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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 in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1975 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y 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 s t u d y . 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 e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date October 1 , 1975 i i 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 written i n manifestly covariant form. By writing 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 directions, the action of the group is linearized and f i e l d equations are written i n rotationally covariant form in 6-dimensional space. It i s then shown that extending the 6-coordinates off the null surface generalizes Minkowski space to a 5-dimensional space. Such a generalization necessitates employing a method of descent to 4-dimensional space from six dimensions which di 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 precisely two species of particle having two different masses. An action principle i s developed i n the 6-space, and a method of f i e l d quantization i s devised. As examples of the method, the special cases of spin-0, s p i n - i , 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 carried out. The formalism presented i n this investigation 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 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 well the pseudoscalar coupling occurs automatically within this gauge invariant formulation. i i i Contents Abstract i i Li s t of Figures v i Acknowledgments v i i Introduction 1 PART I. CONFORMALLY COVARIANT FIELDS IN MINKOWSKI SPACE 1. The Conformal Group in Minkowski Space 1-1 . Definition of conformal transformations 10 1 -2. Infinitesimal conformal transformations 13 1-3» Finite conformal transformations 14 1- 4. Isomorphism with rotations i n 6-dimensional space 16 2. Conformally Covariant Fields 2- 1 • Definition 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. Field variations 24 2-6. Dilation invariance 26 2- 7. Two-particle momentum space representation; experimental consequences 32 3. Conformally Covariant Field Equations 3- 1. Massless Klein-Gordon equation 35 3-2. The Weyl equation 37 3-3* The Maxwell equations 38 3-4. Weak f i e l d approximation of Einstein equations 3-5 • Independence of the generators , 4. Attempts at Including Mass 4-1. Symmetry breaking •• 43 4-2. Conformally covariant mass 44 4-3, Interpretation of dilations 48 4- 4 . Other approaches 49 PART II . 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 for the case L=0 61 5-5. Eigenfunctions of Ji for the case L#) 63 5-6. The po 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 for eigenfunctions of 92 PART I H . ACTION PRINCIPLES Synopsis of Part III 100 ?• Rotationally Invariant Action Principles 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 action principle 111 7- 5* Canonical (anti-)commutation relations 114 8. Special Cases 8- 1 . Scalar f i e l d 118 8-2. Green's function for scalar f i e l d equation •• 128 8-3. Spinor f i e l d 132 8-4. Green's function for 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 Additional literature consulted 163 Appendix 1 170 Appendix 2 172 Appendix 3 • 1?4 Appendix 4 176 Appendix 5 • 181 v i L i s t of Figures Figure 1. Example of an angle preserving conformal mapping. 12 Figure 2. Examples of motions induced on the nu l l surface L=0 in a 3-dimensional space as a result of conformal transformations i n 1-dimensional space. 82 Acknowledgments The author i s indebted to Professor F.A. Kaempffer for advice and cr i t i c i s m . The financial assistance of the National Research Council of Canada i n the form of a Postgraduate Scholarship is gratefully acknowledged. 1 Introduction Within a few years after 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 for 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. After another twelve years had passed, Bessel-Hagen (1921) derived the additional five conservation laws that hold whenever Poincare covariant f i e l d equations are also covar-iant 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 revival of interest in the conformal group among physicists. In that work, Dirac showed how to write conformally covariant f i e l d equations i n manifestly covariant form by exploiting, i n effect, 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 space-li k e and two timelike dimensions, an isomorphism known already to Lie (1893) and appreciated by Pauli (1921) and Weyl (1923). Nevertheless, the study of conformal symmetry remained outside the mainstream of physical f i e l d theory, despite a plea by Schouten (1949), u n t i l recently, when interest i n possible wider physical implications of this symmetry was rekindled, tentatively at f i r s t , as i n the proposals of Yang (1947) and Finkelstein (1955)• but then with increasingly firm 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 , in fact, restricted 2 to massless f i e l d s . This belief has directed, accordingly, a substantial effort into exploring how conformal symmetry i s supposedly "broken" by the actual world of particles (see e.g. the review a r t i c l e by Carruthers 1971)* The present work i s aimed at enlarging the applicability of conformally invariant action principles to the description of massive f i e l d s . A possible avenue to this end i s the systematic study of rotationally invariant action functions i n a f l a t 6-dimensional space with four spacelike and two timelike dimensions. Although the exploration of this avenue involves enlarging the arena of physics to the entire 6-dimensional space with a l l six coordinates accorded f u l l independent status, such a radical approach i s not attempted cavalierly, 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 useful tool i n the treatment of scattering processes. For example, analytical expressions for Feynman dia-grams i n a 5-dimensional formalism have been developed i n recent work by Adler (1972, 1973)• and using this method the calculation of vacuum-polari-zation diagrams i n the 5-dimensional formulation can be put i n a manifestly infrared-finite form. The question of how to incorporate mass into conformally covariant f i e l d theories has periodically attracted some attention, and the concept of a "conformally covariant mass" was introduced i n an effort to render equations with mass formally covariant under the dilations and under the special con-formal transformations which extend the Poincare group to non-linear trans-formations. Such research was summarized by Fulton e t . a l . (1962a), and more recently this theme has been taken up by Barut and Haugen (1972, 1973)• who try to build a covariant character into the mass by employing the 6-dimen-sional 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 arise with both of these related approaches, and some of these problems w i l l be pointed out i n this report, A more illuminating approach i s that favoured by Murai (1958), whose work 3 has been expanded upon by both Ingraham (1960,1971) and Castell (I966). These authors look upon the mass as a property of solutions of f i e l d equa-tions i n 6-space, which when projected into a suitably 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 investigation into and extension of this ap-proach, 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 principles i n 6-dimensional space. As a preliminary step, the conformal group of transformations on Minkow-ski space i s introduced i n Section 1, and the various possible interpreta-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 confor-mal group i s reviewed, some later 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 di 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 restrictions on scattering 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 particles of spin 0 , i , and 1 are indeed con-formally covariant. For a massless spin-2 particle, the d i f f i c u l t i e s 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 Einstein equations. As well, Bracken's recent work on the non-independence of conformal generators at the single-particle level i s considered (Bracken 1973)• At this point begins the original contribution contained i n this report. A demonstration i s given that Bracken's results are, in fact, restricted to the realm of single-part ic le quantum mechanics, and cannot be extended to quantum f i e l d theory. This resul t has the significance that the generators of conformal transformations in quantum f i e l d theory are a l l independent, i n contradistinction to the s ingle-par t ic le case. In section 4, various attempts at including mass in conformally covariant equations without departing from 4-dimensional Minkowski space are examined, and in part icular the idea of the "conformally covariant mass" m is treated. This concept i s introduced when di lat ions are interpreted as uniform changes i n units of length (Barut and Haugen 1972), and i t i s shown here that such an interpretation cannot be maintained i n Lagrangian f i e l d theory, and that a conserved d i la t ion current cannot be constructed for 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 for massless part ic les by considering ro ta -t iona l ly covariant equations on the n u l l hyperquadric in 6-dimensional space. The extension of the 6-dimensional formalism off the n u l l hyper-quadric i n 6-dimensional space i s presented i n Section 5* In th is Section, i t i s shown that one can s t i l l obtain conformally covariant f i e lds from the 6-dimensional formalism, but that the conformal group must be extended to a subgroup of the conformal transformations in 5-dimensional space to achieve a consistent formulation. I t 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 auto-matically brings into play the spin matrices associated with rotations in a 7-dimensional space. This i s due to the fact that the 5 -din»n-sional conformal group i s isomorphic to rotations in a 7-dimensional space. Use of these spin matrices leads to a new coordinate-dependent 5 "descant operator" in 6-dimensional space which serves to eliminate ex p l i c i t coordinate dependence in the f i e l d equations when they are written i n terms of Minkowski space variables. Employing this original procedure, i t i s shown how one can meaningfully ascribe a specific 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 insisting that physical fields be eigenfunctions of the new scale dimension operator one can i n principle incorporate the mass into conformally covariant f i e l d equations. In effect, then, this work forms a bridge between the work of Murai, Ingraham, and Castell, and that of Wyler (1968), who has worked with the 5-dimensional conformal group. This Section i s concluded by the development of the 6-dimensional formalism necessary for treating spinor and vector f i e l d s from this new point of view. Before embarking on the task of treating particles with non-vanishing spin, i t i s instructive to develop a formally simpler model for elucidating the formal structure of the method developed i n Section 5* To this end, in Section 6 the corresponding rotationally invariant theory i s developed for f i e l d s i n a 4-dimensional space with one time axis and three space axes. In this 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 fact that the Lorentz group i s isomorphic to the conformal transformation group in a 2-dimensional Euclidean space. It i s found that one can indeed write down a 4-rotation covariant spinor equation, and solutions for this equation are constructed e x p l i c i t l y . I t i s found that the proposal made in 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 in terms of the 3-dimensional 6 conformal group, one can find solutions which can be ascribed a "mass" in 2-dimensional space* With these results, i t i s found opportune to obtain i n Section 7 a systematic method for the construction of covariant f i e l d equations i n 6-dimensional space, which flow from an action principle which i s not bu 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 rotational and not translational invariance of such an action integral. It i s found that by replacing Gauss' theorem with a type of Stokes' theorem, one can indeed find a different type of Euler-Lagrange equation which yields rotationally but not translationally covar-iant f i e l d equations, and which results from variation of the action integral. By considering the variation of the action under infinitesimal rotations i n 6-dimensional space, an angular momentum tensor i s constructed which gives rise to fif t e e n d i s t i n c t conservation laws. When variations of the boundary are taken into account, these conservation laws can be recovered directly from the action integral by means of a Schwinger action principle for quantum f i e l d s , with the modification that the Lagrangian density i s allowed to depend e x p l i c i t l y on the coordinates themselves. Finally, since the quantum f i e l d theory generator of an arbitrary f i e l d transformation i s thus determined, covariant (anti-)commutation relations result as a self-consistency condition, and by choosing the time as a quantization direction one arrives at equal-time commutation relations• Applications of these ideas to specific f i e l d s are treated in Section 8. F i r s t l y , the simplest case of the scalar f i e l d is studied, and i t i s found that one can extract from the 6-dimensional formalism scalar or pseudoscalar fi e l d s which satisfy i n Minkowski space the usual Klein-Gordon equation with mass. By a slight change of f i e l d variable, i t i s 7 also possible to demonstrate that the f i e l d equation sa t is f ied by the scalar f i e l d can be written as that sat is f ied by a massless f i e l d in 5-dimensional space, and so the conservation laws in Minkowski space which arise from invariance of the action integral can be written simply as the vanishing of the divergences of the four types of current associated with the conformal group in 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 scalar f i e l d s , and the derivative of the f i e l d with respect to the time obeys an equal-time commutation relat ion with the f i e l d i t s e l f . Next, results by Mack and Todorov (1973) for the Green's function for the scalar f i e l d are generalized from the special 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 in a space of any dimensionality for a rotat ional ly covariant f i e l d equation is dependent on the value of a Casimir operator for the group, and that this value can be specif ied uniquely by requiring that the Green's function have a simple form 0 The additional case of 5-dimensional Euclidean space is examined and the results obtained in this case are in agreement with those of Adler (1972). The spinor free f i e l d equation in 6-dimensional space is deter-mined by considering the restr ic t ions imposed on possible candidates for the Lagrangian density by the form of the action principle of Section 7. The result ing equation is compared with a di f ferent spinor equation advanced by Barut and Haugen (1973)• and i t is shown that this other equation cannot be derived from a hermitian Lagrangian density. The or ig ina l spinor equation i s then used to show how one can develop f i e l d equations for massive fermions using the rotat ional ly invariant formalism, and that in principle such an equation can accommodate precisely two 8 specie3 of particle having two different masses. Constructing the angular momentum tensor i n 6-dimensional space, fifteen conservation laws result which can again be written in terms of the conserved currents associated with the conformal symmetry of massless particles i n 5-dimensional space. Choosing the time axis as the direction of quantization, one finds that the spinor f i e l d obeys equal-time anticommutation relations with the adjoint spinor f i e l d . The spinor equation i s then iterated to the form of the scalar wave equation, and using this iteration a Green's function for 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), for the special case of the n u l l hyperquadric, are overcome by the use of the propagator developed here. Finally, 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 equation is obtained. It i s found that the gauge group associated with this 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 for a massive vector boson formally gauge invariant by the introduction of an additional scalar f i e l d . From the analysis of the vector f i e l d , i t i s found that this extra f i e l d component arises i n a natural way i n the rotationally invariant theory, and that the properties of the Stueckelberg formalism are reproduced automatically. However, i t i s found that as well 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 also obtains the additional 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 ield 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 is 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. 1 0 PART I. CONFORMALLY COVARIANT FIELDS IN MINKOWSKI SPACE 1 . The Conformal Group i n Minkowski Space 1 -1 . Definition 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' k(y) . where i n a l l instances raising and lowering of indices i s accomplished by means of a metric S j k describing a f l a t space, (1.2) y k = SkjyJ . y» k = ^jV't, £kj = diag(-1 ,-1 ,-1 ,+1 ) . These mappings are defined by the requirement that the line element (1.3) ds 2(y) = SJkdyJdyk at the point y i s transported by (1.1) into another line element (1.4) ds 2(y«) = SjkdyOdy«k which di f f e r s from the original line element by not more than a spacetime dependent factor, (1.5) ds 2(y«) = e 2 3 ^ ) ds 2(y) . Since the transformation (1.1) yields the relation (1.6) dy' k = Oy'k/*yJ)dy.5 , i t follows that the requirement (1.5) i s met provided (1 .7) SjktV'VWOy^Ay") = «2 $mn • Transformations satisfying this constraint are called "conformal" because the cosine of an infinitesimal "angle" (1.8) cos 9 1 2 » ( d y i ) k ( d y 2 ) k ( d y j 2 d y 2 2 ) - * i s l e f t invariant. The constraint (1 .7) uniquely determines the set of mappings (1.1). Even so, i f i t i s kept i n mind that the same metric i s used for 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 wi l l lead to the same condition (1.7). First ly , 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. This situation is illustrated in Figure 1. This approach is 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 is left invariant. The mathematical advantage of (1.5) is that the unnecessary use of curvilinear coordinate systems is avoided. Secondly, one may again define conformal transformations by (1.5), but assume that Sj^  transforms as a tensor density of weight under changes of the coordinates (Fulton e t .a l . 1962a). Since the determinant of the metric tensor transforms as a scalar density of weight 2, the Jacobian of the transformation (1.1) is hy/dy* | =e"^ r(y). This means that Sjk is left unchanged, but the line element is no longer invariant under coordinate transformations. Alternatively, this type of transformation may also be looked upon as a coordinate transformation which is accompanied by a varia-tion of the metric alone that serves to map the f lat 4-space with curvilinear coordinate system y'^ onto a f lat 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 » dy 'J = e ? (y) dyj , S j k • G ' j k (1.9b) ds 2 » d s ' 2 = G'jkdy'Jdy' 1 1 = e 2 ? (y ) G ^ y ^ y * then in order that ds 2 transform under a change of units as the square of a length, (1.10) ds 2 » d s ' 2 = e 2 ? ( y ) ds 2 , one must have (1.11) ^jk * G'jk S S j k . It should be kept in mind, however, that since $ has the units of momentum 1 2 > 2± Figure 1. Example of an angle preserving conformal mapping. The vectors ds 1 j and ds*2 d i f f e r from ds^ and ds^ in length only by the common scale factor exp(<r). The angle formed between ds/j and d s ^ i s ^ e same as the angle formed between ds_j and dsp. 13 multiplied by length, then changing the units of time and length by the same factor necessarily entails a change i n the numerical value assigned to For any of these interpretations, (1.5) constitutes a generalization 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 this choice w i l l be presented in Section 4. 