^ J = 27  <*-7>  m  6k- 5k = m  [*~ H  1  1  = i[2y (y\-K^)-y^ k  (L/K )^ J 2  K +  K  .  An i n f i n i t e s i m a l r o t a t i o n (1.43)  (5.8)  ^  A  = E  A B  n  B  characterized by the parameters E  A B  as s p e c i f i e d i n  (1.44)  w i l l now have  53  an e f f e c t on the Minkowski coordinates y^, defined by (5.1 a ) , which genera l i z e s i n f i n i t e s i m a l conformal transformations to the case L / O ,  ly  (5.9a)  = ^^^^vJ+o-y^+ayyJy^V+^CL/M- ) 2  k  (5.9b)  -ff-K-2(^y)K  SL = 0  (5.9c)  .  The l a s t term i n the f i r s t equation ( 5 . 9 a ) involves only the parameters of the s p e c i a l conformal transformations and, f o r a p a r t i c u l a r L and X , takes the form of an extra i n f i n i t e s i m a l t r a n s l a t i o n added to <*• . By defining a quantity r = ( L / v ) t 2  the v a r i a t i o n s ( 5 . 9 ) can be looked  ?  upon as transformations from the set of coordinates ( y , r ) to the set k  (y  , k  , r « ) so that  (5.10)  o 3^= < * + € ^ ( t r + 2 ^ y ) y - ^ ( y - r ) ; S r=( <r+2y*y)r . k  k  k  k  2  2  The coordinate r can be interpreted as the radius of a "sphere" i n Minkowski space with center y^, consisting of the set of points 2  ("^-y) =*"  such that  2  (Ingraham 1 9 7 3 ) .  One can define the "angle of i n t e r s e c t i o n "  of two such spheres by (Ingraham 1971) (5.11)  r' +r -fv'-v} 2  2  2  In the i n f i n i t e s i m a l case, y ^=y^+dy^, r r + d r , t h i s reduces to ,  (5.12)  d6  2  = r- (dy -dr ) 2  2  2  , =  ,  and i t i s e a s i l y demonstrated that t h i s generalizes the angle between the l i n e s joining the centers of two c i r c l e s to t h e i r point of i n t e r s e c t i o n . The angle 8 can also be looked upon as the inverse cosine of the inner product of two normalized vectors i n the 6-dimensional space, since one has  (5J3)  cos9 = ( vjA  )/(  L  L')*  54 Inspection of this d e f i n i t i o n shows that the variations (5.8), and hence the variations (5*10), comprise those i n f i n i t e s i m a l transformations which preserve cos0. On the other hand, the conformal transformations i n a 5-dimensional  (k=1 ,2,3,4,7)  space with metric diag(-1 ,-1 ,-1 ,+1 ,-1 ) and coordinates z r e s u l t i n variations with 21 parameters (5.14)  *z =* +fe .B +(<r+2^a)z -; z k  k  J  k  k  k  (k=1,2,3,4,7)  2  .  3  7  7  7 =p =0, then, upon  Now, i f one f i x e s 6 of these parameters by writing k k 9 making the i d e n t i f i c a t i o n s z =y f o r k=1 ,2,3,4 and z'=r, the variations  it  (5.14) with the remaining 15 parameters reduce to the v a r i a t i o n s (5.10). Therefore, one has the option of considering the rotations i n 6-dimensional space as a s p e c i a l case of the conformal transformations i n a 5-dimensional space. Removal of the constraint L=0 also means that the inversion  (5.15a)  v^' = - n f ; ^ 5  =^  k  ;^  6  = ^  amounts to a generalization of the transformation by r e c i p r o c a l r a d i i (1.49),  (5.15b)  y» = -y /(y*-r ) k  k  2  ; vO = - * ( y - r ) 2  .  2  The generalization of the i n f i n i t e s i m a l s p e c i a l conformal transformations occurring i n (5.9) i s recovered from t h e i r f i n i t e versions (5.16)  y' =[y -b (y -r )]/[l-2by+b (y -r )] ; = * [l - 2 b y + b ( y - r ) ] k k when the f i n i t e parameters b are replaced by the i n f i n i t e s i m a l p . As i n k  k  k  2  2  2  2  2  2  *Here one assumes L ^ O . For L < 0 , the quantity r i s imaginary. / = i r , so that W =«< +( yJ+((r+2^y)y -; (y2 /2 K  k  k  k  j  k  +  )  2  2  By defining  . /> = ( < r j S y ) / S  +2  one can see that f o r L<.0, the above v a r i a t i o n s correspond to a subgroup o f the transformations (5.14), but with the 5-dimensional metric replaced by  diag(-1,-1,-1,+1,+1).  55 (5*16) i s decomposable into a r e c i p r o c a t i o n (5*15). a  the case r=0,  t r a n s l a t i o n , followed by another r e c i p r o c a t i o n . The r o t a t i o n (5.8) the  transformation  (5.17)  &X = - ( i ^ E ^ J ^ X  where the m^_  =  ^x/Z)E (m s )X AB  AB+  are given as i n (5.5). and the s  f i n i t e dimensional of  i n 6-dimensional space w i l l induce on the f i e l d Ti-  X-according  kB  are the appropriate  representations of the rotations acting on the indices  to i t s transformation character as s c a l a r , spinor, vector, etc.,  i n that 6-dimensional space, r e s p e c t i v e l y . The operators j^g read e x p l i c i t l y , i n a notation analogous to (2.24),  (5.18a)  p  k  (5J8b)  »  j k  (5.18c) (5.18d)  =  J +7 = iV< 6k s  6k  = J  ? s  j k  T  65  +8  5k  =  i(  y  j  5k>  \-y v)^ k  j  = i ( y \ - H>^)+S  'V^k  S  k  6 5  k. ~= T.-T . - i[2y/>\-(/-r )l 2y.^}  ^ - s ^ )  2  6  5  f  On the other hand, the canonical form of the generators  .  of conformal  transformations i n 5-dimensional space, defined i n analogy to ( 2 . 3 5 ) . i s  (5.19a)  p  k  (5.19b)  i  j k  (5.19c)  y  (5J9d)  kj = i [ 2 » j ^ - * ^  \  (k=1,2,3,4,7,  >  k  =  ^/bz ) k  = i(«^ -VjJ (5)jk +<r  with l a b e l s (5) corresponding  = i  k  =  i[z\-i  ( 5 )  ] 2  n  appended to  5 )  « >« J  ,  , 1  ^(5)jn »(5)J +  to d i s t i n g u i s h them from the  representations i n 4-diraensional space.  56 When one s p e c i a l i z e s the c o n f o r m a l t r a n s f o r m a t i o n s i n t h a t 5 - d i m e n s i o n a l ot^-fl =p?=0 t h e r e m a i n i n g components t h a t g e n e r a t e t h e  space t o t h e case  (5.10) a r e (k=1 ,2,3,4,  transformations  z }„=y ^ +r} ) n r n  K (520a)  p  k  (5.20b)  m  jk  (5.20c)  f =  ( .20d)  c  5  = i^  k  k  (5)jk  i[y\+iV^(5)] it^CyV^rMy ^^  =  d  k  k  i(y^ -y ^)+<r  =  k  2  and t h e y a r e t o be compared w i t h p^, m^,</> »  j  k  a  s  gi  v e  n above i n ( 5 . 1 8 ) .  I n k e e p i n g w i t h t h e remark made f o l l o w i n g (2.28), s u c h a c o m p a r i s o n must be p r e c e d e d by a s i m i l a r i t y t r a n s f o r m a t i o n , c a r r i e d o u t i n t h e i n d e x space of the f u n c t i o n s (5.21)  X,  = u^u"  ;  1  *-=u x.  so t h a t t h e r e i s no " i n t r i n s i c momentum" ^ (5.22)  Up U~  1  k  = U(i^ +Tr )u" k  1  k  = p  k  S s k  = i^  5i  k  + s c  ^j » c  .  Now, t h e v e r y appearance i n (5.20d) o f t h e s p i n o p e r a t o r <T. . t h a t one c a r r i e s o u t t h i s t r a n s f o r m a t i o n f o r t h e g e n e r a t o r s 7 - d i m e n s i o n a l space w i t h m e t r i c  %  requires i na  =diag(-1,-1,-1,+1,-1,+1,-1), because  AB these generalized r o t a t o r s correspond t o t h e e n t i r e s e t o f conformal generators  (5.19) i n t h e 5 - d i m e n s l o n a l s p a c e .  (5*22) i s t h e n p r o v i d e d b y  (5.23)  U = e"  i Z  (k=1,2,3.4,7)  The s o l u t i o n o f t h e problem  57 where TT^ i s the f i n i t e dimensional representation of the t r a n s l a t i o n  7 i n z - d i r e c t i o n , given i n terms of the rotation matrices s i n 7-dimensional AB space as (5.24) TT = s + s . ?  6 ?  5 ?  Indeed, since a l l TT^ commute [see the f i r s t r e l a t i o n (2.23)] , the inverse of the operator (5.23) i s (5.25)  IT  = a  1  *^*  4 1  and, therefore, one has ( f o r a l l k=1,2,3,4,7)  (5.26)  U(i\)U"  = i ^ - i ^ " ^ . ^ ^  1  . . . = i^-TT  k  .  Thus, the operator (5.23) has, i n p a r t i c u l a r f o r k=1,2,3,4, the desired property (5.22). (5.27)  For the case k=7, the equation (5.26) implies the r e l a t i o n  U( * V ) U "  1  = * X -irir  ?  on account of the i d e n t i f i c a t i o n z ^ = r = ( L / v . ) , and an e x p l i c i t c a l c u l a t i o n 2  2  verifies this result. Now one can compute  (5.28) Since, on account of the commutation r e l a t i o n s (2.23),  (5.29)  [A ,m. ] n  =  k  [yS^.fiCy^^jHs^] =  one has simply  (5.30)  m. = m. k  k  and comparison of (5.18b) with (5.20b) y i e l d s the i d e n t i t y (5.3D  <r  )jk- jk s  ( 5  •  58  S i m i l a r l y , f o r the computation of  (5.32)  ^ = Uff  1  =y>-i[z TT ,^]+ ... n  n  one requires evaluation of the commutator, with the r e s u l t  (5.33)  t%.v]=[yV 7^ rlT  i(yk  ^- *V) »6Jfl " - ^ n ^ C [ 6 n 5 n ' 6 5 ] > = ° +  s  + s  s  and again one has simply  (5.34)  y = ?  y i e l d i n g upon comparison of (5.18c) with (5.20c) the i d e n t i t y  (5.35)  ^(5)- ^r = ^ r  + i s  65 •  F i n a l l y , one can compute (5.36)  k j = UkjU"" = - i [ z 7 r . k ] - i [ z r . [ z i r , k ] ] + . . . 1  k  n  m  n  j  n  n  j  m  j  The commutation r e l a t i o n s (2.23) give i n t h i s case  (5.37)  [ X' j]  (5.38)  [ z  z  k  n V  =  - yj 2 i  z r X 7 r  [z\,k.] ] =  n  + i z 2 i r  j  + 2 i z n s  k. = k . 2 z * s . +  n +  2  7 j  + 2 i y  j 65 s  -ky \+2z ^ 2  f  and vanishing commutators of higher order.  (5.39)  jn  s  6 5  One has thus  .  Upon substitution, of (5.18d) and using the i d e n t i f i c a t i o n s ( 5 . 3 0 and (5.35) one can now compare (5.39) with (5.20d) and obtain the i d e n t i t y  To give the generators (5.20) meaning, they must now f i n a l l y be compared with the canonical form (2.35) of the generators of conformal transformations i n 4-dimensional Minkowskian spacetime.  Then i n terms of the quantities  . K . i n 4-dimensional space one i s l e d to the i d e n t i f i c a t i o n s  (5.41b)  X =^  (5.41c)  K  k  -r^  ( 5 )  = H  (  5  )  k  = ^ ,  r  = s  6 k  -s  + i  s  6 5  .  5 k  '5-3. Necessity of the choice of descent operator The operator U, defined by (5.22), was chosen such that i n the new b a s i s , there was no i n t r i n s i c momentum i r = s ^ + s ^ » k  k  k  However, (5*23) i s not  the only possible operator which s a t i s f i e s (5.22), since i n order to remove T  (5.42)  k  from p> f o r k=1,2,3,4, a l l that i s required i s an operator k  = e- y i  W  k i r  k  (k=1,2,3,4)  ,  which has the desired property without making necessary the introduction of the 7-dimensional rotation matrices s^g (A,B=1-7). In f a c t , the matrix W i s the one which i s customarily u t i l i z e d f o r t h i s pupose, whether L i s taken to be zero (Mack and Salam 19&9), or not (Ingraham 1966, 1971; Barut and Haugen 1972, 1973). However, even without the motivation of introducing a f i f t h coordinate r one can demonstrate the necessity of enlarging the dimension of the matrix representation space, when one leaves the hyperquadric L=0,  To see t h i s ,  one must perform the s i m i l a r i t y transformation W( )W~^ on the generators JAB' with the r e s u l t s (5.^3*)  Wpw" =  1\  (5.43b)  wSyr  i( H-y V  (5.43c)  W?W = i ( y \ - * - V )  (5.43d)  Wk.w" = ifa.fy^  1  k  1  =  + S  K  y j  _1  1  -  v  t  J K  + s  6  5  ^ _  i s  W  2  v  - 2 ^ 1 , k^J i " - '  6  j  -s  5  j  ;  - *.  ' M .  6  j  «  5  j  ;  j V  60 Noting that the f i r s t three types of these generators are i d e n t i c a l with the r e s u l t s found using the operator U , there i s no d i f f i c u l t y these operators i n terms of the generators p^,  m  interpreting  j » <pt making use of the k  i d e n t i f i c a t i o n s (5.41 a , b ) . However, the fourth type of generator can not be compared with the canonical form (2.35d) of the generator of s p e c i a l conformal transformations, because of the presence of the term - vC L ( s £ j + s , ^ ) .  On the other hand,  neither can i t be compared with the generator of s p e c i a l conformal t r a n s formations i n 5 dimensions (5«l9d), because of the absence of a term of the form 2z' ^ ( ^ J j ? *  The s o l u t i o n to t h i s problem i s provided by augmenting  the exponential y^TT^ i n W by p r e c i s e l y the extra terra i n terms of  where r i s given  r7T , 7  M- and L as i n s e c t i o n 5-1 . and where IT^ must be given i n terras  of matrices s ^ , S £ which enlarge the s^g algebra to the 7-dimensional space. 7  7  Since the operator Wk^W  i s u s u a l l y not written down, the problem of  i t s interpretation i s u s u a l l y not confronted.  When i t i s c a l c u l a t e d  e x p l i c i t l y (Ingraham 1966), the group of rotations i n 6 dimensions i s taken a p r i o r i as the fundamental symmetry group, and a comparison with conformal transformations, i n 4 or 5 dimensions, i s foregone i n favour of developing a s e l f - c o n s i s t e n t quantum theory based on the If  6-rotations,  one i n s i s t s , however, that contact be maintained with one of the  conformal transformation groups i n either of the Minkowski-type 4- and 5-dimens i o n a l spaces considered above so t h a t , following the programme established i n section 5-1. f i e l d variations take on a form simply related to the canonical form (2,35). then one i s l e d to the operator U, and not the usual operator W, As w i l l be seen, t h i s operator has the e f f e c t of transforming r o t a t i o n a l l y covariant f i e l d equations i n 6 dimensions into manifestly  translationally  invariant f i e l d equations i n Minkowski space, and therefore U (or W) i s referred to as a "descent operator".  61 5-4.  Eigenfunctions of JL f o r the case  L=0 1969)  At t h i s point i n the development, some authors (Mack and Salara introduce the concept of the "physical components" of the f i e l d by requiring that they possess no i n t r i n s i c complementum K y motivated by the observation that, acc ording to (2.23), i f £  X=U  X'  This i s were a  c-nuraber [and not a matrix as defined i n (5.41b)] , then K . would commute with Ji and hence i t s e l f vanish.  Thus, by selecting these components  of solutions of f i e l d equations i n 6-dimensional space, one should be able to arrive at f i e l d s i n Minkowski space that possess a d e f i n i t e value of the scale dimension and, according to the arguments presented  i n section  that transform according to an i r r e d u c i b l e representation of the  2-6,  Lorentz  group. I f one works i n a representation i n which sg^ i s diagonal, then the requirement that X  be an eigenfunction of X  ( i . e . that a l l i t s components  be assigned the same value of the scale dimension i n 4-space) amounts, on account of (5.41b), to the requirement that i t be an eigenfunction of Denoting the corresponding  (5.^) If  K\X  eigenvalue by n, one has the requirement  = nX .  X i s a f i e l d which includes the surface L=0  i n i t s domain of d e f i n i t i o n ,  and i f (*o L2OL=0 i s f i n i t e , then comparison with (5.4) to hold, on the hyperquadric L=0  shows that f o r (5.44)  X must be a homogeneous function of  degree n, i . e .  v^  (5.^5)  A  y> X = n X k  f o r L=0  .  Mack and Salara (I969) work with f i e l d s that are assumed to be defined only on L=0,  so that the second term i n (5*4)  can be neglected, and f o r  these f i e l d s one can assume that (5.45) holds.  The  "physical components"  are obtained by application of a projection operator E acting on the vector space ^  .  of the matrices s^g so that i n the new  vector space ^'=E  7*  62 (5.46)  = EK.E  = 0  (but K .E / 0 i n general)  .  The discussion of section 2-6 indicates that such an operator exists and that i t has the additional property of leaving the spin operators unchanged i n W-' (since a l l the (5.4?)  s«  j k  s^  can be put i n block-diagonal form simultaneously),  k  = Es- E = s . k  in  k  .  For the case L=0, one uses the descent operator W as given i n (5«42), and since  (5.45) implies  v*-\^ commutes with W,  (5.44a) for L=0.  x^X  =  nX  Then the projector E can be shown to diagonalize the operator  isgj  i n ^ty.' with diagonal elements containing only the highest r e a l eigenvalue X  Q  of that operator so that (5«41b) i s replaced by  rXphys " (  n+iEs  65 ^phys E  =  ( n + A  o^  p h  ys  f  °  r  L =  °  '  In other words, /ty- may be characterized as the eigenspace belonging to the 1  is^«  highest r e a l eigenvalue of  These authors then go on to make a further transformation of the f i e l d  with  by m u l t i p l i c a t i o n (5  .49)  y  a K-n X  powers of p  h  y  K,  .  s  so that on L=0  Yis  ordinary f i e l d  i n Minkowski space with X , they assume that instead i t i s  Y  a function of y  k  only.  Now  instead of i d e n t i f y i n g the  that i s to be i d e n t i f i e d with the "physical f i e l d " .  Once the condition  (5.44) has been imposed, the m u l t i p l i c a t i o n (5.49) commutes with the diagonal matrix  ^' as given by (5.48), leading to the assignment of the  same scale dimension  (5.50)  5  J}  ^ U -  X,  to *f as was assigned to n  X  X,  p  h  y  s  )  = (n x )( * - X n  +  0  p  h  y  s  )  .  63 Obviously, however, there i s a point being passed over here, i n that i n t u i t i v e l y one expects the scale dimension of Y to be modified by the scale dimension of v<_", which according to (5.9b) i s given by J[ vC" =nK~ . n  n  n  How one would arrive at (5.50) even without making the s u b s t i t u t i o n (5.44) i s by carrying out the transformation on both sides of (5.41b),  (5.5D  i"ysvO ^X  =*- (vOv+is )K x- X = ( n + K V + i s ) t .  n  n  n  n  65  6 5  so that a f t e r substitutuion of (5.44), Y * i s independent of y- and one regains the i d e n t i f i c a t i o n (5.50). However, there i s no reason to carry out the manipulations involved i n (5.51)  and one should define the scale dimension i  of f by  writing  (5.52a)  IV  = K" (n+ n  *V  + i s 6 5  >*  =(2n+^V+i^)^  so that when (5.^*0 holds, one has  (5.52b)  5-5.  ^=(2n+> )Y o  .  Eigenfunctions of J. f o r the case h^O  When the formalism i s extended to the case L?*0, as was shown i n section 5-3 one can no longer use the descent operator W, but instead one must make use of the operator U, as given i n (5.23).  I t i s again possible to define the  projection operator E having the properties (5.46) and (5**7)» such that E diagonalizes the matrix i s g ^ * H n/  diagonal.  and leaves only >  %  Q  appearing i n the  Indeed, from the point of view adopted i n t h i s work, there i s no  good reason why attention should be r e s t r i c t e d only to the highest eigenvalue of i s ^ . \  Therefore, the f i e l d components belonging to eigenvalues other than  w i l l also be considered, whenever they can be obtained by a p p l i c a t i o n of  additional projection operators E  (5.53)  I = E + % + E + ... + E 2  ±  64 such that  (5.54a)  K ' j = E K jE = 0  (5.54b)  s.  j k  =Es  E =s .  j k  k  when there are i d i s t i n c t values of the eigenvalue  X apart from  \  q  .  Thus, corresponding to a r o t a t i o n a l l y covariant f i e l d i n 6-dimensional space, one can i n general f i n d several f i e l d s i n Minkowski space , each belonging to the subspace of a projector E as i n (5.53). which w i l l  transform  under d i l a t i o n s as eigenfunctions of the scale dimension X i f the condition  (5.44) i s imposed as a requirement. However, f o r L^O, K . ^ that even i f of  (5.55)  ,  X  K  no longer commutes with the descent operator, so  s a t i f i e s (5.44), X i s not n e c e s s a r i l y an eigenfunction  Since one has K  >AJU> "  V[e-^-ia*/*  = i(L2/*0Tr  a= i  7  rir  W  ?  7  ]  =  u,  one finds that  (5.56)  * - V X = i n r X + U vO>^ X ?  and s i m i l a r l y  (5.57)  2L^) X = - i r ¥ X + U 2 L ^ X L  ?  L  .  Combining these two equations one obtains the r e s u l t  (5.58)  I* V -  =  U  ^ ^ A ^  •  and hence f i e l d s X that are homogeneous functions lead to homogeneous functions X ,  6  5-6.  The p o s s i b i l i t y of including mass:  5  eigenfunctions of •£ ^ )  From the discussion of Section 4, i t i s evident that the f i e l d components belonging to s p e c i f i c eigenvalues of JL must correspond i n Minkowski space to massless f i e l d s i f they are governed by f i e l d equations containing no i n t e r a c t i o n terms.  This w i l l be demonstrated  i n l a t e r sections f o r p a r t i c l e s  with spin, but at t h i s point i t i s s u f f i c i e n t to e s t a b l i s h that t h i s i s the case f o r the p a r t i c u l a r l y simple example of a scalar f i e l d  X(>^).  By d e f i n i t i o n of the word "scalar", one has  (5.59)  S A B  X(\)=0  so that there i s no need f o r application of the techniques contained i n the development from (5.17) to (5.48) apart from the i d e n t i f i c a t i o n (5»4lb). Now  suppose X  (5.60)  s a t i s f i e s , i n 6-dimensional space, the wave equation  ^Jfr JL-0  .  A  This choice of f i e l d equation i s , at t h i s stage, governed s o l e l y by the desire f o r s i m p l i c i t y , and i s aimed only at examining whether conformally covariant f i e l d equations i n 6-dimensional space can, i n p r i n c i p l e , describe massive f i e l d s i n 4-dimensional Minkowski space.  [This f i e l d equation i s ,  i n f a c t , the scalar wave equation customarily suggested f o r the manifestly conformally covariant description of spin-0 p a r t i c l e s on L=0  (Dirac1936,  Kastrup 1966, Mack and Salam 1969)]. Using (5.3)  (5.61)  one finds by straightforward c a l c u l a t i o n the i d e n t i t y  f  ?i 2 x-- V V 2  A  + 4 X^>j  L  + 4L"> > L  L  + 12>  L  ,  so that eq.(5.60) can formally be cast into the form of a Klein-Gordon equation  (5.62)  > VjX J  + n 2 x  88 0  •  with  (5.63)  m  2  s  ^M  2  ,  by the introduction of a "mass operator" having eigenfunction X. and  66 eigenvalue  M,  i.e.  (5.64)  4(x^  2  +3+n n X L  L  = M  2  X.  „  However, introducing the mass concept i n t h i s fashion i s seen to be unsatisf a c t o r y from the point of view of conformal covariance i n Minkowski space, since i n order to assign a s p e c i f i c scale dimension to the f i e l d  it  must s a t i s f y (5*44) as w e l l as the f i e l d equation (5.60).  the  operator  vc^  K  Applying  from the l e f t to (5.62), one f i n d s that t h i s i s possible  only i f M vanishes i d e n t i c a l l y ,  (5.65a)  4(ya +3+L% )^ X = 0  (5.65b)  M X = 0  L  K  2  L  .  Therefore, even v i a the agency of a "conformally invariant mass operator"  M.  a "conformally covariant mass" m cannot meaningfully be defined i n Minkowski space. In order to r e p a i r t h i s defect, i t i s apparent from the d i s c u s s i o n f o l lowing (5.18) that one must enlarge the arena of spacetime to include the supplementary variable r , introduced i n order to bring the v a r i a t i o n s (5.9) sions.  