1-2. Infinitesimal conformal transformations To determine those conformal transformations that are continuously con-nected with the identity, i t i s sufficient to consider infinitesimal transformations (1.12a) y' k = y k+ Sy k(y) (1.12b) e2 ? ( y ) = 1+2 S*(y) . The constraint (1.?) reads then* (1.13) &jk( « J B + SyJ.mX S V $yk.n) = (1+2?) Smn . so that (1.14) Vn.m + $y m»n = z * Smn • Taking the trace, v i s found to be (1 . 1 5 ) ? = i $ y » n , and operating on (1.14) with ^ k one obtains (1.16) U §yn.ink + $y m .kn) = Smn ? ' k • By antisymmetrizing i n m and k, (1.17) K Sy m.k " Wk'm^'n = Smn ^ 'k " $kn ^ ' n and operating on this result with b j , (1.18) £( 8 y m . k - S y k . m ) . n j = I mn^'lk " Skn^'jm • one can find a condition on cr by antisymmetrization i n n and j , "For partial differentiations any of the notations ^ kX=X,i c =^XAy k w i l l be used when convenient. The product xkyk=x^y4-x*y. i s denoted by xy, with x k x k denoted by x 2. Also, ]/L and c are set equal to unity unless stated otherwise. 14 0 J9) m^n ^  jk" ^ kn ?-jm" Sjra ?»kn + $jk ?»mn = 0 * By specifying various indices, one can establish that this equation requires (1.20) ^.mn = 0 • Thus & can be written i n the form (1.21) «=• = v +2/? ky k where are five infinitesimal parameters. Substituting into (1.14) yields (1 .22) *y n, m+ \ym,n = 2(<r +2 f k y k ) . These d i f f e r e n t i a l equations for may be integrated to yield, with ten more constants of integration <xk and fc^k=- c k J , (1 .23) Sy k = * k+ f kjyJ+(cr+2 ^ y j ) y k - f k y j y j . The infinitesimal transformations are, then, (1.24a) TRANSLATIONS: Sy k • <*k (1.24b) HOMOGENEOUS LORENTZ TRANSFORMATIONS: ^ y k = e ^ y j (gj*s_£kj) (1,24c) DILATIONS: Sy k = <*-yk (1.24d) SPECIAL CONFORMAL TRANSFORMATIONS: Sy k = 2y k( p y)- ^ y 2 . The number of parameters associated with each of these types of transforma-tion i s 4,6,1,4 respectively, and each type forms a subgroup within the 15 parameter group of variations (1 . 2 3 ) . 1 -3 . Finite conformal transformations The 15 parameter group of f i n i t e conformal transformations that are continuously connected with the identity consists of (1 .25a) TRANSLATIONS: y«k = yk+ ak (1 .25b) HOMOGENEOUS LORENTZ TRANSFORMATIONS: y , k = R kjyJ (1 .25c) DILATIONS: y , k = s y k (1 .25d) SPECIAL CONFORMAL TRANSFORMATIONS: y' k=(y k-b ky 2)/(1 -2by+b2y2) The infinitesimal transformations (1.24) are recovered by writing (1 .26) a k = c t k ; RJj = %^+ ^ ; s=1 + <r ; b k= p k . The transformations inverse to (1.25) are formed using the parameters - a k , -R^j» -s, and -b^. For example, the inverse of (1.25d) i s (1.27) y ^ y ^ + b V 2 ) [l+2(by')+b2(y')2]"1 where, i n keeping with (1.2), the inner products i n (1.27) involving the trans-formed coordinates y' are formed by using the Minkowski metric S j k . The proof of (1.27) follows from the ident i t ies (1.28a) [l+2(by')+b2(y')2]-1 = 1-2(by)+b2y2 (1.28b) y« k +b k (y«) 2 = y * [^(by^y 2]" 1 . The group of special conformal transformations can readi ly be shown to cor-respond to spacetime dependent d i la t ions , since from (1,25d) one finds that the square of the interval between two points y and x i s transformed into (1.29) (y , -x») 2 =^ k (yO-xO)(y ' k -x» k )= ilzlil . [l -2(by)+b2y2] [1 -2(bxj+^x2] Simi lar ly , the inf in i tes imal interval ds formed from the difference dy in the components of the two points y and y+dy, becomes (1.30) ds '^Sijdy^dy'J=ds 2 [l-2(by)+b2y2]"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 trajectories are transformed under special 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' J/dy k)u k(ds/ds')= U j [l-2(by)-t-b2y2]+bJ [4(uy)(by)-2(uy)-2y2(bu)] +yJ [2(bu)-2b2(uy)l 1-2(by)+b2y2 where use has been made of (1.30). Factoring the quantity u^ out of u=UjjV, the tangent vector can be written compactly by defining a quantity such that (1.32) u ' j =u 4 V j . From (1.32) one obtains 16 (1.33) 1-v ' 2 =(1-v?)/(V | f) 2 . Hence the transformation (1.25d) preserves the property that the tra,)ectory has a speed less than, equal to, or greater than the speed of l i g h t , even though from (1.29) one can see that the timelike or spacelike character of the i n t e r v a l between two points i s not necessarily preserved, under a f i n i t e s p e c i a l conformal transformation. This i s possible because i f the sign of ( y - x ) 2 changes, the special conformal transformation must become singular at a point y 0 on the trajectory between y and x; points on either side of y 0 are transformed to i n f i n i t y i n opposite di r e c t i o n s , and the transformed t r a -jectory remains always timelike ( l i g h t l i k e , spacelike), i f the o r i g i n a l t r a j e c t o r y i s timelike ( l i g h t l i k e , spacelike), 1-4, Isomorphism with rotations i n 6-dimensional space The conformal transformations are isomorphic to the rotations i n a 6-dimenT s i o n a l space with metric S^g=diag(-1,-1,-1,+1,-1,+1). This can be seen as follows. Introduce new, diraensionless, coordinates (1.3*0 r f a y * where H i s invariant under translations and homogeneous Lorentz transformations, but transforms under d i l a t i o n s as (1.35) and under special conformal transformations as (1.36) K' = [ l - 2 ( b y ) + b 2 y 2 ] * . I f one introduces further the redundant variable (1.37) then one has the constraint (1.38) \ 0 . In terms of the coordinates >^  = (T» 1 » 1 )» w h e r e (1.39) x ^ - l 3 ? >=-(V-f) so that ^5=-i(H+^); V=i(*-A) t h i s can be written (1.40) SABW *0 . 7 A Now the transformations of the are linear, and the transformation laws corresponding to (1.25) ar© (1.41a) ^' k=rf+a kx; *»=*; x'=2(avj)+a2H+X (1.41b) ^ , k = R k j ^ ; *'=A (1.41c) 1»k=*f; A ^ S - V ; X'=s> (1.41d) ^ i k = ^ . b k A ; K'=-2(bv()+X+b2A; A'= X for translations, homogeneous Lorentz transformations, dilations, and special conformal transformations, respectively. These transformations leave the quadratic form (1.40) invariant, and they are thus properly characterized as rotations i n the space spanned by the coordinates vr\ For infinitesimal transformations one can write (1.41) (1.42a)^k=€kjvvJ-(°<k-pk)yl5+(0<k+(Sk^6 (1.42b) k5M«VrV»lk+<nl6 (1.42c) W6-(0'k*fk>vlk+o' ,l5 or, written more compactly, (1.43) K A = EV B with (1.44) E a V b=-E b a; E a 5=/^ a=.E 5 a; E a 6 ^ V = - E 6 a ; E 5 6 = o ^ E 6 5 . Equation(1.43) i s obviously the 6-dimensional analogue of the rotations (1.24b). Since the transformations (1.41) are orthogonal, they leave invariant the metric i n 6-dimensional space, (1.45) l ' A = W B . so that i f one defines the covariant components of y' as (1.46) y , jn , j / H , = S j A y l , V = ^ k / K » = g j k y ' k , then the Minkowski metric i s also l e f t invariant. Defining the line element (1.47) dS 2 ^n A d"lA=d\ dn.k+d('|6+'|5) d(^ 6-^ 5) then with the restriction (1.40) one finds (1.48) dS2=v£ dy k dy k=v£ ds 2 . 2 1 8 Since dS i s invariant under the rotations (1.41), this relation shows that 2 2 ds transforms under conformal transformations as H . From (1.41), i t follows 2 2 that ds is multiplied by s under dilations, and under special conformal transformations i t s transformation law is given by (1 .30). Among the transformations that cannot evolve continuously from the identity the "transformation by reciprocal r a d i i " (or "reciprocation") defined as (1.49a) y ' ^ y ^ y 2 i s of particular interest. Since i t has the consequence (1.49b) y ' S r - 2 i t amounts, according to (1 .30), to an interchange of H a n d \, (1.49c) K'=-X; X'=-K k 6 so that the coordinates ^  and remain unchanged, and only the coordinate ^ i s inverted, (1.49d) ^k=*i; ^ .5= . ^ . n , 6 = n 6 ^ k V (Alternatively, the transformation (1.49) can be represented by =-^K; Any special conformal transformation can be decomposed into a recipro-cation, a translation, and a reciprocation. Indeed, i f one carries out on the reciprocated coordinates (1.50) y" k=-y k/y 2 k k k a translation y -* y +a one obtains (1 .51) y" k=-(y k+a k)/[y 2 +2(ay) +a 2] k k / 2 so that upon repeated reciprocation y «^  -y /y one has (1.52) y ^ K ^ - a V ) / [l-2(ay) +aY] which i s identical with the special conformal transformation (1.25d) i f a k=b k. 19 2. Conformally Covariant Fields 2-1. Definition of generators A f i e l d 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 2)=T(c 3) implies (2.3b) S(c t) S(c 2)=S(c 3) . Defining the generator T of this operation by (2.4) S ( c ) = e i c P , one has for infinitesimal transformations (2.5) S(c)=I+icr . Introducing the operator d(c) by (2.6) d(c)Y(y)=Y(T-1y) and defining the generator g by (2.7) d(c)=e i c g , then, for infinitesimal transformations, one has (2.8) d(c)=1+icg . The entire variation of ^  can therefore be represented by an operator D(c), (2.9) T(y)=D(c)Y(y)=d(c) s(c)Y(y) . corresponding to a generator g, (2.10) D(c)=e i c* . Then for infinitesimal transformations, (2.11) D(c)=I+icg=(1+icg)(i+icr)=I+ic(g+r) , and hence (2.12) i=g+r . Thus the variation of the f i e l d may be written as (2 .13) S T '=y«(y).r(y)=icIr(y)=ic(g +r)H ;(y) . The quantum f i e l d theory unitary operator U(c) corresponding to D(c) is defined such that applied to f i e l d operators H^(y)t (2.14) u- 1(c) Y(y) U(C)=D(C) V(y) . Then defining the generator G by (2 .15) U(c)=eicC- § and considering infinitesimal 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 for the operators d(c) associated with the con-formal group, one defines operators p^, mj^s-m^j, (p , kj such that (2 .17a) d(a k)=e i( aP) (2.17b) d(wJ k)=e - ( i / 2 ) w J k m J k (2 .17c) d(s)=e i( l n s> (2.1?d) d(b^)=e i( b k) . Using the infinitesimal parameters (1.26), these operators become (2.18a) d(o<.k)=1+iockpk (2.18b) d(^ k)=1 -(i/2)6*M j k (2.18c) d(«r)=1+i«r- 9 (2.18d) d(jJJ)«1+i^kj . Carrying out the infinitesimal transformations T'^y in the argument of the right hand side of (2 . 6 ) , one finds (2 .19a) d ( * k ) Y ( y ) = [ l - * \ ] Y ( y ) (2.19b) d(gJ k) H J ( y ) = [ i H € J k ( y j \ - y k o j ) ] H'(y) (2 .19c) d(<r)H ;(y)=[l-o'y kb k] vf(y) (2 .19d) d(f J) v V ( y ) = [ l - r J ( 2 y ; j y k i k - y 2 ^ ) ] *f(y) . Comparing (2.19) with the definitions (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\ C l (2.20b) m j k = i ( y j i k - y k ^ ) (2.20c) V = i y \ (2.20d) kj = i C Z y ^ j j - y 2 ^ ) = ^ j f - i y 2 ^ • Th© f i rst two generators are recognized as the usual differential forms of the momentum and angular momentum, while the generator of dilations is called the "vir ia l" . Since the special conformal transformations wi l l be shown to form a subgroup which is isomorphic to the translation group in 4-space, the generator kj may be called the "complementum" (Kaempffer 1971). Indeed, under the reciprocation (1 .49 ) , the momentum pj is transformed into the differential generator ky To see this, consider the inverse of the reciprocation, (2.21) y k=-y' k/(y') 2 . Under reciprocation the momentum becomes (2.22) P j * p « j = i W J = i O y ^ y ' ^ / a y 1 1 = i ( 2 y jy na n-y 2^) = kj e 2 - 3 . Algebra of generators The algebra of the self-adjoint generators G=(Pk,Mjk,<|>,K^) defined by (2 .15) . can be inferred from the commutation relations of the differential generators (2 .20 ) , with the result [ P j . P j - 0 ; [M j k ,P n ]= i (S n k P r ^ n j P k ) ; [<Mk]=-iPk ; [Kj.Kj-0 : (2.23) [ K j . p k ] = - 2 i ^ j k * + M j k ) 1 [ V ' ^ ^ ( $ k m V V M ^ k n M j m - S j m M k n ) ; [Mjk] = 0 i h . « J = i ( ^ m V W M - o » frj-fl-aj From this algebra, i t can be seen that the group of special conformal trans-formations 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 lat 6-dimensional space in which an invariant interval is 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 = $ ; = * < ; J6m = ^ V K m ) • the commutation relations (2.23) take then the standard form for rotations (2.25) [ J K L . J M N ] = ^ L M J K N + ^KN^LM" ^LN JK>T ^ K I V L N ^ • From the commutation re la t ion [ $ » P k ] = - i P k follows by i te ra t ion (2.26) [({j.P2] =-2ip2 or [ A . P 2 ] ^ ? 2 with / \ =i(j) . This implies that the only discrete eigenvalue of P 2 i s zero, and that the continuous spectrum of P 2 cannot start at any f in i te value (Wess 1Q60). Thus, exact d i l a t iona l symmetry means that the mass spectrum is 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 ind the generators G defined i n (2.15), one must f ind operators sat isfying the commutation relations (2.23) and having the addit ional properties (2.16), e x p l i c i t l y (2.27a) [Y(y).P k] = P k Y (y ) (2.27b) [Wy).*^] =^ k v V(y) (2.27c) [f(y).$] = y ^ y ) (2.27d) [ ^ y ) , ^ ] = k ^ ( y ) where now the generators g=(p k » m j k « f% k^), defined as in (2.12), contain both the external and the in t r ins ic attributes of the f i e l d that generate the respective conformal transformations. These generators can be found by the method of induced representations, using a representation of the " l i t t l e group" which leaves the point y=0 f ixed to determine representations of the conformal group. The l i t t l e group i s generated by the f inite-dimen-sional (" intr insic") attributes of the f i e l d <r^,-±Jt H ,^ contributing to the generators mj k . If, k j . These matrices are defined by employing (2.27) at y=0. (2.28b) [?((>).Mjk] = <T. kf(0) (2.28c) [H>(0),4)] = -i£ Y(0) (2.28d) [W0).K.j] * K. <f(0) . The ^ k are the spin matrices, and & and Hj are called the "scale dimension" and the "in t r i n s i c complementum". They are defined similarly i n any represen-tation; e.g. for (2.20) one has (2.28c •) [Vj.p] = ( - i ) ( - ^ j ) so that ^=-1 for the differentiation operator. In (2.28b tc,d), one assumes that momentum cannot exist i n in t r i n s i c form, so that the basis i n the index space of *f can always be chosen such that (2.29) VK, =P k v L - i V k S i . Now the entire generator g can be found by using the translation property (2.30) Y(y) = e ^ ^ ) Y(0) e-^Py) and the identity (2.31) fr(y>.G] - e ^ P y ) [^(O).G(-y)] e-i(Py) where G(-y) i s defined by (2.32) G(-y) = B-X*y)Q,H*y) = G-i[(Py),G]-i[(Py),[(Py),G]]+.... On account of the commutation relations (2 .23), one finds (2.33a) P k(-y) = P k (2.33b) M J k(-y) = M j k - i [ P ^ M ^ ] = M j k +(y . P ^ j ) (2.33c) <|)(-y) = H f P ^ . f l = *+y"Pn (2.33d) Kj(-y) = K . - i ^ . K ^ - i [P ny n. [ P ^ . K ^ ^ j ^ ^ y ^ ^ y ^ P y - y ^ j so that, using the definitions (2.28) (2.34a) [T(0),P k(-y)] = [f ( 0).P k] (2.34b) [Y(0).M.k(-yj] = ^ ^ ( 0 ) + [ f(°).(y/ k-y k pj)] (2.34c) [H»(0).$(-y)] = - i i ^ ( 0 ) + [ r (0),y kP k] (2.34d) fV(0),Kk(-y)] = ( K j - a i / y j ^ ^ ^ O ) * [^(0) ,(2y . y ^ - y ^ ) ] . Upon substitution, from (2.31) one obtains (c.f. Mack and Salam 19^9) 24 (2.35a) p k = 1 \ = p k (2.35b) m.k = <yi( y jvy kV = ^  +mjk (2 .35c) y = -ii+iy\ = -ii+ <f (2.35d) ^ - H r 2 i £ y . +2yV. n +i ( 2 y/a k-y 2>.) = V^V^'V** * By setting y=0 in these expressions, i t can be seen that the generators of the l i t t l e group <Tjk, - i i , Kj also satisfy the defining algebra (2 .23 ) . 2 - 5 . Field variations The variation (2.13) of a f i e l d Y due to the most general infinitesimal conformal transformation may now be written (2.36) = (i^ k? k-(i / 2)€J k Sj k+i<r7+ip^kj) with the generators g as given i n (2.35)• The cla s s i c a l transformation law (2.2) takes the form (2.37) r ( y ' ) = [ i-(i/2 )e j ko- J k+(<r+2pJyj^ . A f i e l d i s said to transform as a scalar, vector, tensor, spinor under homogeneous Lorentz transformations i f the spin operators are represented by (2.38a) <Tj k = 0 for scalar Y (2.38b) ( <Tj k ) m r = i(S j BS k p-&j P£ k B) ^ r vector ^ r a (2 .38c) ( 0 - j k r r s = S m r ( «>)V^kO"s for tensor y » (2.38d) ( T j k = (i/4) [ ^ j . y j for spinor Y where the ^ k are the 4x4 Dirac matrices for 4-component spinors Y . Consider now fields Y satisfying Euler-Lagrange equations arising from a Lagrangian density £ (Y,^ k Y). Defining the conjugate momentum T* as the fourth component of the vector (2.39) ¥ k = > 2 / > k Y . the f i e l d equations can be written (2.40) ^ = u / ^ r . Then i f the canonical equal-time commutation (anticoramutation) relations (2.41) | / ( x . t ) , f(y..t ) l . = ; i 5(x-yJ are satisfied, for boson (fermion) f i e l d s , the quantum f i e l d theory generators 25 G, defined in (2.16), take the form (2.42a) P k = t k \ y ) djr (2.42b) M j k = yj_t S.k\y) d Z (2.42c) () = o\y) dy_ (2.42d) K j = y f = t K \ y ) dy. with (2.43a) t k " = T I ^ Y - J k n X (2.43b) S j k « - - i i r n ^ + 7 ^ - 7 ^ . " (2.43c) D n = -TT lS ' + y \ n y (2.43d) K j n = ( e y ^ K ^ t ^ i r ^ J y j ^ ^ S/^+if , f i e lds Here, i t i s understood that a l l independently varied f i e lds are to be summed over, and that the Ji and H. for are the matrices given by (2.28c,d). 3 [c . f . Mack and Salam (1969), who res t r i c t their attention to f i e lds with vanishing K - i n the above currents.] Even though the operators (2.42) generate the f i e l d variations (2.36) whether or not the var iat ion 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 this i s the case do the generators (2.42) sat isfy the defining 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 ind a second rank symmetric tensor u such that (2.44) T T n i ^ - i i r k ( r k n Y = * k u k n ' If this condition i s s a t i s f i e d , upon subtraction of a term whose divergence vanishes ident ica l ly the d i l a t i o n current (2.43c) can bo written in the form (Callan et . a l . 1970) 26 (2.45) D" = y* ekn where the energy-momentum tensor G . k differs from t . k by a term with J J zero divergence, and satisfies (2.46a) e . n . n =0 ; 0 j n = & n . , and also satisfies (2.46b) enn = 0 provided the dil a t i o n current (2.45) i s divergenceless. When the condition (2.44) holds, ©-.k takes the form u (2.47) eJk = T J k + i x ^ , ^ where T^ k i s the symmetrized energy-momentum tensor (Belinfante 1939) (2.48) T J k = T k J = t0k+(i/2)Jm(--nO y-kmv+irm ^ j k v j j ^ ^ m y j and (2.49) X^1"" = - $ ^ J k + $ r a J u n k n r a k u ^ . Callan et. 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 well as sufficient, and the complementum current can be written (2.50) K.n = (2y jy k-y 2S ; j k)e k n . Therefore, the above procedure for finding a traceless energy-momentum tensor can certainly be carried out for massless fi 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 for suitable scale dimension. 2 -6 . Dilation invariance Conservation laws for the currents (2.43) arise from considering the transformation behaviour of the action integral (2.51) I = J / ( V , > k y ) d V under inf in i tes imal conformal transformations. The derivation of the conservation laws for momentum and angular momentum is well known (e .g . Schweber 1961), and these conservation laws take the form (2.52) t k n , n = 0 ; S . k n . n = 0 where t^" and S ^ k n are given by (2.43). Simi lar ly , one can derive the conservation law for the d i l a t ion 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 inf in i tes imal d i lat ions s=1+**, (2.53) S T * = ry* the transformation law (2.37) i s , with the matrix £ defined as in (2.28c), (2.54) vy.(y.) = ( 1 + c r i ) S » ( y ) , so that the variat ion of the f i e l d under pure di lat ions characterized by the inf in i tesimal parameter o~* i s (2.55) ^=<r(^-y\)Y . This terminology amounts to defining the scale dimension of the coordin-ates y k as =+1, on account of the transformation law (2.56) d y » k = s d y k = (1 + <r) dy 1 4 , and enables one to determine the behaviour under di lat ions of other geometrical objects i n terms of that basic scale dimension. Now, i f the integral (2.51) i s to be invariant under d i l a t i o n s , the var iat ion of X. must be a divergence, (2.57) IX = -}k(<ryk.O = <r ( - 4 - ^ ) / i . e . (2.58) ( U / o < r ) = ( - 4 - ^ ) 7 1 . Comparison with (2.55) shows that i f one can meaningfully ascribe a dimen-sion Jl^ to the Lagrangian density X at a l l , then d i l a t iona l invariance, for the case of constant a* , i s guaranteed i f the dimension of i s X^ =-4. (In any event, X/ i s necessarily a c-number since X i s a scalar . ) 28 When this condition is obeyed, one can obtain an expression for the conserved current due to dilational invariance i n the usual fashion by considering cr to be coordinate dependent, so that the dilation has the effects (2.59) Y ^ r ^ ? »ith Y=U-y\)V (2.60) - N n Y * \vr,+OatrOcP+<r*B9 . Then (2.61) UX/^cr) = O W ) K \ ? and (2.62) |S*/>GB<r)] =-rrn9 =vna-y\)f where TT" is defined as in (2,39). Differentiation of (2.62) gives (2.63) ^nf^AOn*')] = O n ^ + A i ? • On account of the f i e l d equations (2.40) one can write (2.63), using (2.61), (2.64) \ D * A a n c r J l = Now, i f one insists on eq. (2.58) as the expression of di 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 for the canonical dilation current vector (2.66) D* = - T V + y k ( v \ f -£KNX ) = -TTW W where t k n i s the canonical energy-momentum tensor (2.43a). The f i r s t term on the right 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 dilation current" vector. In terms of the symmetrized tensor (2.48) the vector (2.66) may be written (2.6?) D ^ - ^ i t ^V-< i / 2 >^m<- T k^ M ? + ^ < r k n 1'-Tr n t f* k" Y ) = =-TTW + y k T k n + i l ^ k > - ( i / 2 n m [ y K ( - F k a > n r a + 1 r m a - k n - T n . The last terra i n this equation is the divergence of a tensor antisymmetric i n (n,m) and can therefore be discarded without affecting the conservation law (2.65) which remains valid i f one defines the dilation current as (2.68) D'n = y \ n - ^ i f + i i r k < r - » f . 29 Aiming at the form (2.45) for the d i la t ion current, introduce now the tensor through (2.47) and write (2.69) V*=^n-hfaiy\»Sm^ + i i f V k n f . The second term on the r ight hand side of (2.69) can be dropped without affecting the conservation of the redef ined d i la t ion current (2.70) D* = y ^ ^ U / ^ + X ^ - i r ^ Y + i ^ < r k n 7 because a simple calculat ion shows that (2.70 ( y k x k n > ) . n > ^ ( x k n ^ x k J - ) . n j m 4 [ y k ( g 3 n U k ' " - s ' ' % J ) ] . n J m = o . Since one has a lso , by def in i t ion (2.49), the identity (2.72) X / ^ X / * * ) = > k u * n the poss ib i l i t y of writing the d i l a t ion current in the form (2.45) i s contin- gent upon the poss ib i l i t y of wr i t ing, as follows from inspection of (2.70), eq.(2.44) (2.73) ir n^H ;-iTr ko- k nH ,= > k u k n • If this condition i s sat is f ied the d i l a t i o n current can be written in the form (2.45), where & k n i s constructed after the prescription (2.47) with (2.49). The conservation of (2.70) implies that & k n i s t raceless, (2.7*) e k * = T ^ * - ^ - T ^ - u ^ = 0 . In other words, whenever the condition (2.73) i s sa t i s f i ed , the trace of the symmetrized tensor (2.48) i s necessarily the divergence (2.75) T k k = >nOkukn> = ^ ( ^ ^ • i ^ ^ H ' ) or, equivalently, the trace of the canonical energy-momentum tensor is the divergence of the in t r ins ic d i la t ion current vector, (2.76) t k k = ^ n ( T T n ^ r ) . Nowhere in the foregoing have the elements of the scale dimension matrix been speci f ied , except in the statement (2.57) that the variat ion of M* must be such that ;£ has the scale dimension -^=-4. Particular matrices Jl arise from , 30 consideration of the momentum Tr (^y) canonically conjugate to M'(y), i . e . from consideration of a particular dynamics for the f i e l d M^ . From these con-siderations, one i s led to matrices which are multiples of the unit matrix, with the multiplier called the "canonical" scale dimension, also denoted byJ?. As an example, consider the neutral scalar massless f i e l d N-7, described by (2.77) 2 = H n n H ' . n S ' . n . The requirement of d i l a t i o n a l invariance of the action i s satisfied i f one ascribes to the f i e l d the canonical scale dimension =-1• To see how this value derives from the canonical momentum densities (2.78) TfJ 2 ^ / M V j Y ) = >dH> . assume that the canonical equal-time commutation relations are obeyed (2.79) [H'W.^YCy)]^^ = i S(x-jr) so that the operator (2.80) fot) = ^ { - M ^ f + r , ^ Y , k - i y ^ , m v f \ i n } dx satisfies the commutation relation (2.81) [M'(y).ftt)] = i(y\->W(y) where J i s as yet unspecified. Similarly, one arrives at (Cheung 1971) (2.82) [Ay),*(t)] =1(^+3^)^(7) . Now i f (j) i s independent of time, then taking d/dt of (2.81) one obtains (2.83) [Ay).*] = i(yk3k-^+1)TT4(y) so that comparing (2.83) and (2.82), one finds that the canonical value Jl=J\ i s necessary i n order that the canonical momentum is ~R^=KYIn fact, this requirement i s simply a statement of the covariance of the commutation relations (2.79) under dilations, since (2.84) S(sx-sjr) = s" 3 $(x-y_) so that, for the example above, (2.85) [*'(«), T T ' ^ s y ) ] ^ ^ = a 2 * " 1 [ T ( X ) , Ay)] implies X =-1 . Similarly, the canonical scale dimensions of noninteracting 31 sp in - j and spin-1 massless f i e lds are _3/2 and -1 repectively. Once one has ascertained the value of Ht the traceless energy-momentum tensor 9^ k can be constructed by employing the condition (2.73) to f ind the tensor u^ k ; e .g . for the Lagrangian density (2.77)• one can use (2.86) u k n = u n k = - I S ^ f * . That definite values of the scale dimension must be assigned to f ie lds H\ in order that the equal-time commutation relations be preserved under d i la t ions , does not exclude the poss ib i l i t y of di f ferent assign-ments of the scale dimension when these commutation relations are no longer necessarily obeyed, and when interact ion terms are added to Lagrangian densities describing free f ie lds (Wilson 1969). However, whenever the defining commutation relations 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 unit matrix ( i . e . each component of the f i e l d must have the same value of i.) for f ie lds which transform according to an irreducible representation of the Lorentz group. This follows, as a consequence of Schur's lemma, from the commutation relat ion (2.87) f<rjk.i] = 0 which must be sat is f ied 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 irreducible set of matrices, Z must act on y as a c-number (and reducible representa-tions of the Lorentz group can be completely decomposed, so that i n each irreducible subspace I i s a multiple of the unit matrix). Therefore, i t is important that conformally covariant f ie lds be assigned a def ini te scale dimension Ji i f they are to transform according to an irreducible representation of the Lorentz group. F ina l ly , this argument can be carried further to show that such f ie lds cannot possess an in t r ins ic compleraentum K'j. This result is arrived at by considering the commutation relat ion 32 (2.88) [ K.,i] = Kj which states that for a general representation of the conformal group Kj must be a singular matrix which can be put i n triangular form with zeroes in the diagonal (Mack and Salam 1969). Hence for an irreducible repre-sentation of the Lorentz group, with J- represented by a c-number, v4 . 3 must vanish identically. 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 situation i s comparable to trying to define the int r i n s i c parity of a right-hand c i r c u l a r l y polarized photon, which i s im-possible since the parity operator changes the polarization. To construct v i r i a l eigenstates, then, one must form superpositions of momentum eigenstates. As well, 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 of the v i r i a l . For this reason, consideration of the v i r i a l eigenstates does not appear to be the most direct means to understanding the physical meaning of the extra conservation laws associated with conformal symmetry. Nevertheless, this does not mean that these conservation laws are devoid of physical consequence. In fact, one can determine directly the r e s t r i c -tions placed on 2-particle scattering amplitudes by conformal symmetry, by looking at the action of products of 2-particle operators on amplitudes. Recently, Chan and Jones (1974a,b) have demonstrated that, i n scattering involving four particles of arbitrary 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 eigen-values — v i z . a particular indefinite metric in Hilbert space. 3 3 severe restr ict ions 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 ( 2 . 3 5 ) are written down for f i e lds with K .=0 and spin s related to the canonical scale dimension X by s= -^-1 ( i . e . scalars, spinors, and second-rank tensors, but not vectors) . Writing the massless quantum f i e l d M'(y) as the Fouxder transform of he l ic i ty \ =-s eigenfunctions w*(p_) multiplied by annihi lat ion and creation operators a * ( £ ) and b + \p_ ) , one can f ind the action of the generators on these operators. This determines another spin-s representa-t ion g" of the conformal group such that for any of the annihi lat ion and creation operators ( 2 . 8 9 ) . [a.G] = f a ; [p,a +] = g* * + . From these operators, i t i s straightforward to f ind the 2 - p a r t i c l e conformal operators. Let two part ic les with momenta p_^  and p_g scatter into two part ic les with momenta 2.y2^* a n ( * suppose that a l l the participants in the process are massless. Assuming that G167=0, then (2.90a) <1jvj.' a|4 E l.' f e) - ( S j ^ E ^ a J V f c ) " where now g ^ - * ^ ^ s a 2 - p a r t i c l e representation of the conformal group. From this equation, one can see that (2.90b) <?J+gJ) = (g 3+g 4) S x where S. i s the S-matrix. S minus the unit matrix contains a 4-dimensional delta function which ensures momentum conservation, and commuting the 2 -part ic le operators through this delta function one obtains their action on the T-matrix T^. As shown by Gross and Wess ( 1 9 7 0 ) , a l l the operators remain unchanged, except for the v i r i a l operator <f , to which an extra 4 i must be added. Then the condition ( 2 . 9 0 b ) places f i f teen d i f fe ren t ia l constraints on T . , and by going to the centre of momentum frame and choosing 34 the z-axis perpendicular to the scattering plane, Chan and Jones have shown that these constraints amount to three possible cases. These are: ( i ) ( X 1 + X 2 - X 3 - X 4 ) = 0 ; ( i i ) ( > 1 + X 2 - > 3 - X 4 ) / 0 and A ^ X g (or \ f \^)t yielding Tx=0; ( i i i ) ( X ,+Xg-X - X )^#> and X t = X £ = - \ y - X ^ , with ? x subject to a d i f fe ren t ia l constraint but not necessarily zero. In summary, then, an experimental consequence of conformal invariance in the exact symmetry case i s h e l i c i t y conservation, A^+A 2 = A^, except in the case ).^ = X£ = - A 3=- A 4. when the amplitude obeys a d i f f e ren t ia l constraint . 35 3. Conformally Covariant F ie ld Equations One can demonstrate e x p l i c i t l y that i n order for free f i e l d equations i n Minkowski space to be covariant under the special conformal transfor-mation subgroup, i t i s necessary that boson f i e lds possess a scale dimension ^=-1, while fermion f ie lds must have i=-3 / 2 . In this section, the ca lcu-la t ion i s carried out for the cases of scalar , spinor, and vector massless f ields,- and d i f f i c u l t i e s encountered with spin-2 f i e lds are pointed out. F i n a l l y , i t i s shown that while the conserved quantities associated with d i la t ions and special conformal transformations can be written in 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 Y i s governed by the Klein-Gordon equation (3.D ^ j S ' ( y ) = 0 , and transforms under inf in i tes imal conformal transformations as (3.2) W ( y ) = ( W ^ - U ^ ) 6J k m j k + icr^ - i ^k-j) t ( y ) . Equation (3.1) i s covariant under conformal transformations provided the f i e l d S"(y) , defined by (3.3) H"(y) = Y(y)+ SY(y) i s also a solution of the same equation, i . e . (3.4) ^ j f «(y) = 0 . I f this i s the case, then one also has that ( y m = > / ^ y , m ) (3.5) g , m n Vm y n r ( y ' ) = o with (3.6) 2 '™ s ( > , y ' m / o y J ) ( ^ y ' n / ^ y k ) g^k so that interpreting the conformal transformations (1.24) either as mappings between dif ferent points or as changes of the coordinate system one i s led to the same resul t . This can be shown for inf in i tes imal special conformal transformations, (3.7) * y k = ZykpJyj-pV . by calculating (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 in j5 , one has (3.9) g*m = \ m 0 ^ 7 ^ . so that (3.4) implies (3.5). I t i s evident that (3.1) is covariant under pure d i l a t ions , as i s also the case for translations and Lorentz transformations. For specia l conformal transformations, (3.4) i s sa t is f ied i f the commutator of the D'Alembertian Y^}j with k n vanishes, i . e . (3.10) [VVj.^ i] = 0 • Substituting for k^ from (2.35). with 1 a c-number, the l e f t hand side of this equation reads, for general sp in , (3.11 ) [V}j.kn] = - 4 C M ff-nJ^44±yn^ij . For a massless scalar f i e l d , ^ j j ^ O , o n e sees immediately that (3 .4) is sa t is f ied only for solutions of (3.1) that are eigenfunctions of the scale dimension with J£=-1. In this case one has (3.12) ^ j V f ' ( y ) = { i - f n ( 2 y n y k ^ k - y 2 ) n + 6 y n ) ] ^ ^ r ( y ) . which shows that V^jH^v) transforms as a scalar f i e l d with scale dimen-sion -3» • The divergenceless probabil i ty current density associated with (3.1) i s (3.13) j k - i Y * > k T * - i t } k v V * . I f the integral of j over y_ i s to be a number, and hence d i l a t iona l ly 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 is given by (2.38d), (3.1*) <r j k - ( i /4) [Yj.Yj . Neutrino f i e lds are governed by the Weyl equation (3J5 ) - i A / = 0 and, using the defining re lat ion of the C l i f fo rd algebra sa t is f ied by the the Dirac matrices Y«5 (3.16) frj.Yk] = 2SJ k . the iterated form of the Weyl equation i s (3.17) ^Yj? = 0 . The set of matrices generated by the <5"jk i s a direct sum of two irreducible representations of the Lorentz group which are related by the pari ty operator. Since the generator of d i lat ions commutes with the pari ty operator,(Mack and Salam 1969), J. acts on ^ as a c-nuraber in each irreducible subspace, and the inf in i tes imal generator of specia l conformal transformations for neutrino f ie lds i s given by (3.18) kj = i ( 2 y ^ k - y 2 ^ _ j - 2 i y j ) + 2 y k ( r ' j k . The commutation re lat ion (3.11) can be conveniently employed to determine the value of ^necessary to ensure that Y H y ) i s a solution of (3.1?) whenever M'(y) i s a solut ion. Substituting (3.1*0 into (3.11). one obtains (3.19) [ V V k n ] " - ^ + 3 / 2 ) 1 ^ + 2 ^ ^ ^ ^ so that, on the space of solutions of the Weyl equation, i n order for 1^ to be a symmetry operator one must have i=-3/2. In this case one finds that tf^jf transforms as a spinor with scale dimension -5 /2 . The divergenceless probabi l i ty current density associated with (3.15) i s 38 (3.20) j k = 9 Y k V with y = . Thus the scale dimension of must indeed be ^=-3/2 i n order that the integral of j have zero scale dimension. 3-3. The Maxwell equations The 4-component vector representation is given by (2.38b). A non-interacting massless vector f i e l d A i s governed by the Maxwell equations (3.21) (S k n >jV-A N )A N = 0 . so that, in this case, covariance of the f i e l d equations i s guaranteed provided (3.22) [ ( ^ i r ^ n ) . ( K V ) % ] » { ( ^ i j)(^)VC^„)^)V - ( k r ) P s ( ^ ^ K ( k r ) P n ( i n J s ) } = o * when acting on a solution of (3.21). Substituting the exp l ic i t form of the spin matrices, one obtains (3.23) [( V ^ V ^ V ^ s ] S 2 1 ( 1 + * ) ^ + 2 1 ( 1 + 4 ) ^ , -- * i ( 1 + 4 ) $ p s \ - M * i y k U P s ^ - W . and comparison with (3.21) shows that J. must be -1 for 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 Einstein equations In the weak f i e l d approximation, the gravitat ional f i e l d may be described by a second-rank symmetric tensor f i e l d h^ k in Minkowski space, which in the free f i e l d case sa t is f ies the equation (Pirani 1964) (3.24) [\nf\\ S f s - A a f r ^ - ^ n S ^ ^ V ^ i ^ r s l = 0 • 39 It i s of interest to ascertain whether eq.(3.24) , as i t stands, i s conformally covariant in the sense outlined i n the preceding sections; i . e . using >' the metric I throughout and assuming that h r s transforms with a speci f ic scale dimension ^ , (3.25) h r s ( y ) * h ' r s ( y ) = <r«-y\)hrs(y) . under inf in i tesimal d i la t ions . In fac t , the f i e l d equations turn out to be not covariant ( c . f . Deser 1973). To study the behaviour of (3*24) under special conformal transformations, one must define the generator kp, with four indices, such that the d i f feren-t i a l part multiplies the f i e l d h^ u with the product of Kronecker deltas Sir-t&Su» i « e « (3.26) ( k p ) V u = i( 2ypy mVy 2^p- 2^p)s r t S s u + 2 y m ( c r p n i ) r s t u with ff' given by ( 2 . 3 8 c ) . Covariance of (3.24) is guaranteed only i f this generator commutes with the d i f fe ren t ia l operator of the f i e l d equation. The result which emerges i s actually of the form (3.27a) [ t t ^ S V v ^ ^ ^ * 0 even when acting on f ie lds h t u which sat is fy the f i e l d equation. The actual calculat ion of this commutator is straightforward but tedious, and has been relegated to Appendix 1. Substituting the tensor form of the spin matrices, the f i n a l result for the l e f t hand side of (3.27a) i s (3.27b) Wyp^n>nSktsV)„>JSkt->u> k5 ; it^Jj'<Stuy2i{2(i^)Sk tSVSJ kStu}p+ - 4 < ^ V ^ P V 5 s P t ^ --2i{(i-i)s p usJ t^r t u-3s J us p t)j k * 40 When operating on a symmetric tensor h^u=hu^, this expression becomes (3.27c) M y p ^ \ S V ^ ^ ^ i $ J V p M S p u h ) . u + 2 i i $ p ^ +2i(2+j?)(h k p AhJ p , k ) where h 3 ^ 1 1 . While the f i r s t term in this expression shows that the entire f i e l d equation transforms i n part as a second-rank symmetric tensor with scale dimension as expected, the remaining terms do not cancel , for any choice of 1, For this reason, the equation is 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 con-servation 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 that, for example, no generator can be written in terms of a l l the other generators. On the one-particle l e v e l , however, i t has recently been shown by Bracken (1973) that for a wide class of representations of the Lorentz group, for f ie lds which sat is fy a massless f i e l d equation of the type (3 .28 ) c- J kb kT V,Y = 0 , the generators y> and k j , when acting on Y , can be written in terms of p"k and mj k . Here, A is f for spinors and 1 for second-rank antisym-metric tensors. For example, for 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) ^ = d/4) * 5 e i J k l 2 ^ ^ . ¥ where ^ = "6 T£ ^ i , and i s the permutation symbol in four dimen-sions, with .^123i+-=+^ * ^ e i terated version of (3«30) reads (3.31) y (y^f) = -i(3+2S j k mJk)Y . For the scalar representation, although there i s no relat ionship of the form (3.30), there i s an equation analogous to (3.31). (3.32) - l(2+m j k m ^ ) ^ . which holds i f one assumes (3.33a) y = i ( y \ + 1 ) (3.33b) » j k = m. k = i ( y j V y k V and (3.34) V ^ C y ) = 0 . One can also f ind an expression for the generators k j , by obtaining k^ from the relat ion (3.35) M = ( n 2 + + X 2 - 1 )Y for general X , where n=(m^ . m ^ i ^ - j ) . Although these relationships show that i t i s possible to express the generators of di lat ions and special conformal transformations in terms of the generators of translations and Lorentz transformations at the l eve l of single-particle quantum mechanics, this is not the case in quantum f i e l d theory. To see t h i s , one can attempt to construct an operator out of the operators Mj k such that $ 2 i s given in terms of M^jJM^ as in (3.32). Such an ident i f icat ion holds only when the operators act on one-particle Fock states, just as the operator P^P^ is zero for massless f ie lds only when acting on one-particle states. 42 To see e x p l i c i t l y how such equali t ies break down at the many-particle l e v e l , consider a complete set of states, made up of combinations of the Fock states which are eigenstates of the momentum operators, labeled by the set of real quantum numbers (j,ra;j 1 ,m' \oL ). They sat i fy the set of equations (Yao 196?) (3 .36a) {j 2-j(j+1)}|j ,m;j \m';o<> = 0 (3 .36b { J , 2 - J ' ( )} j ' ,m' ;^> = 0 (3 .36c) (J^-m) |j ,m; j ' . m ' ;<*> = 0 (3 .36d) (J«3-m»)|j,m;j«,m«;oc> a 0 (3 .36e) ( i<|>-°0|j ,m;j \ni , ;*> = 0 where, defining Ij=(M23»M31 « M 12) and N=(Mji|i .M^.M^-j), J a and J ' & (a=1 ,2,3) are given by (3 .37a) J a = £ ( L a + i N a ) (3 .37b) J ' a = | ( L a - i N a ) . 2 2 1 The f ive operators J , J ' . J^ .J ' ^ . I * can be simultaneously diagonalized in any representation of the conformal group. When acting on this bas is , the pertinent operators have the properties (3 .38a) ( Y 2 +<* 2 ) j j ,m;j«,m';c</ = 0 (3 .38b) { ^ k M ^ - ^ j + I J + j U j ' + O j l j . m ^ ' . m ' ; ^ = o . Comparing eqs . (3 .38) with the ident i f ica t ion (3«32), one can see that the re lat ion (3.39a) (j)2 = - i (2+M. k M j k ) holds only when (J) and Mj k act on the set of states j j,ra; j ' ,m'\<*C] having (3 .39b) o<.2 = 1+2[j(j+1)+j'(j '+1)] . just as the Klein-Gordon equation has the quantum f i e l d theory equivalent P 2 ^ 2 only when P acts on the set of s ingle-par t ic le momentum eigenstates. Hence for a general l inear combination of the basis states, the re la t ion (3 .39a) does not hold. 43 4. Attempts at Including Mass 4 -1 . Symmetry breaking As a f i r s t step in considering the poss ib i l i t y of including the mass concept in conformally covariant f i e l d equations, one can determine the effects of mass terms on the conservation laws established in section 2-5. By straightforward application of the def ini t ions 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 integral (2.50) i s not d i la t iona l ly invariant, but so that instead of (2.57) i t s var iat ion due to di lat ions (2.53). with constant parameter <r, i s of the form (4 .1) - c r ^ k ( y kX ) - c r F where F i s some function. I f that i s the case then the derivation of the conservation law (2.65) is modified, in the last step from eq.