into the form of conformal transformations  Then the requirement that a f i e l d  X  infinitesimal i n f i v e dimen-  transform according to a  representation of the conformal group can be modified, and instead of r e q u i r ing that the condition (5.44) be s a t i s f i e d , one can weaken t h i s condition to the proposal that the f i e l d must be an eigenfunction of the scale dimension i n f i v e dimensions  -^(5).  The e s s e n t i a l consequences of t h i s proposal can be explored by considering the simpler case of f i e l d equations that are covariant with respect to rotations i n a f l a t space of three spacelike and one timelike d i r e c t i o n s . The treatment of t h i s case w i l l be developed i n Section  6.  67 5-7» Dimension of the conformal representation It ^  has been shown t h a t even i n the s i t u a t i o n i n which the squared i n t e r v a l i s non-vanishing a transformation can be c a r r i e d out on f i e l d s that  transform covariantly under rotations i n 6-dimensional space such that the r e s u l t i n g f i e l d s transform covariantly under a conformal transformation group.  This objective i s accomplished by means of the operator U, defined  i n s e c t i o n 5-2. For any f i e l d  X. one can form the corresponding f i e l d  X = U X , even though  i t may be necessary to augment the number of components of able to employ the f i n i t e - d i m e n s i o n a l representations Sy  k  X. i n order to be of rotations  in  7-diraensional space, since several such matrices appear i n the d e f i n i t i o n of U.  Then, i f the representation matrices s ^ g , s ^ are nXn, one can state  unequivocally that the f i e l d  X- transforms according to an nXn represen-  t a t i o n of the conformal group i n the 5-dimensional space spanned by the coordinates ( y , r ) . k  However, i t was shown i n sections  5-5. 5-6 how one can s e l e c t  representations of dimension l e s s than n corresponding to the several eigenvalues of of f i e l d s X .  by making use of projection operators on the space Nevertheless, one must keep i n mind the p o s s i b i l i t y that  the use of these projection operators may prove to be unnecessary.  This  question must be examined i n each case by e x p l i c i t l y considering the transformation properties of subsets of the components of X ,  to see  whether there e x i s t alternative means at one's d i s p o s a l to eliminate c e r t a i n components from X  and so to f a c i l i t a t e the i d e n t i f i c a t i o n of the  remaining components with the usual number of components of a s c a l a r f i e l d , a vector f i e l d , a spinor f i e l d , and so on. Since the case of the scalar f i e l d presents no problem i n i n t e r p r e t a t i o n as i n 6-dimensional space i t consists of a single component, the cases to be treated i n the next sections are those of the vector and spinor f i e l d s .  5 - 8 . Vector f i e l d s i n 6-dimensional space The finite-dimensional representations s^g of rotations characterized by the parameters E ^ as s p e c i f i e d i n ( 1 . 4 4 ) , the indices of a vector ^  (5.66)  (  S a b  )  C d  = K ^  A  C A  (5.8), acting on  i n 6-dimensional space are, as i n (2.38b),  S  Since the variations (5.8)  B D  - S  ^ )  .  C  A D  B  f o r the case L^O can be thought of as a  s p e c i a l subclass of the (pseudo-)orthogonal transformations i n a f l a t space of dimension seven, with metric  &Q^=diag(-1,-1 ,-1 ,+1 , - 1 , + 1 , - 1 ) ,  the interpretation of the f i e l d theoretic generators 3 ^ 3 . given by ( 5 . 1 8 ) , necessitates the enlargement of the dimension of the index space of X to seven, with corresponding spin matrices  (5.66a)  (  S  A  B  )\-i(  SA^BR-SAR^  and (5.66b)  (s^-KS^S^-S^S^)  where now A,B=1,2,3,4,5,6 and Q,R=1 , 2 , 3 , 4 , 5 , 6 , 7 . of the extra component X '  Since the introduction  i s simply a device which serves to bring the  transformation law of the vector f i e l d i n Minkowski space associated with X  into a meaningful form, one can take  X ' to be a function of X  .  so that no new information i s needed to rewrite the transformation law of X  .  Then i t i s permissible to choose X.' i n such a way as to bring  the transformation law of the components corresponding to a 4-vector f i e l d i n Minkowski space into as simple a form as p o s s i b l e . As a p a r t i c u l a r application of the d e f i n i t i o n (5.66a,b)  of the spin  matrices, the operators " t 6 t 5 t » ^=1,2,3,4,7, read i n t h i s case i r  = s  + s  (5.67a)  (TT )  Q  = i( $ + $ )  Q  = i(S  R  R  K  S  R  6  5  - i ( S 6Q S 5Q)S  for  R  +  k Q  k  k=1,2,3,4  and (5.67b)  (7T )  R  ?  J )  R 6  R  +  5  J  7 Q  S Q SQ> * 7 +  -i(  •  R  5  6  From these equations, one sees that a l l the components ("Ifj^Q are zero and  that ^ r , i s a matrix which acts only i n the subspace of components  In accordance with the d e f i n i t i o n ( 5 . 2 3 ) , the descent operator U, which eliminates TT ^ when applied to ^ ,  (5.68)  u = e-  i z t i r  t  = i-i t r -|z z  T  t  t  (zk=yk, z^=r)  s-iT Tr +(i/6)z z z ir Tr t  z  t  s  u  s  t  s  TT +.. U  can be evaluated, because one finds by straightforward c a l c u l a t i o n (5.69a)  (Tr Tr )  (5.69b)  ( ^  =  R  t  s  Q  t  ^ s ^  R  (n) (7r ) R  s  Q  s  = ( S V S V< * 6Q S 5Q>  s  +  Q  - ^t^s) S^u) Q R  =  S  * ts  0  so that f o r vector f i e l d s one has the simple representation  (5.70)  (U)  = I  R Q  I( tQ  -(2 /2)(S 2  I  6  R +  5  R 5  )-z  t  S ( S 6 Q $ 5Q)] " +  R  t  S5 )(S6Q C"5Q)  R+  R  6  +  where (5.71)  The  Z  =  2  Z*Z  operator ( U ~ )  (5.72)  1  U-  1  = e  t  R Q  S  y2. 2 r  .  =  yk-n-  k + r T r 7  #  , defined by ( 5 . 2 5 ) ,  + i z t T T  t  ,  can be formed from the r i g h t hand side of (5.70) by changing the sign of the second term and leaving the f i r s t and t h i r d terms unchanged. Accordingly, one finds that U i s i n t h i s case an orthogonal matrix (5.73a) so that  U  S R  = %QS ; UQ  R  <jSR - % «  S  70  are formed using ( 5 . 7 0 ) , with the r e s u l t s (5.74a)  x  (5.74b)  x  =  (5.74c)  X =X  5  + x  6  )  5  X - ( z / 2 ) ( X + 7L ) 2  6  t  6  6  5  ^ X. -( . /2)(X +X ) 2  t  7  6  5  .  so t h a t , i n p a r t i c u l a r ,  X +X  (5.74d)  6  5  = V ^ 5  •  Although the four components X ^ of the 6-vector X ^ do not transform as a 4-vector i n Minkowski  space, the components  do have the  transfor-  mation properties expected o f a 4-vector under the elements of the poincare group. if  In f a c t , one can by a suitable choice of X  the seventh component X  7  7  show that  i s included i n the transformation law f o r  then the transformation law f o r a l l f i v e components X^(X^.  Xj)  X , k  is  p r e c i s e l y that of a vector i n 5-dimensional space, and that t h i s i s the case f o r a l l the transformations of the conformal group. This can be seen by w r i t i n g down i n components the c l a s s i c a l transforma-  x  R  t i o n law  -> X  | R  ( y ) = X + S X (R=1 R  R  .2,3,4,5,6,7).  According to  the discussion of section 5 - 2 , t h i s transformation law corresponds to the change due to a transformation of the coordinates as i n  (A=1,2,3,4,5,6)  (5.8)  (5.75) together with a transformation i n a seventh d i r e c t i o n which corresponds to the requirement that conformal transformations i n 5-dimensional space be r e s t r i c t e d to those with oi^= (5.76)  E? = 0 A  = 67=0.  This can be w r i t t e n as  71 and one can complete the correspondence with 7-space by introducing a component  (5.77)  v\J given i n terms of the  \  t  i*\J = 0  which s a t i s f i e s  .  By comparison with the r e l a t i o n s (5.78)  Tl  = 0  and the d e f i n i t i o n s  (5.79)  z7  =  r  ; z  =L$/H_  = ^ t y * ,  k  with I<2,  one can t e n t a t i v e l y associate t h i s component (5.80)  a i L  = >cr  .  Then using the i d e n t i f i c a t i o n s (1.44) and (5»?6) i t i s easy to compute the transformation law f o r X  •  (5.81a)  I x = ^ jXJ+^ (x -X ) f (* +* )  (5.81b)  SX = -(* -f )X +<r X  (5.81c)  T*  (5.81d)  IF  k  k  k  5  k  6  5  k  +  k  6  5  6  k  = - ( * + f )X +<r  6  k  X  k  k  = 0  5  •  Making use of the d e f i n i t i o n s (5.74) of the components of X , one finds that the above variations imply the transformation laws  (5.82a)  i  X  k  =j =  K X - X )-( ry )( X -X ) 6  k  5  6  5  [xJ-yJ(X -X )]+/5 ^ 6  5  K  +y [2p X 2pjy.(X -X )] = k  j  6  5  r  =e ^ x W  p J x.+p {-2y x [-i-(y r )(x -x )+y xJ+ j+2  k  j  2+  2  6  5  j  +i(x x5)]], 6  +  72 and  (5.82b)  r*. = r x -* f< x -* )-( r )( * -* ) = 7  7  7  6  5  ?  6  5  z  = 2r/?J [ ^ - y ^ * - * ) ] = 2 r f 6  .  5  These variations are to be compared with those of a vector A*" i n 5-dimens i o n a l space spanned by z^=(y ,r) with e< = £ ^r,= j3=0, i . e . k  (5.83a)  %k  7  7  = (: .k^c-t{5)^zt{5){f^^kK2^{f  JAj)-2^ ( AJ)+  k  k  k  y j  +2 / ( r A ) k  (5.83b)  f A = <T ^ ( ) A + 2 7  7  5  i  ( 5 )  ( ^  J yj  )A7+2r( ^  7  Vj) .  I t i s immediately apparent that (5.82b) i s exactly o f the form (5.83b) i f one makes the choice ^(5)=0.  However, even f o r t h i s choice of  A5).  (5.82a) does not agree with (5.83a), but the discrepancy between the f i n a l terms i n each of these expressions can be eliminated i f one applies a (6-rotation covariant) constraint to X^. t R  (5.84)  ^ X  = v| x -HrX  R  A  = 0  7  A  R  .  This e f f e c t i v e l y supplies a d e f i n i t i o n f o r X , and allows one to recast 7  (5.82a) i n the form  (5.85)  I  x = t j x y f x j+ f (- yj x k  k  J+£  k  k  j  2  j+2r  *> 7  since one has  (5.86)  i^^  i(MK(x +?HK(rV)(x -x ) 6  y  j  6  Thus one finds that from the requirement that the f i e l d  5  . transform  according to the simple law (5.85), one can determine the meaning of the component X' v i a the condition (5.84). This condition i s , i n a sense, a t r a n s v e r s a l i t y condition i n the 7-space R  Y\J-  t  the  ~- o  and one can see that the d e f i n i t i o n of X  c e r t a i n l y holds good f o r  case of the coordinates themselves, since *\J i s defined by  (5.87)  x r ^  7  =  x  2  r  2  = L =  *( i \ ) k  k  .  73 Here i t should be noted that whenever the 6-vector x . i t s e l f s a t i s f i e s a A  t r a n s v e r s a l i t y c o n d i t i o n , so t h a t  *\ X  vanishes, then X ^ i s zero by  A  A  definition. One can a l s o r e w r i t e the d e f i n i t i o n (5•84) i n terms of the components X . R  S u b s t i t u t i n g f o r X=U~ X. w i t h the help of the matrix ( I T ) 1  1  R n  , one  f i n d s simply t h a t (5.88)  0 = ^  %  R  R  S |K(X +rt  •  6  Now, f i n a l l y , one can examine the eigenfunctions of -^(5) by e x p l i c i t l y w r i t i n g out the matrix i s ^ . (is )  (5.89)  R  6 5  Q  = ^  /  This matrix i s -  ^  M  5Q  so t h a t (is5^) a f f e c t s only the 5.6 components o f X .  Thus, by the  j u d i c i o u s use o f the c o n s t r a i n t (5.84), i t i s p o s s i b l e t o avoid completely any use o f p r o j e c t i o n operators, and s t i l l t o be able t o reduce the number of components i n v o l v e d i n the transformation law o f a vector f i e l d X_^, which i t s e l f forms an eigenfunction o f -^(5).  5-9. Spinor f i e l d s i n 6-dimensional space The s p i n operators (2.38d) i n Minkowski space, governing the t r a n s f o r mation behaviour o f a 4-component spinor f i e l d under r o t a t i o n s , are 1  (5.90)  <r = ( i / 4 ) [ * J , x ] jk  k  w i t h 4X4 Dirac matrices s a t i s f y i n g the anticonmutation r e l a t i o n s  (5.91)  \i\t } k  = y * +a j  k  k  * j = 2$J*  and given i n Weyl r e p r e s e n t a t i o n as  '0  (5.92)  2T *5  where H  (5.93)  1  2  X  O V  V  '0 I  y  =  f o r a=1 , 2 , 3 ;  'il  0  >  0 -ii  0  ^  j  3 ^ ^ # and the 2X2 P a u l i matrices have the standard form o  r  r  o - i ^  0- = 2  i  0  'l <r = 3  0^  - lof  1 1J  7* This representation suggests searching f o r a f i n i t e - d i m e n s i o n a l represent a t i o n of the rotation operators s i n analogy to  (5.90),  (5.94)  AB  <r  =  with operators p  (5.95)  A  i n 6-dimensional space, constructed  [/5 ,y3 ] A  B  s a t i s f y i n g the anticommutation r e l a t i o n s  fy\/ }=2S B  .  AB  The simplest representation f o r such operators, admitting; r e f l e c t i o n s , are the s i x 8x8 matrices(Murai 1958)  f = k  (5.96a)  (5.96b)  = y5 a  p  (5.96c)  f  = IH  6  following sense.  where s  {l  o  3  <r =  0* =  k  o  }  u  U*  5  o  0- i i  2  0  jS A transform as the components of a 6-vector i n the I f one defines a 6-vector V  [v\<r ] = (s )A BC  B C  V  ii  The s i x matrices  (5.97)  °*1  ,k  o  k  B C  D Y  A  by the transformation law  D. (5»66), then the y9 form a 6-vector  i s the vector representation  A  since they s a t i s f y t h i s lav: i f the <j-  AB  are defined as i n  on account o f the anticommutation relations  (5.94).  Indeed,  (5*95) one has  [p ,<r ] = i ( S f > - $ Y ) . A  (5.98)  BC  A B  c  A C  B  This f a c t enables one to form covariants i n 6-dimensional space using the 6-vector  ^  i n analogy to the formation of covariants in 4-dimensional  Minkowski space using the 4-vector  tf . k  The c l a s s i f i c a t i o n of such invariants into tensors and pseudotensors of various kinds requires consideration of discrete symmetry operations a f f e c t i n g the space-like coordinates.  The invariance of the quadratic form  75 (5.99)  dB= p dv =  f d'  A  k  l A  l  k  +  f5  d  l  5  ^6 ^  +  d  6  can be used as a c r i t e r i o n f o r the construction of operators discrete symmetry operations.  representing  The discrete transformations a f f e c t i n g the  spacelike coordinates can be conveniently c l a s s i f i e d into two types, as f o l l o w s .  (5.100)  Consider the "inversion"  d ^ » - d v ^ fora=1,2,3 a  d\ *  ;  a  h  - K i ^ f o r b=4,5,6 b  .  The operator P representing t h i s transformation must s a t i s f y (5.101)  0 and [ ? . / ] = 0  .  b  {P.p "i = a  This i s accomplished by choosing  (5.102)  P = f**f p 5  =  6  r^t  5  H<^2  •  Next, consider the"reciprocation"  (5.103)  - d ^  5  J  +d^  A  for  .  The operator R representing t h i s transformation must s a t i s f y  (5.104)  jR,jj5} =  and | R , ^ ] = 0 A  0  for  .  This i s accomplished by choosing  (5.105)  R = £ ^Zpp^p  6  = X  5  S <T  2  .  Thus, the operations P and R separately interchange the 4-component spinors X  u  and X^ which compose any 8-component spinor X  (5.106)  X =  *u L dJ X  whereas the product operator  (5.107)  PR = - x ^ B I  represents a symmetry operation i n the subspaces of X  u  and X.^.  Any operator S i n the space spanned by the spinor (5.106) w i l l be c a l l e d a "scalar" i f i t i s invariant under the rotations generated by (5.94), under inversions generated by (5.102), and under reciprocations  by (5.105),  generated  76 [ s , cr* ] = 0 ; [s,P] = 0 ; [s,R] = 0 .  (5.108)  3  Obviously, the i d e n t i t y operation i s a s c a l a r . c a l l e d a "pseudoscalar"  An operator Q w i l l be  i f i t i s invariant under rotations, but changes  sign under inversions and reciprocations,  [Q, < r ] = 0 ; {Q,P} = 0 ; {Q.R} = 0  (5.109)  AB  An example of a pseudoscalar  (5.110)  f  = (1/6!) i  .  i s the operator  f f f f f f k  f i B C J E F  B  c  D  E  F  =.u  ^  H  which s a t i s f i e s the conditions (5.109) on account of (5.95). Now, i n analogy to the corresponding  d e f i n i t i o n of the adjoint  4-component spinor y=y <f**' +  i n Minkowski space, the adjoint spinor be defined by the requirement that  x i n 6-dimensional space can  X X transform as a s c a l a r .  Thus,  a rotation  (5.111)  X*  X' = DX.  generated by an operator D defined i n terms of c o e f f i c i e n t s E^B as i n (1.44), (5.112)  D = l-(i/2)E <r  A B  A B  should be accompanied by the transformation  (5.113)  X *  X' =  where now (5.114)  D-  1  = I+(i/2)E <r  .  A B  A B  Writing  X i n the form  (5.115)  X =X A +  constitutes a d e f i n i t i o n of the "adjunction operator" A.  I f one subjects A  to the condition (5.116)  A  = I  2  i.e. A  - 1  =A  then i t can be constructed, as follows. (5.117)  X  1  Taking the adjoint of (5.111) gives  = (DX ) A = X D A = * V D A = XAD+A +  +  +  1  +  and, by comparison with (5.113), y i e l d s f o r A the condition  77 (5.118)  D" = AD A 1  or  +  D = AD' A +  .  1  Substitution of (5.112) and (5.114) shows that this amounts to requiring  (5.119)  (<r ) A3  +  = A<r-AB. *A  .  A  Thus, one can a l t e r n a t i v e l y  define the adjunction operator A by the  conditions  (5.120)  AjS A = - ( / ) A  A  ;  +  A = I  ,  2  because the r e l a t i o n (5.119) follows then,  (5.121)  -AB\+ = _ t * l k \ t t A BBB_ * B ok*+. A  (C  )  = -(iA)(/ / -/yY=(iA)(yS  A +  / -y« / )=A<r A B +  B +  A +  A B  The operators (5.92) have the r e a l i t y properties  (5.122)  /  = -(y8 )  a  a  f o r a=1,2,3  +  ; / = / ^ ; y3 = - / 5  4  5 +  5/ = j 8 ^ 6  .  The conditions (5.120) are thus obviously met by the construction  (5.124)  A = -i^  4  jJ  6  a <r .  =  3  In terms of the 4-component spinors  X  u  and X ^ defined i n (5.106) the  adjoint spinor has the form  (5.125) with  X= ( X . -K) u  Xu,d g i  Using jl 7  v e  n V h  d  X^.d* •  the set of 8x8 7-dimensional r o t a t i o n matrices orQR can be  1  defined by augmenting the set <r  (5.126)  <r = U/4) 7A  A  A  by the matrices  B  ]  .  In accordance with the p r e s c r i p t i o n (5.23) write now X=\J~'X_ where the operator U~^ i s given by  (5J27)  * y * ^ 7  U"' -  . ^ ^ - ^ ) - ^ ^ - ^ )  Now, with the representations (5.94) and (5.96) one has here (5.128a)  0 - ^ - ^ =  -(i+y )y 5  o  k  (i/2)  ( i - V ) o-  0  5  and with the representation (5*110) one has also (5.128b)  - ? 6  5? -  (i/2)  1+1*5 v.  o  k  _  These operators are nilpotent with exponent 2 , (t=1,2,3,4,7)  (5.129)  (cr -<r ) 6t  5t  2  =0 ,  and therefore one has the simpler representations  (5.130)  u-  =i+iz^o^W* )  1  ; u = i-i (<r -<r ) . 6 t  5 t  Z t  The matrix i <f^^ i s given e x p l i c i t l y by  (5.131)  i <T  = £ d i a g ( I , -I,  65  -I, I)  ,  I=diag(1,1)  ,  so that grouping the components of X into four sets of 2-component  spinors X 2 , 3 , 4 » 1t  (5.132)  X=  f > there are two 4-component f i e l d s , ¥> and  h  (5.133)  KJ  Y=  <p  A  (n T -  which correspond to d i f f e r e n t eigenvalues o f i °*5^, v i z , X = f , 0  respectively.  X] -j =  As i n the discussion of the vector f i e l d , one may inquire  whether i t i s possible to dispense with the need f o r projection operators A  E, E corresponding to the subspaces to which these two f i e l d s belong, i n order to write the transformation law f o r each type of f i e l d s o l e l y i n terms of components i n i t s own subspace. In each case, comparison i s to be made with the transformation law f o r a 4-spinor i n 5-dimensional space, spanned by the representations  °"(5) . ^ k7  (t=1 , 2 , 3 , 4 , 7 ) , where  (k, j=1 ,2,3,4) can be of either of  the forms  (5.134a)  <T *7 = <i/<o  (5.134b)  <r-  y5)] , ^ k j  (5)  k (5)  7 = ( i / 4 ) [>J ,(+tf5)] k  f  £-  ( 5 )  =  \j k j ] >  k j = (i/4) [ >  tK  t  yj]  .  79 Substitution of the representations (5.9*0 and (5.126) into the c l a s s i c a l field  variation  I  (5.135)  -(i/2)E V  X=  A  A B  £  and application of the operator U, leads to the transformation laws f o r  9 and  (p  (5.136a)  Tp = - ( i / 2 ) € < r  (5.136b)  19=  jk  -(i/2) e  j k  (5)jk  (f  f+(T  (|)^+i^ [-2i(|)z +2 t k  k  Z  <^+(T(-|)^+ij5 [-2i(4)z +2z o k  ( 5 ) j k  < r ( 5 ) k t  t  i  k  + 2 i  f ^(5)k7 ? k  ](^  ( 5 ) k 1  .]f +  *  Hence i t i s possible to avoid the use of a. projection operator E , since the transformation law f o r y involves only <f> i t s e l f . 5  for  — A  S (f> contains  However, e q . ( 5 J 3 6 b )  <j0 i n the f i n a l term, and so the use o f a projection  A  operator E f o r non-vanishing components <p cannot be avoided without devising a covariant constraint on X which has the e f f e c t of eliminating four of the eight components of X .  A constraint of t h i s type can be  written (5.137a)  {\ f -I?^)%  =0  k  k  ,  which i s the generalization of the same condition on L=0 (the s p e c i a l case L=0 i s treated i n e . g . Todorov 1973).  In terms of the f i e l d X , t h i s  constraint reads (5.137b)  |u(|J +/J ) 6  5  -y_=  0  or simply (5.137c)  <f = 0  .  Thus i t i s possible to e n t i r e l y do away with the necessity of using projection operators, but only at the price of eliminating h a l f the f i e l d components.  80  6 Two-Dimensional Conformal Group 0  6-1 . Four-dimensional  analogue of manifestly covariant formalism  An e x p l i c i t example of an equation i n 4-dimensional Minkowski space that i s rotationally covariant but not translationally invariant w i l l serve to elucidate the discussion of the preceding of a f i e l d with non-vanishing  sections f o r the case  spin. The value of such an exercise l i e s i n  the fact that the Lorentz group i s isomorphic to the analogue of the conformal transformation group i n a space with two spacelike dimensions and no timelike variable. In fact, the conformal group associated with any f l a t space, having a metric with (n-1) minuses and 1 plus i n i t s signature i s easily shown to be isomorphic to (pseudo-) rotations i n a f l a t (n+2)-space having a metric with n minuses and 2 pluses i n i t s signature. In 2-dimensional space with coordinates y^ Jk=1,2;  %j =diag(-1,-1)], k  the "conformal group" i s the 6-parameter r e s t r i c t i o n of the conformal group i n Minkowski 4-space, (6.1)  S y = ot + 4 ^+<ry +2y (|3 k  k  k  k  k  y)- ^ y k  2  (j=1,2; k=1,2)  characterized by the 6 infinitesimal parameters  , et , £ =~ £  , 2  .cr . fi , p -  Then defining (6.2)  ^ = *y , k  k  ^  =  (L-* -x. y )/(2K) 2  2  2  v£ = ( + * - * y ) / ( 2 *-)  .  2  2  L  the infinitesimal rotations (6.3a)  ^  (6.3b)  E^ = £  = E  A  k  ^  A B  j k  B  ,  S =diag(-1,-1,-1,+1) , A,B=1 ,2,3,4 AB  , E^ = c A 3  ,E ^ =  f  E  3*  =  ^  generalize (6.1) to the variations (6.4)  Sy  k  = « +$ ^+<ry*+2y\fSy)-i3 (y -><- l) k  k  k  Z  2  .  