(2.64), by substitution of (4.2) Otfr <r) = - \ ( y k / ) - F instead of (2.58), leading to the equation <*.3) = F for the divergence of the d i la t ion 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) X = Hi^J Vf - m 2 Y 2 ) . I f the scale dimension of Y is taken to be -^=-1, and m is considered to be a  parameter invariant under d i la t ions , then the variat ion of jC due to the d i la t ion (2.53) can be computed, using the transformation law (2.55) 44 (4.5) V * t+<K-i-y\)Y with the resul t , for the case of constant G-', (*.6) %£ = - c y y k ^ ) - ( r m 2 V 2 . Thus, the parameter m i s seen to break the d i la t iona l invariance in that i t gives r ise to a function F as in (4.1), (4.7) F = m 2 ^ 2 . Now the d i la t ion current (2.45 ), which reads in this case (4.8) D* = y n { (2 / 3 ) \ Y ^ T - O / S H ^ n j + ( S 7 3 ) ( g n k V V M n a k Y ) + i s conserved only i f m=0. Equation (4,3) can be worked out in terms of ^ t and i s seen to hold as an identi ty on account of thef ie ld equation (*.10) V ^ - h ^ Y = 0 that flows from the action principle with (4,4) as Lagrangian density. 4-2. Conformally covariant mass I f one regards the conformal transformations (1.25c,d) as (spacetime dependent) changes of uni ts , then, as was pointed out in section 1-1, i t is necessary that the numerical value of $ must vary, i n a spacetime depen-dent fashion. 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 in order, that e .g . the speed of l igh t c written in terms of the Compton wavelength ^ , c =$/(XnO. be l e f t unchanged. Here one invokes the assumption that the numer-i c a l value of the mass m is unaffected by changes of the length uni 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 inf in i tesimal variat ion (c . f . Fulton e t . a l . 1962a) 45 (4.11a) Sra = < f b m x ' in and (4 .11b) o B s 2 p J y j j ^ m with - ^ m = - 11 for dilations and special conformal transformations respectively, for a mass m which is "conformally covariant" i n this sense. For a scalar particle with mass , satisfying the equation (4 .12) ( Y h j + m 2 ) ^ ) = 0 , ' the transformation property of Y under dilations is assumed to be (* .13) Y ' ( y ' ) = <riY(y) + Y(y) . Employing the definition (4 . 11a) , one finds that (4.14) [<V>y,d)My'j)+a,2]vP(y») = = (1-2o-+/ <r)> j^Y(y)+(1+24^+^Om 2f(y) = = V-ZS+JcX^+m^VM = 0 , where the second equality results from setting ^^=-1 • Then one can say that i n the new system of units, the f i e l d equation (4 .12) i s again sa t i s f i e d . I t i s i n the sense of eq.(4.14) that the anomalous transformation laws (4 ,11) are usually employed (e.g. Hamilton 1972). However, i n Lagrangian f i e l d theory, one must use the complete transforma-tion property of the f i e l d , (M5) Y»(y> - Y(y) + o - ( i - y \ ) Y(y) . Using this transformation law for the f i e l d , but continuing to use the variation (4 .11a) for the mass, one finds that i t is not possible to retain covariance of the Klein-Gordon equation (4 .12) under dilations, as long as the mass i s non-vanishing, since (4 .16) y ^ Y ' t y ^ Y ' t y ) = ( i + i c r - 2 < r - 2 ^ y k ^ ^ f ( y ) + +(1+^+24or--(r'y^ k)m 2f(y) = - C y \ ^ W 7 ) = «m2o-y\Y(y) . 46 These calculations have been based on the behaviour of eq.(4.12) under d i l a t ions , 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 factor O+^j0')* However, the transformation law (4,11b) of the mass under special conformal transformations involves multiplying m by a spacetime dependent factor 0+2f*Vj4n^* Indeed, the p o s s i b i l i t y of a coordinate dependent d i la t ion parameter cr was essential for the der iva-t ion 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 retain the d i la t iona l covariance of the f i e l d equation even when the mass term i s present. In this case, the anomalous transformation laws (4.11) must be replaced by variations which in analogy to (2.35c,d) are given by (4.17a) W = <T ( i ^ - y ^ m (4.17b) Sm = H<f fyyXVfX^ty^j)}" for di lat ions and special conformal transformations. The transformation law (4.17a) serves the purpose of restoring the d i l a -t ional invariance of the action in tegra l , since i f one chooses -4n=-1 then the scale dimension of X. i s J.^ =-4, and the variat ion of X. i n the case of constant <T i s a divergence (4.18) S t = -<r\(y k£) s - « - y y k/)-mW - m ^ 2 S m - r m Y 2 y k i k i n . However, although the action i s invariant, the f i e l d equation i s no longer sa t is f ied i f Y(y) i s replaced by Y ' ( y ) and m by m'=m+Sm given by (4.17a). In fac t , for the Klein-Gordon equation (4.12) one finds +(1+icr'+2i^-2<r'y kd )(m 2Y)+ m k + * - y \ ( * 2 v O . In addit ion, the d i la t ion current D", as defined in (2.43c) (4.20) D n = - ^ f + y \ n , i s no longer conserved, and for the scalar f i e l d one has 47 (4.21) D n , n = m2 ^ +my k(^ km)H ; 2 . The reason for this is that in the derivation of the current (4.20) i t i s now necessary to include the variat ion &m of the mass, so that (4.22) O ^ / W ) = O^AYr? + *n1$ + ( ^ A * ) m where (4.23a) 9= d - y \ ) Y (4.23b) m = (4 - 3 ^ ) m . Combining (4.22) with the resul t (4.24) [fct/aftn*-)] =^T n Y . with the help of the f i e l d equation (4.12), one obtains (4.25) [ } c r ) ] . „ = C * £ A < r - ) - U ^ m ) m . Hence, whenever (4.18) holds, so that (4.26) = - \ ( y k . £ ) . one has (4.27) D n , n = U/ />m)(Vy\) » • The transformation laws (4.17) are seen to elevate the status of the mass to that of an independent f i e l d var iable, and indeed unless this 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 ight hand side of (4.27) by zero, i n disagreement with the exp l ic i t calculat ion (4.21), However, the approach leading to (4.22) necessitates as a concomitant circumstance the existence of an additional f i e l d equation for the mass, flowing from the action pr inc ip le . For the Lagrangian corresponding to the scalar f i e l d , this equation is simply (4.28) (SjC/Sm) = U X A m ) = -m Y 2 = 0 . Clear ly , this requires either m=0, thus reverting to the d i la t ion 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 lat ions The various interpretations of conformal transformations outlined in section 1-1 can now be evaluated i n the l ight of the above considerations. F i r s t l y , i t i s evident that the interpretation of di lat ions as uniform changes of the unit of length means that equations with mass terms possess symmetry under changes of un i ts , i n the sense that the f i e l d equation with Y , n, and y replaced by v f ' , m 1 , and y* is again s a t i s f i e d . However, one cannot establ ish a conservation law in Minkowski space associated with this symmetry*, since when (4.13) and (4.11a) are obeyed, one has for a scalar f i e l d with Jt=-\ [using (4.18) with ^ = 0 , and (4.25) with m=-m] (4.29) D n , n = m 2 Y 2 . 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. Because unit transformations are ruled out by (4.29) ( c . f . Coleman and Jackiw 1971, p.555), one may interpret d i la t ions as mappings from points in Minkowski space onto dif ferent points i n spacetime— the f i r s t interpretation suggested in section 1-1 — and interpret special conformal transformations as spacetime dependent d i l a t ions . This interpretation i s analogous to "di lat ions" in non- re la t iv is t ic macroscopic physics, in which the space and time variables are allowed to transform separately. For example, in the Kepler problem the orbit of an object is transformed into another possible orbit provided the period i s transformed with the 3/2 power of the parameter specifying the d i la t ion 'One can force (4.29) into the form of a conservation law by inventing a vector f i e l d gj spec i f i ca l l y for this purpose (Hamilton 1973), such that but this seems somewhat a r t i f i c i a l in view of the fact that according to (4.16) the f i e l d equation s t i l l can not be made covariant under the to ta 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 orbi t e l l i p s e . In fact i t i s possible to determine even larger groups of transformations of this sort , corresponding to non-re la t iv is t ic versions of the conformal transformations with space and time transforming d i f ferent ly , which are symmetries of the Schrodinger equation including mass terms (Barut 1973). 4-4. Other approaches The conclusion result ing from (4.28), that mass m breaks the d i la t iona l invariance of a massless f i e l d for the case of constant <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 addit ional terms containing m, and possibly the derivatives ^ m , in expressions other than 2 ? m f . However, such a procedure would necessarily be somewhat arbitrary, due to the absence of any guiding physical pr inciples 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 s t i l l 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 in 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 suitably 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, insuf f ic ient experimental evidence for the actual existence of such f ie lds invented for this purpose, the treatment of compensating f i e lds in 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 ie lds in higher dimensional spacetimes with projections onto 4-dimensional space sat isfying equations resembling the equations governing conformally covariant f ie lds in Minkowski space. The examination of this poss ib i l i t y is the purpose of the remaining sections in this report. PART II. FIELDS IN HIG HER-DIMENSION AL SPACES 50 5. Generalization of Minkowskian Spacetime 5-1. Introduction D i f f i cu l t i es associated with attempts at encompassing the mass concept within conformally covariant f i e l d theories were outlined in the previous sections. The remainder of this report consists of an examination of f i e l d theories in spaces which are generalizations of Minkowskian spacetime, in an effort to extend the app l icab i l i ty of conformal symmetry to the case of massive par t ic les . There are many possible approaches to this problem (Murai 1958. Caste l l 1966, Ingraham 1971, Wyler 1971. Barut and Haugen 1972), but i t seems reason-able here to formulate a theory that extends the theory of conformal symmetry 1 in 4-dimensional space yet does not a rb i t ra r i l y depart from i t , in the sense that the f i e l d equations derived in higher-dimensional spaces should be expressable in terms of f i e lds in Minkowski space which transform according to a transformation law which closely resembles that associated with a representation of the conformal group, as given in (2,36). This means, in part icular , that when written in terms of Minkowski space var iables, in keeping with the discussion of section 2-6 these f ie lds should possess a def ini te value for the scale dimension, so that equal-time commutation relat ions can be imposed on free f ie lds and a quantization procedure can be carried 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 inear ized by the introduction of the conformal coordinates *^ A , as defined in (1.34) and (1.39). There, the variable L=Ylk\k was assumed to vanish ident ica l ly . For this case, Mack and Salam (1969) have 51 shown that by defining f i e lds over the coordinates v^t manifestly conformally covariant f i e l d equations can be formulated in 6-dimensional space for f ie lds which correspond to massless scalar , spinor, and vector f ie lds in 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 six independent variables y k , vt , L related to the by ( 5 J a ) y k = ^ k / (^ 6 -^ 5 ) ; * = ^ - ^ 5 ; L = ^ A \ f * 2 y 2 + K ( ^ 6 ) (5.1b) v ^ x y * ; ^ ^ ( L - ^ - ^ y 2 ) ^ ^ ; \6=1$={L+ * 2 - ^ y 2 ) / 2 " with the understanding that the constraint (1.40) does not hold. By this proposition one conceives a l l physical f ie lds 7~ as functions of s ix variables « \ A , (5.2) M\) = X « ( y k . * , L ) defined in the entire 6-dimensional space and considered, in part icular , 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 special case L=0 i s assumed. The di f ferent iat ion operators j f ^ ^A^ can be expressed in terms of the di f ferent iat ion operators V » a s (5.3) 7A = ( o y k / M A ) ^ k + ( ° K / H A ) V + O L / H A ) > L . E x p l i c i t l y one has (5.3a) M l/*)V42n * = 0 / * ) ^ ,+2Ky^ T , / * )2>J+2xyJ> 52 (5.3b) * 5 K 1U )y\-V + 2 ^ l = ( i / * ) y \ - V - [ ( 1 / * ) - * - * y 2 ]> L =-^ 5 (5.3o) ^ 6 = - ( 1 / K ) y \ + ^ + 2 ^ L = - ( 1 / H )y ka R+^ + [ < L / K ) + X - n y 2 ] ^ 6 so that (5 .*) i\AJf = V + 2 L b L . The d i f fe rent ia l generators of the rotations in 6-dimensional space, defined as (5.5) « f f l » i<TA -WA> in analogy to the corresponding operators (2,20b) in 4-dimensional spacetime, read e x p l i c i t l y (5.5a) m. k = i < y j o k - y k t y (5.5b) m 5 k = i[(n5/x- J V v ^ n V V l (5.5c) ^-ipW^VvVrk***] (5.5d) m 6 5 = i ( y \ - > O j and are seen not to contain o , One has also ii (5 .6) m 6 k + m 5 k = i ^ . i . e . \ = ^fo ^ Jf v ^ - ^ ^ and, using the third re lat ion (5.1a) <*-7> m 6k- m 5k = 1 [ * ~ 1 H V V 'vV 2 7 ^ J = = i [ 2 y k ( y \ - K ^ ) - y ^ K + ( L / K 2 ) ^ K J . An inf in i tesimal rotation (1.43) (5 .8) ^ A = E A B n B characterized by the parameters E A B as specif ied in (1.44) w i l l now have 53 an effect on the Minkowski coordinates y^, defined by (5.1 a) , which gener-al izes inf in i tesimal conformal transformations to the case L /O, (5 .9a) lyk = ^^^^vJ+o -y^+ayyJy^V+^CL/M- 2) (5 .9b) - f f - K - 2 ( ^ y ) K (5 .9c ) SL = 0 . The las t term in the f i r s t equation (5.9a) involves only the parameters of the special conformal transformations and, for a part icular L and X, takes the form of an extra inf in i tesimal translat ion added to <*• . By defining a quantity r = ( L / v t 2 ) ? the variations (5.9) can be looked upon as transformations from the set of coordinates ( y k , r ) to the set ( y , k , r « ) so that (5.10) o 3^= < * k + € k ^ ( t r +2^y)y k -^ k (y 2 -r 2 ) ; S r=( <r+2y*y)r . The coordinate r can be interpreted as the radius of a "sphere" in Minkowski space with center y^, consisting of the set of points such that 2 2 ("^-y) =*" (Ingraham 1973). One can define the "angle of intersection" of two such spheres by (Ingraham 1971) (5.11) r ' 2 + r 2 - f v ' - v } 2 In the inf in i tesimal case, y,^=y^+dy^, r , = r + d r , this reduces to (5.12) d6 2 = r - 2 ( d y 2 - d r 2 ) , and i t i s easi ly demonstrated that this generalizes the angle between the l ines joining the centers of two c i rc les to their point of in tersect ion. The angle 8 can also be looked upon as the inverse cosine of the inner product of two normalized vectors in the 6-dimensional space, since one has ( 5 J 3 ) cos9 = ( vjA ) / ( L L ' ) * 54 Inspection of this def in i t ion shows that the variations (5.8), and hence the variations (5*10), comprise those inf ini tesimal transformations which preserve cos0. On the other hand, the conformal transformations in a 5-dimensional space with metric diag(-1 ,-1 ,-1 ,+1 ,-1 ) and coordinates z (k=1 ,2,3,4,7) result i n variations with 21 parameters (5.14) *z k=* k+fe k.B J+(<r+2^a)z k-; kz 2 (k=1,2,3,4,7) . 3 7 7 7 Now, i f one fixes 6 of these parameters by writing =p =0, then, upon k k 9 making the ident i f icat ions z =y for k=1 ,2,3,4 and z'=r, the variations i t (5.14) with the remaining 15 parameters reduce to the variations (5.10). Therefore, one has the option of considering the rotations in 6-dimensional  space as a special case of the conformal transformations in a 5-dimensional  space. Removal of the constraint L=0 also means that the inversion (5.15a) v^' 5 = -nf ; ^ k = ^  ; ^ 6 = ^ amounts to a generalization of the transformation by reciprocal r a d i i (1.49), (5.15b) y » k = - y k / ( y * - r 2 ) ; vO = - * ( y 2 - r 2 ) . The generalization of the inf in i tes imal special conformal transformations occurring in (5.9) i s recovered from their f i n i t e versions (5.16) y ' k =[y k -b k (y 2 - r 2 ) ] / [ l -2by+b 2 (y 2 - r 2 ) ] ; = * [l -2by+b 2(y 2-r 2)] k k when the f in i t e parameters b are replaced by the inf in i tesimal p . As in *Here one assumes L ^ O . For L<0, the quantity r is imaginary. By defining / = i r , so that W K = « < k + ( k j y J + ( ( r + 2 ^ y ) y k - ; k ( y 2 + / 2 ) . S/> =(<r + 2 jSy)/ one can see that for L<.0, the above variations correspond to a subgroup of the transformations (5.14), but with the 5-dimensional metric replaced by diag(-1,-1,-1,+1,+1). 55 the case r=0, (5*16) is decomposable into a reciprocation (5*15). a translation, followed by another reciprocation. The rotation (5.8) i n 6-dimensional space w i l l induce on the f i e l d Ti-the transformation (5.17) &X = - ( i ^ E ^ J ^ X = ^x/Z)EAB(mAB+skB)X where the m^ _ are given as i n (5.5). and the s are the appropriate f i n i t e dimensional representations of the rotations acting on the indices of X-according to i t s transformation character as scalar, spinor, vector, etc., in that 6-dimensional space, respectively. 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 = J6k+75k = iV< s6k + 85k> S 'V^k (5J8b) » j k = J j k = i ( y j \ - y k v ) ^ j k (5.18c) ? s T65 = i ( y \ - H > ^ ) + S 6 5 (5.18d) k. ~= T6.-T5 . - i[2y/>\-(/-r2)lf2y.^} ^ - s ^ ) . On the other hand, the canonical form of the generators of conformal transformations i n 5-dimensional space, defined i n analogy to (2 .35) . i s (5.19a) p k = i \ (k=1,2,3,4,7, > k = ^/bzk) (5.19b) i j k = i(«^k-VjJ+<r(5)jk (5.19c) y = i [ z \ - i ( 5 ) ] (5J9d) kj = i [ 2 » j ^ n - * 2 ^ 5 ) « J > « , 1 ^ ( 5 ) j n + » ( 5 ) J with labels (5) appended to , to distinguish them from the corresponding representations i n 4-diraensional space. 56 When one s p e c i a l i z e s the conformal transformations i n t h a t 5-dimensional space to the case ot^-fl =p?=0 the remaining components that generate the tran s f o r m a t i o n s (5.10) are (k=1 ,2,3,4, z n}„=y k^ +r} ) K n k r (520a) p k = i ^ k (5.20b) m j k = i(y^ k-y k^)+<r ( 5 ) j k (5 .20c) f = i[y\+iV^ ( 5 ) ] ( 5 . 2 0 d ) cd = i t ^ C y V ^ r M y 2 ^ ^ and they are to be compared w i t h p^, m ,^</> » k j a s g i v e n above i n (5.18). I n keeping w i t h the remark made f o l l o w i n g (2.28), such a comparison must be preceded 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 out i n the index space of the f u n c t i o n s X , (5.21) = u ^ u " 1 ; *-=u x. so t h a t there i s no " i n t r i n s i c momentum" ^ k S s 5 i c + s^j c» (5.22) Up kU~ 1 = U(i^ k+Tr k)u" 1 = p k = i ^ k . Now, the v e r y appearance i n (5.20d) of the s p i n operator <T. . r e q u i r e s t h a t one c a r r i e s out t h i s t r a n s f o r m a t i o n f o r the generators i n a 7-dimensional space w i t h m e t r i c % =diag(-1,-1,-1,+1,-1,+1,-1), because AB these g e n e r a l i z e d r o t a t o r s correspond to the e n t i r e s e t of conformal generators (5.19) i n the 5-dimenslonal space. The s o l u t i o n of the problem (5*22) i s then provided by (5.23) U = e " i Z (k=1 , 2 , 3 . 4 , 7 ) 57 where TT^  is the f in i t e dimensional representation of the translat ion 7 in z -d i rec t ion , given in terms of the rotation matrices s i n 7-dimensional AB space as (5.24) TT? = s 6 ? + s 5 ? . Indeed, since a l l TT^  commute [see the f i r s t re lat ion (2.23)] , the inverse of the operator (5.23) i s (5.25) IT 1 = a 4 1 * ^ * and, therefore, one has (for a l l k=1,2,3,4,7) (5.26) U ( i \ ) U " 1 = i ^ - i ^ " ^ . ^ ^ . . . = i ^ - T T k . Thus, the operator (5.23) has, in part icular for k=1,2,3,4, the desired property (5.22). For the case k=7, the equation (5.26) implies the relat ion (5.27) U( * V ) U " 1 = * X - i r i r ? on account of the ident i f icat ion z^=r=(L/v. 2 ) 2 , and an exp l ic i t calculat ion ver i f ies this resu l t . Now one can compute (5.28) Since, on account of the commutation relations (2.23), (5.29) [ A n , m . k ] = [ y S ^ . f i C y ^ ^ j H s ^ ] = one has simply (5.30) m. k = m. k and comparison of (5.18b) with (5.20b) y ie lds the identi ty (5.3D < r ( 5 ) j k - s j k • 58 Simi lar ly , for the computation of (5.32) ^ = U f f 1 =y>-i[znTTn,^]+ ... one requires evaluation of the commutator, with the result (5.33) t%.v]=[yV r l T 7^ i ( y k ^- *V) + »6Jfl " - ^ n ^ C [ s 6 n + s 5 n ' s 6 5 ] > = ° and again one has simply (5.34) y = ? yielding upon comparison of (5.18c) with (5.20c) the identi ty (5.35) ^ (5 ) - r ^r = ^ + i s 6 5 • F ina l l y , one can compute (5.36) kj = UkjU""1 = k j - i [ z n7r n . k j]-i [ z nr n . [ z m i r m , k j ] ] + . . . The commutation relations (2.23) give in th is case (5.37) [ z X ' k j ] = - 2 i y j z r X 7 r n + i z 2 i r j + 2 i z n s j n + 2 i y j s 6 5 (5.38) [ z n V [ z \ , k . ] ] = -kyf\+2z2^ and vanishing commutators of higher order. One has thus (5.39) k. = k. + 2z*s . n + 2 7 j s 6 5 . Upon substitution, of (5.18d) and using the ident i f icat ions (5 .30 and (5.35) one can now compare (5.39) with (5.20d) and obtain the ident i ty 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 transforma-tions in 4-dimensional Minkowskian spacetime. Then in terms of the quantities . K . in 4-dimensional space one i s led to the ident i f icat ions (5.41b) X = ^ ( 5 ) - r ^ r = ^ , + i s 6 5 (5.41c) K k = H ( 5 ) k = s 6 k - s 5 k . '5-3. Necessity of the choice of descent operator The operator U, defined by (5.22), was chosen such that in the new basis , there was no in t r ins ic momentum i r k =s^ k +s^ k » However, (5*23) i s not the only possible operator which sat is f ies (5.22), since i n order to remove T k from p>k for k=1,2,3,4, a l l that is required is an operator (5.42) W = e - i y 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 fac t , the matrix W i s the one which i s customarily u t i l i z e d for this 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 imi lar i ty transformation W( )W~^  on the generators JAB' with the results (5.^3*) Wpkw"1 = 1\ (5.43b) wSyr1 = i ( y j H - y K V + S J K (5.43c) W?W _ 1 = i ( y \ - * - V ) + s 6 5 (5.43d) Wk.w"1 = ifa.fy^ - v t ^ _ i s W v 2 - 2 ^ 1 , k^-J i " - ' j V 6 j - s 5 j ; - * . ' M . 6 j « 5 j ; 60 Noting that the f i r s t three types of these generators are ident ica l with the results found using the operator U , there is no d i f f i c u l t y interpreting these operators in terms of the generators p^, m j k » <pt making use of the ident i f icat ions (5.41 a ,b) . However, the fourth type of generator can not be compared with the canonical form (2.35d) of the generator of special 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 special conformal t rans-formations in 5 dimensions (5«l9d), because of the absence of a term of the form 2z' ^ ( ^ J j ? * The solution to this problem is provided by augmenting the exponential y^ TT^  i n W by precisely the extra terra r7T 7 , where r i s given i n terms of M- and L as i n section 5-1 . and where IT^ must be given i n terras of matrices s ^ 7 , S £ 7 which enlarge the s^g algebra to the 7-dimensional space. Since the operator Wk^ W is usually not written down, the problem of i t s interpretation i s usually not confronted. When i t i s calculated e x p l i c i t l y (Ingraham 1966), the group of rotations in 6 dimensions i s taken a p r i o r i as the fundamental symmetry group, and a comparison with conformal transformations, in 4 or 5 dimensions, i s foregone i n favour of developing a sel f -consistent quantum theory based on the 6 -rotat ions, If one i n s i s t s , however, that contact be maintained with one of the conformal transformation groups in either of the Minkowski-type 4- and 5-dimen-sional spaces considered above so that, following the programme established in 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 led to the operator U, and not the usual operator W, As w i l l be seen, this operator has the effect of transforming rotat ional ly covariant f i e l d equations in 6 dimensions into manifestly t ranslat ional ly invariant f i e l d equations in Minkowski space, and therefore U (or W) i s referred to as a "descent operator". 61 5-4. Eigenfunctions of JL for the case L=0 At this point i n the development, some authors (Mack and Salara 1969) introduce the concept of the "physical components" of the f i e l d X=U X' by requiring that they possess no i n t r i n s i c complementum K y This i s motivated by the observation that, acc ording to (2.23), i f £ were a c-nuraber [and not a matrix as defined in (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 fields in Minkowski space that possess a definite value of the scale dimension and, according to the arguments presented i n section 2-6, that transform according to an irreducible representation of the Lorentz group. If one works in a representation in which sg^ is 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 in 4-space) amounts, on account of (5.41b), to the requirement that i t be an eigenfunction of . Denoting the corresponding eigenvalue by n, one has the requirement ( 5 . ^ ) K \ X = nX . I f X i s a f i e l d which includes the surface L=0 in i t s domain of definition, and i f (*o L2OL=0 i s f i n i t e , then comparison with (5.4) shows that for (5.44) to hold, on the hyperquadric L=0 X must be a homogeneous function of degree n, i . e . (5.^5) v^ A y > k X = n X for L=0 . Mack and Salara (I969) work with fields that are assumed to be defined only on L=0, so that the second term in (5*4) can be neglected, and for these fields 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 in 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 in W-' (since a l l the s^ k can be put i n block-diagonal form simultaneously), (5.4?) s « j k = Es- kE = s . k in . For the case L=0, one uses the descent operator W as given i n (5«42), and since v*-\^ commutes with W, (5.45) implies (5.44a) x ^ X = n X for L=0. Then the projector E can be shown to diagonalize the operator isgj in t^y.' with diagonal elements containing only the highest real eigenvalue X Q of that operator so that (5«41b) i s replaced by rXphys " ( n + i E s65 E ^phys = ( n + A o ^ p h y s f ° r L = ° ' In other words, /ty-1 may be characterized as the eigenspace belonging to the highest real eigenvalue of i s^« These authors then go on to make a further transformation of the f i e l d X, by multiplication with powers of K, ( 5.49) y a K - n X p h y s . so that on L=0 Yis a function of y k only. Now instead of identifying the ordinary f i e l d i n Minkowski space with X , they assume that instead i t i s Y that i s to be identified with the "physical f i e l d " . Once the condition (5.44) has been imposed, the multiplication (5.49) commutes with the diagonal matrix ^' as given by (5.48), leading to the assignment of the same scale dimension J} to *f as was assigned to X , (5.50) 5 ^ U - n X p h y s ) = ( n + x 0 ) ( * - n X p h y s ) . 63 Obviously, however, there is a point being passed over here, i n that int u i t i v e l y one expects the scale dimension of Y to be modified by the scale dimension of v<_"n, which according to (5.9b) is given by J[ vC"n=nK~n. How one would arrive at (5.50) even without making the substitution (5.44) i s by carrying out the transformation on both sides of (5.41b), (5.5D i"ysvO n^X =*-n(vOv+is6 5)K nx- nX =(n+KV+is 6 5)t . so that after substitutuion of (5.44), Y * i s independent of y- and one regains the identification (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 > * = ( 2 n + ^ V + i ^ ) ^ so that when (5.^*0 holds, one has (5.52b) ^=(2n+ > o ) Y . 5-5. Eigenfunctions of J. for 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). It i s again possible to define the projection operator E having the properties (5.46) and (5**7)» such that E diagonalizes the matrix isg^ * n /H% and leaves only > Q appearing in the diagonal. Indeed, from the point of view adopted in this work, there is no good reason why attention should be restricted 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 application of additional projection operators E (5.53) I = E + % + E 2 + ... + E± 64 such that (5.54a) K ' j = E K jE = 0 (5.54b) s . j k = E s j k E = s . k when there are i d is t inct values of the eigenvalue X apart from \ q . Thus, corresponding to a rotat ional ly covariant f i e l d in 6-dimensional space, one can in general f ind several f i e lds in Minkowski space , each belonging to the subspace of a projector E as in (5.53). which w i l l transform under di lat ions as eigenfunctions of the scale dimension X i f the condition (5.44) i s imposed as a requirement. However, for L^O, K . ^ K no longer commutes with the descent operator, so that even i f X sa t i f ies (5.44), X is not necessarily an eigenfunction of , Since one has (5.55) >AJU> " K V [ e - ^ - i a * / * W 7 ] = = i ( L 2 / * 0 T r 7 a = i rir? u , one finds that (5.56) * - V X = i n r ? X + U vO>^ X and s imi lar ly (5.57) 2L^) L X = - i r ¥ ? X + U 2 L ^ L X . Combining these two equations one obtains the result (5.58) I * V - = U ^ ^ A ^ • and hence f ie lds X that are homogeneous functions lead to homogeneous functions X , 6 5 5-6. The possibility of including mass: eigenfunctions of •£ ^ ) From the discussion of Section 4, i t i s evident that the f i e l d components belonging to specific eigenvalues of JL must correspond in Minkowski space to massless fields i f they are governed by f i e l d equations containing no interaction terms. This w i l l be demonstrated i n later sections for particles with spin, but at this point i t is sufficient to establish that this i s the case for the particularly simple example of a scalar f i e l d X(>^). By definition of the word "scalar", one has (5.59) S A B X ( \ ) = 0 so that there is no need for application of the techniques contained i n the development from (5.17) to (5.48) apart from the identification (5»4lb). Now suppose X s a t i s f i e s , i n 6-dimensional space, the wave equation (5.60) ^ J f r A J L - 0 . This choice of f i e l d equation i s , at this stage, governed solely by the desire for simplicity, and i s aimed only at examining whether conformally covariant f i e l d equations in 6-dimensional space can, i n principle, describe massive fields i n 4-dimensional Minkowski space. [This f i e l d equation i s , in fact, the scalar wave equation customarily suggested for the manifestly conformally covariant description of spin-0 particles on L=0 (Dirac1936, Kastrup 1966, Mack and Salam 1969)]. Using (5.3) one finds by straightforward calculation the identity (5.61) f ?iA 2 x--2 V V + 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) > JVjX + n 2 x 88 0 • with (5.63) m2 s ^ M 2 , by the introduction of a "mass operator" having eigenfunction X. and 66 eigenvalue M2, i . e . (5.64) 4 ( x ^ +3+nLnLX = M2 X. „ However, introducing the mass concept i n this fashion is seen to be unsatis-factory from the point of view of conformal covariance i n Minkowski space, since i n order to assign a specific scale dimension to the f i e l d i t must satisfy (5*44) as well as the f i e l d equation (5.60). Applying the operator vc^ K from the l e f t to (5.62), one finds that this i s possible only i f M vanishes identically, (5.65a) 4 ( y a K+3+L% L ) ^ L X = 0 (5.65b) M2 X = 0 . Therefore, even via 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 repair this defect, i t i s apparent from the discussion 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 infinitesimal variations (5.9) into the form of conformal transformations i n five dimen-sions. Then the requirement that a f i e l d X transform according to a representation of the conformal group can be modified, and instead of requir-ing that the condition (5.44) be satisfied, one can weaken this condition to the proposal that the f i e l d must be an eigenfunction of the scale dimension i n five dimensions -^(5). The essential consequences of this 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 directions. The treatment of this case w i l l be developed i n Section 6. 67 5-7» Dimension of the conformal representation It has been shown that even i n the si tuat ion i n which the squared interval ^ i s non-vanishing a transformation can be carried out on f i e lds that transform covariantly under rotations i n 6-dimensional space such that the result ing f i e lds transform covariantly under a conformal transformation group. This objective i s accomplished by means of the operator U, defined i n section 5-2. For any f i e l d X. one can form the corresponding f i e l d X=UX, even though i t may be necessary to augment the number of components of X. i n order to be able to employ the f inite-dimensional representations Syk of rotations i n 7-diraensional space, since several such matrices appear in the de f in i t ion 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-tat ion of the conformal group in the 5-dimensional space spanned by the coordinates ( y k , r ) . However, i t was shown in sections 5-5. 5-6 how one can select representations of dimension less than n corresponding to the several eigenvalues of by making use of projection operators on the space of f i e lds X . Nevertheless, one must keep in 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 in 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 exist alternative means at one's disposal to eliminate certain components from X and so to f a c i l i t a t e the ident i f ica t ion of the remaining components with the usual number of components of a scalar 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 in interpreta-t ion as in 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 ie lds in 6-dimensional space The finite-dimensional representations s^g of rotations ( 5 . 8 ) , characterized by the parameters E ^ as specif ied in (1 .44) , acting on the indices of a vector ^ A in 6-dimensional space are, as in (2 .38b), (5.66) ( S a b ) C d = K ^ A C S B D - S A D ^ B C ) . Since the variations (5.8) for the case L^O can be thought of as a special subclass of the (pseudo-)orthogonal transformations in 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 .18) , necessitates the enlargement of the dimension of the index space of X to seven, with corresponding spin matrices (5.66a) ( S A B ) \ - i ( S A ^ B R - S A R ^ and (5.66b) ( s ^ - K S ^ S ^ - S ^ S ^ ) where now A,B=1,2 ,3,4,5,6 and Q,R=1 ,2 ,3 ,4 ,5,6,7. Since the introduction of the extra component X ' i s simply a device which serves to bring the transformation law of the vector f i e l d in 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.' in such a way as to bring the transformation law of the components corresponding to a 4-vector f i e l d in Minkowski space into as simple a form as possible. As a part icular application of the def in i t ion (5.66a,b) of the spin matrices, the operators " i r t = s 6 t + s 5 t » ^=1,2,3,4,7, read in this case (5.67a) (TT K ) R Q = i ( $ 6 R + $ 5 R ) S k Q - i ( S 6Q+ S 5Q)S k R for k=1,2,3,4 and (5.67b) ( 7 T ? ) R Q = i (S 6R+ J 5 R ) J 7 Q - i ( S 6Q + S5Q> * 7 R • 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 in the subspace of components In accordance with the def in i t ion (5 .23 ) , the descent operator U, which eliminates TT ^ when applied to ^ , (zk=yk, z^=r) (5.68) u = e - i z t i r t = i - i z t T r t - | z t z s - i T t T r s + ( i / 6 ) z t z s z u i r t T r s TT U+.. can be evaluated, because one finds by straightforward calculat ion (5.69a) ( T r t T r s ) R Q = ( n ) R s ( 7 r s ) s Q = ( S V S V< * 6Q+ S 5Q> * ts (5.69b) ( ^ t ^ s ^ R Q - ^ t ^ s ) R S ^ u ) S Q = 0 so that for vector f i e lds one has the simple representation (5.70) (U) R Q = I I tQ( I 6 R + 5 5 R ) - z t S t R ( S 6Q + $ 5Q)] " -(2 2 /2 ) (S 6 R + S5 R ) (S6Q + C"5Q) where (5.71) Z 2 = Z * Z t S y 2 . r 2 . = y k - n - k + r T r 7 # The operator (U~ 1 ) R Q , defined by (5 .25 ) , (5.72) U- 1 = e + i z t T T t , can be formed from the r ight hand side of (5.70) by changing the sign of the second term and leaving the f i r s t and third terms unchanged. Accordingly, one finds that U i s in this case an orthogonal matrix (5.73a) U R S = % QS ; UQR <jSR - % «S so that 70 are formed using (5 .70 ) , with the results (5.74a) x 6 + x 5 ) (5.74b) x 5 = X t - ( z 2 / 2 ) ( X 6+ 7L5) (5 .74c) X 6 = X 6 ^ X . t-( 7 . 2 /2)(X 6+X 5) . so that, in part icular , (5.74d) X 6 +X 5 = V ^ 5 • Although the four components X ^ of the 6-vector X ^ do not transform as a 4-vector in Minkowski space, the components do have the transfor-mation properties expected of a 4-vector under the elements of the poincare group. In f a c t , one can by a suitable choice of X 7 show that i f the seventh component X 7 i s included in the transformation law for X k , then the transformation law for a l l f ive components X^(X^. Xj) i s precisely that of a vector in 5-dimensional space, and that th is i s the case for a l l the transformations of the conformal group. This can be seen by w r i t i n g down in components the c l a s s i c a l transforma-t ion law xR -> X | R ( y ) = XR+ SX R (R=1 . 2 , 3 , 4 , 5 , 6 , 7 ) . According to the discussion of section 5-2, this transformation law corresponds to the change due to a transformation of the coordinates (A=1,2,3,4,5,6) together with a transformation i n a seventh direct ion which corresponds to the requirement that conformal transformations in 5-dimensional space be restr icted to those with oi^= = 67=0. This can be written as as in (5.8) (5.75) (5.76) E ? A = 0 71 and one can complete the correspondence with 7-space by introducing a component v\Jt given i n terms of the \ which satisfies (5.77) i*\J = 0 . By comparison with the relations (5.78) Tl = 0 and the definitions (5.79) z7 = r =L$/H_ ; z k = ^ t y * , one can tentatively associate this component with I<2, (5.80) a L i = >cr . Then using the identifications (1.44) and (5»?6) i t i s easy to compute the transformation law for X • (5.81a) Ix k = ^ kjXJ+^ k (x 6-X 5) +f k(* 6+* 5) (5.81b) SX 5 = -(*k-fk)Xk+<r X 6 (5.81c) T* 6 = - ( * k + f k)Xk+<r X 5 (5.81d) I F = 0 • Making use of the definitions (5.74) of the components of X , one finds that the above variations imply the transformation laws (5.82a) i X k = j K X6- X5)-( ryk)( X 6-X 5) -= [xJ-yJ(X6-X5)]+/5K^ +yk[2pjX r2pjy.(X6-X5)] = = e ^ x W p J x.+p k{-2y jx j + 2 [- i-(y 2 +r 2)(x 6-x 5)+y jxJ+ + i (x 6 + x5) ] ] , 72 and (5.82b) r*.7 = r x7-*7 f< x6-*5)-( r z ?)( *6-*5) = = 2r/?J [ ^ - y ^ * 6 - * 5 ) ] = 2 r f . These variations are to be compared with those of a vector A*" i n 5-dimen-sional space spanned by z^=(y k,r) with e<7= £ ^r,= j3 7=0, i . e . (5.83a) %kk = (:k.k^c-t{5)^zt{5){f^^kK2^{f J A j ) - 2 ^ k ( y j A J ) + +2 / k ( r A 7 ) (5.83b) f A 7 = <T ^ ( 5)A 7+2 i ( 5 ) ( ^ J y j)A7+2r( ^  V j ) . I t i s immediately apparent that (5.82b) i s exactly of the form (5.83b) i f one makes the choice ^(5)=0. However, even for this 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^.Rt (5.84) ^ R X R = v | A x A - H r X 7 = 0 . This effectively supplies a definition for X 7 , and allows one to recast (5.82a) i n the form (5.85) I xk = tkj x J + £y k fj x j+ f k(-2yj x j + 2 r* 7> since one has (5.86) i ^ ^ y j i ( M K ( x 6 + ? H K ( r V ) ( x 6 - x 5 ) . Thus one finds that from the requirement that the f i e l d transform according to the simple law (5.85), one can determine the meaning of the component X' via the condition (5.84). This condition i s , i n a sense, a transversality condition i n the 7-space R ~- o Y\J-t and one can see that the definition of X certainly holds good for the case of the coordinates themselves, since *\J is defined by (5.87) x r ^ 7 = x 2 r 2 = L = *( k i \ k ) . 73 Here i t should be noted that whenever the 6-vector x . A i t s e l f s a t i s f i e s a tra n s v e r s a l i t y condition, so that *\ AX A vanishes, then X ^  i s zero by d e f i n i t i o n . One can also rewrite the d e f i n i t i o n (5•84) i n terms of the components X R . Substituting f o r X=U~1X. with the help of the matrix (IT 1 ) R n , one finds simply that (5.88) 0 = ^ R % R S | K ( X 6 + r t • 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 ^ . This matrix i s (5.89) ( i s 6 5 ) R Q = ^ / - ^ M 5Q so that (is5^) affects only the 5.6 components of X . Thus, by the judicious use of the constraint (5.84), i t i s possible to avoid completely any use of projection operators, and s t i l l to be able to reduce the number of components involved i n the transformation law of a vector f i e l d X_^, which i t s e l f forms an eigenfunction of -^(5). 