2  81 To v i s u a l i z e the e f f e c t s of these transformations i n the  4-dimensional  space, i t i s convenient to suppress one spacelike coordinate, say y , by 2  setting y - 0 .  Then the transformations ( 6 . 3 ) read e x p l i c i t l y  2  = -fcV^+OcV)^  (6.5a)  ^1  (6.5b)  ^ 3= ^1^1)^1+^  (6.5c)  = (* + J )* 4w3 1  1  For pure d i l a t i o n s  S1 =o , i  (6.6)  a  .  1  l  (  l  «^=/3l=0, one has f  =^4 ^ 4 = ^3 t  r  and the motions i n y^-space correspond to motions i n 4-space on the hyperquadrics "\ 1 L along the intersections with the planes *{] =constant. A  3  A  S i m i l a r l y , when<r=0, o j - ^ = 0 ,  the motions are along the i n t e r s e c t i o n s  with the planes ^-j^onstant, and, f i n a l l y , when<r=0, ^ + ^ = 0 , the motions are along the c i r c l e s that delineate the intersections with the planes  "Inconstant. The simplest case 6-2.  L=0 i s i l l u s t r a t e d i n Figure 2 .  Four-rotation covariant spinor equation  A "Lorentz" covariant spinor f i e l d equation can now be postulated i n the 4-space spanned by the coordinates »\ (A=1,2,3,4) by u t i l i z i n g the A  usual spinor representation i n terms of the Dirac matrices ^  (6.7)  B  =  (i/4)[s\*B]  ,  A  ,B=1,2,3.4 .  Since the equation f o r a 4-component spinor X("|) need not be t r a n s l a t i o n a l l y i n v a r i a n t , the coordinates  may appear e x p l i c i t l y i n the f i e l d equation,  and as an example one may consider the analogue, i n the 4-space under consideration, of the spinor equation i n De S i t t e r space (Dirac 1935). which i n t h i s 4-space assumes the form* *This equation has relevance i n the usual spinor theory i n Minkowski space y (k=1-4) because one finds that K  2 * - % ^ =  iUJyj)(* * )%iy > r k  k  k  k  so that f o r wave functions that are homogeneous functions of y , the Weyl equation f o r neutrinos can be written i n the form (6.8). k  Figure 2. Examples of motions induced on the n u l l surface L=0 i n a 3-dimensional space as a r e s u l t of conformal transformations 1-dimensional space.  in  83 ((rABy  (6.8)  uV  iX)  *  =0  ( . A  B=1  ^ V * !  . .3,*. 2  8  )  where X i s a c-number which can be a f u n c t i o n o f L without a f f e c t i n g the r o t a t i o n a l covariance o f the equation. W r i t t e n i n terms of the d i f f e r e n t i a l operators (6.9)  mAB  t h i s equation becomes (6.10)  (tf tf m -i4i;\ ) X = 0 A  B  A B  .  The Dirac matrices can be w r i t t e n i n Weyl r e p r e s e n t a t i o n as  tf S  *  0  - £  <r  0  0 I 1 0  f o r a=1,2,3 ; & 4 _  a  (6.11) tf  5  = tf  1  tf  tf  2  tf  3  4  ii  =  0  0- i i where the 2X2 P a u l i matrices have the standard form (6.12)  1  s  0  1  0  1  0  i  -i  "3*  0 J  . I =  .° - . 1  'i  o  .0  1.  ,  S u b s t i t u t i o n o f the r e p r e s e n t a t i o n (6.11) i n t o eq.(6,10) y i e l d s an equation of the form  r  G+2iX  (6.13)  0  = 0  H+2iX  *2  ****  where  A  X j £ are 2-component f i e l d s over \ , and G and H are the 2X2 matrices - 12 43 -m^+m^-im^-im^ i m  + m  m^^ +im^2'-iJ 22'^' 4l  im^-m^  - 12~ 43  -m -m^-ira +im  n  (6.14a)  G =  (6.14b)  H =  i m  m  1  m  31 - 4 2 " i m  i r a  31  23"  I I 1  4l  2  23  + m  43  42  84  (6.13)  The form of the f i e l d equation  can be s i m p l i f i e d by defining  the 3-vectors  (6.15)  1 = (n^.m-^ .m^) i n  (m^ » 42 43) m  =  ,m  and the 3-vectors  (6.16)  2  = i(i+in) • 4' = i d - " ) 1  J± = J1+J2 • 4  w i t h  =  Jf+J  1 2  s a t i s f y i n g the commutation r e l a t i o n s  0a"3b]  1  f  Pl'Ji]  =  abc  1  abc -c  f  In terms of these operators, the matrices (6.14) can be written  <-2iJ  G =  (6.18)  6-3.  -2ij  -2ij_  3  2ij  +  Infinite-dimensional  ^  '  H 3  =  -2iJl  -2ij|  2ij»  representation  Since the operator i(m g+ <r g)(m + c r ^ ) i s a Casimir operator f o r the AB  A  A  Lorentz group, one can f i n d basis functions f o r i n f i n i t e - d i m e n s i o n a l i r r e d u c i b l e representations by considering eigenfunctions of -fra^gm^. •These w i l l be functions X ^ ^ e , ^ , « R) t  6,<p,<x,R  defined by ( e . g . Xursunoglu f  "j I  1  2  of the spherical coordinates  1962, p.256)  = r sin6 coslj) = r sin© sin?  (6.19) R cosh** R sinh < r =JR sinh«t (R c o s h * for  4v *| ?/0, where the f i r s t and second alternatives correspond to timelike  (R  0) and spacelike (R < 0) i n t e r v a l s , r e s p e c t i v e l y .  2/  the ranges (K 9$ TT , 0  2  These variables have  2 T T , 0$ * < «* , and 0<CR «»a .  Written i n terms of 1 and n, the two Casimir operators of the Lorentz group f o r the i n f i n i t e - d i m e n s i o n a l representation are  85  (6.20a)  fi^gm*  = 2(f  8  + j/ )  = (l  2  -  2  n ) 2  and  (6.20b)  -(i/8) £ ^ m ^ Q  =f - i  ,  2  = i l - n = 0  .  Hence, to specify completely an i r r e d u c i b l e i n f i n i t e - d i m e n s i o n a l representat i o n one need consider eigenfunctions of the f i r s t invariant operator only.  (6.19),  In terms of the variables  f o r the case R > 0, t h i s operator reads 2  explicitly  (6.21)  i ^ A g i / = -sinh c<  l  l  9)[sin6  5  2  2  + (} / 2  o<. ) + 2 c o t h *  .  2  where  (6.22)  2  = - s i n - 6 (> /I 1  O  0)]  +  sin" e 2  ( } / 2  > (f ) 2  .  The eigenvalues of the square of the angular momentum operator are denoted by < r ( « + N - 2 ) i n an N-dimensional Euclidean space with <r* r e a l , so that i n -  t h i s case with <y complex one seeks solutions of the equation  (6.23)  X = <r((T+2)X .  In the system of coordinates (6.19), according to (6.21) the variables separate i n the Casimir operator, and eigensolutions X are independent of R. Thus, writing  (6.24a)  X = Q ( * ) Y (6.9) , lra  where Y ^ problem  (6.24b)  m  i s an ordinary s p h e r i c a l harmonic, one can solve the eigenvalue  (6.23)  by making the substitution  Q(< ) S s i n h - 2 *  P(<* )  ,  where P( <<.) i s some other as yet unspecified function of o( .  Changing to  the variable z=coshed, one finds that P( ot ) must solve ( c . f . V i l e n k i n  (6.25)  { 0 - z  2  ) O  2  A  z )-2*U 2  h  8)+(fnr)(3/2 +<r ) - [ ( - l - l ) / ( 1 - z ) ] } p = 0 2  so that comparing with the equation  (6.26)  { ( 1 - z ) 0 / ^ z ) - 2 z O />0+^(^+D-^ /(1-z )}p^ (z) = 0 2  2  2  1964)  2  2  2  86 one finds that P(°0 (6.2?)  i s the associated Legendre function  P(* ) = P T ^ f (cosh*)  and i s r e a l f o r r e a l <T.  /  Then the normalized eigenfunctions are given by  (Kalnins e t . a l . 1973)  (1=0,1,2,...; m=-l,-l+1  1)  .  These eigenfunctions are transformed among themselves by the action of the d i f f e r e n t i a l  operators 1 and n .  In Naimark's notation (Naimark 1964, p.117),  a number c i s defined by (6.29)  <r (<r+2) = -1+c  2  and a number l =0 corresponds to the remaining Casimir operator. o  With l ^ l ^ + i l g ,  n + ^ + n g , the transformations are given by (6.30)  1  X^  ±  = [(l m 1 )(l m)] +  +  T  ^X  ;  c l a  = *X  1 X 3  clm  clln  and (6.31a)  n X +  c l m  [(l-m)(l-m-1)]* c  =  +  (6.31b)  n.X  c l m  X  x  [(1+»M)(1-M«2)]*C  = -[(l m)(l nu1)]* C +  +  x  -[(l-m 1)(l.m 2)]ic +  (6.31c)  n  3  X  c l m  +  c  m +  W  X  l + 1  C  f  B  X  1+1  = [(l-m)(l+m)]*  -[(l*1)(l.  >  i  W  (  X ^  1)]^  1 + 1  f  1  +  C t l + 1 f m + 1  B  . ,  -  e j l + l t B u l  -  X ,i 1, c  +  r a  where (6.31d)  Cx = i [ ( - c + l ) / ( 4 l 2 - l ) ] i 2  2  .  For r e a l c , the representation i s unitary i f the number i n the square brackets i s p o s i t i v e , f o r 1=0,1,2,...;  i . e . the representation i s unitary i f c 4 1 . 2  87 6-4. Solutions of the spinor equation The spinor equation (6.8) can be i t e r a t e d to give an equation containing AB the operator m gin  , by using the operator i d e n t i t y ( v a l i d only when A has  A  the range 1 ,2,3,4) (6.32)  [<r- ^ ^ -i(X+1)[ A B  A  B  [<r n EF  E  5  "  (1  /8)* m +A( AB  AB  X+1)  •  Thus a 2-component s o l u t i o n X ^ , as defined by (6.13), may be found by reversing the order of the factors i n (6.32) and writing  (6.33)  ^-clra  = [G-2i(A+1)]  0  so t h a t , by virtue of the defining r e l a t i o n (6.23) and the i d e n t i f i c a t i o n (6.29), one has (6.34)  [G+21A] X J  =  [l-c +4>(> +1)] % 2  .  Hence one finds a s o l u t i o n (6.33) of the spinor equation (6.8) f o r a l l values 1 = 0 , 1 , 2 , . . . and m=-l,-l+1 , . . . , 1 corresponding to the eigenvalue c provided one makes the choice (6.35)  c  = 1+4A(A+1)  2  .  S i m i l a r l y , one finds solutions  (6.36)  X  2  = [H-2i(X+1)]  Xclm 0  provided the eigenvalue c i s given by (6.35)« Using the r e l a t i o n s (6.30) and (6.31), the (unnormalized) s o l u t i o n (6.33) of the spinor f i e l d equation can be written e x p l i c i t l y as (6.37)  X = '-i[m+2(A +1  )]X +[(l-m)(l«m)] ^CjC^yf[(l+m+1  )(l-m+1)]*C  eln  - i [(l-m)(l+m+1)] ^ w T + Ol-mJd-m-l)] ^  X  ^ '  1+1  X  <l+1|B  ^ + Ijl-Kn+I )(l+m+2)]i '  'Cl+lXcl+lia+1  8 8  Substituting into the f i e l d equation, one does indeed f i n d that t h i s spinor function s a t i s f i e s  ( 6 . 8 )  Another form of  ( 6 . 3 3 )  fy  ( 6 . 3 8 )  =  [G-2i(  provided the value  £-2(*  + 1  3W  c l m - 1  of c i s s e l e c t e d .  may be found by making the alternate choice  A +  1)]  c  v.  Cigl4*)(l-»+1 ) ] ^  ( 6 . 3 5 )  c,l,m+1 J  ^l m)(l m-1 ^ C ^ ^ ^ - f w i +  )(l-m 2^  +  [U-*)U+*>]Vc.1-1 , m [< +  1+I0+1  X -** 1  C  +  1  )J l 1 \,1 1 i c  +  +  ^  ^  J  ,m  0 J  0 but using the ( 6 . 3 9 )  relation  Jo" lo*  ^  ^  J?  e l  . ,  d  B  *  *9 «  Su.  one f i n d s that (6.40)  |(sin0  so that f o r  X*  ^  dG dy = -4[(l-m)(l+m+1)]2  (\+1)  A^-1 these solutions are not independent?  The quantum f i e l d  operator can thus be written as the most general l i n e a r combination of types of solutions exhibited above, m u l t i p l i e d by creation and  all  annihilation  operators. *****  The above computations serve to show that solutions equation can be found i n terms of the solutions problem f o r the scalar Casimir operator m^gm^ . 5  "X. of the spinor of the eigenvalue  Since these eigensolutions  are independent of R, one f i n d s that the spinor solutions X (6.41)  i.e.  R( V > R R  =  \  that spinor solutions  k  \ Z = 0  satisfy  ,  are homogeneous functions of degree z e r o .  *The analogue of t h i s s i t u a t i o n occurs when solving the Weyl equation (i«r'V-i> )^ = 0 by t h i s method. One f i n d s t  (i£*l-i\)[(i£*I+i* ) t  provided P ^ - | p _ | t and the f i r s t =  Ox)]5 =0=(i!*I-i\){(i!"I+i* )  e  t  s o l u t i o n i s proportional to the second  i f e i t h e r P 4 | £ | or P4 -|p_| i n both. =  =  jipx^  89 This fact can now be used as a guiding p r i n c i p l e i n attempting to f i n d solutions i n terms o f the more f a m i l i a r variables y , x , L . K  6 - 5 . Spinor solutions i n "Minkowski" coordinates In terras o f the Euclidean coordinates y Minkowski coordinates — and the variables  K  (k=1,2) — analagous to the  X and L , given by (6.2), the  f i e l d equation (6.8) may be rewritten as an equation f o r the f i e l d (6.42)  X = u X ' ( y . H ,L) k  .  Substituting the 4-space analogues of the r e l a t i o n s (5.3) between the differentiation  (6.43)  operators  (T^^A^B  52  $  o-  and the operators  A  J k y  one finds (j,k=1,2)  j ^ k +i(o- J+«r J)> j + 6  5  + i ( < r J - c r ) [( * " L - y ) ^ 6  + cr  6 5  2  5j  (y } k  k  2  -  k  -y X V ] +  •  The operator 0 i s given by the 2-space analogue of  (6.44)  u  =  e  -iy ir k  k  i(L/H )H5  (5«23).  2  +  and i n the representation  ( k = 1 f 2 )  (6.11) the matrices  TT  ,n ,  TT  are written i n  terms of the 5-dimensional spin matrices  (6.45)  ^rs  =  L*r» * s ]  (r.s=1,2,3,4,5)  as  0  0 0 TT  1  = 0 - ^ - ^  =  0 1  i  0 0 0 0 0 0 1 oJ  0 0 -i  OJ  1 0 (6.46)  0 0 L 0 -1  0 0  90 T T , are nilpotent to the power 2,  One finds that a l l f i v e (6.47)  (-rr ) = 0 ; (TT5) = 2  2  0  k  so that one has the simpler representations (6.48)  U= I-iy TT -ix- L2Tr k  ;  1  k  5  U"  1  = I + i y T r + i * . - L 2 TT k  .  1  k  The operator (6.44) has the e f f e c t o f transforming the f i e l d equation into a form that i s t r a n s l a t i o n a l l y invariant i n the coordinates y ^ .  The f i n a l  terra i n the exponential i n (6.44) has the e f f e c t of ensuring that f o r Lj^O the generator ( ^\ " jy ' m  m  i  ^i^'  ¥  i  *  fy^)  s  transformed into a form which can be  interpreted as a generator of s p e c i a l conformal transformations. Keeping i n mind that U depends upon y , X - , so that k  (6.49)  ^k< " ) = u  1  i i r  k  •  = -iX- L^7T  ,  1  5  a p p l i c a t i o n of the indicated s i m i l a r i t y transformation to the f i e l d equation yields (6.50)  0(<r  A B  v(  A  2 +iA)uB  1  =  i(<r +<r )^ 4k  + 43 Vx-" L^ir vA.V(i/2)( 1  K  ( r  5  3k  k  +*- LTT ^ 2  k  -^- i><r 1  k  5 k  }  k  <r +<P )TT +$*~h* 7f +i A kk  k  k  5  where use has been made of the commutation r e l a t i o n s (6.5D  [TT .<r. ]=0 ;  fo^O;  5  [-Tr .6- ]=iTT 5  43  5  ;  k  [iT^ ^  =-i ^  [ V ^ - ^ J ^ i ^  ;  , \jT  ( <A+ <r ) T l 1  y  3j  J=2  I f one labels the components of X by suffixes 1,2,3,4, *1 (6.52)  X  2  X =  then the f i e l d equation can be written i n the form of two coupled equations f o r 2-component f i e l d s  if and (f, defined by  91 A  (6.53)  .  CO  •  T  *3  =  i n the form  (6.54a)  (V- L(-<r -b + <r * ) - < r (L* <V 4 ) ] <£+ 2  2  l  1  2  + [(|H- L^(<r 1 +(T ^ )+i(X-iK^ )]^  =o  1  1  [i*- i£(<r i +tr ^2>  (6.54b)  1  1  3  a  r  ^  e  n  e  P  a u  2  ^  + 1 +  2  K  £*^K)]  o )</> =  2  2 3  + i  2  +i(<r > where 0~1  1  +  0  2  l i matrices as given i n  (6.12).  Writing out the matrix i <T^ i n the representation 3  (6.11) ( t h i s matrix  corresponds to the matrix i s ^ i n the 6-space version of the theory),  (6.55)  i ^  =  3  <p  Thus the f i e l d and  i ^ >5 = 3  diag(|,-i,-£.£)  .  corresponds to the eigenvalue  A  <p corresponds to the eigenvalue  .  A^=-f.  one has  X =2 Q  of the matrix  The space of f i e l d s  itr^,  X  comprises the two parts belonging to the orthogonal projectors corresponding to these two eigenvalues. If,  i n analogy with the procedure of section 5-2, one makes the  identifi-  2 cation  r=(L/K ) , then looking upon 2  (6.56)  i  (6.57)  r b =--  (  i.e.  3  a parameter i n the d e f i n i t i o n of r  (5.35) i s 43 = i<r  the analogue of since one has  L as  )  r  KV  ,  one goes over from using y k , H , L as coordinates to using y , r , L . k  Since r i s a variable which i s homogeneous of degree zero i n the 4-space just as yk i s homogeneous of degree zero,  (6.58)  A n  £  A  (r)  S (vO„  +2Lb ) L  (L*/K) =  0  ,  then from the discussion of solutions of the spinor equation i n spherical  92 coordinates, one expects to be able to form solutions X written i n terms 1 2 of y , y and r o n l y , i e , solutions i n 3-dimensional space, 0  A  Hence, one can say that each of the two f i e l d s  <P and 0° belongs to  a d e f i n i t e value X,i of the scale dimension i n 3-dimensional Euclidean space, spanned by the coordinates (y , r ) , and therefore the connection with representations of the conformal group i s not l o s t by generalizing the coordinate variations to the case L^O ( a l b e i t one must now consider representations o f the conformal transformation group i n three, and not two, dimensions), 6-6, Introduction of the mass concept f o r eigenfunctions of -^(3) So f a r t h i s example has served to indicate how one should proceed when considering the transformation proerties of f i e l d s with non-vanishing s p i n . I t remains, f i n a l l y ,  to demonstrate that f o r such f i e l d s one can incorporate  the mass concept within the framework of conformally covariant f i e l d s , i n 3-dimensional space, which are eigenfunctions of the scale dimension -^(3)» To t h i s end consider eq.(6.54), written out component by component i n terms of the variables y of  k  and r .  The four equations f o r the four components  X can be recombined into the following simpler set of equations f o r  combinations of components o f X :  (6.59a) (6.59b)  Kir V r } ) ) ( 2 +i(ir^ +X) ( X i r * ) =0 2  2  2  i(i V^2>  r  r  <Xi-irX )  3  " t o r>*2 =  3  0  (6.59c) (6.59d)  i(-ir } 2  r } )X3 + i(|iO +A) 2  1 +  2  r  U^+ir^)  = 0  .  Each of these equations can be i t e r a t e d by operating with ( i ^ ^ + ^ ) from 2  the l e f t , with the r e s u l t s  {-r ^ ^  (6.60b)  ^- 2 ^ \ ( r ^ ) 2 . 2 r > - 4 A ( X 1 ) } X  (6.60c)  {-r  (6.60d)  $-r ^ ^ + ( r ^ ) 2 . 4 r > - 4 A U« ) } ( *  2  k  (r> ) -4r> ^X(  X+2)}(x -irX ) = 0  (6.60a)  2  k +  r  r  1  =0  k  r  +  r  r  +  2  > V^V - 'V^ACX+O^ 2  k  2  2  2  =0  k  k  r  r  3  4  +  irX ) = 0  .  2  Now sine© one has (6.61a)  r^ (rX)  =  (6.61b)  (r^ ) (rX)  r  2  r  rX+r ^ X 2  r  .  = r[l+2rir+(r>r) ]x 2  e q s . ( 6 . 6 0 a , d ) read (6.60e)  {-r ^ ^ +(r^ ) -4r> -4X(X+2)} X, 2  k  2  k  r  r  - i r {-r  (6.60f)  ^ _  2  2  r  ^ _2, 2 _ r  X  f-r ^ ^ +(r> ) -4ro -4X(X+2)} 2  k  2  k  r  r  +ir £ - r ^ ^ 2  k  +  (  r  ^  r )  2_  2 r  _j  8 X  ^  3  X  k  >^  =  0  + 2 _ 8 -3} * X  2  = 0  .  Multiplying e q s . ( 6 . 6 0 b , c ) by i r and subtracting from or adding to ( 6 . 6 0 f ) or ( 6 . 6 0 e ) r e s p e c t i v e l y , one finds that the above equations s i m p l i f y to  (6.62a)  {-r ^ \+(r^ )2j*r} -4XU+2)5  (6.62b)  {-r  2  k  r  2  r  X - i r ( - 4 X -3) ^  =0  ]  ^ > +(r^ )2.4r> -4X(>+2)]^+ir(-4A-3)X k  k  r  r  2  =0  .  Hence, making the choice (6.63)  X = -3/* . A  the i t e r a t e d equations can be written i n terms of the f i e l d s defined by ( 6 . 5 3 ) . as  <f and (f ,  94 (6.64a)  $-r  (6.64b)  {-r > ^ +(r^ ) -2r^ +(3/4)}^ = 0  >>  2  k  2  + ( r ^ ) - 4 r W ( 1 5 / 4 ) } y>  =0  2  k  r  k  2  k  r  r  .  As w e l l as on the grounds of s i m p l i c i t y , the value by another c r i t e r i o n .  A = -3/4  i s dictated  The second Casimir operator of the Lorentz group,  s i m i l a r to (6.20b), does not vanish i n the 4-component spinor representat i o n , but instead one (6.65) (where  -(i/8)£  Si234 ^)» =+  has  A B C D  (m  ) s -b- {<r 5  A B + ( r A B  A B  7  A  a B  (3i/4)}  ^he free f i e l d equation amounts to the statement that  when applied to the f i e l d the Casimir operator ( 6 . 6 5 ) i s equal to the constant m u l t i p l i e r zero times the f i e l d . since acting on X X *\ +  This m u l t i p l i e r must be zero  t h i s operator forms a pseudoscalar when m u l t i p l i e d by  whereas X J f ^ X +  i s a scalar.  In eqs.(6.64), using as t r i a l solutions the f a c t o r i z a t i o n s  (6.66a)  fF(r)  if  ^(y)  |F'(r) X ( y ) 4  (6.66b)  f  F(r)  =  X (y) 3  lF'(r) X (y)J 2  one finds that the variablesr and y are separate, since upon substitution of  (6.66a,b)  the iterated equations become  (6.67a)  [ ^ ^ ( y j J / X ^ y ) = {[(r> ) -4r* +(15/4)] F(r)}/[r F(r)]  (6.67b)  kka X (y)]/X4(y) = i[(ri ) -4r^ +(15/4)]F«(r)]/[r F'(r)l  (6.67c)  |^ X (y)|/X (y) =  (6.67d)  [^ X (y)]/X (y) = ^[(ri ) -2rt +(3/4)]F'(r)}/[r F(r)l  2  2  r  r  2  2  k  k  4  3  r  r  3  ffiri ) -2r^+(3/4)jF(r)V[r F(r)] 2  2  r  2  2  k  2  2  r  r  .  In each of these equations, on the left hand side there is a function depending only on y, and on the right hand side a function depending only on r. Therefore consistency requires that one has separate validity of  95 the equations (6.68a)  O VK )>M(y) = 0  (6.68b)  {r2K2+(r> ) -4r^ +(15/4)}F(r)  (6.68c)  (^ +K' )X (y) = 0  (6.68d)  ^r2K'2+(ri )2-4r) +(15/4)}F'(r) = 0  (6.68e)  O ) +K ) X (y)  (6.68f)  [r K2 ( > )2.2ri +(3/4)}F(r) = 0  (6.68g)  (^ ) +K )^ (y) =  (6.68h)  }r2K'+(r} )2-2rV(3/4)}F'(r) =  2  k  =0  2  Jr  r  2  k  4  r  k  r  = 0  2  k  3  2  +  r  k  r  r  | 2  k  0  2  •  0  r  A A  where K , K ' , K , K * are four possibly complex constants. The equations f o r the functions of r can now be cast i n the form (6.69)  {(z\) +z 2  2  v  2} ^ z  (  z  )  =  o  §  vrhich i s the equation f o r a c y l i n d e r function Z ^ ( z ) ;  i.e.  Z  y  i s any  l i n e a r combination* (6.70)  Z  = aj  w  + bY  y  w  of Bessel functions of the f i r s t kind J  , and of the second kind Y  y  y  .  To do t h i s , one uses the r e l a t i o n s (6.71a)  r & [ r P F(r)]  (6.71b)  ( r i ) [ P F(r)]  r  = rP(p+r* ) F ( r )  2  r  (p= some r e a l number)  p  r  = rP[p2+2pri +(r^)2]F(r) r  so that e . g . with p=2, defining a new quantity G(r) (6.72a) then G(r) (6.72b)  F(r)  = r  by  G(r)  2  satisfies r2^K r2+Kr[cVa(Kr)] 2  2  4} (r) = 0 v  V  = \  2  i.e.  ;  v^=+J  .  G  Hence G(r) i s a c y l i n d e r function Z ( K r ) , (6.72c)  related to F ( r )  with order  v given by  or-f  However, since i n ( 6 . 7 0 ) the numbers a and b are a r b i t r a r y ,  The form a j +hj_ i s not used since J endent f o r a l l values of v . y  u  y  and  and since  are not l i n e a r l y indep-  96 J+i,  Y+i happen to bo expressable  (6.73)  Ji(z) =  sin(z) = Y . i ( z ) ;  (£TTZ)-£  Yi(z) = - ( i l T z ) "  i n terms of elementary functions  cos(z) =  2  -J_A(Z)  ,  then any Z i ( z ) can be w r i t t e n as some Z i ( z ) , so that considering only v =-| yields a l l solutions.  X of the second-order equations (6.64)  Thus solutions  are given by*  (6.74a)  X, = r2  (6.74b)  A  (6.74c)  X  2  Z  i( r) K  .A  7((y)  = r Z'i(K'r)  X3 = r L(Kr)  (6.74d)  X,(y)  2  y(y) 3  X(y)  = r Z'i(K'r) 2  4  Substitution of these solutions of the iterated equations (6.64) into the first-order equations (6.59) then yields conditions on the K*s and the Z's in order that functions of the form (6.74) actually solve the field equations,. With A= -3/4, the field equations become (6.75a)  ir r^ X (i/ )(ri -3/ )(X -irX ) = 0  (6.75b)  ir^ ( -irX )+(i/2)(-ri +i)X * 0  (6.75c)  - K k } ( X4+irX )+(i/2)(-rV4)*  2  2  k  k  2 +  2  Xl  r  1  3  k  3  r  2  2  = 0  3  (6.75d) -ir r i X +(i/2)(ri -3/2)(X +irX ) = 0 2  k  k  3  r  /+  2  where by convention extra zeroes are added to the x ' s to make up 2-spinors  f  F  (6.76a)  X,=  0  ^ ,  Xo -  L2  .0 .  X  ,  X4 -  ""In the massless case the r-dependence is given by (ar^/ +br / ) for <p and (ar3/ +br2) for p . 2  3  2  2  4J  97 and (6.76b)  r  1  =  , r =ir , , 2  2  Then multiplying (6.75c,b) by p.r, and adding to (6.78a,d) respectively, one arrives at the simpler coupled equations (6.77a)  (i/2)r T }  (6.7?b)  (i/2)r r i  k  k  k  k  V(i/2)(ra -3/2) X = 0 1  r  X f(i/2)(r^ -3/2) r  r  = 0  .  Now making use of the formula f o r the d e r i v a t i v e of a cylinder function, one has (6.78a)  rd [r Zi(rK)]  = 2r Z±(rK)+r3lc[-(2Kr)-1 Zi(rK)+Z.i(rK)]  (6.78b)  r ^ r ^ Z ^ r K ' ) ] = 2r Z'i(rK )+r3K«[-(2K r)- Zi(rK )+Z»i(rK«)J  2  2  r  2  ,  ,  1  ,  ,  so that e q s . ( 6 . 7 7 ) read (6.79a) 0 = r 3 [ Z i ( r K ) r\x (6.79b)  (y)+K'z:i(rK') X ( y ) } + ( 2 - i - 3 / 2 ) r Z i ( r K ' ) ^ ( y ) 2  t  4  0 = r3^l(rK')r ^ X4(y)+KZ.A(rK)X (y)]+(2--|-3/2)r ZA(rK)^ (y) , k  2  k  1  1  To f i n d solutions of these equations they must be reduced to a simpler form.  From the discussion following e q . ( 6 . 5 5 ) , one should look f o r a form  which allows one to write ( 6 . 7 9 ) as a single equation f o r the 2-spinor field  </>(y), defined by (  (6.80)  ?(y)  Aiming at an equation i n the form of the analogue of the Dirac equation i n the 2-diraensional Euclidean space, (6.81)  ( - i P ^ +m) p(y) = 0 k  k  .  my? being a "mass" term when m i s positive and r e a l , then from  eqs.(6.68a,c)  one must put (6.82a)  K'  2  = K  2  According to ( 6 . 7 9 ) .  = m  2  .  there must be a r e l a t i o n s h i p between Z and Z* f o r  98 (6.81) to r e s u l t . (6.82b)  Taking K +K=m, t h i s i s ,=  Z«_A(rm) = iZ^(rm)  ,  or equivalently (6.82c)  Z'i(rm) = - i Z i(rm) 2  ~2  Defining c o e f f i c i e n t s a,b,c,d by (6.83)  Zi(rm) = aJi(rm)+bYi(rm) , Z»i.(rm) = cJi(rm)+dYi(rm)  ,  then from (6.73), (6.82c) implies (6.84)  d = ia  (The case K  ,  c = -ib  .  -K= -m goes the same way, but with the r e s u l t d= -a, c=  , =  So one arrives at the r e s u l t that i n order to f i n d f i e l d s X  -b.)  which are  solutions of the Dirac-type equation (6.81), then (6.85)  Zi.(rm) = a«U(rm)+bYi(rra) ; Z»i(rm) = -ibJi(rm)+iaYi(rm) , 2  2  2  2  where a and b are any complex numbers.  2  2  In' f a c t , up to t h i s point there  seems to be no reason, on the basis of an examination of the c y l i n d e r functions, to r e s t r i c t the number m to the p h y s i c a l l y i n t e r e s t i n g case i n which i t takes on values on the non-negative r e a l a x i s . In any event one cannot expect functions defined over 2-dimensional Euclidean space to bear any physical s i g n i f i c a n c e , although the methods developed above can be c a r r i e d over without change to the more complicated and r e a l i s t i c cases treated i n the following chapters, where i t w i l l be assumed that m i s a r e a l non-negative number.  At t h i s stage, i t s u f f i c e s to  note that an examination of the remaining equations (6.75) leads to another Dirac-type equation f o r a f i e l d  <jP(y), with another "mass", m, which need  not be equal to m. So i t appears that more than one mass can be accommodated by solutions to a s p i n - i equation i n t h i s Euclidean-space model.  The expectation  that t h i s feature i s carried over to the" case of f i e l d s i n Minkowski  space as w e l l w i l l be borne out i n Section 8.  One could then  identify  the two species of p a r t i c l e with the muon and the e l e c t r o n , or i n the massless case with the muon and electron neutrinos, thus completing the l i s t of known leptons.  100 PART III.  ACTION PRINCIPLES  Synopsis of Part  III  7. Rotationally Invariant Action P r i n c i p l e 7-1. Conservation laws on hyperboloids The 5-dimensional i n t e g r a l over a region TL^ on the surface ^ " | = L i s A  A  written i n terms of the boundary i n t e g r a l over si^ by using Stokes* theorem. Taking the regions JTL^  and xi^  to be elements of 1- and 2-parameter fam-  i l i e s of surfaces i n 6-dimensional space, t h e i r corresponding area elements are found to contain of  S ( * | ^ - L ) when the i n t e g r a l s are written i n terms A  A  d V 6  7-2. Alternative Euler-Lagrange equations The action i n t e g r a l i s therefore also defined to contain to f a c i l i t a t e  using the Stokes  1  theorem.  %(^  A  ^ -L)d A  An alternative Euler-Lagrange  equation i s found which r e s u l t s from using conjugate momenta d i f f e r e n t from the usual ones. 7-3• Angular momentum tensor i n 6-diraensional space An angular momentum tensor with four indices i s found which s a t i s f i e s a generalized divergencelessness c o n d i t i o n .  Integrals over J l ^ y i e l d  f i f t e e n conserved q u a n t i t i e s . 7-4. Modified Schwinger action p r i n c i p l e The f i f t e e n conserved quantities are i d e n t i f i e d with the generators of the group of transformations by varying the boundary of the action i n t e g r a l i n the usual f a s h i o n , except that t r a n s l a t i o n a l invariance of the Lagrangian density i s not assumed. 7-5 • Canonical (anti-)commutation  relations  Employing the quantum f i e l d theory generator derived i n section 7-4 f o r an a r b i t r a r y f i e l d v a r i a t i o n y i e l d s the canonical r e l a t i o n s as s e l f - c o n s i s t e n c y c o n d i t i o n s .  (anti-)commutation  ,  101 8, Special Cases 8-1. Scalar f i e l d The wave equation f o r the s c a l a r f i e l d s i o n a l action p r i n c i p l e .  X i s derived from the 6-dimen-  The single a r b i t r a r y number C which appears i n  t h i s equation i s equal to the eigenvalue of the only non-vanishing Casimir operator of the rotations i n 6-dimensional space.  Solutions are found i n  terms of powers of the a u x i l i a r y coordinate r and cylinder functions index depends upon C .  whose  These solutions s a t i s f y the Klein-Gordon equation  with mass i n Minkowski space. Provided C takes on the value -15/4, the f i e l d equation can be brought into the form of a massless wave equation i n 5-dimensional space f o r a f i e l d variable Y which i s r e l a t e d to  X by m u l t i p l i c a t i o n with powers of  r.  This connects the 6-dimensional theory with Wyler's approach to the mass concept, and i t i s shown that the conserved quantities which r e s u l t from evaluating the angular momentum tensor can be i d e n t i f i e d with those associated with conformal invariance i n 5-dimensional space.  At t h i s point i t  is  not necessary to have C= -15/4, but the next section j u s t i f i e s t h i s v a l u e . 8-2. Green's function f o r s c a l a r f i e l d equation Mack and Todorov's r e s u l t s on covariant Green's functions of the form [( ^ ^ • ^ ) ]  are extended to the case " ( ^ =^0, where they take the form  4 1  A  2  ) ] . 2  q  The Green's function i n 5-dimensional Minkowski-type space  which results from t h i s generalization i s compared with the standard r e s u l t given by Gel'fand and S h i l o v , which has q= - 3 / 2 .  To see how t h i s number  r e s u l t s from the 6-dimensional formulation, A d l e r ' s work on r o t a t i o n a l l y covariant Green's functions off the n u l l hyperquadric i n 5-dimensional space i s extended to 6-dimensional space.  It  i s found that one i s led simul-  taneously to q=-3/2 and to C=-15/4, as i n section 8-1, i n order that the Green's function be of the simple form which generalizes the r e s u l t of Mack  102 and Todorov. Thus, the point made i n t h i s section i s that one can f i n d the Green's function f o r the scalar wave equation f o r the case studied here, "|  ^^$0,  and that i t has a simple form, 8-3. Spinor f i e l d The free f i e l d equation f o r the spinor f i e l d dimensional action p r i n c i p l e .  It  T C i s derived from the  6-  i s shown that this spinor equation  d i f f e r s from an a l t e r n a t i v e suggested by Barut and Haugen i n that i t can be derived from a r e a l Lagrangian d e n s i t y .  It  i s shown how solutions of  the spinor equation can be found by i t e r a t i o n provided the single number X i n the equation i s given by A = - 5 / 4 .  arbitrary  Then solutions are given  i n terms of powers of r and c y l i n d e r functions of r which depend upon parameters ra and m, multiplied by functions of the Minkowski space v a r i a b l e s . Using the methods developed i n Section 5» i t i s shown that there are two types of s o l u t i o n , each of which s a t i s f i e s an equation i n Minkowski space which i s p r e c i s e l y the Dirac equation when m and m are i d e n t i f i e d . w i t h p a r t i c l e masses.  The value  A= -5/4 i s j u s t i f i e d by showing that i t  two  results  from setting one of the Casimir operators of the transformation group equal to a number when acting on  X•  The conserved quantities are derived from the angular momentum tensor, and the canonical anticommutation r e l a t i o n s of the f i e l d are derived from the r e s u l t s of Section ? . Thus, i t  i s shown i n t h i s section that two types of massive  f i e l d s can be grasped simultaneously by a single equation i n 6-dimensional space, so that the 6-dimensional formalism allows one to describe the observed leptons i n a simple f a s h i o n .  103 8-4. Green's function f o r spinor f i e l d  equation  The spinor free f i e l d equation i s i t e r a t e d to the form of the s c a l a r wave equation, so that the Green's function i s given i n terms of the spinor f i e l d equation m a t r i x - d i f f e r e n t i a l operator acting on the scalar Green's This i t e r a t i o n allows one to derive the value of X from the  function.  value C= -15/4 found i n section 8-2, and one has again A = -5/4; j u s t i f i e s setting  = -5/4 i n section 8 - 3 .  A  i . e . this  To make sure that t h i s spinor  Green's function f o r "| ^*} f0 makes sense, the analogous expression i s conk  structed on the u n i t hyperquadric  i n 5-dimensional Euclidean space, and i t  i s found to agree with Adler's r e s u l t i n that case.  The  6-dimensional  Green's function i s then compared with Adler's two guesses f o r the case *IA"1  A=0, a  n  d  i  t  s  n  o  w  n  t  h  a  t  S  o i n  g  t  o  1A^ ^° l A  e  i  m  i  n  a  t  e  s  some o f  t  h  e  diffi-  c u l t i e s that Adler encounters i n developing covariant scattering theory i n 6-dimensional space. 8 - 5 . Minimal coupling;  gauge invariance with massive vector boson  The minimal coupling i s c a r r i e d out i n 6-dimensional space of the spinor f i e l d to a vector or pseudovector f i e l d which s a t i s f i e s a free f i e l d equation found by considering the vector representation of a Casimir operator of the transformation group.  The gauge group of the vector f i e l d  k B  i s i d e n t i c a l , i n Minkowski space, with that of the gauge function i n the  theory of the Stueckelberg f i e l d , which provides a way of r e t a i n i n g gauge invariance even with a massive vector boson.  The a d d i t i o n a l s c a l a r or  pseudoscalar f i e l d component which corresponds to the Stueckelberg f i e l d B arises automatically from the methods developed i n Section 5 f o r the vector field.  I t i s found that as well as the usual vector and pseudovector type  couplings  <V *  pseudoscalar  k v  f B  k  and <? *  type couplings  k  B^, one also arrives at the s c a l a r and  Y ^ B and ^  B i n Minkowski space.  104  PART I I I . ACTION PRINCIPLES 7. R o t a t i o n a l l y I n v a r i a n t A c t i o n P r i n c i p l e I n order t o develop a method by which f i e l d equations can be derived from a Lagrangian density X. i n 6-space by means of an a c t i o n p r i n c i p l e , i t i s necessary to determine the form t h a t conservation laws assume when only r o t a t i o n a l and not t r a n s l a t i o n a l invariance i s the required property of the a c t i o n i n t e g r a l . To t h i s purpose one must f i n d an analogue i n s i x dimensions of Gauss' theorem, which can be a p p l i e d t o s i t u a t i o n s f o r which the v a r i a t i o n s of the coordinates v ^ take points on the hyperboloid *y "\A k k  A  =  ^- ^° points n  on the same hyperboloid. Then by studying the v a r i a t i o n s of f i e l d s on t h i s hyperboloid, one can construct an a l t e r n a t i v e t o the Euler-Lagrange  equations  which i s a p p l i c a b l e t o s i t u a t i o n s i n which t r a n s l a t i o n a l i n v a r i a n c e does not o b t a i n *  7-1 • Conservation laws on hyperboloids Since one i s i n t e r e s t e d i n a c t i o n i n t e g r a l s on the surface  ^ \ =L A|  A  i n s i x dimensions, i t i s necessary t o o b t a i n an i d e n t i t y r e l a t i n g a 5-dimens i o n a l i n t e g r a l over some region JTl^ of t h i s surface t o a 4-dimensional i n t e g r a l taken around the(olosed) boundary Si^ of sx.y i s given by the g e n e r a l i z a t i o n of Stokes to the case o f s i x dimensions.  If  1  fxj^Xjklt  This r e l a t i o n  theorem (e.g. Anderson 196?) i s  a  fourth-rank tensor  i n 6-dimensional space, then one can form a s c a l a r q u a n t i t y ^Ai A2A3A4 d * t 1 A  A 2 A  3 4 by l a b e l i n g points contained i n -n.4 by means o f A  f o u r parameters X-j, X , A3, A^ and d e f i n i n g the fourth-rank tensor* 2  (7.1)  df 1 2 3 4 = A  A  A  A  ^ V L f . ^ ^ d X ,  d> d> dA . 2  *Here, use has been made of the generalized Kronecker d e l t a , which  3  4  A s i m i l a r element of "area" d t 1 2 3 4 5 may be defined on the surface A  A  A  A  j l ^ by describing the coordinates i n terms of f i v e variables  A  o  that surface  n  parametrically  "X.j 2,3 4 5*  As a consequence of the d e f i n i t i o n (7.1), one can form the hypersurface integral  which i s a s c a l a r under coordinate mappings as well as under parameter changes.  arbitrary  In terms of these s t r u c t u r e s , Stokes' theorem can  be written  (7-3)  "L,  *A,A A A„ 2  3  ' W j V A j  d t ^ W i •  As i n the 3- and 4-dimensional cases, t h i s equation may be s i m p l i f i e d by introducing the quantities  •('/*') (7.4c)  d3  A)  ^  £  =(t/5D  W  j  A ] A 2 A 3 W 6  A  «  6  ^  A  d^2W5A6  With the introduction of these q u a n t i t i e s ,  (7»3)  .  can be rewritten i n the  more convenient form (7.5)  d  by making use of the ( 7  . , 6  £  A  '-  A  ^  +  = 2 L  S A B  5  F^.B  dS  A  identity ,  -  A  6  t  B  L  ...B A R  ...A6 ' -') 5  r  t  l  6  1  •  has the following p r o p e r t i e s : a ) i t i s completely antisymmetric i n superscripts and subscripts; b) i f the superscripts are d i s t i n c t from eachother, and the subscripts are the same set of numbers as the supers c r i p t s , the value i s *1 depending on whether the superscripts are an even or odd permutation of the s u b s c r i p t s ; c) otherwise the value i s 0. Also, i s assumed to be simply connected.  By construction, F  A B  i s an antisymmetric tensor density, and dS^g i s  also antisymmetric. The r i g h t hand side of ( 7 . 5 ) ay be cast into a more useful form ffl  by selecting an area element d S such that derivatives of F ^ occur only 3  A  i n terms of the d i f f e r e n t i a l  generators of rotations  3 i( =  M A  ^ B " ^ B ^A^*  AR  Then i n those cases i n which F current density j (7.7a)  can be written i n terms of a vector  as  A  F^ = \  A  j  - r^BjA  B  one has (7.7b)  mjjfM  = 2i  \^ r ( A  lA  3 jB).(i„.  ^)(  B  .  This form corresponds to the conservation law suggested by Boulware e t . al.  (1970)  (7.8)  (m^-i S  f o r currents j (7.9a)  A B  )j  =0  B  that separately obey the condition  A  3 J =0 A  A  and the t r a n s v e r s a l i t y condition (7.9b)  n ^  A  =  »  0  A  because one has i n f a c t (7.10)  m  ^  = 2v  A l  ( m A B  .  i  S  A G  ) jB.  8 i V l  ^A  i n those cases f o r which F ^ can be written i n the form ( 7 . 7 a ) . The surface element d S which brings the integrand of the 5-dimensional A  i n t e g r a l into the form n ^ g F ^ i s found by converting the r i g h t hand side of ( 7 . 5 ) into an i n t e g r a l over a region  i n 6-dimensional space.  This can be accomplished by defining the 1-parameter family of surfaces f(v^)=constant, so that i n terms of f(v^) the element of area d S on the A  surface f=f  (7.11)  Q  i s given by ( c . f . Fubini e t . a l . 1973)  ds = d A  6  ^ S(f-f ) 0  0>fA^ ) A  f=fo  •  107 Then choosing the surface f=- <^ , on the hyperquadric 2  ^ = L one has 2  the surface element  (7.12)  S(^ -L)2^  d S = -d\  ,  2  A  A  where the i n t e g r a t i o n over d S i s to be c a r r i e d out over  The proof  A  that t h i s area element corresponds to (7.4c), with the parameters ( Ai 2,3,4,5) (Z.y t -) i =  / +  v <  h  a  s  t  For the area element d S  A B  been relegated to Appendix 2 . , one must s e l e c t another 1-parameter set *\ =L. i n order to pick out a set of  of hypersurfaces, on the surface 4-spaces -A.^.  2  There are many possible choices, but i t i s both convenient  and u s e f u l to choose the surfaces h=(-*\^/vi )=constant. to choosing the time t=  This corresponds  as the d i r e c t i o n i n which "charges" associated  with conservation laws remain unaltered, i . e . the d i r e c t i o n of quantization (c.f.  Fubini e t . a l .  1973)•  This choice also amounts to s e l e c t i n g the  remaining parameters ( ) j , )_2» X^; X  ^)(y.» =  x ) f °  r  parametrization of  _rv  "^«  Since d S ^ i s to be antisymmetric, one can define i t a s *  (7.13)  ds  = d ^ fbf(n)l  *h( A )  6  A B  .  >•  S(f+L)  o"(h+t)  .  h+t=0  f+L=0  E x p l i c i t l y , one has  (7.14)  *h(l)/H  B  " - x-" S* 1  +  K  B  "  2 ( S  B " ^B ) I 6  5  and s u b s t i t u t i o n of t h i s equation and of that f o r  (7.15)  dS  A B  = d\  2 v| K[A  1  4  f ( \ )  into (7.13) y i e l d s  {S f+t(S -S )} S(-*" 5  B  B]  6  B]  1  ^+t)  S(-f+L)  As was the case f o r d S , one can e x p l i c i t l y check (7.15) by comparison A  with the d e f i n i t i o n (7.4b), and again the proof i s given i n Appendix 2 . Using these area elements, the i n t e g r a l i d e n t i t y (7.5) can f i n a l l y be rewritten as *The square brackets have the meaning of antisymmetrization, ^(A ^B]  g  ^A ^B_ ^B ^A  108 (7.16)  21 j m F  U^ -L) d ^ =  A B  2  A B  6  The r i g h t hand side i s the form that the i n t e g r a l over the boundary jx^ takes when expressed as an i n t e g r a l over d ^ .  Hence (7«16) represents  a type o f conservation law i n 6-space, and whenever the boundary i n t e g r a l vanishes, one has the divergence condition  (7.17)  m  A  B  F  A  B  = 0  .  7-2. Alternative Euler-Lagrange equations As a next step, one can determine how the form which conservation laws must take i n 6-dimensional space, (7.17). a f f e c t s the form of the E u l e r Lagrange equations.  To t h i s end, one must propose an action i n t e g r a l  over the region -fig which takes i n t o account the f a c t that only rotations, and not t r a n s l a t i o n s , are included i n the group of motions i n 6-space. Thus the transformation behaviour of any set of coordinates >^ , f o r which A  »\2=L, i s r e s t r i c t e d such that f o r  ^  | A  , one s t i l l has ^ = L . ,2  Hence  the integrand i n an action i n t e g r a l must be constrained such that *^ l i e s A  on  <^ =L, and t h i s can be accomplished by defining the a c t i o n I as I = L , £  (7 .18a)  $(^ -L) d 1 6  2  .  The form of the Lagrangian density t. suggested by the spinor equation adopted i n section 6-2 i s (7.18b)  X  =  *(^.X,m  A B  X)  where m g i s the representation by d i f f e r e n t i a l operators of the generators A  of r o t a t i o n s .  Here, i t should be noted that since X. need not be trans-  l a t i o n a l l y invariant, the coordinates ^ possible e x p l i c i t arguments of £ .  A  have been included among the  109 Taking the v a r i a t i o n  (7.19a)  SX=  (7.19b)  U\ ) =0  S I of I w i t h f i x e d boundary o f  defined by  iXyQ-ZW ,  A  one has (7.20)  %I  - J{(i£/&*>SX + SCm^X.) & ^ ( m  A B  X)]]  W\ -L) d ^ 2  6  since according t o (7.19b) the coordinates themselves are unaffected by the v a r i a t i o n . (7.21)  Thus one can w r i t e S I as  SI = ^ ( ^ X K Z ^ C S x ^ W m ^ T ? *  3  ) ^ }  Stf-L) d "\ 6  where the analogues i n 6-dimensional space o f the conjugate momenta i n Minkowski space are d e f i n e d by (7.22)  (m x")  v-^ =  AB  .  According t o (7.16) the i n t e g r a l of the second term i n s i d e the brackets i s equal t o an i n t e g r a l over the boundary, and one can assume t h a t t h i s i n t e g r a l vanishes since the boundary i s taken t o be s p a t i a l ( y - ) i n f i n i t y S e t t i n g the v a r i a t i o n  S i equal t o zero f o r a r b i t r a r y v a r i a t i o n S X ,  one a r r i v e s a t the a l t e r n a t i v e Euler-Lagrange (7.23)  frlfrZ)  0  equations  - m^TfAB = o •  I t i s immediately apparent t h a t the form o f t h i s equation i s i n p r i n c i p l e v e r y d i f f e r e n t from t h a t o f the u s u a l Euler-Lagrange equation. 7-3. Angular momentum tensor i n 6-dimensional space One can now u t i l i z e the f i e l d equation (7.23) t o educe the manner i n which conservation of angular momentum i s contained i n the a c t i o n principle• Consider the a c t i o n i n t e g r a l (7.29a) and i t s transformation p r o p e r t i e s under the i n f i n i t e s i m a l r o t a t i o n s (5.75)» (7.24)  Il =E A  A  , B  \  B  110 which induce on the components of the f i e l d  tZil) "  (7.25) where s  -(i/2)E  s XHE (^  A B  ^ -7  A B  A B  X. the  A  B  transformation  *>  B  a  are the spin operators of the f i e l d X. .  A E  If  the i n t e g r a l (7.18a)  i s to be invariant under (7.24) and (7.25), the v a r i a t i o n i n X must be a "divergence" (7.26)  k£  = -(i/2)E  A B  m g £ = m ^-(i/2)E .£ }  .  AB  A  AR  or  (7.27)  0 £ A E B ) = -(i/2)m A  A B  £  .  When t h i s condition i s s a t i s f i e d , one can obtain an expression f o r a tensor density which s a t i s f i e s the divergence condition (7.17) as a r e s u l t of r o t a t i o n invariance i n the usual fashion by considering E  A B  to be coordinate  dependent, so that the r o t a t i o n has the e f f e c t s (7.28a)  %  (7.28b)  m  *>% + E 3 X A  C D  * »  A B  where X  A G  -C / )(  =  i  m X-KiD EAS ^ 4EAB C D  C D  )  A B  B t D  2  n  + S A B  ^  AB  )  .  