5-9. Spinor f i e l d s i n 6-dimensional space The spin operators (2.38d) i n Minkowski space, governing the transfor-mation behaviour of a 4-component spinor f i e l d under rotations, are 1 (5.90) <r j k = (i/4) [ *J ,x k ] with 4X4 Dirac matrices s a t i s f y i n g the anticonmutation relations (5.91) \i\tk} = y j * k + a k * j = 2 $ J * and given i n Weyl representation as '0 V 2T 0 (5.92) '0 Iy ' i l 0> 0 - i i ^ j for a=1 ,2,3; = *5 1 2 3 ^ where H X ^ # and the 2X2 P a u l i matrices have the standard form (5.93) O V o r 0-2= ro - i ^ <r3= ' l 0^ - f 1 i 0 lo 1J 7* This representation suggests searching for a finite-dimensional represen-tation of the rotation operators s in 6-dimensional space, constructed in analogy to (5.90), (5.94) <rAB = [/5A,y3B] with operators p A satisfying the anticommutation relations (5.95) fy\/B}=2SAB . The simplest representation for such operators, admitting; re f lec t ions , are the six 8x8 matrices(Murai 1958) ,k (5.96a) (5.96b) (5.96c) f k = k °*1 3 p = y5 a <r} = o V {lk o o u 5 U * o f6 = I H 0*2 = 0 - i i i i 0 The s ix matrices jS A transform as the components of a 6-vector in the following sense. I f one defines a 6-vector V A by the transformation law (5.97) [v \<r B C ] = ( s B C ) A D Y D . where s B C i s the vector representation (5»66), then the y9A form a 6-vector since they sat is fy this lav: i f the <j-AB are defined as i n (5.94). Indeed, on account of the anticommutation relations (5*95) one has (5.98) [pA,<rBC] = i ( S A B f > c - $ A C Y B ) . This fact enables one to form covariants in 6-dimensional space using the 6-vector ^ in analogy to the formation of covariants in 4-dimensional Minkowski space using the 4-vector tfk. The c lass i f i ca t ion of such invariants into tensors and pseudotensors of various kinds requires consideration of discrete symmetry operations affecting the space-like coordinates. The invariance of the quadratic form 75 (5.99) d B = p A d v l A = f k d ' l k + f 5 d l 5 + ^ 6 d ^ 6 can be used as a cr i ter ion for the construction of operators representing discrete symmetry operations. The discrete transformations affecting the spacelike coordinates can be conveniently c lass i f ied into two types, as fol lows. Consider the "inversion" (5.100) d ^ a » -dv^ a fora=1,2,3 ; d\h* - K i ^ b for b=4,5,6 . The operator P representing this transformation must sat is fy (5.101) {P.p a"i = 0 and [ ? . / b ] = 0 . This i s accomplished by choosing (5.102) P = f * * f 5 p 6 = r^t5 H<^ 2 • Next, consider the"reciprocation" (5.103) - d ^ 5 J + d ^ A for . The operator R representing this transformation must sat is fy (5.104) jR , j j5} = 0 and | R , ^ A ] = 0 for . This i s accomplished by choosing (5.105) R = £ ^ Zpp^p6 = X5 S <T2 . Thus, the operations P and R separately interchange the 4-component spinors Xu and X^ which compose any 8-component spinor X * u (5.106) X = whereas the product operator L X d J (5.107) PR = - x ^ B I represents a symmetry operation in the subspaces of X u and X.^. Any operator S in the space spanned by the spinor (5.106) w i l l be cal led 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 generated by (5.105), 76 (5.108) [ s , cr*3] = 0 ; [s,P] = 0 ; [s,R] = 0 . Obviously, the identity operation i s a scalar. An operator Q w i l l be called a "pseudoscalar" i f i t i s invariant under rotations, but changes sign under inversions and reciprocations, (5.109) [Q, <r A B] = 0 ; {Q,P} = 0 ; {Q .R} = 0 . An example of a pseudoscalar i s the operator (5.110) f = (1/6!) i f i B C J E F f k f B f c f D f E f F = . u H ^ which satisfies the conditions (5.109) on account of (5.95). Now, i n analogy to the corresponding definition of the adjoint 4-component spinor y=y+<f**' in Minkowski space, the adjoint spinor x i n 6-dimensional space can be defined by the requirement that X X transform as a scalar. Thus, a rotation (5.111) X * X' = DX. generated by an operator D defined i n terms of coefficients E^B as i n (1.44), (5.112) D = l - ( i / 2 ) E A B < r A B should be accompanied by the transformation (5.113) X * X' = where now (5.114) D-1 = I+(i/2)E A B<r A B . Writing X i n the form (5.115) X = X +A constitutes a definition of the "adjunction operator" A. If one subjects A to the condition (5.116) A 2 = I i . e . A - 1 = A then i t can be constructed, as follows. Taking the adjoint of (5.111) gives (5.117) X 1 = (DX ) +A = X +D +A = *V 1D +A = XAD+A and, by comparison with (5.113), yields for A the condition 77 (5.118) D"1 = AD+A or D + = AD' 1A . Substitution of (5.112) and (5.114) shows that this amounts to requiring (5.119) (<rA3)+ = A<rA*A . Thus, one can alternatively define the adjunction operator A by the conditions (5.120) AjSAA = - ( / A ) + ; A 2 = I , - B. because the relat ion (5.119) follows then, -AB\+ = _ t * l k \ t t A B B_ * B ok*+. (5.121) ( C ) = - ( i A ) ( / A / B - / y Y = ( i A ) ( y S A + / B + - y « B + / A + ) = A < r A B A The operators (5.92) have the rea l i ty properties (5.122) / a = -(y8 a ) + for a=1,2,3 ; / 4 = / ^ ; y3 5 = - / 5 + 5 / 6 = j 8 ^ . The conditions (5.120) are thus obviously met by the construction (5.124) A = - i ^ 4 j J 6 = a <r3 . In terms of the 4-component spinors X u and X^ defined in (5.106) the adjoint spinor has the form (5.125) X = ( X u . - Kd) with Xu,d g i v e n hV X ^ . d * • Using jl 71 the set of 8x8 7-dimensional rotation matrices orQR can be defined by augmenting the set <r A B by the matrices (5.126) <r7A = U /4) A ] . In accordance with the prescription (5.23) write now X=\J~'X_ where the operator U~^  is given by (5J27) U"' - * y * ^ 7 . ^ ^ - ^ ) - ^ ^ - ^ ) _ Now, with the representations (5.94) and (5.96) one has here - ( i + y 5 ) y k o 0 ( i - V 5) o-k and with the representation (5*110) one has also 1+1*5 o v. (5.128a) 0 - ^ - ^ = ( i /2) (5.128b) - 6 ? 5? - ( i /2) These operators are nilpotent with exponent 2, (t=1,2,3,4,7) (5.129) (cr 6 t-<r 5 t) 2 = 0 , and therefore one has the simpler representations (5.130) u-1 = i + i z ^ o ^ W * ) ; u = i - i Z t ( < r 6 t - < r 5 t ) . The matrix i <f^ ^ is given e x p l i c i t l y by (5.131) i <T65 = £ d i a g ( I , -I, -I, I) , I=diag(1,1) , so that grouping the components of X into four sets of 2-component spinors X 1 t 2,3,4» (5.132) X = f > h A (n -K J T there are two 4-component f i e l d s , ¥> and <p (5.133) Y = which correspond to dif ferent eigenvalues of i °*5^, v i z , X 0 =f , X]=-j respectively. As in the discussion of the vector f i e l d , one may inquire whether i t i s possible to dispense with the need for projection operators A E, E corresponding to the subspaces to which these two f ie lds belong, in order to write the transformation law for each type of f i e l d sole ly in terms of components in i t s own subspace. In each case, comparison i s to be made with the transformation law for a 4-spinor in 5-dimensional space, spanned by (t=1 ,2,3,4,7) , where the representations ° " (5 ) k 7 . ^ (k, j=1 ,2,3,4) can be of either of the forms (5.134a) <T ( 5 )*7 = <i/<o y5)] , ^ k j = \j>kt K j ] (5.134b) <r- ( 5 ) k7 = ( i /4) [>Jk,(+tf5)] f £ - ( 5 ) k j = ( i /4) [ > t y j ] . 79 Substitution of the representations (5.9*0 and (5.126) into the c l a s s i c a l f i e l d variat ion (5.135) I X = - ( i / 2 ) E A V A B £ and application of the operator U, leads to the transformation laws for 9 and (p (5.136a) Tp = - ( i / 2)€ j k<r ( 5 ) j kf+(T ( | ) ^ + i ^ k [ - 2 i ( | ) z k + 2 Z t < r ( 5 ) k t ] ( ^ (5.136b) 19= - ( i / 2 ) e j k(f ( 5 ) j k < ^ + ( T ( - | ) ^ + i j 5 k [ - 2 i(4 ) z k + 2 z t o i - ( 5 ) k 1 . ] f + + 2 i f k ^ ( 5 ) k 7 ? * Hence i t i s possible to avoid the use of a. projection operator E , since the transformation law for y5 involves only <f> i t s e l f . However, e q . ( 5 J 3 6 b ) — A for S (f> contains <j0 in the f i n a l term, and so the use of a projection A operator E for non-vanishing components <p cannot be avoided without devising a covariant constraint on X which has the effect of eliminating four of the eight components of X . A constraint of this type can be written (5.137a) { \ k f k - I ? ^ ) % =0 , which i s the generalization of the same condition on L=0 (the specia l case L=0 i s treated in e . g . Todorov 1973). In terms of the f i e l d X , this constraint reads (5.137b) | u ( | J 6 + / J 5 ) -y_= 0 or simply (5.137c) <f = 0 . Thus i t i s possible to ent irely do away with the necessity of using projection operators, but only at the price of eliminating hal f the f i e l d components. 80 6 0 Two-Dimensional Conformal Group 6-1 . Four-dimensional analogue of manifestly covariant formalism An ex 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 sections for the case of a f i e l d with non-vanishing 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; %jk=diag(-1,-1)], the "conformal group" i s the 6-parameter restriction of the conformal group in Minkowski 4-space, (6.1) S y k = ot k+ 4k^+<ryk+2yk(|3 y)- ^ ky2 (j=1,2; k=1,2) , characterized by the 6 infinitesimal parameters , et , £ =~ £ .cr-. fi , p Then defining (6.2) ^ k = *yk , ^  = (L-* 2-x. 2y 2)/(2K) . v£ = (L+ * 2- * 2y 2 ) / ( 2 *-) the infinitesimal rotations (6.3a) ^ A = E A B ^ B , SAB=diag(-1,-1,-1,+1) , A,B=1 ,2,3,4 (6.3b) E^ k = £ j k , E^3 = c A , E ^ = f E 3 * = ^ generalize (6.1) to the variations (6.4) Syk = «k+$k^+<ry*+2y\fSy)-i3k(yZ-><-2l) . 2 81 To visualize the effects of these transformations i n the 4-dimensional space, i t i s convenient to suppress one spacelike coordinate, say y 2, by setting y 2 - 0 . Then the transformations (6 .3) read e x p l i c i t l y (6 .5a) 1^ = -fcV^+OcV)^ (6.5b) ^ 3 = ^ 1 ^1)^1+^ (6 .5c) = (*1+(J1)*l14wl3 . For pure dilations «^ =/3l=0, one has (6 .6) Sa1 = o , i f =^4 t ^ 4 = r ^ 3 and the motions i n y^-space correspond to motions i n 4-space on the hyper-quadrics "\A1A3L along the intersections with the planes *{] =constant. Similarly, when<r=0, o j - ^ = 0 , the motions are along the intersections with the planes ^-j^onstant, and, f i n a l l y , when<r=0, ^ + ^ = 0 , the motions are along the circles that delineate the intersections with the planes "Inconstant. The simplest case L=0 i s i l l u s t r a t e d i n Figure 2 . 6 - 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 (A=1,2,3,4) by u t i l i z i n g the 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 for a 4-component spinor X("|) need not be translationally invariant, 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 Sitter space (Dirac 1935). which in this 4-space assumes the form* *This equation has relevance i n the usual spinor theory i n Minkowski space y K (k=1-4) because one finds that 2 * - % ^ = iUJyj)(* k* k)%iy k > k r so that for wave functions that are homogeneous functions of y k , the Weyl equation for neutrinos can be written in the form (6.8). Figure 2. Examples of motions induced on the nul l surface L=0 in a 3-dimensional space as a resul t of conformal transformations i n 1-dimensional space. 83 (6.8) ((rAByu V i X )* = 0 ( A. B = 1. 2.3,*. ^ V * ! 8 ) where X i s a c-number which can be a function of L without affecting the r o t a t i o n a l covariance of the equation. Written i n terms of the d i f f e r e n t i a l operators (6.9) m AB t h i s equation becomes (6.10) ( t f A t f B m A B - i 4 i ; \ ) X = 0 . The Dirac matrices can be written i n Weyl representation as tf S * a 0 - £ <r 0 for a=1,2,3 ; & 4 _ 0 I 1 0 (6.11) tf5 = tf1 tf2 tf3 tf4 = i i 0 0 - i i where the 2X2 P a u l i matrices have the standard form (6.12) 1 0 1 0 - i s 1 0 i 0 J "3* ' i o , . I = .° -1. .0 1. Substitution of the representation (6.11) into eq.(6,10) y i e l d s an equation of the form r G+2iX 0 (6 . 1 3 ) H+2iX *2 = 0 **** A where Xj £ are 2-component f i e l d s over \ , and G and H are the 2X2 matrices (6.14a) G = - i m 1 2 + m 4 3 -m^+m^-im^-im^ m^ ^ +im^2'-iJn22'^'14l im^-m^ (6.14b) H = -i m 1 2 ~ m 4 3 -m 3 1-m^-ira 2 3+im 4 2  m31 - i m42 " i r a 2 3 " I I 1 4 l 2 + m 4 3 84 The form of the f i e l d equation (6.13) can be simplif ied by defining the 3-vectors (6.15) 1 = (n^.m-^ .m^) i n = (m^ »m42 ,m43) and the 3-vectors (6.16) 2 = i(i+in) • 4' = id- 1") w i t h J± = J1+J2 • 4 = Jf+J2 1 satisfying the commutation relations 0a"3b] 1 fabc Pl'Ji] = 1 fabc -c In terms of these operators, the matrices (6.14) can be written < ^ ' -2iJ 3 -2ij_ (6.18) G = -2ij + 2ij 3 H = -2ij | - 2 i J l 2 i j » 6-3. Infinite-dimensional representation Since the operator i(mAg+ <rAg)(mAB+ cr^ ) is a Casimir operator for the Lorentz group, one can f ind basis functions for infinite-dimensional irreducible representations by considering eigenfunctions of -fra^gm^. •These w i l l be functions X ^ ^ e , ^ , « tR) of the spherical coordinates 6,<p,<x,R defined by (e .g . Xursunoglu 1962, p.256) f "j 1 = r sin6 coslj) I 2 = r sin© sin? (6.19) R cosh** R sinh < r =JR sinh«t (R c o s h * 4 v for *| ?/0, where the f i r s t and second alternatives correspond to timelike 2 2 (R / 0) and spacelike (R < 0) in terva ls , respectively. These variables have the ranges (K 9$ TT , 0 2TT, 0$ * < «* , and 0<CR «»a . Written in terms of 1 and n, the two Casimir operators of the Lorentz group for the infinite-dimensional representation are 85 (6.20a) f i^gm* 8 = 2 ( f + j / 2 ) = ( l 2 - n 2 ) and (6.20b) - ( i /8 ) £ ^ m ^ Q = f - i , 2 = i l - n = 0 . Hence, to specify completely an irreducible infinite-dimensional representa-t ion one need consider eigenfunctions of the f i r s t invariant operator only. In terms of the variables (6.19), for the case R 2> 0, this operator reads exp l ic i t l y (6.21) i ^ A g i / 5 = -sinh2c< l 2 + ( } 2 / o<.2) + 2coth* . where (6.22) l 2 = - s i n - 1 6 (> /I 9)[sin6 O 0)] + sin"2e ( } 2 / > (f2) . The eigenvalues of the square of the angular momentum operator are denoted by <r (« - +N-2) in an N-dimensional Euclidean space with <r* r e a l , so that in this 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 in the Casimir operator, and eigensolutions X are independent of R. Thus, writing (6.24a) X = Q ( * ) Y lra(6.9) , where Y^ m i s an ordinary spherical harmonic, one can solve the eigenvalue problem (6.23) by making the substitution (6.24b) Q(< ) S s i n h-2* P(<* ) , where P( <<.) is 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 lenkin 1964) (6.25) { 0 - z 2 ) O 2 A z 2 ) - 2 * U h 8)+(fnr)(3/2 +<r ) - [ ( - l-l ) 2 / (1 -z 2 ) ] }p = 0 so that comparing with the equation (6.26) { (1 -z 2 )0 2 /^z 2 ) -2zO />0+^(^+D-^ 2/(1-z 2)}p^ (z) = 0 86 one finds that P(°0 is the associated Legendre function (6.2?) P ( * ) = P T ^ f ( c o s h * ) / and is real for real <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 fe rent ia 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 o=0 corresponds to the remaining Casimir operator. With l ^ l ^ + i l g , n+^+ng, the transformations are given by (6.30) 1 ± X ^ = [(l+m+1 )(lTm)] ^ X c l a ; 1 3 Xclm = *X c l l n and (6.31a) n + X c l m = [(l-m)(l-m-1)]* c x X c > W f B f 1 + + [(1+»M)(1-M«2)]*C 1 + 1 X C t l + 1 f m + 1 (6.31b) n . X c l m = - [ ( l + m)(l + nu1)]* C x X C i W ( B . , -- [ ( l - m + 1 ) ( l . m + 2)]ic l + 1 X e j l + l t B u l (6.31c) n 3 X c l m = [(l-m)(l+m)]* ^ -- [ ( l * 1 ) ( l . m + 1 ) ] ^ 1 + 1 X c , i + 1 , r a where (6.31d) Cx = i [ ( - c 2 + l 2 ) / ( 4 l 2 - l)]i . For real c , the representation i s unitary i f the number in the square brackets is posi t ive , for 1=0,1,2,...; i . e . the representation is unitary i f c 2 4 1 . 87 6-4. Solutions of the spinor equation The spinor equation (6.8) can be iterated to give an equation containing AB the operator mAgin , by using the operator identi ty (val id only when A has the range 1 ,2,3,4) (6.32) [<r- A B^ A^ B-i(X+1 ) [ [<r E F n E 5 " ( 1 /8)*ABmAB+A( X+1) • Thus a 2-component solution X^, as defined by (6.13), may be found by reversing the order of the factors in (6.32) and writing (6.33) = [G-2i(A+1)] -^clra 0 so that, by virtue of the defining relat ion (6.23) and the ident i f icat ion (6.29), one has (6.34) [G+21A] X J = [l-c 2+4>(> +1)] % . Hence one finds a solution (6.33) of the spinor equation (6.8) for 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 2 = 1+4A(A+1) . Simi lar ly , 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 relations (6.30) and (6.31), the (unnormalized) solution (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 )]Xeln+[(l-m)(l«m)] ^ CjC^yf[(l+m+1 )(l-m+1)]*C 1 + 1X < l + 1 | B - i [(l-m)(l+m+1)] ^ w T + Ol-mJd-m-l)] ^ X ^ ^ + 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 ind that this spinor function sat is f ies ( 6 . 8 ) provided the value ( 6 . 3 5 ) of c i s selected. Another form of ( 6 . 3 3 ) may be found by making the alternate choice cc,l,m+1 J ( 6 . 3 8 ) fy = [G-2 i ( A + 1) ] v. J Cigl4 * ) ( l - » + 1 ) ] ^ c l m - 1 ^ l + m)( l + m-1 ^ C ^ ^ ^ - f w i ) ( l - m + 2 ^ C ^ ^ J £ - 2 ( * + 1 3 W [U-*)U+*>]Vc.1-1 ,m + [< 1 + I 0 + 1 X 1 - * * 1 ) J i c l + 1 \ , 1 + 1 ,m 0 0 but using the re lat ion ( 6 . 3 9 ) Jo" lo* ^ ^ J ? e l . B , d * *9 « Su. one f inds that (6.40) | ( s i n 0 X * ^ dG dy = -4[(l-m)(l+m+1)]2 ( \+1) so that for A^-1 these solutions are not independent? The quantum f i e l d operator can thus be written as the most general l inear combination of a l l types of solutions exhibited above, multiplied by creation and annihi lat ion operators. ***** The above computations serve to show that solutions "X. of the spinor equation can be found in terms of the solutions of the eigenvalue problem for the scalar Casimir operator m^gm^5. Since these eigensolutions are independent of R, one finds that the spinor solutions X sa t is fy ( 6 . 4 1 ) R( V > R R = \ k \ Z = 0 , i . e . that spinor solutions are homogeneous functions of degree zero. *The analogue of this situation occurs when solving the Weyl equation ( i « r 'V - i> t )^ = 0 by this method. One finds Ox) ( i £ * l - i \ ) [ ( i £ * I + i * t ) e ]5 =0=(i!*I-i\){(i!"I+i*t) jipx^ provided P^ = - |p_|t and the f i r s t solution i s proportional to the second i f either P 4 = | £ | or P4 =-|p_| in both. 89 This fact can now be used as a guiding principle in attempting to f ind solutions in terms of the more famil iar variables y K , x , L. 6-5 . Spinor solutions in "Minkowski" coordinates In terras of the Euclidean coordinates y K (k=1,2) — analagous to the Minkowski coordinates — and the variables X and L, given by (6.2), the f i e l d equation (6.8) may be rewritten as an equation for the f i e l d (6.42) X = u X ' ( y k . H ,L) . Substituting the 4-space analogues of the relations (5.3) between the di f ferent iat ion operators $ A and the operators one finds (j,k=1,2) (6.43) (T^^A^B 5 2 o - J k y j ^ k +i(o- 6J+«r 5J)> j + + i ( <r 6J-cr 5 j) [( * " 2 L - y 2 ) ^ k - y X V ] + + c r 6 5 ( y k } k - • The operator 0 i s given by the 2-space analogue of (5«23). (6.44) u = e - i y k i r k + i ( L / H 2 ) H 5 ( k = 1 f 2 ) and in the representation (6.11) the matrices TT , n , TT are written i n terms of the 5-dimensional spin matrices (6.45) as ^ r s = L*r» * s ] (r.s=1,2,3,4,5) TT 1 = 0 - ^ - ^ = 0 i 0 0 0 0 - i OJ (6.46) 0 1 0 0 0 0 0 0 1 o J 1 0 0 0 L 0 -1 0 0 90 One finds that a l l f ive T T , are nilpotent to the power 2, (6.47) (-rrk)2 = 0 ; (TT5)2 = 0 so that one has the simpler representations (6.48) U = I - i y k T T k - i x - 1 L 2 T r 5 ; U" 1 = I + i y k Tr k + i*.- 1L2 TT . The operator (6.44) has the effect of transforming the f i e l d equation into a form that i s t ranslat ional ly invariant i n the coordinates y^. The f i n a l terra in the exponential in (6.44) has the effect of ensuring that for Lj^ O the generator (m^\i"mjyi'¥ ^i^' fy^) * s transformed into a form which can be interpreted as a generator of special conformal transformations. Keeping i n mind that U depends upon y k , X - , so that (6.49) ^k< u" 1) = i i r k • = - i X - 1 L ^ 7 T 5 , application of the indicated s imi lar i ty transformation to the f i e l d equation y ie lds (6.50) 0 (< r A B v ( A 2 B +iA)u- 1 = i(<r4k+<r3k)^k + * - 2 L T T k ^ k - ^ - 1 i > < r 5 k } k + ( r 4 3 K V x - " 1 L ^ i r 5 v A . V ( i / 2 ) ( <rkk+<Pk)TTk+$*~h* 7f5+i A where use has been made of the commutation relations (6.5D fo^O; [TT5.<r.k]=0 ; [ iT^ ^  =-i ^ , [-Tr5.6-43]=iTT5 ; [ V ^ - ^ J ^ i ^ ; \jTy ( <A1+ <r3j) T l J=2 I f one labels the components of X by suffixes 1,2,3,4, *1 (6.52) X = X 2 then the f i e l d equation can be written in the form of two coupled equations for 2-component f ie lds if and (f, defined by (6.53) A . CO = * 3 • T 91 in the form (6.54a) (V- 2L(-<r 2-b l + <r1 * 2 ) -<r (L* <V 4 ) ] <£+ + [(|H-1L^(<r1 11+(T2^2)+i(X-iK^K)]^ = o (6.54b) [ i * - 1 i £ ( < r 3 i 1 + t r 2 ^ 2 > + i ^ + 1 + £ * ^ K ) ] + +i ( < r 2 > o2)</> = 0 where 0~1 2 3 a r e ^ n e P a u l i matrices as given in (6.12). Writing out the matrix i <T 3^ in the representation (6.11) ( this matrix corresponds to the matrix i s ^ in the 6-space version of the theory), one has (6.55) i ^ 3 = i ^  >53 = diag(|,-i,-£.