Therefore  O*/"*  (7.29)  E ) = Of. A X )  X^+TT^QX^  (-tu^ )] = ^  X  A B  and  (7.30) where f f (7.3D  frtft C D  8  i s defined by (7.22). m ^ f ^ ^ E ^ ) ]  C D  A B  Application of m  CD  to eq.(7.30) gives  -(i-bDTf^^AB+Tf^n^AB  •  On account of the f i e l d equations following from the action p r i n c i p l e based on X , (7.23), one can write eq.(7.31) as  (7.32)  ^ [ U / M m c ^  8  ) ]  = (^/^)^  +TT \ C  A B  D  X  A  B  =  = 0.£AE B) . A  I n s i s t i n g on (7.26) as the expression of rotation invariance, eq.(7.31) takes the form of a conservation law  111  (7.34)  =" - i f f  C D S a b  Inserting  C D  (m  + s A B  AB^  ^  ^B ^  •  D  (7.44) into the i n t e g r a l i d e n t i t y (7.16) one finds that  f o r each A,B there e x i s t s a vanishing boundary i n t e g r a l of Taking the boundary to be made up of two p i e c e s a n d to times t and t ' ,  J  ~ \ , corresponding L  together with s p a t i a l (y_-)infinity, then  the meaning that the f i f t e e n  (7.33) has  quantities  (7.35) are conserved i n time ( f o r given L ) , where dSQp i s the surface element defined by  (7.15). It i s also possible to interpret the r e l a t i o n s (7.33)  as current conservation laws.  This f a c t w i l l emerge more c l e a r l y below  when p a r t i c u l a r cases are examined. 7-4. Modified Schwinger action p r i n c i p l e It  i s possible to place the conservation laws  (7.33) i n a more general  setting by examining the e f f e c t of a r b i t r a r y v a r i a t i o n s  on the a c t i o n  i n t e g r a l I without r e s t r i c t i n g attention to the case of f i x e d boundary S l ^ , Thus, one expects to f i n d that the conserved "charges" M^g can be i n t e r preted as the quantum f i e l d theory generators of rotations v i a a Schwinger action p r i n c i p l e (Schwinger  1951). modified by allowing the p o s s i b i l i t y  of e x p l i c i t coordinate dependence of the Lagrangian density t- . In the present case the v a r i a t i o n of the action i s given by  (7.36)  SI  =  \ * A  »AB^ + SC B >«1 X  6  A+ T  A  • o[(^rt) -(L+rL)]} d ^' - I 2  6  ^]-  112  under the v a r i a t i o n  \ -> Y  (7.37)  A  A  Substituting f o r d ^  (7.38)  =  1  1  A +  ^  •  by means of the Jacobian  J ( ^ ) =  eq.(7.36) can be written (7.39)  SI « f  £ • ( X}^ X ,^)  Since the v a r i a t i o n  (7.40)  J(f.l)  t  AB  $(^ -L') d ^ - I , 2  6  .  J * ^ i s i n f i n i t e s i m a l , one has  j(nM> =  1 +  [}< Ti )Ai ] • A  A  Expanding £ • about X as  (7.41)  X' = X + ( ^ ^ A x ) S X + f f S ( a X ) - K U / ^ ^ ) T\ AB  A  A  AB  and r e s t r i c t i n g attention to only those transformations  .  %*]^- which preserve  the value of  (7.42)  ( V T^)  2  = <f  .  L» = L  ,  eq.(7»39) becomes (7.43)  SI =[^ {(^^/^)SX+TT mAB(S%)+^ §( T f ) / M ]  In t h i s equation, the d e f i n i t i o n (7*19) of the v a r i a t i o n used to commute  A  AB  +  6  S  has been  S and ni^g.  In the present case, one i s interested only i n those transformations s a t i s f y i n g (7,42).  In Appendix 2 i t i s shown that i n t h i s case  one c a n w r i t e  % I as t h e sum o f two i n t e g r a l s  L i(^/^)- ^ } m  (7.44)  SI= +  6  L  A3  $ ( * t - D d >[ + 2  AB  6 ^D^  C Q  S* (i/2)X[>( T f ) / ^  6  +  D  ]]  S(*| -L) 2  d  Hence, p r o v i d e d t h e f i e l d e q u a t i o n s  (7.23) a r e s a t i s f i e d , so t h a t t h e  f i r s t term v a n i s h e s , t h e n whenever  S i i s zero, the i n t e g r a l i d e n t i t y  states that the quantity  (7.45)  F =  l/(2i) J ^  C D  i X + ( i / 2 ) £ [V S ^ ) / ^ 1 ] ] d S C  D  C D  i s i n d e p e n d e n t o f t h e t i m e t . F o l l o w i n g S c h w i n g e r , one c a n i d e n t i f y t h i s i n t e g r a l w i t h t h e quantum f i e l d t h e o r y g e n e r a t o r  of rotations.  From t h e d e f i n i t i o n  (7.46a)  H x U  X'(\)  2  F i s defined by  (7.46b)  U- = e 1  i  F  so t h a t f o r i n f i n i t e s i m a l r o t a t i o n  (7.46c)  Fa  parameters E  A B  , one f i n d s  ^  i f M^g i s d e f i n e d b y  (7.46d)  [X.M ] = ( m AB  A B +  s )X A B  .  Thus, w r i t i n g (7.4 a)  =  7  ,S  C A  D  B  dS  C  D  one has  (7.47b)  o> = - | f t % c  SiB  A B +  s  A B  )x  i n agreement w i t h t h e p r e s c r i p t i o n  +  |S  C A  (7.3*0•  S  E B  t .  6  ^  .  (7.^6)  7-5.  Canonical (anti-)commutation  relations  Interpreting the operator F as the generator of variations of a quantum f i e l d X , one must have  (7.48)  = i[F,x]  f o r any v a r i a t i o n  eX.  If,  in particular,  one chooses an  o X accompanied by no coordinate v a r i a t i o n ,  %  arbitrary  - 0, then (7.48) can  be looked upon as a consistency condition which, together with the d e f i n i t i o n (7.45), can y i e l d the commutation relations f o r the I m p l i c i t i n the d e f i n i t i o n (7.45) i s a summation over a l l f i e l d components  (7.49)  <t  JTt  C D  %X«  dS  C D  .  on the surface defined by d S ^ y ^ ' ) ,  (7.50a)  %X  (7.50b)  ^TT/ (^) = i[l/(2i) lf\  f i  o  n  ( ^ [ ) = i[l/(2i) J u ^ C ^ ' H X<(y)dS B  Making use of the  independent  X.  F = l/(2i)  Thus, f o r any  field.  C D  (^)  S^)dS  C D  e  C D  raus  *  n  a  v  e  ( *|«) , X  (y) ,  )]  Tf/ ^)] 3  identity  these r e l a t i o n s can be written  (7.52*)  (7.52b) If  ^/^) =iJ[ff.(V) [s^(^) , CD  -  ^ (^) =i J{nP(^) [u^y) , ff^)]. AB  one assumes,further, that ( c . f .  Roman 1969)  115  [x^) .  (7.53a) for a l l  <* ,j3  (?.53b)  \ X f i %  and a l l  9  0  vj, tf|* on the surface  0= D x j - ) ' ) . ^ ) ] .  and since the v a r i a t i o n i s  (7.53c)  0 = [x^)  f o r ^ . " \ ' on  +  then  [?.(-(').  *  arbitrary,  , X (v\)]_ fi  -OL^.  Hence, ( 7 « 5 2 a ) becomes  * X^) - - i ^ t y  (7.54) for  . ft  C D  (f)]-  S x ^ ' ) ds ( ') CD  a  ^ , v^' on -^-j^. The simplest s o l u t i o n f o r ( 7 . 5 4 ) i s found by writing  (7.55) where  (7-56)  [x^) ^  C D  .  Tt  C D  (f)].  S  = -2  S  ( | - ^ ' ) i s defined f o r two points  ^ l~V y  )  dS  ^"V>  V^."]' o n - ^ - ^ by*  Y  Jj^'gCV)  C D  CD l = (/ ,}  and i s thus antisymmetric i n C and D. The simplest way to determine  S  C D  ( ^ - ^ J ) i s by writing dS^g, given 1  by ( 7 . 1 5 ) , i n the form (7.57)  d S ^ = n [A  n» -j B  dS(y,, K )  This singular function i s introduced i n analogy to the function %k(y-y*), which i s defined by (Gourdin 1969)  $g(y') * (y-y') d < r k  f o r y , y ' on cr ; e . g . i f  k  ( «) = y  g  ( ) y  the surface i s t=constant,  then  116 where n^, n '  A  are two orthogonal unit vectors  (7.58a)  n  A  = *| /(Kr)  (7.58b)  n  A  n  ,  A  A  = n'  A  n'  n' = 1  A  A  ;  S 5- $ / )  = ^/+t( n  n'  A  A  A  = 0  )(n ^n'g-j ) = 2  .  S " ( K ~ >*^-t)  .  ; (n  Then one has  (7.59)  (2r) S ( v \ - L )  dS( ,K) = d \  $(H.- ^-t) =  2  £  1  = (-Vl3/2)dV dK. dL (2r) Uv\ -L) 2  1  Since the product of d S ( Y^* ) and & (v^-\q') must be simply CD  CD  (7.60)  ^  C D  (^-^') dS  C D  ( v | « ) = S(v-y.«)  S(t-t»)  £(x.-X»)  S(L-L')'  •dV d K dL , then one must have  (7.61)  ^  A B  ( ^ - ^ « ) = (-2rK )" 3  $(v.-yj)  1  S U - M . ' ) n&n« 3 B  .  Hence the (anti-)commutation r e l a t i o n (7.55) becomes  (7.62)  [?p(V  . ff/ (^')]_ = % B  .(X LH  ^ [  2  S(l-Z')  A  S(K-M.«)*  { ^ +t ^ K  JB]6^  ^  For example, taking the s p e c i f i c set of values {A,B} =^(6,k)-(5,k)^ , one has  (7.63)  , ff, (^')- ? f . 6k  fe (vp p  =  ^«  5k  (v\')].  Sd-Z .) 1  =  $(*--*.') (KL)"  1  f o r equal times t ' = t . The f i n a l factor can also be written i n terms of r=H~^L'2, since one has  (7.64a)  $ ( * - v c » ) = ( r « ) IT* 2  S(r-r»)  and hence (7.64b)  (XL)"  1  ^(K-VC«) = r r S(r-r«) 3  2  This completes the discussion of quantization of the 6-dimensional f i e l d theory v i a the action p r i n c i p l e .  The consequences of the  r e l a t i o n s (7.62) are investigated below f o r s p e c i f i c examples.  118 8. S p e c i a l Cases 8-1. S c a l a r f i e l d The analogue o f the Euler-Lagrange equation, (7.23), can now bo used to construct a f r e e f i e l d equation i n 6-dimensional space f o r a s c a l a r field X ( * | ) .  Such a f i e l d equation must flow from the a c t i o n p r i n c i p l e  based on a Lagrangian d e n s i t y i o f the form (8.1)  X = £ (1 ,^,m  where m,  A B  X)  are the d i f f e r e n t i a l operators defined by (5.5). and given  i n terms of y , H ,L i n (5»5a,b,c,d). The form of •£ d i c t a t e s the form o f a l l p o s s i b l e i n v a r i a n t s which are admissible c o n t r i b u t i o n s to X . and i n order f o r the f i e l d equation to be l i n e a r i n 7- one need o n l y consider those i n v a r i a n t s which are b i l i n e a r i n the f i e l d and i t s f i r s t d e r i v a t i v e s , i f only equations o f up t o second order are permitted.  Thus, the set o f a l l r o t a t i o n a l i n v a r -  i a n t s having the appropriate form i s r e s t r i c t e d t o the two f u n c t i o n a l s  (8.2)  I j = ( m 7 )(m X )  ; I =X  AB  AB  2  2  Thus the most general second-order l i n e a r f i e l d equation based on a ; Lagrangian d e n s i t y (8.1) has the form (8.3)  a  m A B B ^ X  +bX  =  0  where both a and b can be f u n c t i o n s o f L = ^  (  V A  A  without a f f e c t i n g the  r o t a t i o n a l covariance o f (8.3), since (8.4)  m  AB  a(L) = m  =  i  Afi  (L) > a ( L ) = L  ^A^B-1  =0  B  *A>  (  L )  V<  L)  "  119 The value of the r a t i o b/a i s connected with the value of the second-order Casimir operator C for the 6-dimensional rotation group. In a l l , there are three such operators (Murai 1953). and d e f i n i n g ,  for  general s p i n , the generators of rotations (8.5a)  m  = m  AB  +  AB  s  A B  they can be written (8.5b)  C = |S  "AB 5  (8.5c)  D = 1 ^ABCDSF *  (P. C.A\ (8.5d)  v - A E - 2  A B  AR  *°  A B  D  J ™AB sCD ^ABCDEF m  j.GHIJEF ~ ~ °GH I J  1  *  m  The s c a l a r f i e l d  X- i s defined, i n ( 5 * 5 9 ) . by the property  (8.6)  =0  S  A  B  X  ,  and i t i s easy to determine that i n the s c a l a r representation the only surviving Casimir operator i s C.  Then i n order that  X- transform  according to an i r r e d u c i b l e representation of the 6 - r o t a t i o n s , be a constant number when acting on X .  C must  This w i l l be assumed to be the case  in the f o l l o w i n g , so that the f i e l d equation ( 8 . 3 ) becomes simply (8.7a)  fm^ m  A B  X  - CX = 0  .  Thus the Lagrangian density j£ i s given by (8.7b)  X - -i(m  A B  X)(m  corresponding to an action (8.7c)  1=  L  6  l-Km  A B  X) - C X  2  .  integral  A B  X)U  A B  X)-CX ] 2  S ( < ( - L ) <A( 2  "  .  AR The problem of finding eigenfunctions of the operator m ^ n r  0  occurs  whenever one wishes to f i n d a set of basis functions which enable one to investigate harmonic analysis on hyperboloids i n any pseudo-Euclidean (i.e.  f l a t ) space.  Provided the squared i n t e r v a l L i s non-vanishing,  one can always s e l e c t a set of " s p h e r i c a l " coordinates such as  (6.19)  120  such that the basis f o r solutions of the eigenvalue problem of the form ( 8 . 7 a ) consists of homogeneous functions of degree zero, no matter what the dimensionality of the f l a t space under consideration, and no matter how many spacelike and timelike situation-is clearly different  axes one uses (Strichartz  1973)•  The  on the hyperquadric L=0, since i n that  case one cannot u t i l i z e L as a variable with which to construct a set of s p h e r i c a l coordinates a f t e r the fashion of s p h e r i c a l polar coordinates i n 3-dimensional Euclidean space. In f a c t ,  a set of coordinates i n which the variables separate  in  ( 8 , 7 a ) has been e x p l i c i t l y exhibited, and a set of basis functions i n terms of spherical coordinates i n 6-dimensional space can be given i n terms of hypergeometric functions (Winternitz 1971; Limic e t . a l . However i t  i s not only tractable  of ( 8 . 7 a ) i n terms of y ^ . H . L c a l c u l a t i o n the  (8.8)  im  A B  1966, 1967)•  but indeed i s simpler to f i n d solutions  Using ( 5 « 5 ) .  one finds by  straightforward  identity  mAB = - L 2  Evidently, eq.(8.8)  ^  A  A  Jf  A +  ^  ^  ^B  .  i s not applicable to the case L=0 due to the presence  of L i n the f i r s t term.  In f a c t , the s c a l a r wave equation ( 8 . 7 a ) i s not  the equation which i s customarily employed on L=0 f o r the  manifestly  conformally covariant d e s c r i p t i o n of s p i n - 0 p a r t i c l e s f o r just reason.  this  Instead, use i s made of the simpler equation ( 5 . 6 0 ) .  In terms of the coordinates y , w . , L , using (5*3) one finds f o r k  operator  (8.9)  <$ $  A  A  the i d e n t i t y  J }  * ' V* V ^ ^ L + ^ L H + ^ L  k2  2  A  and for the operator  (8.10)  v^ # A  A  (5«6l),  S  ^  A  ^  A  the i d e n t i t y  + 2L^  L  .  (5.^  ).  •  the  121 Substituting the above two expressions, one finds that the operator (8.8) can be written  (8.11)  KB ^  -*~ L\l>  s  2  k +  ^  W  2  +5 * V  Now going over from the set of coordinates (y ,'K k  .  ,L) to (y ,r,L), k  with  r given by (8.12a)  r = * " L^  ,  1  one has (8.12b)  * \  = -r>  r  so that (8.11) becomes Im^  (8.13)  m  - C = -r ^ +(r^ ) -4r^ -C  AB  2  k  2  r  r  .  Inspection of (8.13) shows that the v a r i a b l e s separate i n ( 8 ? a ) , so that 0  i t i s permissible to write X = F ( r ) X(y)  (8.14) with  X as a product  X(y) any s o l u t i o n o f  U ^ +m )X(y) = o  (8.15) where m  k  2  k  2  i s r e a l and non-negative by assumption. Thus eq.(8.?a) reduces  to the eigenvalue problem (8.16)  lr m +(r^ ) -4r^ -c}F(r) = 0 2  2  2  r  r  .  This equation can be cast i n the form (8.17)  {(z> ) +z -v }z (z) = 2  2  2  z  0  y  by w r i t i n g (8.14a)  F ( r ) = rP G(r)  (p=some r e a l number)  and using the r e l a t i o n s (8.18a)  r ^ ^ ^ r ) ]  = rP(pfr> )F(r)  (8.18b)  ( r } ) [ > F ( r ) ] = rP[p +2pr> + ( r } ) ] F ( r ) 2  r  so that with p=2, (8.16) becomes  r  2  2  r  r  122 (8.19)  r {m r +mr[V^(nir)] - 4 - C ] G ( r ) = 0 2  2  2  2  Thus solutions y  (8.20)  have the simple form  X. = r  where X ( y )  .  2  Z+  ( 4 + c )  | (mr)  *(y)  i s a s o l u t i o n of the usual Klein-Gordon equation (8.15).  C l e a r l y , the wave equation (8.7a) describes spin-0 p a r t i c l e s with mass. Since according to (8.6) a l l the spin matrices s^g vanish, then from (5.35)  and (8.12b) one has (8.21) so that  i  (  5  )  X =0  X i s an eigenfunction of  -£(5)  corresponding to the eigenvalue 0;  the postulate stated after (5«65), that i n order that f i e l d s with mass transform according to a representation of a conformal group they must be eigenfunctions of  ^(5)  has indeed been v e r i f i e d i n t h i s case, since  X does transform according to the s c a l a r representation of the conformal group i n f i v e dimensions, with '^(5) 0» =  Now one can go on to investigate the remaining features of the quantum f i e l d theory which r e s u l t s from the action p r i n c i p l e based on ji .  These  include the conserved quantities associated with r o t a t i o n a l invariance, the current conservation laws, and the canonical commutation r e l a t i o n s f o r the f i e l d . F i r s t l y , the Lagrangian density (8.7b) can be expressed as a function of y k , x , L , with the r e s u l t  (8.22a)  ^ =M" L\X 2  ^ X-(X^) -CX k  2  .  2  Keeping i n mind that the Jacobian J ( ^ ; y , K »L)=f^( \)/^(yxL)f=^/2 v  included, the action (8.7c) becomes  must be  (8.23b)  123  J  I =  U ^ - L ) (-K /2) £_ 2  3  d*V  d x dL »  ^j[-iHL^X^X+iK (^\X) +iH CX ]dK]d y =  =  3  2  3  2  4  To demonstrate that the Lagrangian density gives the correct f i e l d equation (8.7a) with (8.11), one can compute  bt nx]  (8.23c)  O ^ K ^ O k * ) ] -  K  \  O x * > ]  -  = - > i 3 | _ - 2 ^ ^ k+( x ^ J ^^>< -C] X = 0 2+  K  L  k  i n agreement with (8.13)• The conjugate momenta are given by (8.24a)  Tf  A B  = ^£/Mm  X ) = -n* X. 3  A B  so that the angular momentum tensor (7.47b) i s simply (8.24b)  S  C D A B  = imCDX(m X) iS A B  +  C A  S  D B  I  .  By a straightforward c a l c u l a t i o n involving commutations of the m^'s, one f i n d s that i n t h i s case the current conservation law (7«33) takes the form (8.24c)  0 =  S  C D A B  s (|m m -C)X CD  CD  m^X  and i s thus s a t i s f i e d i d e n t i c a l l y f o r solutions ,X of the f i e l d equation. For example, i f one chooses the set of indices {A,B\ = f(6,k)+(5,k)}, then (8.24c) has the form of a current conservation law (8.25a)  t J,j = 0 k  with t ^ J given by  (8.25b)  t ^ - ^ X  ^ X-S S-i> X k  k  J  n  } XHX(a X)} n  2  provided the mass i s given by the equation (8.15), which, using (8.16), can also be written (8.26)  r- [-(r> ) +4r^ 4c]>l 2  2  r  r  = mX 2  .  124 S i m i l a r l y , the choice {A,B^=Lj,M y i e l d s  (8.27)  s  0 =  n j k  ,  n  =  (yjV-y^n)^  As expected, the choice {A,B} = (6,5}  .  does not lead to a conservation law  i n Minkowski space f o r non-vanishing m, but instead one has 0 = ( * ^ +m )X K y * } +r> )X  (8.28a)  k  2  k  r  or (8.28b)  D .  =  n  n  [(^^ X)(iX)-X(i X)]n ^ *„[0 -O*] n  n  2  +  - ^ D a ^ X ^ X4^ (Xm2X)]-X(m2X ) k  K  where D* i s given by 1  (8.28c)  D* =  >  i XiX+y t n  k  n k  i s given by (5.41b),  and  (8.28d)  A =* V  =-r^  .  r  When Ji- -1, eq.(8.28b) takes the form o f the broken conservation law (4.3) with symmetry breaking function mr X. . Noting that the generator of d i l a t i o n s i s given by (5.20c), which reads (8.29)  f  " i(y ^ +r"i ) k  k  r  ^(5) ^» =  i n the present case, since value m (8.30a)  2  o  n  e  c  a  n  a  ^  s o  determine how the eigen-  appearing i n (8.26) i s affected by a d i l a t i o n y  = (1+«r)y  , k  k  ; r« = (1 + <r)r  which induces en X the transformation (8.30b)  x*(y ,r ) = ( I + i r - y ) X(y,r)  .  Since one can evaluate the commutator  (8.3D  [A + " i-(^r) r  2  2 + 4 r  k  ^ r i • f] " 2i[i\+r- {-(rt ) +4rt +c}] + C  2  2  r  r  .  then one can see that i f X i s a s o l u t i o n of the f i e l d equation (8.7«)» which reads i n terms of y^.r (8.32)  ^ ^ X+r- {-(r^ ) +4r^ -K;^X = 0 k  2  k  then so i s X ' »  2  r  Consequently,  r  if X  ,  i s an eigenfunction of the operator(8.26)  125 — c a l l t h i s operator M ,  say — belonging to the eigenvalue m , then  2  2  one has  (8.33)  r- {-(r^ ) +4r> +c] 2  = M *,' =  2  r  2  r  = M * + i<r [M ,</>] X 2  2  + i c r ^ ( M X ) = m (1+i<r<p )X-2<rm X = 2  2  2  » m (1-2<r)X-' 2  where i n t h i s equation the f a e t that o r i s assumed to be i n f i n i t e s i m a l was used.  Thus one has complete agreement with the usual transformation  property (4.11a) of the mass i n Minkowski space, outlined i n Section 4, and one arrives again at the broken conservation law (4.3) space.  i n Minkowski  However, despite the f a c t that the eigenvalue of M changes under 2  d i l a t i o n s , and that exponentiation of the i n f i n i t e s i m a l  transformation  w i l l enforce a continuous spectrum m as defined in (8.26), one can s t i l l argue that only mass r a t i o s are observable.  Then since every mass must  undergo th© transformation indicated by (8.33) these mass r a t i o s w i l l remain unaltered by transformations (8.30). Notwithstanding the f a c t that some of the usual Minkowski space currents are not conserved, there s t i l l are f i f t e e n conserved quantities M^g, given by (7.35).  For the Lagrangian density ( 8 . ? b ) , i t i s somewhat tedious but  straightforward to evaluate these q u a n t i t i e s .  For example, one finds  that the quantum f i e l d theory generator of translations i n the y ^ - d i r e c t i o n i s given by choosing the indices iA,Bl = f(6,4)+(5,4)}, on the surface t = t , L = L , 0  (8.34)  P4 M S  6 4  +M  5 / +  =  S(t-t ) c  S ( L - L ) ( X L ) M V d * dL* 0  •ii(^X) +i(vx)-(v X M L - M [ C X 2  1  2  2  + (~} X) ]} 2  K  .  By a s l i g h t change of f i e l d v a r i a b l e , however, a l l the conservation laws (8.24c) can be brought into the form of conservation laws i n 5-dlmensional space.  This i s accomplished by writing the f i e l d equation  0  126 ( 8 . 7 a ) i n the form of a f i e l d equation i n 5-dimansional space governing a massless spin-0 p a r t i c l e .  To t h i s end, consider a f i e l d  variable  r e l a t e d to X. by (8.35a)  H> = r "  X  n  (n=some r e a l number)  so that written i n terms of V the f i e l d equation ( 8 . 7 a ) becomes (8.35b)  0 = = r  ^ + r - [ - ( r ^ )2-f4r^ - K % " T = k  2  p  [\  n  )> +r-2 [_ 2 } k  k  r  y  r  ^  r  +  3  r  ^^  Z  n  } . 2  r  r  n  + 4 n + c  ] }^  .  C l e a r l y , t h i s can be brought into the form ( c . f . Wyler 1969)  (8.36)  =  0  w  i  t  h  z  t =  (y .r), k  t=1,2,3,4,7  simply by choosing the value (8.37a)  n = 3/2  ,  provided the Casimir operator C i s assumed to have the value obtained from (8.37b)  0 = -n +4n+C = (15/4)+C  ;  2  i.e. C = -15/4  .  Then one does indeed have from (8.24c) the four conservation laws  (8.38a)  t  ,  (8.38b)  S  (8.38c)  D( ) ,  (8.38d)  K  u (  5  (  )  s  5  )  s  \  t  u  ,  u ( 5 ) t  u  * 0  = 0  u  5  = 0  u  u  =0  where the four types of current tensors are given by (8.39a)  t  ( 5 ) s  (8.39b)  S  (  (8.39c)  D  i  (8.39d)  K  ( 5 ) s  =-^Y> Y - S ( - n Y *r> u  u  v 5  5  )  s  s  s  u  f  = z t  = z * \  u  v  s  ( 5 ) u  n 5  )  s  -* t u  ( 5 ) s  -(3/2) f  t  *  V»f  = (2z z - Ssvz2)t(5)yu v  s  3  * xp^uy + ( 3 / 2 ) Y s  2  .  127 However, the approach of these authors i s considerably d i f f e r e n t  from  that presented here, i n that features such as relations resembling (8.40) which appear i n t h e i r work do not arise from action p r i n c i p l e s developed i n the manner of Section 7, i n which stress i s placed on the absence of t r a n s l a t i o n a l invariance i n order to determine the minimum requirements f o r an action p r i n c i p l e based s o l e l y on r o t a t i o n a l invariance [ e . g . compare the divergence theorem constructed i n Ingraham (19^0) with the r o t a t i o n a l l y invariant Stokes' theorem constructed i n section 7-l3. interpretation  Neither i s an  given i n t h e i r work i n terms o f eigenfunctions of the  scale dimension i n f i v e dimensions, or i n terms of massless f i e l d s i n 5-dimensional space, as i n (8.36).  