£) . Thus the f i e l d <p corresponds to the eigenvalue X Q = 2 of the matrix i t r ^ , A . and <p corresponds to the eigenvalue A^=-f. The space of f i e l d s X comprises the two parts belonging to the orthogonal projectors corresponding to these two eigenvalues. I f , i n analogy with the procedure of section 5-2, one makes the i d e n t i f i -2 -cation r=(L/K ) 2 , then looking upon L as a parameter in the def in i t ion of r the analogue of (5.35) i s (6.56) i ( 3 ) = i<r 43 since one has (6.57) r br =-- K V , i . e . one goes over from using y k , H,L as coordinates to using y k , r , L . Since r i s a variable which i s homogeneous of degree zero in the 4-space just as yk i s homogeneous of degree zero, (6.58) n A £ A (r) S (vO„ +2LbL) ( L * / K ) = 0 , then from the discussion of solutions of the spinor equation in spherical 92 coordinates, one expects to be able to form solutions X written in terms 1 2 of y , y and r only, i 0 e , solutions in 3-dimensional space, A Hence, one can say that each of the two f ie lds <P and 0° belongs to  a def ini te value X,i of the scale dimension in 3-dimensional Euclidean space, spanned by the coordinates (y , r ) , and therefore the connection with repre-sentations of the conformal group is not lost by generalizing the coordinate variations to the case L^O (albeit one must now consider representations of the conformal transformation group in three, and not two, dimensions), 6-6, Introduction of the mass concept for eigenfunctions of -^(3) So far this example has served to indicate how one should proceed when considering the transformation proerties of f ie lds with non-vanishing sp in . I t remains, f i n a l l y , to demonstrate that for such f ie lds 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 this end consider eq .(6.54), written out component by component in terms of the variables y k and r . The four equations for the four components of X can be recombined into the following simpler set of equations for combinations of components of X : (6.59a) K i r 2 V r 2 } 2 ) ) ( 2 + i ( i r ^ r + X ) ( X r i r * 3 ) = 0 (6.59b) i(i V^2> < X i - i r X 3 ) " t o r>*2 = 0 (6.59d) (6.59c) i ( - i r 2 } 1 + r 2 } 2 ) X 3 + i ( | i O r + A ) U ^ + i r ^ ) = 0 . Each of these equations can be iterated by operating with ( i ^ ^ + ^ 2) from the l e f t , with the results (6.60a) { - r 2 ^ k ^ k + ( r > r ) 2 - 4 r > r ^ X ( X+2)}(x 1- irX 3) = 0 (6.60b) ^- r2 ^ k \ + ( r ^ r ) 2 . 2 r > r - 4 A ( X + 1 ) } X 2 = 0 (6.60c) {-r2 >k V ^ V 2 - 2 ' V ^ A C X + O ^ = 0 (6.60d) $-r 2 ^ k ^ k + ( r ^ r ) 2 . 4 r > r - 4 A U « ) } ( * 4 + i r X 2 ) = 0 . Now sine© one has (6.61a) r ^ r ( r X ) = r X + r 2 ^ r X (6.61b) ( r ^ r ) 2 ( r X ) = r [ l + 2 r i r + ( r > r ) 2 ] x . eqs.(6.60a,d) read (6.60e) { - r 2 ^ k ^ k + ( r ^ r ) 2 - 4 r > r - 4 X ( X + 2 ) } X , -- i r {-r 2 ^ _ 2 r ^ r_2, X 2 _ 8 X _3j ^ = 0 (6 .60f ) f - r 2 ^ k ^ k + ( r > r ) 2 - 4 r o r - 4 X ( X + 2 ) } Xk + +ir £ - r 2 ^ ^ k + ( r ^  r ) 2 _ 2 r > ^ 2_8 X -3} * 2 = 0 . Multiplying eqs. (6 .60b,c) by i r and subtracting from or adding to (6 .60 f ) or (6.60e) respectively, one finds that the above equations simplify to (6.62a) { - r 2 ^ k \ + ( r ^ r ) 2 j * r } r - 4 X U + 2 ) 5 X] - i r ( - 4 X -3) ^  = 0 (6.62b) {-r 2 ^ k > k + ( r ^ r ) 2 . 4 r > r - 4 X ( > + 2 ) ] ^ + i r ( - 4 A - 3 ) X 2 = 0 . Hence, making the choice (6.63) X = - 3 / * . A the iterated equations can be written in terms of the f ie lds <f and (f , defined by (6 .53) . as 94 (6.64a) $-r 2 > k> k+(r^ r ) 2 - 4 r W (15/4)} y> =0 (6.64b) { - r 2 > k ^ k + ( r ^ r ) 2 - 2 r ^ r + ( 3 / 4 ) } ^ = 0 . As well as on the grounds of simplicity, the value A = -3/4 i s dictated by another c r i t e r i o n . The second Casimir operator of the Lorentz group, similar to (6.20b), does not vanish i n the 4-component spinor representa-tion, but instead one has (6.65) - ( i / 8 ) £ A B C D ( m A B + ( r A B ) s -b- 5{<r A B 7 A aB - (3i/4)} (where Si234 = +^)» ^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 (6.65) i s equal to the constant multiplier zero times the f i e l d . This multiplier must be zero since acting on X this operator forms a pseudoscalar when multiplied by X + * \ whereas X +Jf^X is a scalar. In eqs.(6.64), using as t r i a l solutions the factorizations fF(r) ^ ( y ) (6.66a) if |F'(r) X 4(y) (6.66b) f = F(r) X 3(y) lF'(r) X2(y)J 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>r)2-4r*r+(15/4)] F(r)}/[r2F(r)] (6.67b) kkakX4(y)]/X4(y) = i[(rir)2-4r^r+(15/4)]F«(r)]/[r2F'(r)l (6.67c) |^ kX 3(y)|/X 3(y) = ffirir)2-2r^+(3/4)jF(r)V[r2F(r)] (6.67d) [^ kX 2(y)]/X 2(y) = ^[(ri r) 2-2rt r+(3/4)]F'(r)}/[r 2F(r)l . 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 k V K 2 ) > M ( y ) = 0 (6.68b) {r2K2+(r> J r) 2-4r^ r+(15/4)}F(r) = 0 (6.68c) ( ^ k + K ' 2 ) X 4 ( y ) = 0 (6.68d) ^r2K'2+(ri r )2-4r) r +(15/4)}F ' (r ) = 0 (6.68e) O k ) k +K 2 ) X 3 ( y ) = 0 (6 .68f ) [ r 2 K2 + ( r > r )2 .2r i r +(3 /4)}F ( r ) = 0 (6.68g) (^ k ) k +K | 2 )^ 2 ( y ) = 0 (6.68h) }r2K'+(r} r )2-2rV (3/4)}F ' (r) = 0 • A A where K,K' ,K,K* are four possibly complex constants. The equations for the functions of r can now be cast in the form (6.69) {(z\)2+z2- v 2} z ^ ( z ) = o § vrhich i s the equation for a cylinder function Z ^(z ) ; i . e . Z y i s any l inear combination* (6.70) Z w = a j y + b Y w of Bessel functions of the f i r s t kind J y , and of the second kind Y y . To do t h i s , one uses the relations (6.71a) r& r [ rP F(r)] = rP(p+r* p) F(r) (p= some rea l number) (6.71b) ( r i r ) 2 [ r P F(r)] = rP[p2+2pri r+(r^)2]F(r) so that e . g . with p=2, defining a new quantity G(r) related to F(r) by (6.72a) F(r) = r 2 G(r) then G(r) sa t is f ies (6.72b) r2^K 2r2+Kr[cVa(Kr)] 2 4 } G ( r ) = 0 . Hence G(r) i s a cylinder function Z v ( K r ) , with order v given by (6.72c) V 2 = \ ; i . e . v^=+J o r - f However, since in (6.70) the numbers a and b are arbitrary, and since The form a j y +hj_u is not used since J y and are not l inear ly indep-endent for a l l values of v . 96 J+i, Y+i happen to bo expressable in terms of elementary functions (6.73) J i ( z ) = (£T T Z ) - £ sin(z) = Y . i (z ) ; Yi (z ) = - ( i l T z ) " 2 cos(z) = - J _ A ( Z ) , then any Z i (z ) can be written as some Z i ( z ) , so that considering only v =-| yields a l l solut ions. Thus solutions X of the second-order equations (6.64) are given by* (6.74a) X, = r2 Z i ( K r ) X,(y) (6.74b) (6.74c) (6.74d) A .A X 2 = r Z'i(K'r) 7(2(y) X3 = r L(Kr) y3(y) = r 2 Z'i(K'r) X4(y) 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) i r 2 r^ kX 2 +(i/ 2)(ri r-3/ 2)(X 1-irX 3) = 0 (6.75b) ir^ k( X l-irX 3)+(i/2)(-ri r+i)X 2 * 0 (6.75c) - K k } k ( X4+irX 2)+(i/2)(-rV4)* 3 = 0 (6.75d) -ir 2r ki kX 3+(i/2)(ri r-3/2)(X / ++irX 2) = 0 where by convention extra zeroes are added to the x's to make up 2-spinorsf (6.76a) X , = F0 ^ L X2 , X o -. 0 . , X4 -4J ""In the massless case the r-dependence is given by (ar^/2+br3/2) for <p and (ar3/2+br2) for p . 97 and (6.76b) r 1 = , r 2 = i r 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 k } k V(i / 2)(ra r -3 /2 ) X 1 = 0 (6.7?b) ( i / 2 ) r r k i k X r f ( i / 2 ) ( r ^ r - 3 / 2 ) = 0 . Now making use of the formula for the derivative of a cylinder function, one has (6.78a) rd r[r 2Zi(rK)] = 2r2Z±(rK)+r3lc[-(2Kr)-1 Zi(rK)+Z.i(rK)] (6.78b) r ^ r^Z^rK')] = 2r 2Z'i(rK ,)+r3K«[-(2K ,r)- 1Zi(rK ,)+Z»i(rK«)J , so that eqs . (6 .77) read (6.79a) 0 = r3[Zi(rK) r\x t (y)+K'z:i(rK') X 4(y ) } + ( 2-i - 3 / 2)r 2Zi(rK')^(y) (6.79b) 0 = r 3 ^ l ( r K ' ) r k ^ k X 4 ( y ) + K Z . A ( r K ) X 1 ( y ) ] + ( 2 - - | - 3 / 2 ) r 2 Z A ( r K ) ^ 1 ( y ) , To find solutions of these equations they must be reduced to a simpler form. From the discussion following eq . ( 6 .55 ) , one should look for a form which allows one to write (6.79) as a single equation for the 2-spinor f i e l d </>(y), defined by ( (6.80) ?(y) Aiming at an equation i n the form of the analogue of the Dirac equation in the 2-diraensional Euclidean space, (6.81) (-i P k^ k+m) p(y) = 0 . my? being a "mass" term when m i s positive and real, then from eqs.(6.68a,c) one must put (6.82a) K'2 = K 2 = m2 . According to (6 . 79 ) . there must be a relationship between Z and Z* for 98 (6.81) to result. Taking K,=+K=m, this i s (6.82b) Z«_A(rm) = iZ^(rm) , or equivalently (6.82c) Z'i(rm) = -iZ i(rm) 2 ~ 2 Defining coefficients 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 = i a , c = - i b . (The case K , = -K= -m goes the same way, but with the result d= -a, c= -b.) So one arrives at the result that in order to find f i e l d s X 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 2 2 where a and b are any complex numbers. In' fact, up to this point there seems to be no reason, on the basis of an examination of the cylinder functions, to r e s t r i c t the number m to the physically interesting case i n which i t takes on values on the non-negative real axis. In any event one cannot expect functions defined over 2-dimensional Euclidean space to bear any physical significance, although the methods developed above can be carried 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 real non-negative number. At this stage, i t suffices to note that an examination of the remaining equations (6.75) leads to another Dirac-type equation for 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 spin-i equation in this Euclidean-space model. The expectation that this feature i s carried over to the" case of fields in Minkowski space as wel l w i l l be borne out in Section 8. One could then ident i fy the two species of part icle with the muon and the electron, or in 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 Principle 7-1. Conservation laws on hyperboloids The 5-dimensional integral over a region TL^ on the surface ^ A " | A =L i s written in terms of the boundary integral 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, their corresponding area elements are found to contain S(*| A^ A-L) when the integrals are written in terms of d 6 V 7-2. Alternative Euler-Lagrange equations The action integral i s therefore also defined to contain %(^ A ^ A - L ) d , to f a c i l i t a t e using the Stokes 1 theorem. An alternative Euler-Lagrange equation i s found which results from using conjugate momenta d i f ferent 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 sa t is f ies a generalized divergencelessness condit ion. Integrals over J l ^ y ie ld f i f teen conserved quant i t ies. 7-4. Modified Schwinger action pr inciple The f i f teen conserved quantities are ident i f ied with the generators of the group of transformations by varying the boundary of the action integral in the usual fashion, except that translat ional 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 in section 7-4 for an arbitrary f i e l d variat ion yields the canonical (anti-)commutation relat ions as self-consistency condit ions. 101 8, Special Cases 8-1. Scalar f i e l d The wave equation for the scalar f i e l d X i s derived from the 6-dimen-sional action pr inc ip le . The single arbitrary number C which appears in this equation i s equal to the eigenvalue of the only non-vanishing Casimir operator of the rotations in 6-dimensional space. Solutions are found in terms of powers of the auxi l iary coordinate r and cylinder functions whose index depends upon C. These solutions sat is fy the Klein-Gordon equation with mass in 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 in 5-dimensional space for a f i e l d variable Y which i s related to X by mult ipl icat ion 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 resul t from evaluating the angular momentum tensor can be ident i f ied with those assoc-iated with conformal invariance in 5-dimensional space. At this point i t i s not necessary to have C= -15/4, but the next section jus t i f i es this value. 8-2. Green's function for scalar f i e l d equation Mack and Todorov's results on covariant Green's functions of the form [( ^ ^ • ^ 2 ) ] 4 1 are extended to the case " ( ^ A=^0, where they take the form which results from this generalization i s compared with the standard result results from the 6-dimensional formulation, Adler 's work on rotat ional ly covariant Green's functions off the nu l l hyperquadric in 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 in section 8-1, i n order that the Green's function be of the simple form which generalizes the resul t of Mack ) 2 ] q . The Green's function in 5-dimensional Minkowski-type space given by Gel'fand and Shilov, which has q= -3 /2 . To see how this number 102 and Todorov. Thus, the point made in this section i s that one can f ind the Green's function for the scalar wave equation for 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 for the spinor f i e l d T C i s derived from the 6-dimensional action pr inc ip le . It i s shown that this spinor equation d i f fe rs from an alternative suggested by Barut and Haugen in that i t can be derived from a real Lagrangian density. It i s shown how solutions of the spinor equation can be found by i terat ion provided the single arbitrary number X i n the equation i s given by A = -5 /4 . Then solutions are given in terms of powers of r and cylinder functions of r which depend upon parameters ra and m, multiplied by functions of the Minkowski space var iables. Using the methods developed i n Section 5» i t i s shown that there are two types of solut ion, each of which sa t is f ies an equation i n Minkowski space which i s precisely the Dirac equation when m and m are ident i f ied.with two part ic le masses. The value A= -5/4 i s just i f ied by showing that i t 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 relations of the f i e l d are derived from the results of Section ? . Thus, i t i s shown in this section that two types of massive f i e lds can be grasped simultaneously by a single equation in 6-dimensional space, so that the 6-dimensional formalism allows one to describe the observed leptons in a simple fashion. 103 8-4. Green's function for spinor f i e l d equation The spinor free f i e l d equation i s iterated to the form of the scalar wave equation, so that the Green's function i s given in terms of the spinor f i e l d equation matrix-differential operator acting on the scalar Green's function. This iteration allows one to derive the value of X from the value C= -15/4 found i n section 8-2, and one has again A= -5/4; i . e . this j u s t i f i e s setting A = -5/4 i n section 8 -3. To make sure that this spinor Green's function for "| ^*} kf0 makes sense, the analogous expression i s con-structed on the unit hyperquadric i n 5-dimensional Euclidean space, and i t i s found to agree with Adler's result i n that case. The 6-dimensional Green's function i s then compared with Adler's two guesses for the case *IA"1A=0, a n d i t s n o w n t h a t S o i n g t o 1A^A^° e l i m i n a t e s some of t h e d i f f i -culties that Adler encounters in developing covariant scattering theory i n 6-dimensional space. 8 -5. Minimal coupling; gauge invariance with massive vector boson The minimal coupling i s carried 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 satisfies 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 identical, 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 retaining gauge invariance even with a massive vector boson. The additional scalar 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 for the vector f i e l d . It i s found that as well as the usual vector and pseudovector type couplings <V * k v f B k and <? * k B^, one also arrives at the scalar and pseudoscalar type couplings Y ^ B and ^  B i n Minkowski space. 104 PART I I I . ACTION PRINCIPLES 7. Rotationally Invariant Action P r i n c i p l e In order to 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 action p r i n c i p l e , i t i s necessary to determine the form that 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 action 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 applied to situations for which the variations of the coordinates v^ A take points on the hyperboloid *yk "\A=k ^-n^° points on the same hyperboloid. Then by studying the variations of f i e l d s on t h i s hyperboloid, one can construct an alternative to the Euler-Lagrange equations which i s applicable to situations i n which t r a n s l a t i o n a l invariance does not obtain* 7-1 • Conservation laws on hyperboloids Since one i s interested i n action integrals on the surface ^ A |\ A=L i n s i x dimensions, i t i s necessary to obtain an i d e n t i t y r e l a t i n g a 5-dimen-si o n a l i n t e g r a l over some region JTl^ of t h i s surface to a 4-dimensional i n t e g r a l taken around the(olosed) boundary Si^ of sx.y This r e l a t i o n i s given by the generalization of Stokes 1 theorem (e.g. Anderson 196?) to the case of s i x dimensions. I f fxj^Xjklt i s a fourth-rank tensor i n 6-dimensional space, then one can form a scalar quantity ^Ai A2A3A4 d*t A 1 A 2 A 3 A 4 by labeling points contained i n -n.4 by means of four parameters X-j, X 2 , A3, A^ and defining the fourth-rank tensor* (7.1) d f A 1 A 2 A 3 A 4 = ^ V L f . ^ ^ d X , d > 2 d > 3 d A 4 . *Here, use has been made of the generalized Kronecker d e l t a , which A similar element of "area" d t A 1 A 2 A 3 A 4 A 5 may be defined on the surface j l ^ by describing the coordinates o n that surface parametrically in terms of f ive variables "X.j 2,3 4 5* As a consequence of the def in i t ion (7.1), one can form the hypersurface integral which i s a scalar under coordinate mappings as well as under arbi trary parameter changes. In terms of these structures, Stokes' theorem can be written (7-3) *A,A 2A 3A„ " L , ' W j V A j d t ^ W i • As in the 3- and 4-dimensional cases, this equation may be simpl i f ied by introducing the quantities •('/*') ^ W j A 6 « A ^ (7.4c) d3 A ) = ( t / 5 D £ A ] A 2 A 3 W 6 d ^ 2 W 5 A 6 . With the introduction of these quantit ies, (7»3) can be rewritten in the more convenient form (7.5) d S A B = 2 L 5 F^ .B d S A by making use of the ident i ty ( 7 . 6 , £ A ' - A ^ + , - A 6 t B L . . . B R A r t l . . . A 6 5 ' 6 - ' ) 1 • has the following p r o p e r t i e s : a ) i t is completely antisymmetric in superscripts and subscripts; b) i f the superscripts are d is t inc t from eachother, and the subscripts are the same set of numbers as the super-s c r i p t s , the value i s *1 depending on whether the superscripts are an even or odd permutation of the subscripts; c) otherwise the value i s 0. Also, is assumed to be simply connected. By construction, F A B is an antisymmetric tensor density, and dS^g is also antisymmetric. The r ight hand side of (7.5) fflay be cast into a more useful form by selecting an area element dS A such that derivatives of F^ 3 occur only in terms of the d i f fe rent ia l generators of rotations M A 3 = i ( ^ B " ^ B ^A^* AR Then in those cases in which F can be written in terms of a vector current density j A as (7.7a) F ^ = \ A j B - r^BjA one has (7.7b) mjjfM = 2 i \^ A r l A ( 3 B j B ) . ( i „ . ^ ) ( . This form corresponds to the conservation law suggested by Boulware et . a l . (1970) (7.8) (m^- i S A B ) j B = 0 for currents j A that separately obey the condition (7.9a) 3 A J A = 0 and the transversali ty condition (7.9b) n A ^ A = 0 » because one has in fact (7.10) m ^ = 2 v l A ( m A B . i S A G ) j B . 8 i V l ^ A in those cases for which F ^ can be written in the form (7 .7a ) . The surface element dS A which brings the integrand of the 5-dimensional integral into the form n^gF^ i s found by converting the right hand side of (7 .5) into an integral over a region in 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 dS A on the surface f=fQ i s given by ( c . f . Fubini e t . a l . 1973) (7.11) d s A = d 6 ^ S ( f - f 0 ) 0>fA^ A) f = f o • 107 Then choosing the surface f=- <^2, on the hyperquadric ^ 2 =L one has the surface element (7.12) dS A = -d\ S ( ^ 2 - L ) 2 ^ A , where the integration over dS A i s to be carried out over The proof that this area element corresponds to (7.4c), with the parameters ( Ai t 2 ,3 ,4 ,5 ) = (Z .y / + t v < - ) i h a s been relegated to Appendix 2. For the area element d S A B , one must select another 1-parameter set of hypersurfaces, on the surface *\2=L. i n order to pick out a set of 4-spaces -A.