8-2. Green's function f o r scalar f i e l d equation The condition (8.37), which y i e l d s a numerical value f o r the only non-vanishing Casimir operator C i n the s c a l a r representation, was motivated by the desire to be able to cast eq.(8.32) i n t o a simple form.  It  i s also  possible to derive t h i s condition by examining the structure of a Green's function f o r the 6-dimensional s c a l a r equation (8.7a). Covariant Green's functions f o r r o t a t i o n a l l y covariant f i e l d equations have been developed by Adler (1972) f o r the case of f i e l d s on the u n i t hypersphere i n 5-dimensional Euclidean space  and on the n u l l hyperquadric  i n a 6-dimensional space with one time a x i s , by Fubini e t . a l . (1973) f o r the case of f i e l d s on hyperspheres i n 4-dimensional Euclidean space, and by Mack and Todorov (1973) f o r the case of manifestly conformally covariant f i e l d s i n 6-dimensional space on the hyperquadric L=0, Also, i n the 6-dimensional case d i f f e r e n t  r e s u l t s have been advanced  by Ingraham (1971) and Castell-(1966), who advocate the determination of Green's functions d i r e c t l y i n terms of the coordinates y , r , L by  128 Thus one can maintain that f i e l d s associated with p a r t i c l e s with mass i n Minkowski space can be looked upon as "projections", onto the 4-dimensional world, of massless p a r t i c l e s i n 5-dimensional space, spanned by the coordinates y , r . k  As w e l l , one can say that both the conservation  laws and the broken conservation laws, associated with the transformation behaviour of p a r t i c l e s with mass under the conformal group i n Minkowski space, are a c t u a l l y manifestations of the f i f t e e n unbroken conservation laws associated with the transformation behaviour of massless p a r t i c l e s i n 5-dimensional space, under the transformations of the r e s t r i c t i o n (5.10) of the 5-dimensional conformal group. F i n a l l y , i t remains to check that the commutation r e l a t i o n s (7.63) i n 6-dimensional space a c t u a l l y correspond to meaningful commutation r e l a t i o n s i n Minkowski space.  Using the d e f i n i t i o n (8.24a) of the conju-  gate momenta, and assuming that bosons obey commutation r e l a t i o n s and not anticommutation r e l a t i o n s , the equal-time commutation r e l a t i o n s (7.63) read (8.40)  [X(y.r) , - i ^ X (y« ,r«)]  t = t  ,  =  %&  $(y- «) P £  $(r-r»)  L  '  Z  for a quantum f i e l d X which i s a l i n e a r sum of creation and a n n i h i l a t i o n operators multiplying products (8.4U)  X  £  (  B  -  *(y)  r Z (rm) 2  y  of solutions X ( y ) of the Klein-Gordon equation (8.15) and c y l i n d e r functions Z  y  (rm), where V i s given by  (8.41b)  V = +(4+C)i =  = t\  •  These r e s u l t s are to be compared with s i m i l a r work by C a s t e l l (1966) and Ingraham (1971).  This s i m i l a r i t y i s due to the f a c t that f o r a scalar  f i e l d , none o f the techniques involving the operator U need be applied, and i n p a r t i c u l a r the consequences  of the difference between the operators  U and W, pointed out i n section 5-3»  do not manifest themselves.  129 convolution of the usual Green's function i n Minkowski space with functions of r . Here, the covariant approach to finding Green's functions w i l l be extended by leaving the hyperquadric L=0.  If  ^  and ^  A  A  a  r  e  2  the coor-  dinates of two d i f f e r e n t p o i n t s , each of which l i e s on v| =L, then one looks 2  f o r a formal solution of the wave equation which can be written i n terras of ( " l i - " ^ ) ' 2  (8.42a)  i »  f o r ^ -j ^ ^ £  « t  e  (imi^m,*  o  n  e  s  «®ks a number q such that  -C)[(^ -. )2]q = o  5  1  ,  l 2  where (8.42b)  m  = i^l  l A B  Then f o r ^ j = * [  2  0/^  A  B 1  )-i'\l  B  0/~>  •  there appear delta function contributions on the r i g h t  hand side of (8.42a). To take these contributions into account, consider f o r the moment the s p e c i a l case L=0.  (8.43)  y  ii  2  = \2 l  For t h i s case one has 2  =  0  »  V U  5  ~i*1 2( 1- 2> K  y  V  2  f  o  r  L  =  where y , H are defined separately f o r *\ ^ and * J . For a f i e l d k  2  0  ^(y)  describing massless p a r t i c l e s i n Minkowski space, define the Feynman function  (8.44a)  A ( ^ ( y y ) = - i <0| T ^ ( y , ) r  2  where T denotes the time-ordered product.  Y(y )}|0> 2  This singular function i s  given by the simple expression (Mack and Todorov 1973) (8.44b)  Am(y) = c  (y +i0) 2  q  = l i m ( y + i € )<* 2  ,  q real  where q=-1 f o r free f i e l d s , and (8.44c)  c  = -i2~ l(4Tr)2 t  q  2  T ( - q ) / r(q+2)  .  Then i f one defines the corresponding function i n 6-dimensional space by m u l t i p l i c a t i o n by powers of K ,  ,  130  (8.45a)  ^(^,T )-=(x,x ) 2  A  q  2  F 4 )  (y y ) r  2  then comparison with (8.43) shows that (8.45b)  A C ^1.^2)  =  F  (-2 lr*l2  c  y  q  + i 0 K  v 1  <2  •  ) q  These r e s u l t s are generalized to the case L^O by considering the general form of (8.43),  S  (^1-^2^  (8.46a) for ^  v  j =^\ =L. 2  2  X {-LH - X - (M  M  2  1  2  2  1  2  -H ) +(y y ) ^ 2  1  2  2  r  2  I f one defines the supplementary v a r i a b l e s r^  2  2  = L 2 / K J  2  ,  then one f i n d s that (8.46b)  = LX j ~ X ( X  z  Z  1 - X2 )  _2  2  (T)-r )  2  2  so that (8.46a) can be written (8.46c)  (*l /} ) r  = X X (z -z )  2  L  2  2  1  2  2  where (8.46d)  z  = z  2  z  t  = y  t  ^  2  .  , t=1,2.3,4,7  2  Therefore, w r i t i n g i n the general case  (8.47a)  A <ni'n F  2)  c  =  q[ i (  1-1  *i*2]  2 ) 2 + 1 0  q  corresponding to the 5-dimensional propagator  (8.47b)  A*  F 5 )  (y -y2 1  ; K  1-  v t  2  )  H  ( i M  then  can be written simply as  (8.47c)  A*(5)( 1- 2) = q [ ( 1 - 2 z  Z  c  z  z  M  ) 2  ) 2 + i 0  "  ]  q  q  A  F  HT12>  •  *  This corresponds, up to a conventional f a c t o r 2 i , to the usual expression f o r the elementary s o l u t i o n i n f l a t 5-space with one time axis f o r the wave equation (8.36) describing massless p a r t i c l e s (Gel'fand and Shilov 1964, p.280), provided  q i s given by q=-3/2.  Obviously, t h i s i s the Green's  131 function corresponding to the f i e l d ^ , defined by (8,35a) and (8,37a) for the case i n which C i s given by ( 8 . 3 7 b ) ,  X f o r the case L^O i s given by  corresponding to the f i e l d  A[  (8.48)  5 )  (  Z I  .Z ) £ ( r ^ ) ^  A(5)<V 2> Z  2  =  q < 1 2>" r  c  Thus the Green's function  r  =  ~  L  [(*1-* ) i<>]  q  2+  ^  q  11 • I-2 y  )  =  q  2  IP  so that when C= the  q  -15/^t  -3/2,  =  i s indeed a formal s o l u t i o n to  f i e l d equation (8.32) f o r X  when Z j ^ z . 2  To see that t h i s value o f C i s the unique value which allows one t o write a Green's function f o r the wave equation i n the simple form (8.42a), suppose that  ^•j^'tg  a  n  d  consider, i n any D-dimensional f l a t space, the  expression (8.49a)  • l  A  B  « h  A  [o^-^) ]*  B  2  ^ ( q - O ^ I f one assumes now that (8.49b)  m  l A B  )  2  y /  [^  2 =  (  ?1  2  •  L . t h i s becomes  2 = 2  m AB[ v -/j )2]<l 1  {-8^ q(q-1)-4q(D-1)^^  ^ ^  2  ^  5  =  2  = |-8q(q-1 )L-4q(D-1 )(*| j» ^ )} [ ( L - 1 * "| 2 ] " ^ )  2  q  +  2  +8q(q-1)(L-^ ^ ) [2(L-'| ^ )] 2  1  q  2  1  Thus one can ensure that f o r ^ ^ ^ i s proportional to ("| j - ^  (8.50)  2  .  2  2  the r i g h t hand side of t h i s equation  by requiring  -8q(q-1) = 4q(D-1 )  ;  i . e . -q = (D-3)/2  .  This choice has the s i g n i f i c a n c e that f o r {\^ \2' Y  (8.51a)  iauB*/  8  [( *|  r  •) ) Z  2  ] ^  )  I  Z  y  * -*(D-1 )<D-3>[(*1 » | ) }<D-3)/2 2  r  2  132 (8.51b)  C = -£(D-1)(D-3)  .  For D=6, t h i s y i e l d s C=-1 5/4 again* so that i n s i s t i n g on the form (8.49a) f o r the Green's function automatically j u s t i f i e s the value of C arrived at h e u r i s t i c a l l y , by the requirements of s i m p l i c i t y , i n (8.37b). Since i t should be possible to recast f i e l d equations f o r p a r t i c l e s with spin into a form resembling the scalar f i e l d equation (8.7a), e . g . by i t e r a t i o n , the constraint (8.51b) on the value of C can be used to advantage i n specifying the form of f i e l d equations f o r f i e l d s having nonvanishing s p i n .  This procedure w i l l be employed below f o r th© f i e l d  equation f o r s p i n - i p a r t i c l e s .  8-3. Spinor f i e l d In section 5-9 i t was shown that spinor f i e l d s i n 6-dimensional space ar© 8-component objects X ( ^ ) ,  which transform under rotations according  to th© 8X8 representation ( 5 .  94).  One can now determine the form of the l i n e a r free f i e l d equation which i s to be employed f o r a d e s c r i p t i o n of s p i n - i p a r t i c l e s i n 6-dimens i o n a l space.  If  such an equation i s to be of not higher than f i r s t  order i n th© derivatives of the f i e l d , and s a t i s f i e s the requirement of r o t a t i o n a l covariance i n 6-dimensional space, thon according to the program proposed i n Section 7. the most general such equation i s to be determined by considering th© most general r o t a t i o n a l l y invariant Lagrangian density jL which i s of th© form  (8.52)  i. = t  (^.X,m  A B  X )  and i s constructed out of expressions b i l i n e a r i n X and expressions which are products of  X and  m  ABX.  *For D=5, one has C=-2 i n agreement with the paper by Adler (1972).  133 Now, by forming products of the matrices p , as defined i n (5.96) and (5.110), and using <r  as defined i n (5.94), one can construct  AB  sixty-four l i n e a r l y independent matrices (Drew 1972)*  (8.53)  I ; />A ; r * 8 ; f>kff  C a l l i n g these  pi,  Ufafc) ; f  X  k  ; f1  i=1,2,...,64, the number of d i f f e r e n t  1,6,15,20,15.6,1, r e s p e c t i v e l y . covariants  ; fp  .  of each type i s  Then one can form seven types of b i l i n e a r  T^Tt. where X i s the a d j o i n t s p i n o r d e f i n e d by  (5*115)  and  ****  (5.124). These b i l i n e a r combinations of X. and K then have simple transformat i o n properties i n 6-dimensional space, e.g.  (8.53)  Scalar S = XX; Pseudoscalar  Q  =  xfiX;  Vector V  A  = xp X  .  A  Keeping i n mind that the desired Lagrangian density can be an e x p l i c i t f u n c t i o n of ^ since t r a n s l a t i o n a l invariance i n 6-dimensional space i s not demanded, one can form four l i n e a r l y independent expressions of the kind s p e c i f i e d abovet*  (8.54)  There are two scalar invariants  I] = ^ < r m X  ;  A B  A B  I  2  = XX  and two pseudoscalar i n v a r i a n t s , obtained from (8.54) by replacing X. by xft.  However, the two pseudoscalars must be eliminated from consideration  on the following grounds.  Including the f i r s t pseudoscalar i n  amounts to  The l i n e a r independence of these matrices follows from the observation that, except f o r I , they are traceless and that f o r each nijfr there e x i s t s a pj such that pi ^ pi, n d given pi and r j f o r i ^ j there e x i s t s a p n / i such that pi pJ= f » Then, i f ^ c ^ i ^ , taking the trace y i e l d s r  a  n  ci=0 and taking the trace a f t e r m u l t i p l i c a t i o n with P ^ I y i e l d s Cj=0.  The invariant  X\ v  k  f^X  i s omitted since i t can be replaced by a  divergence plus excluded terms, by v i r t u e of the i d e n t i t y  5 x^ ^x k  5  i »AB( x V p X ) + B  +L( xp 2 x k  k  -( x \  k r  + 2 xp >i) k  k  i*] X B  +  I^B^AP**")  •  134 considering conjugate momenta T T 8 A  f  given by (7.22), which multiplied into  give r i s e to covariant expressions whose transformation behaviour i s i n part that of a tensor density.  But i t i s p r e c i s e l y these conjugate momenta which  appear i n the anticommutator on the l e f t hand side of (7.55)•  Since the  r i g h t hand side of (7.55) i s given, up to a f a c t o r , by the tensor quantity gAB(^.y)  f  defined by ( 7 . 5 6 ) , there i s a c o n t r a d i c t i o n unless the c o e f f i c i -  ent o f the f i r s t pseudoscalar invariant i n X. i s zero.  As w e l l , since X.  appears i n the expression (7.47b) f o r the angular momentum tensor S gCD A  t  i n order f o r the angular momentum M^R to transform as a tensor X. must c o n s i s t of scalar invariants only, so that the c o e f f i c i e n t of the second pseudoscalar term i n X must also vanish.  The remaining set of permissible  invariants consists of the two terms i n (8.54), so that the density to be considered i s the l i n e a r  (8.55)  =  alj+bl£  a (  Lagrangian  combination  » b complex,  where the a r b i t r a r y c o e f f i c i e n t s a,b can be functions of L without a f f e c t i n g the r o t a t i o n a l invariance of X. • The expression (8.55) i s not n e c e s s a r i l y hermitian as i t stands, but since the Lagrangian density should be r e a l up to a m u l t i p l i c a t i v e constant, i n order f o r i t to be used i n the hermitian generator (7.46c) of the u n i t a r y operator (7.46b), £ i s brought into a r e a l form by adding the hermitian conjugate  £ , From the r e a l i t y property (5*119) and the d e f i n i t i o n of m +  one has simply that  (8.56)  (X> m X) =-m ^ BC+A+£= . C  BC  <r  since from (5.124) A i s hermitian.  (8.55a)  ^ ^ A ^ . ^  +  BC  + ^ )=4[a x V % +  B  %BC% r  Therefore the r e a l part of / i s *  - * AB ? V X + ( b + b * ) % X.} . a  m  B  Now, the second term can be expressed i n terms of the f i r s t , since one has the i d e n t i t y  (8.57)  m^Xff-^X  =m ( X AB  cr^Z  )-Z^^X  .  AB  135 so that suppressing i n  the inconsequential divergence forming the  f i r s t term on the r i g h t hand side of (8.57), one arrives at a Lagrangian density of the form  (8.58)  £• =  X(T  A B  m x -2 X U  ,  A B  X real  .  The f i e l d equation derived from the corresponding action p r i n c i p l e i s found from (7.23), where the conjugate momenta (7.22) are  (8.59)  tf  A B  =  "*2/ (m x)= 0  A3  X«rAB  .  ****  Hence, the spinor f i e l d equation f o r  (8.60a)  -m  j? <r -2X X = 0 AB  A B  and taking the adjoint,  (8.60b)  X reads  the f i e l d equation f o r  ( T ^ m ^ X -2A X*  =0  X is  .  As i t should, the second equation,(8.60b), also r e s u l t s from taking the v a r i a t i o n of (8.58) with respect to  X.  Making use of the  antisymmetry  of o""* , t h i s equation can also be put i n t o the form 8  (8.60c)  fi-^A  2 X B  +  iXX  = 0  ,  which i s the form of the 6-dimensional spinor equation suggested by Dirac (1936) and revived by Hepner (1962). This f i e l d equation f o r spinors i n 6-dimensional space i s to be compared with that suggested by Barut and Haugen (1973a,b.c.d). These authors propose extending further the group of motions i n 6-space to include 6-translations,  along with the 15-parameter group of r o t a t i o n s , i n order  to write a spinor equation i n a form d i r e c t l y analogous to the Dirac equation i n 4-dimensional space  (8.61) It  (i/i  A  Jf -M)¥ = 0 A  '.  i s then asserted that the parameter M can be looked upon as a "conformally  invariant mass", giving r i s e to a "conformally covariant mass" nM i n  136 Minkowski space.  However, the above equation does not admit s o l u t i o n s , f o r  nonvanishing M, which belong to a s p e c i f i c value of the scale dimension i n 4-dimensional space (Drew 1973), even i f one accepts these authors' proposal that Minkowski space i t s e l f be looked upon as corresponding to a surface l y  =  >  < 0  i n 6-dimensional space (Barut and Haugen 1972).  Moreover,  X acts on solutions i n Minkowski space of e q , ( 8 , 6 l ) as a d i f f e r e n t i a l operator and not as a matrix, so that one cannot assert that the equation r e s u l t i n g from (8,61) i s covariant under the 4-dimensional conformal group. As w e l l , since the operator W, and not the operator U defined by (5.23), was used i n considerations of e q , ( 8 , 6 l ) , i t cannot be maintained that Wt' transforms according to a representation of any conformal group, so that i s not meaningful to demand only that Wt  it  belong to an eigenvalue of  Furthermore, the a l t e r n a t i v e course of using the operator U instead of the operator W i n w r i t i n g e q . ( 8 . 6 l ) i n terms of Minkowski space v a r i a b l e s r e s u l t s i n a complicated equation i n 4-space which c e r t a i n l y has no simple r e l a t i o n s h i p with the Dirac equation. Therefore, i n v e s t i g a t i o n of t h i s equation w i l l not be pursued further here, f i r s t l y because of the above mentioned problems, and secondly because of the f a c t that (8.61) cannot be derived from a r e a l Lagrangian d e n s i t y . The proof i s relegated to Appendix 3. where i t i s shown that only f o r the case M=0 can one construct a r e a l Lagrangian d e n s i t y . Returning then to the spinor f i e l d equation of Dirac and Hepner, (8.60), as a f i r s t step i n rewriting the f i e l d equation i n terms of y ^ . H . L the operators m^g can be replaced by t h e i r equivalents i n these coordinates, given by (5«5).  (8.62)  fcJVj  The equation which r e s u l t s from t h i s replacement i s  V*  « \ - » T t T [ ( H- L-y2)> k  k  2  k + y k  y J > . - y x>  + / ( y V > 0 > J A U =0 , 5  +i  k  J +  137 where the i d e n t i f i c a t i o n s of  o 5j +with  K  -  c  have been employed i n (8.62).  k  and TT^, r e s p e c t i v e l y ,  Now, the operator U"^ i s given e x p l i c i t l y  i n (5»127), and a p p l i c a t i o n of the s i m i l a r i t y transformation U( )U~^ to the matrix d i f f e r e n t i a l  operator i n (8.62) y i e l d s the equation f o r the  f i e l d variable X =UX .  Keeping i n mind that i H  i s not independent of y , X , k  so that  (8.63)  V  u  ~  1  >  =  i  T  r  k  ^\(U~ ) = - i x - 1 2 T  '  1  L  ?  ,  one has  (8.64)  U ( < r n 3 +i A )U"1 = i K \ + n - L T T ^ A B  k  2  k  B  A  -K- L^<r 1  ^ +w-lL2ir X^ k  7 k  ?  +(i/2) K TT +  +(3/2)x- L^TT +iX  -  .  1  > <  * \  k  k  ?  At t h i s stage i t i s convenient to go over to the coordinates ( y , r , L ) k  defined by the r e l a t i o n s (8.12).  Then the s o l u t i o n of the f i e l d equation  following from (8.60) and (8,64) proceeds exactly as i n the s o l u t i o n of the corresponding equation, (6.50), i n the 4-dimensional c a s e .  Again i t  is  found that only f o r a p a r t i c u l a r value of A , i n t h i s case  (8.65)  A=-5/4  ,  does the i t e r a t e d equation y i e l d solutions i n Minkowski space. elaborated i n Appendix 4, and t h i s value of  The proof i s  A i s also suggested by the f a c t  that i n the spinor representation the Casimir operator D, given by (8.5c), takes the form*  (8.66)  D = (9/2) ^{<r  AB  \ ^ -i(5/4)] B  .  Using (8.65), i t i s shown i n Appendix 4 that i n terras of the four 2-spinors defined i n (5«132), the spinor f i e l d equation has solutions  *In the d e r i v a t i o n of the r e l a t i o n (8.66), use was made of the  identity  138 (8.67a)  X. = r 3 i ( r m ) X ^ y )  (8.67b)  X  Z  2  = r Z|(rm) X ( y )  (8.67c) (8.67d)  = r Z_i(rm) X ^ y )  2  2  3  X  = -r Z_i(rm) X^(y)  .  3  4  where Z,Z a r e two p o s s i b l y d i f f e r e n t c y l i n d e r f u n c t i o n s o f o r d e r f , and A.  i t i s n o t n e c e s s a r y t o assume e q u a l i t y o f ra and m.  A  The f i e l d s  X ( y ) , X(y)  s a t i s f y the f o l l o w i n g equations (8.68a)  S  \X^yJ+imX^y) = 0  (8.68b)  ^ ^ X (y)+imX (y) = 0  k  k  k  4  1  and (8.69a)  Zi(rm){V ^ X (y)+iSX (y)}-imZi(rm)X^(y) = 0  (8.69b)  Z ^ r m ) {^ b X (y)+3SiX3(y)}+iraZ A(rm)X (y)  k  k  2  k>  k  The 4 - s p i n o r f i e l d s  (8.70)  3  ^(y) =  2  -  1  =0  p ( y ) , ( y ) are defined by ( 5 . 1 3 3 ) . A  Xjty) \ ( y )  V(y) =  X (y) 3  w4  In terms of these v a r i a b l e s , e q s . ( 8 . 6 8 ) read (8.71)  - i * H ^ > (y)+m^(y) = 0 k  so that (f>(y)  ,  s a t i s f i e s the Dirac equation with mass m.  A .  A  <p(y)  On the other hand ~  s a t i s f i e s the Dirac equation with mass m only i f  X. i s constrained to  obey the supplementary condition ( 5 * 1 3 7 ) , so that the components  <p(y)  vanish, (8.72)  - i * * > ^ ( y ) + m p(y) = 0 k  if  y>(y)=0  .  Hence there are three separate procedures, which one can employ at t h i s p o i n t , that r e s u l t i n massive spinor equations.  Firstly,  the f i e l d  X can  139 be constrained to obey the supplementary condition (5•137)• so that the components and 9 =0. nents ^ or  (j)  are eliminated.  Then  <p s a t i s f i e s  the Dirac equation (8.72)  Secondly, the projectors E and E , which project out the compo-  y> and <j> r e s p e c t i v e l y , can be applied to X  to eliminate  either  T h i r d l y , the spinor f i e l d equations, (8.68) and (8.69), can be  <p .  interpreted as equations f o r two d i f f e r e n t f i e l d s whose behaviour i s governed by coupled equations and whose components are mixed by the* s p e c i a l conformal transformations.  Of these three methods, the f i r s t seems the simplest,  since i t r e s u l t s i n a f i e l d which transforms according to an i r r e d u c i b l e representation of the Lorentz group, and according to a representation of a conformal group, without having to declare c e r t a i n components "unphysical" by the use of projection operators.  However t h i s i n t e r p r e t a t i o n rules out  the p o s s i b i l i t y of encompassing both the electron and the muon at once i n a single equation.  For t h i s reason, the view that projection operators  should be employed w i l l not be ruled out i n the f o l l o w i n g ,  [A  similar  analysis of the spinor equation (8.60) was c a r r i e d out by Drew(1972), but instead of using U the operator W defined by (5.42) was employed i n that work to rewrite the free f i e l d equation.  Then eigenfunctions of Ji were d e t e r A  mined by making use of projection operators E and E as outlined i n section 5-5. These f i e l d s then could be associated only with massless p a r t i c l e s i n Minkowski space.] For any of these i n t e r p r e t a t i o n s , y  according to (5.131) the f i e l d s<f>and  belong, r e s p e c t i v e l y , to the eigenvalues +§ and -•§• of the scale dimen-  sion ^  s  o  that the postulate that f i e l d s with mass i n Minkowski space  must be eigenfunctions of -^(5) has been v e r i f i e d i n t h i s case. With these r e s u l t s , one can now turn to consider the quantum f i e l d theory based on the free spinor f i e l d equation (8.60).  With the d e f i n i t i o n  (8.59) of the conjugate momenta, the angular momentum tensor i n 6-dimensional space (7.47b) i s given by  140 (8.73)  = -* ? ^ ( " B D  +  A  and f o r solutions identically.  *AB>* H  SAC S B D Z  £ of the f i e l d equation, the Lagrangian density vanishes  To write t h i s tensor i n terras of X_, i t i s necessary to  perform the s i m i l a r i t y transformation U( )tP  on the operators ( A3 0~AB^*  1  m  +  and the relations (5.22) to (5.39) can conveniently be used f o r t h i s purpose. The conserved quantities M  (8.74)  M  = f  A B  A  |  SAB  C  D  dS  are given i n terms of S  A B  •^-^cSfABl^-^^cCs^j  0  1  -s  6  Substituting the e x p l i c i t form (8.73), with  [ A B  by  u A  B  S( K- *f-t)  = J^dfy 2  C D  CD  S( ^ - L ) 2  f5)]  £=0 f o r solutions X , one  finds that  (8.75a)  ^ ( S g ^  0  =  - * ! ^ ^  6  = - x ( - > ^ ( r - 5 - L ^ T ) U(m +<r )U- X 6  7  1  AB  AB  and  (8.75b)  1C[AB]^ ' ^[yjr MK 4(^" L.yV ]("i VAB S  C  j  =  2  4  4  S  )?  AB  = - i x X [n^-2r ^ + 2 r TT ^-2y^ (T 5-2y rtr ] U C m ^ <T )U" X 4  2  6  4  7  1  7  by using the  AB  identities  (8.76a)  Ucr U-  (8.76b)  U T T V = TT  (8.76c)  UK lT  (8.76d)  U (y^V  jk  1  =  1  k  =  1  1  ( r ^ T T ^ T T j k  K -2y (r 5+2y - J+2ro- +2y (y k  k  6  k  k  j 0  = < r ^ . ^ T T +r n  k  7  J-rTr )-(y -r )TT 7  j T r  2  2  7  k  For example, choosing the set of indices {A,B}= i(6,k)+(5.k)] , one finds that the generator of translations i n the y - d i r e c t i o n i s given on the k  surface t = t , L=L Q  .  0  i n terms of the 4-component f i e l d s  <p and <f> by  k  141 (8.7?)  P  M +M  5 k  6k  5k  = f  dn  t = t  dLdV  S(t-t ) Q  S(L-L ). Q  L=L°  £^ $  .{KU  +* * i 3  k  jVo ? + i K k  2  L2i(  f  +f  )} •  Now i f the t r a n s v e r s a l i t y c o n d i t i o n (5*137a) i s imposed on the f i e l d % , then according t o (5.137c), the f i e l d components (f> v a n i s h .  I n the  expression (8,77), t h i s means t h a t a l l the terms i n the braces vanish save the f i r s t term, which i s p r o p o r t i o n a l to the f o u r t h component of the energymomentum tensor (8.78a)  V  = i ^ J i  k  ( p  . A  On the other hand, i f the f i e l d components ip are required to v a n i s h , then the second term i s p r o p o r t i o n a l to the f o u r t h component of (8.78b)  t ' ^ = i J> V^ (p  .  k  k  The l a t t e r s i t u a t i o n occurs i f instead of the c o n d i t i o n (5.137) one makes use of the p r o j e c t i o n operators E and E which separate the spaces belonging to the components y and ^>, as explained i n s e c t i o n 5-5• As i n the case of the s c a l a r f i e l d , one can r e c o n c i l e the existence of f i f t e e n conservation laws w i t h the presence of mass by r e w r i t i n g the f i e l d equations as equations f o r massless f i e l d s i n 5-dimensional space. To t h i s end, consider once again the f i r s t order f i e l d equation f o r the f i e l d  p .  I t i s shown i n Appendix 4 that, making the s l i g h t change of f i e l d v a r i a b l e (8.79)  y  s  r(5/2)vy  #  t h i s equation reads (8.80)  U a +S } )f k  7  k  =0  7  *^7~\  » k=1 2,3.^ t  w i t h  •  I f on the other hand the c o n d i t i o n (5.137) i s used t o e l i m i n a t e f, then w i t h (8.81) the f i e l d $  (f> = r ( 3 / 2 ) y satisfies  ,  142  (8.82a)  U^k+tf ^)^ 7  =  0  •  i  f  r  °  5  •  with (8.82b)  £ s )$5  .  7  +  Thus one can say that 4-component spinor f i e l d s with mass are the manifestations i n Minkowski space of 4-component massless spinor f i e l d s i n 5dimensional space. In addition, there are p r e c i s e l y two such types of f i e l d possible, with masses m and m, as i n (8.71) and ( 8 . 7 2 ) . As a f i n a l step, i t remains to evaluate the commutation r e l a t i o n s (7»62) f o r spinor f i e l d s .  For the Lagrangian density (8.58) as i t stands, there  are four non-vanishing components of the conjugate momentum X-V* on the l e f t hand side of (7«63)« obey anticommutation  appearing  I f one assumes that fermion f i e l d s  r e l a t i o n s , then applying the operator U, these compon-  ents y i e l d the anticommutation r e l a t i o n s s a t i s f i e d by the f i e l d components ^  +  with t h e i r conjugates,  (8.83a)  $Xy« . r ' ) }  j£(y.r) ,  t = t  , =  $  e.<M')  r 3 L  "  2  S(r-r')  .  On the other hand, combining the r e l a t i o n s found f o r ^A,B} = f(6,4)+(5,4)} i n (7.62) with the r e s u l t s from the remaining indices ( i n order to cancel contributions containing y^ e x p l i c i t l y ) , one also has  (8.83b)  }£<y.r)  ,  ^ (y».r')}  (8.83c)  {^(y.r)  ,  ^ ( y « ,r« ) }  +  t = t  ,  +  t = t  = 0  , = 2  J  g (y- ») r V  2  £  $ (r-r«)  This discussion has shown that viewing spinor f i e l d s describing p a r t i c l e s with mass i n Minkowski space as "projections" of 5-dimensional massless f i e l d s , one can define quantum f i e l d theory generators of the transformation group, conserved currents i n 4-space, and canonical a n t i -  .  143 commutation r e l a t i o n s of f i e l d components.  Therefore, i t i s not without  i n t e r e s t to consider further a covariant Green's function f o r the spinor f i e l d equation,  8 - 4 , Green's function f o r spinor f i e l d equation The problem of constructing a Green's function f o r the s c a l a r wave equation has already been treated i n section 8-2,  In order to f i n d  a corresponding Green's function f o r s p i n o r s , i t i s only necessary to perform an i t e r a t i o n of the spinor f i e l d equation, i n order to bring  it  into the s c a l a r wave equation form. In order to be able to make comparison with the 4-dimensional model constructed i n Section 6, t h i s i t e r a t i o n w i l l be c a r r i e d out f o r spinors i n any D-dimensional f l a t space with |3 -matrices s a t i s f y i n g  (8.84)  {|3  , ;  A  B  $ = 2S  A B  f o r any f l a t metric, A,B=1,2,,..,D, and a l l other symbols being defined i n complete analogy with the 6-dimensional case.  In the D-dimensional space,  one then finds  ( 8 . 85a)  mj&P = -2 v^ ^  * (2D-4) \  2  (8.85b)  o^a^  ( 8 . 85c)  (TupP  5  A  +  k  * 2^ A+  A  * ( ^  B  A  JJ ) B  = ±D(D-1)  W  ''^J )^ k  B  +  ^  A  *A  so that (8.85d)  ( <r m )(<r m ) = | m CD  AB  AB  CD  A B  m  A B  -(0-2)0-^^  .  Now the f i e l d equation analogous to (8,60) can be i t e r a t e d , with the r e s u l t  (8.86)  (<T  ^^V /^ 1  )  =  -C^m^HfiCD^V+A]  ^AB"*8  144 v a l i d f o r any number y " (8.87a)  y> = - \  -KD-2)  the i t e r a t e d equation is  [  (8.87b)  A B ( r  In p a r t i c u l a r , f o r  0  of the form of the s c a l a r wave equation,  [<r \V ] = C  v ViX-(i/2)(D.2)J l A  = -(l/8)m m  A B  A B  + \  iX  [>4(D-2)]  according to ( 8 . 5 1 ) the above equation has a Green's f u n c t i o n of  Now, the form  [(l,-, ) ]- 2  (8.88a)  (D  3)/2  2  only f o r (8.88b)  4 A [A+KD-2)]  For example, f o r D=4, with ( 6 . 6 9 ) , which was For D=6,  = -KD-1)(D-3)  (8.87  for  -5/4,  and the l a t t e r number agrees  -1.  X are  Then according to  has  D  8  -KD-2)+(i>  and again the l a t t e r value i s the  («-ABlu2,B-i){p%c)li -ii|  1i^l2»  =  (8.66),  the p o s s i b l e values o f  ) and ( 8 . 5 1 ) , one  (8.89)  \  a r r i v e d at by an e n t i r e l y d i f f e r e n t route.  same as that i n ( 8 . 6 5 ) and For D=5,  -3/4,  one has > =  X = -3/4,  one has  ; i.e.  [(H-V ]-'} • 0 2  that the spinor Green's function i n 5-dimensional  0  space  i s given formally by (8.90)  s ^ v ^ . v ^ ) = [r ti l2 =  [Allowing f o r  as i n (8.4?a).]  (8.9D  <T  so that with  v  A B  lc  lD  [0\i-*] >- ]  •  2  2  1-I2)  2  w  ^ t h a limit,  To evaluate t h i s expression, one needs the r e l a t i o n  a ^B^ll- l2) ,  2 = v 1  l ^ -(i/2)]  corresponds to replacing (*[  A  t  CDv  ^  2 = 2  L,  one  2 q  =-iq(M )  finds  2  2 ( q  -  1 )  n/f  A W  B-VW  145  (8.92)  S ( 5 )  <lj.V  I f one chooses L=1, then t h i s formal solution i s equivalent to the form given by Adler f o r f i e l d s on the u n i t hyperquadric i n 5-dimensional Euclidean space (Adler 1972). An i d e n t i c a l procedure can now be c a r r i e d out f o r the case D=6, with the r e s u l t  (8.93)  * , -(3/4)] [ ( n i - t 2 ) ] "  S(«| H a ) =  2  3/2  B  _(3i/2)h, f A  =  t2 f -L] B  A  B  hrv ] 2  5/2  This expression i s t o be compared with the two alternative  guesses proposed  by Adler f o r the s p e c i a l case L=0 f o r the spinor f i e l d equation with X= -1 (Adler 1972),  (8o 4a)  S-CMz) - 1, f  (8.94b)  S^,,^)  9  \  A  A  B z  = (^l, l2 r A  f  ("h 1 C  B  2 c  r  3  •  2  A  The f i r s t of these i s rejected by Adler since i t provides no c o n t r i b u tions to amplitudes.  This r e s u l t follows from defining a Feynman rule  f o r a vertex where a current with p o l a r i z a t i o n index A acts at coordinate  n. by (8.95)  e r  A  (  n  )  s . ^ B  ^  and noting that according to (5.74) the contribution to a 4-vector X from a 6-current proportional to ^  A  vanishes.  The second propagator  i s c a l l e d a "pseudopropagator" because i t i s not a formal s o l u t i o n t o the f i e l d equation. Since the propagator S(>^ ,^2) defined by ( 8 . 9 5 ) i s c e r t a i n l y a formal  solution of the f i e l d equation, the argument against eq.(8.94b) cannot be raised here.  (8.96)  N  *A  ^ f  As for the argument against (8.94a), using  = ^  B  B  2  - L  .  one has  (8.97)  rA(^)  S ( ^  - L ^ l f  )  2  r ( ) = -6i( B  )(l2p)f  l2  +  r5[ 2 L  f  f^^-  k  ^ A 7 2 ^ l f -l2 ) B  B  1 l i 2  f  so that this propagator-vertex chain does not give zero contributions A  to currents since the f i r s t two terms do not contain  A  or ^  .  At this point, one could go on to define  (8.98)  S(y H l t  l S  y . x ) = U(^) 2  SOf,,^)  2  V~\*lz)  and construct Minkowski space scattering amplitudes in terms of the formulation in 6-dimensional space, but further developments in this direction w i l l not be pursued here.  8-5. Minimal coupling; gauge invariance with massive vector boson The Lagrangian density £  giving rise to the spinor f i e l d equation  (8.60) is given by (8.58), and by inspection X~ is invariant under phase transformations of the f i e l d X\ , (8.99)  X * *  for constant A •  «p(iAg)  ,  However, as i t stands £ i s not invariant under the  vy -dependent transformation (8.100)  x + VX  ,  X*  XT  1  ,  V=exp[i A(^)g]  A Lagrangian density which is invariant under this transformation can  147 be constructed by considering a 6-tensor compensating f i e l d F g ( ^ ) , which A  undergoes a gauge transformation (8.101)  F  A B  *  F  A B  -i  m A B  A  whenever X. transforms as i n (8.100). The Lagrangian density (8.102)  i}  = X o - ^ A B + g F ^ ) * : -2X X X  i s i n this case invariant under the transformation (8.100). The simplest such F  A R  can be formed from a 6-vector compensating f i e l d A g ^ ) , which  undergoes a gauge transformation (8.103)  A * A B  B  +? A0\) B  whenever X. transforms as i n (8.100), by forming the antisymmetric product of Ag with n , A  (8.104)  F ^ *  n ( A  A  B ]  In this case the Lagrangian density (8.102) becomes (8.105)  = X<r (m AB  AB  +g ^ y x  .2 XX A  The entire Lagrangian density describing the mutually interacting \ f i e l d s and A f i e l d s must also include a term X^ which w i l l y i e l d upon variation of the action the f i e l d equations s a t i s f i e d by the free compensating f i e l d .  One should then expect that jL^ i s gauge invariant  i n 6-dimensional space, and that by choosing a special family of gauges the form of the f i e l d equation should reduce to that of the scalar wave equation (8.7a). The simplest approach to the derivation of a f i e l d equation f o r A B i s provided by an examination of the structure of the Casimir operator C, given by (8,5b),  As for the case of the scalar f i e l d , setting the action  on A of this scalar operator equal to a c-number C means that A w i l l R  B  148 s a t i s f y a second-order  l i n e a r p a r t i a l d i f f e r e n t i a l equation, and t h i s  equation w i l l not be very d i f f e r e n t from the form ( 8 . 7 a ) . For the vector representation ( 5 . 6 6 a ) , one has e x p l i c i t l y  (8.106a)  |(s  (8.106b)  (s )  A B  )  D  C  AB  (s  C  A B  (m ) AB  D  )  • 5  D E  S  D E  %\  = 2im  ,  C E  so that the operator C i s given i d e n t i c a l l y by (8.107)  ( O  =  C E  = (im  AB  je^.^cj  [ AB D  m  +2im  +5) %  AB  M  C S  S  E  +  (  S  *B  )  D  E  J  .  .  C E  Thus the vector f i e l d equation may be w r i t t e n (8.108)  (|m  m  AB  AB  +5 -C)  A  D  +2im  D £  A  8  * 0  As i t stands, t h i s f i e l d equation i s not gauge i n v a r i a n t under the f u l l gauge group ( 8 . 1 0 3 ) .  However, one can extract from i t s structure the  family of permissible gauges A  under which i t i s i n v a r i a n t .  To t h i s  purpose, one can make use of the commutators (8.1 09a)  [m ,  (8.109b)  [m  J] c  AB  = - i S A c ) B *± $ C B ^ A  m ,  = - 4 ^  AB  AB  Then replacing A  by A +  fi  g  f  -10  ^  )f A » as i n ( 8 . 1 0 3 ) , one finds that i f A  fi  is a  s o l u t i o n of ( 8 . 1 0 8 ) , then so i s Ag+ j5gA provided  (8.110a)  0 = (im^^-C^A+aii^g^A  i . e . provided  (8.110b)  (im  A is  A B  m  AB  2 ^  {(im^B-OA^)}  a s o l u t i o n of the scalar wave equation ( 8 . 7 a ) ,  -c) A("p  = 0  .  From the analysis of t h i s equation, i t i s known that  A( {) must be formed Y  149 from a product of functions not involving y A = f(L) r  (8.110c) where  Z+  2  ( i + + c )  i (rm)  k  and a function  A(y)  A(y)  A(y) s a t i s f i e s O ^ W )  (8.1l0d)  A(y) = 0  m real  ,  .  At t h i s point, one can remark that eq.(8.110d) i s reminiscent of the gauge condition which i s imposed on the function A ( y ) i n the theory o f the, Stueckelberg f i e l d B(y) (Stueckelberg 1938).  In t h i s theory, the  extra f i e l d variable B(y) i s invented i n an ad hoc manner so as to provide an a d d i t i o n a l compensating f i e l d which renders the Lagrangian f o r a massive vector boson formally i n v a r i a n t under gauge transformations (8.111a)  B  k  *  } A(y)  B+ k  k  for functions /\(y) constrained by (8.1I0d).  whenever B  B(y) -» k  by  that B(y) undergo the transformation  the requirement (8.111b)  This i s accomplished  B(y)+mA(y)  transforms as i n (8,111a).  The remaining  relations  governing the free f i e l d s are given by l» -Hn )B (y) = 0  (8.112a)  O  (8.112b)  (} ^ +m )B(y) = 0  (8.112c)  "i ^(yJ+mBCy)  k  k  2  k  k  2  k  = 0  where the a u x i l i a r y condition (8.112c) i s akin to the Lorentz c o n d i t i o n . Then the equation i n Minkowski space f o r the spinor f i e l d  Y(y)  interacting  with Bj^y) i s w r i t t e n  (8.113)  -i^ (^ -igB )r(y)+m V(7) = 0 v  k  k  k  F  This equation i s then covariant under spinor f i e l d (8.114)  ^ ( y ) -t»  transformations  t ( y ) exp[ig A(y)]  provided such replacements are accompanied by the gauge transformation (8.111). Since adoption of the f i e l d equation (8.108) f o r the vector  field  i n 6-dimensional space leads one automatically to the family of gauges  150 A ( y ) s a t i s f y i n g ( 8 . 1 1 0 d ) , one may attempt to determine under what conditions the remaining r e l a t i o n s ( 8 , 1 1 1 ) and ( 8 , 1 1 2 ) follow from the 6-dimensional free f i e l d equation. In p a r t i c u l a r one seeks an a u x i l i a r y condition on A  which reproduces i n Minkowski space the Lorentz-like  R  condition ( 8 , 1 1 2 c ) ,  An obvious choice f o r such a gauge condition i s that  which brings ( 8 , 1 0 8 ) into the form of a vector analogue of the s c a l a r wave equation ( 8 , 7 a ) , v i z , (8.115a)  A  (8.115b)  2im A'  B  ^  A'  D  E  E  A +^ A'(n)  =  B  B  B  5A» = 0 D  +  .  I t i s shown i n Appendix 5 that t h i s c o n d i t i o n can indeed be  implemented  by a gauge transformation, and that ( 8 . 1 1 5 ) i s preserved under subsequent gauge transformations by r e s t r i c t i n g the gauge group ( 8 . 1 1 0 b ) f u r t h e r to functions  A"(*\) which are homogeneous of degree  (8.116)  5*ID & A " -  2 0  A" = 0  (2C/5)»  i«e.  .  Therefore, since L i s homogeneous of degree 2 , f o r A" (^ ) the f u n c t i o n f(L)  in (8.110c) is  (8.117)  f(L) = L  ( C  /  .  5 )  With the condition ( 8 . 1 1 5 ) , eq.(8.108) reduces to the simple form (8.118)  (im  mAB-OA = 0 0  A B  .  Now the p r e s c r i p t i o n ( 5 « 7 ^ ) can be used to write (8.118) as a s e t of equations f o r A , keeping i n mind the f a c t that the operator k  (U)^Q.  given  i n ( 5 * 7 0 ) , brings into consideration the extra f i e l d v a r i a b l e A 7 , where the  component A7 i s defined i n terms of ^ j ^ A by  (8.119a)  8  A  ?  = -L"2 I R A  8  (5.84),  .  According to ( 5 . 8 8 ) t h i s d e f i n i t i o n leads to (8.119b)  A  6  + A  5  = 0  .  In Appendix 5 i t i s shown that one can also f i n d an appropriate gauge  151 transformation with gauge function A ( ^ ) n  (8.120)  A  - A  6  5  = 0  such that  ,  provided the number C has the value (8.121)  C=-(15/4)  ,  i . e . the same value as that arrived at i n the study of the s c a l a r wave equation i n section 8-1 • The gauge transformation induced on the f i e l d A the form (8.111a) by i d e n t i f y i n g the f i e l d B (8.122a)  B  k  = xA  k  can be brought into  k  with  k  so that (8.122b)  B  5  *A  •  7  In Appendix 5 i t i s shown that the f i e l d s Bj^y) and B(y) s a t i s f y the f i e l d equations (8.112a,b), and that the meaning of the a u x i l i a r y c o n d i t i o n (8.115) i s that the Lorentz-like c o n d i t i o n (8.112c) i s s a t i s f i e d i n Minkowski space. That B i s a s c a l a r f i e l d under Poincare transformations follows from (5«82b) and the f a c t that X - i s a Poincare scalar, since Ag i s assumed to transform as a v e c t o r .  A l t e r n a t i v e l y , one can assume that Ag transforms  as a pseudovector, i n which case B i s a pseudoscalar f i e l d under Poincare transformations. F i n a l l y , one can return to the minimally coupled spinor f i e l d equation s p e c i f i e d by  To determine what a l t e r a t i o n s occur i n the form of the  equation s a t i s f i e d by the spinor f i e l d  X = U x when i n t e r a c t i o n with Ag  i s included, one must perform the transformation U( )U~ on the matrix 1  <r l$ ,  with the r e s u l t  (8.123)  Utcr^gA^in  BCY  c  +(x  tr  65  1  = (K- LTT +iMK +L2<r )A +  +li TT  1  k  K  k  7  7  )rA -K 7  M  <r W T T 7 ) ( A + A 5 ) 6  6  k  .  152 In the gauge i n which A^=A^=0, the l a s t term vanishes.  Hence the equation  s a t i s f i e s follows from t h i s r e l a t i o n and from (8.64),  which X (8.124)  { i  K (o k-igBkJ+i- TT ( ^ - i g B ^ - r <r ( k  2  k  k  ?  <r (-rh +igrB)+  h  65  +r F ( - r ^ + i g r B ) 4 K T r + ( 3 / 2 ) r T T + i > } X k  ?  k  r  Just as i n eq.(8 113). the replacement 0  7  =0  r  .  ^ - i g B ^ occurs i n the  f i e l d equation, and the f u n c t i o n o f the corresponding  replacement  r i j ^ r ^ - i g r B i s to ensure the gauge invariance of the relevant terms. r  The extra terms i n X. which r e s u l t from including B are (8.125)  - 2 g r % ( < r + r T r ) B X = gr( ppp+ 6 5  7  ^5^)B+2gr ( ^ 5 ^ ) 2  where the 4-component spinors (p and <f> are defined i n (5.133) •  B  .  This  shows that i n the s i t u a t i o n i n which one imposes the t r a n s v e r s a l i t y condition (5»137) and the f i e l d components <f> vanish, then i f B^ i s a A  pseudovector the spinor f i e l d  (p i n t e r a c t s with the pseudoscalar f i e l d  B with the customary meson-nucleon i n t e r a c t i o n (Schweber 1961). A l t e r n a t i v e l y , one can replace (8.100) by the phase transformation e x p ( i jS?Ag)«  In t h i s case the replacement  ^  k  - i g  occurs i n the  f i e l d equation, and the extra term i n X. which r e s u l t s from including B a. A i s of the form  <p p B . As w e l l , any combination of these two phase  transformations can be considered. The complete system of equations describing the X - and B - f i e l d s i n mutual i n t e r a c t i o n c o n s i s t s of (8.124) together with the equations f o r Bjj and B including terms f o r the fermion current.  C l e a r l y , exact solutions  of t h i s system of equations w i l l be exceedingly d i f f i c u l t to obtain, as i s the case f o r coupled spinors and vectors i n the usual formulation i n Minkowski space.  153 Therefore i t would be appropriate at t h i s point to embark upon a treatment of a perturbational approach to these equations, using the covariant Green's functions developed i n sections 8-2 and 8-4 to construct a set of Feynman rules f o r scattering amplitudes due to the minimal coupling i n t e r a c t i o n . This course w i l l not be followed here. be noted that while  However, i n conclusion i t should  (8.100) i s the simplest gauge group which one can  consider, by no means need attention be r e s t r i c t e d to t h i s c a s e . If  one were to allow more general groups of gauge transformations, then  as a r e s u l t of gauge invariance there would appear not only mutiplets of massive vector or pseudovector bosons, but corresponding to each there would also appear automatically multiplets of massive scalar or pseudoscalar bosons.  154 9. Concluding remarks The task of incorporating the mass concept into conformally invariant action p r i n c i p l e s has been c a r r i e d out f o r the general case i n Section 5. and f o r the p a r t i c u l a r cases of s p i n - 0 , spin--§-, and spin-1 p a r t i c l e s Section 8.  in  Not enly does the formalism used here accommodate the mass  i n a reasonably straightforward manner, but also provides a means by which one can apprehend a massive compensating f i e l d within the confines of a gauge invariant  theory.  The necessary preparation f o r the study of the perturbation  theory  associated with a minimally coupled spinor f i e l d i s developed i n t h i s investigation.  Making use of the r e s u l t s presented here, such a study has  p o t e n t i a l f o r providing i n t e r e s t i n g insights into quantum electrodynamics and the problems associated with the theory of the intermediate boson, which i s believed to mediate the weak i n t e r a c t i o n .  vector  As w e l l ,  the  i n t r u s i o n of the strong i n t e r a c t i o n i n t o ordinary gauge theory i s adumbrated by the automatic appearance i n Minkowski space of the pseudoscalar meson-nucleon coupling as a r e s u l t of minimal coupling i n 6-dimensional space. It  also seems u s e f u l here to l i s t some other of the as yet incompletely  explored d i r e c t i o n s i n t h i s f i e l d .  F i r s t l y , i t may be p l a u s i b l e that  the d i f f i c u l t i e s associated with finding a conformally covariant spin-2 f i e l d equation may by bypassed by the use of the 6-dimensional formalism. One p o s s i b i l i t y would be to use a Petiau-Duffin-Kemmer formalism f o r a u n i f i e d d e s c r i p t i o n of boson f i e l d s (Petiau 1936, Duffin 1938, Kemmer  1939).  In f a c t , Kemmer has shown that there are p r e c i s e l y three non-  t r i v i a l r e a l i z a t i o n s of his matrix algebra i n a space with s i x dimensions (Kemmer 1943), so that i n a l l the representations are 1,7,21, and 35 dimensional.  A l l these representations can be accommodated simultaneously  155 by using a d i r e c t product of the 8X8  y3-matrices introduced i n Section 5,  so that using the r e s u l t i n g 64x64 matrices i n a Dirac-Hepner type equation, as i n section 8-3, one could conceivably develop f i r s t - o r d e r equations f o r spins 0,1,  and 2 d i r e c t l y from 6-dimensional space.  A l s o , since the r o t a t i o n a l l y invariant action p r i n c i p l e developed i n Section 7 i s not dependent i n i t s e s s e n t i a l s on the dimensionality of the space concerned, i t could be used i n the 5-dimensional DeSitter space harbouring r o t a t i o n a l l y invariant actions whose projections i n t o a s u i t a b l y chosen 4-dimensional space can be interpreted as those corresponding to p a r t i c l e s i n a Riemannian space of constant curvature R.  Then i n the l i m i t  of large R Poincare invariant actions could be recovered. In the same v e i n , i t might also be worthwhile to consider the p o s s i b i l i t y of adopting l o c a l r o t a t i o n a l symmetry as the guiding p r i n c i p l e i n 6-dimensional space.  In t h i s case the method of Utiyama (1956) only prescribes how to  augment a t r a n s l a t i o n a l l y invariant Lagrangian density by the introduction of compensating f i e l d s so as to render the action i n t e g r a l  manifestly  invariant under transformations whose parameters depend upon the coordinates of some underlying (possibly curved) reference manifold.  Since only  r o t a t i o n a l and not t r a n s l a t i o n a l invariance i s required as a prerequisite f o r candidates f o r any action function i n 6-dimensional space, i t seems l i k e l y that one could not simply generalize the vierbein formalism to the 6-dimensional case, so that the use of a modification of Utiyama's method might shed new l i g h t on the theory of  gravity.  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Conformal transformation of symmetric tensor equation The i n f i n i t e s i m a l generators of s p e c i a l conformal transformations  by (3.26).  for second-rank tensors are defined  In order to study the  transformation properties of the weak f i e l d approximation of the E i n s t e i n equations, one must evaluate the commutator (3.27a), which c o n s i s t s of four terms.  (A1.1)  F i r s t l y , one has  ^ ^S  k  n  r  9 (k ) s  =V ( p>Vu n  r  p  k  - (k )  s t  u  r  (k >Vu^  "  +  s  n  p  -4i{-(1 i>yy V«} £t^u  S tS r  J  k  p  S  V = n  u  " ^°V tu^  +  k j  p  m  •  Secondly,  (A1.2)  ^\S $ (VYu k  n  r  "  s  ( k  P r sVV tS ) k  r  J  n u  "  = A ( P > V u - ( p)VnAi " k  k  = { 2i(1 -i)S p  V  2  i  VAuV  ^V  ) k n  <  t^\- y ^pm) 2  1  ^^p.V^iSp^Miy^^J-  sV  ^ K \ ^ y % \ ^ u \ ^ }  + 2  2 i  m  2  k J t n  ^p^ tuV ^  \^  n  2  ) tu^ k n  P n  j +  •  The t h i r d term is  (Al .3)  >\S  3 r  ^(V7u-<VVs^A^u "  • vvv V. - <vVtA, • where the l a s t equality follows from the i d e n t i t y  <*.»>  ( ^ n t ^ V ^ t n  •  Thus the t h i r d term i s given i n terms of the second by interchanging j and k.  171 For the fourth terra one has  (AI.5)  A w y Y u = A  V  (  n  -  <vVs^\ = u  t n u - ( V V s ^ « t u  =  2i{( 1 - j > ) s j M p V ^ - K 1 - i J S p ^ p V ^ J + z y ^ « - S ^ J  k  J  p  p  -y  V -rV ) k  J  ^tu ( +2  ^ p m ^ t u ^ Several d i f f e r e n t  2  «V  ) n t u ^ ( c rp  k  n  ^ <V  k  J  m£tu^  ) b^ n  ntu  *  n  types of terms occur i n the commutator (3»2?a).  Collecting  terms of one type, (A. .6)  «  6^)«  t a  ^(«r- J) p  t a  ^ J(r- _) n  t a  I  t u  .3-2(<r  k p  )*  ^  , t  1 1  .  The remaining terms involving the spin matrices are  (AI .?)  2y {-(^ n  p n  )^ V3 H^ u  n  • s \ U p ^ V V . - s V . ^ p  +  5  i  p a  )  k J  t 5 p  t n  k  VM^ j^ A  ^ ' -  p  s  J  '  u  I  '  k  >  «  ( )^ >"  n +  i  V  p  u  +  t  F i n a l l y , c o l l e c t i n g a l l the contributions to the commutator, one arrives at the r e s u l t (3.27b), so that the d i f f e r e n t i a l  operator which acts on the tensor  field  generator i n the weak of f isepl edc iapproximation a l conformal transformations of the E i n s t e i n i nequations Minkowskidoes space. not commute with the  172 Appendix 2 . V e r i f i c a t i o n of area elements and an i n t e g r a l i n 6-space The d e f i n i t i o n (7-12) of d S , A  (A2.1)  dS = - d ^ 6  U ^  A  2  2*^  - L )  ,  can be compared e x p l i c i t l y with the area element (7.4c) i f one chooses the parameters  A] 2 3 4 5  (A2.2)  \j  = y ,  o  \  1  ^he surface ( f + L ) = 0 to be  n  = y ,  X  2  2  = y3,  3  = y\  >  5  = K.  .  Then from (7.4c) one has e x p l i c i t l y (A2.2a)  dS  A l  t  =  ^^/^\0''^ [ ' ^5 ^''' 5 v  k]B]BzB3BkB5  B5/  b  )d  dX  •  and defining the s i x t h independent parameter by (A2.3)  A  = L  6  ,  then one can produce the Jacobian J ( * ^ , X ) i n (A2.2a) by forming (A2.3a)  ("b^A/>X )ds = -£ B 1 # # . B 6 (>1 /^A 1 )...0'9 /^A 6 )B1  6  B6  A  •d ) j =  > (  »d  =  - J ( Y ^ , A )d A,j . . . d A ^  with J ( ^ , X ) determined by (A2.4)  = J(v^, X ) dy  d\  Using the r e l a t i o n s (5*1) (y^.HtL),  (A2.5)  d * . dL  k  .  which connect the sets of coordinates (v^ ) A  one f i n d s f o r the determinant of the J(v^.A)=-H /2  an(  j  6x6 matrix "i(v^ )A(y»*»L) A  .  3  Now checking the r e s u l t (A2.3a) with the d e f i n i t i o n (A2.1) of d S , one A  uses  (A2.6)  Mm k  =° . H /* 6 5  6  x  =  M  6 / n  6 • /< O 1  2v  to obtain (A2.7)  (^ /U )dS A  6  = - X ( » - L ) d r = -Jty.X) S ( ^ - L ) d V d H dL 2  A  l  2  6  l  so that with the d e f i n i t i o n (A2.2) this agrees with (A2.3a) provided integration over the area element (A2.1) i s c a r r i e d out over • ( » rL  >  In the case of dS^g, given by (7.15), the d e f i n i t i o n (7.4b) reads  (A2.8)  dS  A l A 2  ^A B B B3B O l  =  r  2  2  1  B l  5  /^^)CM  '(>>*\ lh\ )0\ />>\ )d B  ^  B5  3  5  B 2 / i X  dX  2 ' )  dA  2  d \  3  .  5  Then one has (A2.9)  ar\ M )(M /> V A  B  d S  4  AB  =  -  J ( r  \ '  X  ) d >  d  1  A  d  >  2  3  d  X  5  •  On the other hand, one finds by straightforward computation that since (A2.10)  SV- y (^ +S ) A  ^ /*Ai* = A  H  the d e f i n i t i o n (7.15) of d S (A2.11)  A B  4  6  .  yields  (^^ /o> )(>>1 /U )dS A  A  5  B  6  4  = -d 1 6  A B  S(-^ +L) 2  S(-K- /|^t) 1  and according to (A2.4) t h i s agrees with (A2.9) provided i n t e g r a t i o n over the area element (7.15) i s c a r r i e d out over -H.^. To see how to transform the f i n a l two terms i n the integrand of (7.43) v i a Stokes' theorem i n the form (7.16), one must evaluate  (A2.12)  ^ {(i/2) t(  f J  D  S | {) [#«l J 1 ) J " f l D  t  D  +  Now from (7.42) one has (A2.13)  ^  ^  = 0  so that the l a s t term i n (A2.12) vanishes. (A2.14)  v|  D  gr^ =  ivf  Also, for linear  transformations,  ,  so that (A2.12) becomes (A2.15)  ^  C D  \(V2)/(  f ?"\ )\ = \U c  ) •  Therefore eq,(7.43) can be written as the sura of the two terms given i n (7.44).  174 Appendix 3 . I n a d m i s s i b i l i t y of 6 - t r a n s l a t i o n invariant spinor Lagrangian Consider the Lagrangian density Xpostulated  by Barut and Haugen  (1973b) f o r use i n an a c t i o n p r i n c i p l e associated with e q . ( 8 . 6 l ) , (A3.D  2  B  = itf*  H  V-Myf  A  .  Since these authors include translations among the permissible motions i n 6-diraensional space, t h e i r v a r i a t i o n a l equation takes the form of an ordinary Euler-Lagrange equation  (A3.2)  ^l M  Now, using the r e a l i t y  (A3.3)  (?p  JA^Ba/ac  -  m  V  A  * T>  •  = 0  A  property  ^  -  f Y  ^  A  •  which follows from (5.120), the r e a l part of (A3.1) can be w r i t t e n  (A3.*»>  i  ,  B H " * < ^ B H  +  ^ B H  )  = -M V f + ( i / 2 )  f f ^ + ( i / 2 ) ^ qj^ V = A  A  = -M9f+(i/2) « f A ( 9 p Y ) A  and i t s imaginary part i s (A3.*b>  2!  = -(i/2)(^  BH  B H  -^ ) H  = i(9f  A  ^ - ^ 9 ^ )  •  Since the second term on the r i g h t hand side of (A3.4a) i s a divergence,  it  cannot a f f e c t f i e l d equations of the form (A3.2), so that by i n s p e c t i o n one has that the r e a l and imaginary parts of ^  (A3.5a)  Mt = 0  (A3.5b)  f ^  =  k  B B  - y i e l d separately the equations  0  r e s p e c t i v e l y , i n agreement with (8,61) only f o r the case M=0.  The converse  statement i s also t r u e , that the f i e l d equation (8,61) cannot be derived from a r e a l Lagrangian d e n s i t y , since i f  (A3.6a)  /  (  i  i  i  ) B  H  = a S ^ A ^  b9f  175 then the real part is (A3.6b)  ^  i  v  ) B  H  =K  ^  (  i  ^ ^  = i( a 9 JB  A  (  i B  j  )  +  )  =  -a* \ Y f * )+K b+b*) H Y = A  7  = Ka+a^YfA^^b+b^Yr-l^a^pA^)  #  leading to field equations of the form (A3.6c)  | 3 3 t+NM* = 0 , A  A  in disagreement with (8.61) except for the inadmissible case of imaginary M.  176 Appendix 4 . Solution of spinor f i e l d equation The s o l u t i o n of the f i e l d equation (8.60) i n terms of the f i e l d variable X  i s expedited by writing (8.64) i n terms of the four sets of 2-component  spinors ^  2 J k defined i n (5*132)•  With the representations (5.94),  (5*126) of the s p i n matrices, eq.(8.62) can be written as the s e t of four equations (A4.1a)  r2^ X +(i/2)r^  (A4.1b)  i^ Xi-(i/2)r^ X  (A4.1c)  -i^ Xi -(i/2)ri X  (A4.d)  -r X i y +(i/2)r^ X^i>X -|rX\X +ir^ X -(3i/2)r^  k  2  k  X +i>X Hr^ X  r  l  r  k  2  +  r  1  + i 2  + 3  (  2 +  k  +ir ^ X3-(3i/2)rX * 0 2  i +  r  >)^2^ ^k 3 r  X  i ( 2 + > )X $n%X y  3  =  0  =0  2  k  k  3  r  4  1  r  2  2  =0  As was the case i n the comparable s i t u a t i o n which occurred i n section 6-6, t h i s set of equations can be greatly s i m p l i f i e d by recombining the four equations into equations f o r combinations of components of X: (A4.2a)  ir2^ X +(i/2)r^ (X +rX )+i>(  (A4.2b)  i^ (y +rX ).(i/2)r^ X +i(2+X)X  (A4.2c)  - i ^ ( X + r ^ ) - ( i / 2 ) r i X + i ( 2 + \ ) Xj = 0  (A4.2d)  -ir ^ X +(i/2)r^ (X  k  2  k  r  1  k  1  3  3  4  r  2  k  2  r  2  3  r  i +  Xj+rX-j) = 0  2  =0  3  +rX )+i>(X 2  4 + r  X ) 2  = 0  .  This arrangement greatly f a c i l i t a t e s performing the i t e r a t i o n of the f i e l d equation f o r X , since operating from the l e f t on each of these equations with  tf i k>  k  has the simple r e s u l t s  (A4.3a)  [r  (A4.3b)  [ ^ ^ ^ - ( r ^ ^ r ^ X O ^ ) ]  2  & > - ( r > ) + 6 r ^ +4 X ( X +3)] ( + r Q>) = 0 2  k  r  r  <p = 0  177  where  (A4.4)  are the 4-component spinors defined i n (5»133) •  *3  (f =  .*2 Now commuting r through the operator i n square brackets i n (A4.3a), and then subtracting (A4.3b) multiplied by r , eq.(A4.3a) s i m p l i f i e s to (A4.5)  { r 2 ^ - ( r i ) + 6 r ^ - f 4 > ( +3)}^+(4X+5)y 2  Hence, X (A4.6)  r  k  A  r  = 0  .  must have the value > = -5/4  i n order that the equations f o r y> and y? can be solved. In t h i s case, the i t e r a t e d equations can be w r i t t e n as (A4.7a)  {r ^ -( ^ )2 6ri -(35/4)}y>  (A4.7b)  ^^-(r^Z^r-OSAOlf  =0  2  k  r  r  +  r  Since the variables separate, ip and f F(r) ^ ( y f  A  (A4.8)  .  are products >(r) X ( y ) " 3  tn —  •  F'(r) ^ ( y )  "0  7  F'(r) X,(y)  where X ( y ) ani 7C(y) are solutions of the usual Klein-Gordon equations 14 32 (A4.9a)  ( V< V  (A4.9b)  (f Vm )X  m 2  )*1,4(y>  2  3 f 2  (y)  B  =  0  0  •  Since there i s no compelling reason f o r equating m and ra, t h i s assumption w i l l not be made. The p o s s i b i l i t y of encompassing two masses within one spinor equation i s therefore l e f t open.  Substituting the conditions (A4,9)  into (A4.7), one finds that the equations f o r F,F',F,F' can be written i n the form (8,17) f o r cylinder functions Z  y  178 (A4.10a)  F ( r ) = r3 Z (rm) , F»(r) = r  (A4.lOb)  F ( r ) = r Z ( r m ) , F»(r) = r  v  2  2  y  A  3  Zj,(rm)  Z'(rm)  A  where Z . Z ' . Z . Z * are four possibly d i f f e r e n t cylinder functions  and v i s  given by (A4 10c) o  v  - \  Z  ; i.e.  However, since J _ i = - Y i ,  v = +_• or  ^= - i .  any l i n e a r combination  Y_A.=J.|.,  Z „ A  of  J_A.  and Y_i.  can be written as a d i f f e r e n t l i n e a r combination Z A of J I and Y i , so that Z  2  considering of X are  only ^-j y i e l d s a l l s o l u t i o n s .  (A4.Ha)  X, = r3zi(rra)X (y)  (A4.11b)  X  2  = i^Z'^rni) * ( y )  (A4.11c)  X  3  =r^rm)  (A4.11d)  X  Now  2  E x p l i c i t l y , the components  1  2  X (y) 3  = r3z«i(rm)X (y)  4  4  X= -5/4,  rewriting eqs.(A4.2) with  one has  (A4.12a)  ix <^  k X 2 + ( i  (A4.12b)  |X^ X  1 +  (A4.12c)  -i^ > X -ir^ d X +(i/ )(-ri +3/2)X  (A4.12d)  -ir X ^ X (i/2)(r^ -5/2)X'^(ir/2)(r^-3/2)X  2  k  /2)(  .  r  r  -5/2)X +(ir/2)(ri -3/2)X  i r V^ X +(i/2)(-ri +3/2)X k  k  r  3  2  k  r  2  3  3  =0  k  k  3 +  r  =0  =0  2  2  k  4  k  p  1  2  = 0  .  Then multiplying (A4.12c,b) by r , and adding to (A4.12a,d) r e s p e c t i v e l y , one arrives at the simpler coupled equations (A4.13a)  ir^ ^ X +(i/2)(ri -5/2)^ =0  (A4.l3b)  -ir^ a X  k  k  r  1  k  k  4 +  (i/2)(ri -5/2)X r  1  =0  .  Making use of the formula f o r the derivative of a cylinder function, one has  179 (A4.14)  = 3r Zi(rm)+Ai j-(ljferm)Z^(rm)+Z_i(rm)] 3  r^ [r Zi(rm)j 3  r  ,  so that eqs.(A4.13) are simply (A4.15a)  0 = r { Z | ( r m ) Y a * i (y)+imZ '_i(rm) X ^ y f l + i ( 3 - 1 - 5 / 2 ) r 3 z ' i(rm)  (A4.15b)  0 = r^-Z'i(rm)^ A ^( )+imZ i(rm)X (y)Vi(34-5/2)r3Zi(rm)X (y) .  4  ^(y)  k  k  k  k  y  -  1  1  Therefore, i f t h i s i s to reduce to a simple form the fuctions Z,Z' must be related by (A4.l6a)  Z'^Crm) 3 Z|(rm)  or equivalently (A4.l6b)  -Z'i(rm) = Z_i(rm)  .  In terms of the 4-component f i e l d  , with (A4.16), the Dirac equation i s  s a t i s f i e d by ^ ( y ) , (A4.1?)  (-i* ^ +m) p(y) = 0  .  k  k  S i m i l a r l y , f o r the remaining components X ^ , o n e  finds  (A4.18a)  0 = ^(rm)y^ X (y)-imZ' A(rm)X (y)+ |(rm)y^ X (y)  (A4.l8b)  0  k  3  -  2  Z  k  1  =-zi(rfi)rt X (y)-imZ_A(rm)X (y)-Z'i(rm)y X X^y) k  k  2  3  k  A A  so that these equations take on a simpler form only i f the functions Z,Z' are related by (A4.19a)  -Z'_|(rm) S  z^( ) rm  or equivalently (A4.19b) -Z'|(ro) 5 -Z_|(rm)  . A  In that case the 4-spinor f i e l d the f i e l d  y(y)  ^ ( y ) s a t i s f i e s the Dirac equation only i f  i s simultaneously constrained to vanish.  To e s t a b l i s h the connection with the 5-dimensional Minkowski space, one and M* defined by  must change the f i e l d v a r i a b l e s to 4-spinors (A4.20)  ps <5/2)y r  .  £s <3/2>$ r  .  Then i n terms of 2-spinors Y^ 2  a n  Y-j  d  A.  (A4.21)  * = v.  J  the f i e l d equations (A4.13) and (A4.12b,c) read  (A4.22a)  r *  b ^ \ +ir > > Y  (A4.22b)  -r Y > Y +ir *  r  (A4.23a)  r  ^ + r V > Yi  (A4.23b)  - r ^ ^ z - i r - b ^ i - r ^ ^ z  k  k  2  = 0  7 ]  =0  r  K  K  2  and  respectively.  k  fy-irb  k  r  2  k  =0 - 0  These equations immediately y i e l d the 5-dimensional  equations (8.80) and ( 8 . 8 2 ) .  181 Appendix 5» Gauge properties o f the vector f i e l d To show t h a t the c o n d i t i o n (8,115b), (A5.1)  (2im° +5 S  D  E  )[A +  /  E  E  A » ( ^  )]  s o  ,  can be implemented by a gauge f u n c t i o n A ( ^ ) belonging t o the f a m i l y 1  (8,110b), one must f i n d a A (A5.2)  s a t i s f y i n g both (A5«1) and the wave equation  1  (imAB mAB . c) A « ( n > =  .  0  Contracting (A5«1) w i t h ^ J J , (A5.3)  n (2H» E 5 D  +  S )AE = - 7 (2imD 5 D  D  E  E+  D  D 0  E  ) ( J  S - ( n D m^E . 5 « ] ^ ) A «  A') -  .  D  E  E  and comparing w i t h (A5.2), i t i s seen t h a t i t i s indeed p o s s i b l e t o f i n d such a gauge f u n c t i o n A* by equating (A5.4)  [c-(5/2) »| jp] A ' = -1^(211^2+5 S ) ^ D  D  B  •  Then any subsequent gauge transformation w i t h gauge f u n c t i o n A " preserves the c o n d i t i o n (A5»1) provided  (A5.5)  2im  D  ^ A"+5^ A E  E  D  M  s  0  •  From the second e q u a l i t y i n (A5»3), t h i s means t h a t i n order f o r subsequent gauge f u n c t i o n s t o s a t i s f y the d e f i n i t i o n (8,110b) of the gauge group o f the f i e l d equation f o r A , B  (A5.6)  (im  AB  mAB-c) A"("\) = 0 .  w h i l e a l s o preserving the c o n d i t i o n (A5»1)» they must be r e s t r i c t e d f u r t h e r t o the s e t o f f u n c t i o n s homogeneous i n v| of degree (2c/5)»  (A5.7)  5 "{D ^ A " - 2 C A D  n  = 0  .  Consider now the behaviour of the components Ag and A r , under a gauge F i r s t l y , using the formulae (5«3) f o r the d e r i v a t i v e s  transformation.  the gauge transformation f o r the f i e l d A i s k  (A5.8a)  A  k  = A - y ( A + A ) -» A - y ( A + A ) + ^ A - 2 y H * A = k  k  • A +* k  I)  6  - 1  5  o A k  k  k  6  5  k  k  L  182 and that f o r A^+A^ i s simply  A6+A5 = A5+A3  (A5.8b) For  the f i e l d A A  (A5.8c)  A6+A +2K^ A  .  L  5  one has  7  = A + ^ - L ^ ( A 6 + A ) = -IT* ^  A  1  7  5  ?  -»  B  B  + v ^ L ^ A ^ )  A -L-i(K^^+2Li )A+H- Li(2K^ A ) = 1  L  7  =  A  7  - K L - T ^  A  K  L  .  and the combination of components A5-A3 vanishes by v i r t u e of the d e f i n i t i o n of A . 7  In f a c t , t h i s i s a gauge i n v a r i a n t statement since  (A5.8d)  0 = A6-A5 = A6-A +2y A +2  L a A - ( y - H.-2L) (A6+A5)  k  5  2  ?  k  A -A 6  5  = 0  .  From (A5.8b), ( A ^ + A i j ) can be made to vanish by a gauge transformation provided one can f i n d a f u n c t i o n A " ( \ ) s a t i s f y i n g (A5«6) and (A5.7), K  as w e l l as  (A5.9)  2M* A"  -(A +A ) = -(A +A )  5  6  L  5  6  .  5  To show that the gauge c o n d i t i o n (A5«9) i s not a t variance with (A5«6), m u l t i p l y (A5«9) by the wave operator, (A5.10)  -KimrjE m ^ - O U ^ ) s  M  ^ ^(ini L  Now,  D E  m  D E  = (_»_)_ H P ^ « C ) ( H ^ _ A )  =  n  -C)A } +H-1> > A B  K  , ,  K  +5Mo A  , , +  L  2V<2^ ,i A'» >  L  .  the l e f t hand side vanishes by the f i e l d equation (8,118), so that A "  s a t i s f i e s (A5,6) only i f (A5.11)  VT  1  ^  K  >  K  A "  +  5 ^  L  A " + 2 K  2  o  K  *  L  A  N  = 0  .  = 0  ,  But (A5«6) reads e x p l i c i t l y (A5.12)  (-K- Lo o +H ^> +5H> K  2  2  k  K  M  -C)A"  so that (A5.11) i s s a t i s f i e d i f  (A5J3)  [K\(R^I +2L^ )+5LO +4H^-C]A" = 0 k  l  l  .  183 Now,  (A5«7) reads e x p l i c i t l y  (A5.1<0  ( H V + 2 L « ) A " = (2C/5)A"  .  L  Therefore a gauge f u n c t i o n A " s a t i s f y i n g  both (A5.6) and (A5»7) can be  found provided the number C i s given by  (A5.15)  c = -(15/*0  so that (A5«13) i s v e r i f i e d  identically.  The f i e l d equation (8.118) w r i t t e n i n terms of the coordinates y^, L, and r = M - 1 l i " y i e l d s (A5J6)  \-r2^k^ J k  l r  -  ( ^ )2.c}A -2r2> (A6+A )= 0  i r +  r  r  k  k  5  ,  so that i n the gauge A6=A5=0, with C given by (A5«15). one has (A5.17)  {-r2>k> ^ > k  Solutions A (A5d8a)  r +  ( ), )2 r  r  + l 5  /^}  A  k  - o  #  are products  k  A  r  k  = r  2  Zi(rm) A ( y ) k  with (A5.18b)  (> o +m )A (y) = 0 k  k  The l a t t e r equation e n t a i l s  (A5.19)  .  2  k  \ SHA  (8.112a) i f one i d e n t i f i e s B  with  K  .  k  To see what f i e l d equation ky s a t i s f i e s , consider the i d e n t i t y  (A5.20a)  0 =v( (im mAB.c)AD = ( K B ^ - 0 ^ A ) - 3 ^ - 2 ^ ( V | A B ) 2 I ^ A B  D  D  D  D  AB  B  +  B  B  so that (A5.20b)  0 =%L^(im m B-C)A A  A B  -3L"i  D  faufP-C){u\p)-  =  ^BA -2L-^ ^ ( *[ gA ) D  B  B  D  .  Now the c o n d i t i o n (8.115b), when m u l t i p l i e d by L~2" and contracted with *Vp, reads (A5.21 )  3L-* ^ - 2 1 *  L - i 1 cF( V ) B  D  = 0  so that from the d e f i n i t i o n of A^, i n the gauge A5=A^=0 eq.(A5.20b) states that A 7 s a t i s f i e s the f i e l d  equation  184 (A5.22)  {-r i > - 4 r ^ + ( r ^ ) + 1 5 / 4 } A k  2  = 0  2  k  r  r  7  with solutions (A5.23a)  A  = r  ?  2  Z'i(rm) A ( y ) ?  where (A5.23b)  (o  k  H  + m 2 )  M  y )  =  '  0  The l a t t e r equation i s just (8.112b) i f (A5.24)  B = H  A  .  ?  As a l a s t step i n the study of the free B - f i e l d s , one must work out e x p l i c i t l y the form that the condition (A5<>21) assumes i n the gauge i n which A^ and A^ vanish, with the r e s u l t (A5.25)  0 = -3A -2r I ^ ^ r } A 7  r  7  =  = (-3+4-1 J r Z'i(rm) A (y)+2mr 2  7  3  Z'^rm) A (y) ?  - 2r3 Zi(rm) * A*(y) k  .  This s i m p l i f i e s to the form (8.112c) i f the cylinder functions Z,Z' are r e l a t e d by (A5.26a)  Z».i(rm) = -Zi(rm)  ,  i n which case (A5.26b)  0 = -2r3 Zi(rm) j^ A (y)+mA (y)} k  k  7  .  

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