^. There are many possible choices, but i t i s both convenient and useful to choose the surfaces h=(-*\^/vi )=constant. This corresponds to choosing the time t= as the direct ion in which "charges" associated with conservation laws remain unaltered, i . e . the direct ion of quantization ( c . f . Fubini e t . a l . 1973)• This choice also amounts to selecting 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 as* (7.13) d s A B = d 6 ^ f b f ( n ) l *h( A ) >• . f+L=0 h+t=0 S(f+L) o"(h+t) . E x p l i c i t l y , one has (7.14) *h ( l ) / H B " - x-"1 S*B + K " 2 ( S B 6 " ^B 5 ) I 4 and substitution of this equation and of that for f ( \ ) into (7.13) y ields (7.15) d S A B = d\ 2 v|[AK-1 {SBf+t(SB]5-SB]6)} S(-*"1 ^+t) S(-f+L) As was the case for d S A , one can e x p l i c i t l y check (7.15) by comparison with the def in i t ion (7.4b), and again the proof i s given in Appendix 2. Using these area elements, the integral identi ty (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 jm A BF A B U^ 2-L) d 6 ^ = The right hand side i s the form that the integral over the boundary jx^ takes when expressed as an integral over d ^ . Hence (7«16) represents a type of conservation law i n 6-space, and whenever the boundary integral  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). affects the form of the Euler-Lagrange equations. To this end, one must propose an action integral over the region -fig which takes into account the fact that only rotations, and not translations, are included i n the group of motions i n 6-space. Thus the transformation behaviour of any set of coordinates >^A, for which »\2=L, i s restricted such that for ^ | A , one s t i l l has ^ , 2=L. Hence the integrand i n an action integral must be constrained such that *^A l i e s on <^  =L, and this can be accomplished by defining the action I as (7 .18a) I = L , £ $ ( ^ 2 - L ) d 6 1 . 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 mAg i s the representation by d i f f e r e n t i a l operators of the generators of rotations. 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 ^ A have been included among the possible e x p l i c i t arguments of £ . 109 Taking the v a r i a t i o n S I of I with f i x e d boundary of defined by (7.19a) S X = iXyQ-ZW (7.19b) U \ A ) = 0 , one has (7.20) %I - J{(i£/&*>SX + SCm^X.) & ^ ( m A B X ) ] ] W\2-L) d 6 ^ since according to (7.19b) the coordinates themselves are unaffected by the v a r i a t i o n . Thus one can write S I as (7.21) SI = ^ ( ^ X K Z ^ C S x ^ W m ^ T ? * 3 ) ^ } Stf-L) d6"\ where the analogues i n 6-dimensional space of the conjugate momenta i n Minkowski space are defined by (7.22) v-^ = (m A B x") . According to (7.16) the i n t e g r a l of the second term inside the brackets i s equal to an i n t e g r a l over the boundary, and one can assume that t h i s i n t e g r a l vanishes since the boundary i s taken to be s p a t i a l ( y - ) i n f i n i t y 0 Setting the v a r i a t i o n S i equal to zero f o r a r b i t r a r y v a r i a t i o n SX, one arrives at the alternative Euler-Lagrange equations (7.23) frlfrZ) - m^TfAB = o • I t i s immediately apparent that the form of t h i s equation i s i n p r i n c i p l e very d i f f e r e n t from that of the usual 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) to educe the manner i n which conservation of angular momentum i s contained i n the action p r i n c i p l e • Consider the action i n t e g r a l (7.29a) and i t s transformation properties under the i n f i n i t e s i m a l rotations (5.75)» (7.24) I l A = E A B , \ B 110 which induce on the components of the f i e l d X. the transformation (7.25) tZil) " - ( i / 2 ) E A B s A B X H E A B ( ^ A ^ B -7 B *a> where s A E are the spin operators of the f i e l d X. . If the integral (7.18a) is to be invariant under (7.24) and (7.25), the variat ion in X must be a "divergence" (7.26) k £ = - ( i / 2)E A B m A g £ = m A R ^-( i /2)E A B.£ } . or (7.27) 0 £ A E A B ) = - ( i / 2 )m A B £ . When this condition i s sa t i s f i ed , one can obtain an expression for a tensor density which sat is f ies the divergence condition (7.17) as a resul t of rotation invariance in the usual fashion by considering E A B to be coordinate dependent, so that the rotat ion has the effects (7.28a) % *>% +E A 3X A B where X A G = - C i / 2 ) ( n A B + S A B ) (7.28b) m C D * » m C D X - K i D C D E A S ) ^ A B 4 E A B B t D ^ . Therefore (7.29) O*/"* E A B) = Of. A X ) X^+TT^QX^ and (7.30) frtft (-tu^8)] = ^ C D X A B where f f C D i s defined by (7.22). Application of m C D to eq.(7.30) gives (7.3D m ^ f ^ ^ E ^ ) ] - ( i - b D T f ^ ^ A B + T f ^ n ^ A B • On account of the f i e l d equations following from the action pr inciple based on X, (7.23), one can write eq.(7.31) as (7.32) ^ [ U / M m c ^ 8 ) ] = ( ^ / ^ ) ^ A B + T T C \ D X A B = = 0.£AE AB) . Insisting on (7.26) as the expression of rotation invariance, eq.(7.31) takes the form of a conservation law 111 (7.34) S a b C D =" - i f f C D ( m A B + s A B ^ ^ ^ B D ^ • Inserting (7.44) into the integral identi ty (7.16) one finds that for each A,B there exists a vanishing boundary integral of Taking the boundary to be made up of two p i e c e s a n d J ~ L \ , corresponding to times t and t ' , together with spat ia l (y_-)infinity, then (7.33) has the meaning that the f i f teen quantities defined by (7.15). It i s also possible to interpret the relat ions (7.33) as current conservation laws. This fact w i l l emerge more c lear ly below when part icular cases are examined. 7-4. Modified Schwinger action principle It i s possible to place the conservation laws (7.33) in a more general setting by examining the effect of arbitrary variations on the action integral I without rest r ic t ing attention to the case of f ixed boundary S l ^ , Thus, one expects to f ind that the conserved "charges" M g^ can be in ter -preted as the quantum f i e l d theory generators of rotations via a Schwinger action principle (Schwinger 1951). modified by allowing the poss ib i l i t y of exp l i c i t coordinate dependence of the Lagrangian density t- . In the present case the variat ion of the action i s given by (7.35) are conserved in time (for given L ) , where dSQp i s the surface element (7.36) S I = \ A * 6 »AB^+SCABX>«1A + T ^ ] -• o[(^rt)2-(L+rL)]} d 6^' - I 112 under the var iat ion (7.37) \A -> Y A = 1 A + ^ • Substituting for d ^ 1 by means of the Jacobian (7.38) J ( ^ ) = eq.(7.36) can be written (7.39) SI « f £ • ( X}^ABXt,^) J ( f . l ) $ ( ^ , 2 - L ' ) d 6 ^ - I . Since the variat ion J * ^ i s in f in i tes imal , one has (7.40) j(nM> = 1 + [}< Ti A )Ai A ] • Expanding £ • about X as (7.41) X' = X + ( ^ ^ A x ) S X+f f A B S (a A B X ) - K U / ^ ^ A ) T\A . and rest r ic t ing 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 =[^ 6{(^^/^)SX+TT A BmAB(S%)+^ §( T f ) / M A ] + In th is equation, the def in i t ion (7*19) of the var iat ion S has been used to commute S and ni^g. In the present case, one i s interested only in those transformations sat isfying (7,42). In Appendix 2 i t i s shown that in this case one can w r i t e % I as the sum of two i n t e g r a l s (7.44) S I = L6i(^/^)-mAB^A3} $ ( * t 2 -D d6>[ + + L 6 ^ D ^ C Q S* +(i/2)X[>( T f ) / ^ D ] ] S ( * | 2 - L ) d 6 ^ . Hence, provided the f i e l d equations (7.23) are s a t i s f i e d , so that the f i r s t term vanishes, then whenever S i i s ze r o , the i n t e g r a l i d e n t i t y (7.^6) s t a t e s t h a t the q u a n t i t y (7.45) F = l / (2 i ) J ^ C D i X + ( i / 2 ) £ [V S ^ C ) / ^ 1 D ] ] d S C D i s independent o f the time t . Fo l l o w i n g Schwinger, one can i d e n t i f y t h i s i n t e g r a l w i t h the quantum f i e l d t heory generator of r o t a t i o n s . From the d e f i n i t i o n (7.46a) X ' ( \ ) 2 H x U F i s d e f i n e d by (7.46b) U-1 = e 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 parameters E A B , one f i n d s (7.46c) F a ^ i f M^g i s d e f i n e d by (7.46d) [X.M A B] = (m A B +s A B)X . Thus, w r i t i n g (7.4 7a) = , S A B C D d S C D one has (7.47b) SiBo> = - | f t c % A B + s A B ) x + | S A C S B E t . i n agreement w i t h the p r e s c r i p t i o n (7.3*0• 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 ] for any variat ion e X . I f , in part icular , one chooses an arbitrary o X accompanied by no coordinate var ia t ion, % - 0, then (7.48) can be looked upon as a consistency condition which, together with the def in i t ion (7.45), can y ie ld the commutation relations for the f i e l d . Implicit in the def in i t ion (7.45) is a summation over a l l independent f i e l d components X.<t (7.49) F = l / (2 i ) J T t C D %X« d S C D . Thus, for any on the surface defined by d S ^ y ^ ' ) , o n e raus* n a v e (7.50a) % X f i ( ^ [ ) = i [ l / (2 i ) J u ^ C ^ ' H X < ( y ) d S C D ( *|«) , X )] (7.50b) ^TT / B ( ^ ) = i [ l / ( 2 i ) l f \ C D ( ^ ) S ^ ) d S C D ( y ) , T f / 3 ^ ) ] Making use of the ident i ty these relations can be written (7.52*) /^^ ) =iJ[ff.CD(V) [s^ (^ ) , -(7.52b) ^AB(^) =i J{nP(^) [u^ y) , ff^)]. -I f one assumes,further, that ( c . f . Roman 1969) 115 (7.53a) [x^) . \ X f i % 9 0 for a l l <* ,j3 and a l l vj, tf|* on the surface then (?.53b) 0= D x j - ) ' ) . ^ ) ] . + [ ? . ( - ( ' ) . * and since the variat ion i s arbitrary, (7.53c) 0 = [x^) , Xfi(v\)]_ for ^ . " \ ' on - O L ^ . Hence, ( 7 « 5 2 a ) becomes (7.54) * X ^ ) - - i ^ t y . f t C D ( f ) ] - S x ^ ' ) ds C D ( a ' ) for ^ , v '^ on -^-j^. The simplest solution for (7.54) is found by writing (7.55) [ x ^ ) . T t C D ( f ) ] . = -2 S S C D ^ " V > where ^ C D ( Y | - ^ ' ) i s defined for two points V^."]' on-^-^ by* (7-56) Jj^'gCV) ^ y l ~ V ) dSCD(/l,} = and i s thus antisymmetric in C and D. The simplest way to determine S C D ( ^ - ^ J 1 ) is by writing dS^g, given by (7 .15 ) , i n the form (7.57) d S ^ = n [A n»B-j dS(y,, K ) This singular function is introduced in analogy to the function %k(y-y*), which i s defined by (Gourdin 1969) $g(y') *k(y-y') d < r k ( y « ) = g ( y ) for y , y ' on cr ; e . g . i f the surface is t=constant, then 116 where n^, n ' A are two orthogonal unit vectors (7.58a) n A = * | A / ( K r ) , n ' A = ^ / + t ( S A 5 - $ / ) (7.58b) n A n A = n ' A n ' A = 1 ; n A n ' A = 0 ; (n )(n ^n'g-j ) = 2 . Then one has (7.59) d S ( £ , K ) = d \ (2r) S (v \ 2 -L ) $ ( H . - 1 ^ - t ) = = (-Vl3/2)dV dK. dL (2r) Uv\ 2-L) S" ( K ~1 >*^-t) . Since the product of dS C D ( Y^* ) and &CD(v^-\q') must be simply (7.60) ^ C D ( ^ - ^ ' ) d S 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 ( ^ - ^ « ) = (-2rK3)"1 $(v.-yj) S U - M . ' ) n&n«B3 . Hence the (anti-)commutation relat ion (7.55) becomes (7.62) [?p (V . ff/B(^ ')]_ = % S(l-Z') S(K-M.«)* . ( X 2 L H ^ [ A { ^ + t ^ K J B ] 6 ^ ^ For example, taking the specif ic set of values {A,B} =^(6,k)-(5,k)^ , one has (7.63) fep(vp , ff,6k(^')- ?f. 5 k (v\')]. = = ^ « Sd-Z1.) $(*--*.') (KL)"1 for equal times t '=t. The f i n a l factor can also be written in terms of r=H~^L'2, since one has (7.64a) $(*-vc») = (r«) 2 IT* S(r-r») and hence (7.64b) ( X L ) " 1 ^ ( K - V C « ) = r 3r 2S(r-r«) This completes the discussion of quantization of the 6-dimensional f i e l d theory via the action principle. The consequences of the relations (7.62) are investigated below for specific examples. 118 8. Special Cases 8-1. Scalar f i e l d The analogue of the Euler-Lagrange equation, (7.23), can now bo used to construct a free f i e l d equation i n 6-dimensional space f o r a scalar f i e l d X ( * | ) . Such a f i e l d equation must flow from the action p r i n c i p l e based on a Lagrangian density i of the form (8.1) X = £ ( 1 ,^,m A B X ) where m, 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 •£ dictates the form of a l l possible invariants which are admissible contributions to X . and i n order for the f i e l d equation to be l i n e a r i n 7- one need only consider those invariants 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 derivatives, i f only equations of up to second order are permitted. Thus, the set of a l l r o t a t i o n a l invar-iants having the appropriate form i s r e s t r i c t e d to the two functionals (8.2) I j = (m A B 7 )(m A BX ) ; I 2 = X 2 Thus the most general second-order l i n e a r f i e l d equation based on a ; Lagrangian density (8.1) has the form (8.3) a m A B B ^ X +bX = 0 where both a and b can be functions of L = ^ A V ( A without a f f e c t i n g the r o t a t i o n a l covariance of (8.3), since (8.4) m A B a(L) = m A f i (L) > La(L) = = i ^ A ^ B - 1 B *A> ( L ) V<L) " = 0 119 The value of the rat io b/a is 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 def ining, for general spin , the generators of rotations (8.5a) m A B = m A B + s A B they can be written "AB (8.5b) C = |S A R 5 A B (8 .5c) D = 1 ^ ABCDSF * A B * ° D (P. C.A\ v - A J ™AB sCD j.GHIJEF ~ ~ (8.5d) E - 2 ^ABCDEF m 1 °GH m I J * The scalar f i e l d X- i s defined, in (5*59) . by the property (8 .6) S A B X = 0 , and i t i s easy to determine that i n the scalar representation the only surviving Casimir operator i s C. Then in order that X- transform according to an irreducible representation of the 6-rotat ions, C must be a constant number when acting on X . This w i l l be assumed to be the case in the fol lowing, so that the f i e l d equation (8.3) becomes simply (8 .7a) f m ^ m A B X - C X = 0 . Thus the Lagrangian density j£ is given by (8.7b) X - - i ( m A B X ) ( m A B X ) - C X 2 . corresponding to an action integral (8 .7c) 1 = L 6 l - K m A B X ) U A B X ) - C X 2 ] S ( < ( 2 - L ) <A( " . AR The problem of f inding eigenfunctions of the operator m^nr 0 occurs whenever one wishes to f ind a set of basis functions which enable one to investigate harmonic analysis on hyperboloids in any pseudo-Euclidean ( i . e . f l a t ) space. Provided the squared interval L is non-vanishing, one can always select a set of "spherical" coordinates such as (6.19) 120 such that the basis for solutions of the eigenvalue problem of the form (8.7a) 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 axes one uses (Strichartz 1973)• The s i tuat ion- is c lear ly different on the hyperquadric L=0, since in that case one cannot u t i l i z e L as a variable with which to construct a set of spherical coordinates after the fashion of spherical polar coordinates in 3-dimensional Euclidean space. In fact , a set of coordinates in which the variables separate i n (8,7a) has been e x p l i c i t l y exhibited, and a set of basis functions in terms of spherical coordinates in 6-dimensional space can be given i n terms of hypergeometric functions (Winternitz 1971; Limic e t . a l . 1966, 1967)• However i t i s not only tractable but indeed i s simpler to f ind solutions of (8.7a) in terms of y ^ . H . L Using ( 5 « 5 ) . one finds by straightforward calculat ion the ident i ty (8.8) i m A B mAB = -L 2A ^ A J f A + ^ ^ ^ B . Evidently, eq. (8.8) is not applicable to the case L=0 due to the presence of L in the f i r s t term. In fac t , the scalar wave equation (8.7a) i s not the equation which i s customarily employed on L=0 for the manifestly conformally covariant description of spin-0 part icles for just this reason. Instead, use i s made of the simpler equation (5 .60 ) . In terms of the coordinates y k , w . , L , using (5*3) one finds for the operator <$A $ A the identi ty ( 5 « 6 l ) , (8.9) J A }k 2 * ' 2 V* V ^ ^ L + ^ L H + ^ L • and for the operator ^ A ^ A the identity ( 5 .^ ) . (8.10) v^A # A S + 2 L ^ L . 121 Substituting the above two expressions, one finds that the operator (8.8) can be written (8.11) K B ^ s - * ~ 2 L \ l > k + ^ 2 W +5 * V . Now going over from the set of coordinates (y k,'K ,L) to (y k,r,L), with r given by (8.12a) r = * " 1L^ , one has (8.12b) * \ = - r > r so that (8.11) becomes (8.13) Im^ m A B - C = - r 2 ^ k + ( r ^ r ) 2 - 4 r ^ r - C . Inspection of (8.13) shows that the variables separate i n (8 0?a), so that i t i s permissible to write X as a product (8.14) X = F(r) X(y) with X(y) any solution of (8.15) U k ^ k + m 2 ) X ( y ) = o where m2 is real and non-negative by assumption. Thus eq.(8.?a) reduces to the eigenvalue problem (8.16) l r 2 m 2 + ( r ^ r ) 2 - 4 r ^ r - c } F ( r ) = 0 . This equation can be cast i n the form (8.17) {(z> z) 2+z 2-v 2 } z y(z) = 0 by writing (8.14a) F(r) = rP G(r) (p=some real number) and using the relations (8.18a) r ^ ^ ^ r ) ] = rP ( p f r > r ) F ( r ) (8.18b) ( r } r ) 2 [ > F ( r ) ] = rP[p 2+2pr> r+(r} r ) 2 ] F ( r ) so that with p=2, (8.16) becomes 122 (8.19) r2{m2r2+mr[V^(nir)] 2 -4 -C]G(r) = 0 . Thus solutions y have the simple form (8.20) X . = r 2 Z + ( 4 + c ) | (mr) * (y) where X(y ) i s a solution of the usual Klein-Gordon equation (8.15). Clear ly , the wave equation (8.7a) describes spin-0 part ic les 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) i ( 5 ) X = 0 so that 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 lds with mass transform according to a representation of a conformal group they must be eigenfunctions of ^(5) has indeed been ver i f ied i n th is case, since X does transform according to the scalar representation of the conformal group in f ive dimensions, with '^(5) =0» Now one can go on to investigate the remaining features of the quantum f i e l d theory which results from the action pr inciple based on ji . These include the conserved quantities associated with rotat ional invariance, the current conservation laws, and the canonical commutation relat ions for 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 yk ,x ,L , with the resul t (8.22a) ^ = M " 2 L \ X ^ k X - ( X ^ ) 2 - C X 2 . Keeping in mind that the Jacobian J ( ^ ;y, K »L)=f^( v\)/^(yxL)f=^/2 must be included, the action (8.7c) becomes 123 (8.23b) I = J U ^ 2 - L ) (-K3/2) £_ d*V d x dL » = ^ j [ - i H L ^ X ^ X + i K 3 ( ^ \ X ) 2 + i H 3 C X 2 ] d K ] d 4 y = To demonstrate that the Lagrangian density gives the correct f i e l d equation (8.7a) with (8.11), one can compute (8.23c) btKnx] O ^ K ^ O k * ) ] - \ O x * > ] -= -> i3|_ K-2 L^ k^ k+( x ^ J 2 +^^>< -C] X = 0 i n agreement with (8.13)• The conjugate momenta are given by (8.24a) Tf A B = ^ £ / M m A B X ) = -n*3X. so that the angular momentum tensor (7.47b) i s simply (8.24b) S A B C D = i m C D X(m A B X ) + i S A C S B D I . By a straightforward calculation involving commutations of the m^'s, one finds that i n this case the current conservation law (7«33) takes the form (8.24c) 0 = S A B C D s (|mCDmCD-C)X m^X and i s thus satisfied identically for 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 k J , j = 0 with t^J given by (8.25b) t ^ - ^ X ^ k X-S k J S-i> n X } n X H X ( a 2 X ) } provided the mass i s given by the equation (8.15), which, using (8.16), can also be written (8.26) r - 2 [ - ( r > r ) 2 + 4 r ^ r 4 c ] > l = m2X . 124 Similarly, the choice {A,B^=Lj,M yields (8.27) 0 = s j k n , n = (yjV-y^n)^ . As expected, the choice {A,B} = (6,5} does not lead to a conservation law in Minkowski space for non-vanishing m, but instead one has (8.28a) 0 = ( * ^  k+m2)X K y * } k+r> r)X or (8.28b) D n. n = [ ( ^ ^ n X ) ( i X ) - X ( i 2 X ) ] n n ^ * „ [ 0 + - O * ] -- ^ D a ^ X ^ KX4^ k (Xm2X)]-X(m2X ) where D*1 i s given by (8.28c) D* = > i n X i X + y k t k n and i s given by (5.41b), (8.28d) A = * V = - r ^ r . When Ji- -1, eq.(8.28b) takes the form of the broken conservation law (4.3) with symmetry breaking function mr X. . Noting that the generator of dilations i s given by (5.20c), which reads (8.29) f " i ( y k ^ k + r " i r ) i n the present case, since ^(5) = ^» o n e c a n a ^ s o determine how the eigen-value m2 appearing i n (8.26) i s affected by a dilation (8.30a) y , k = (1+«r)yk ; r« = (1 + <r)r which induces en X the transformation (8.30b) x*(y ,r ) = ( I+ir - y ) X(y,r) . Since one can evaluate the commutator (8.3D [ A k + r " 2 i-(^r) 2 + 4 r^r + C i • f] " 2i[i\+r- 2{-(rt r) 2+4rt r+c}] . then one can see that i f X i s a solution of the f i e l d equation (8.7«)» which reads i n terms of y^.r (8.32) ^ k^ kX+r- 2{-(r^ r) 2+4r^ r-K;^X = 0 , then so is X'» Consequently, i f X is an eigenfunction of the operator(8.26) 125 — c a l l this operator M2, say — belonging to the eigenvalue m 2 , then one has (8.33) r - 2 { - ( r^ r ) 2 +4r> r+c] = M 2 * , ' = = M 2 * + i<r [M2 ,</>] X + ic r^ (M 2X ) = m2(1+i<r<p )X-2<rm2X = » m2(1-2<r)X-' where i n this equation the faet that o r is assumed to be in f in i tes imal was used. Thus one has complete agreement with the usual transformation property (4.11a) of the mass i n Minkowski space, outlined in Section 4, and one arrives again at the broken conservation law (4.3) i n Minkowski space. However, despite the fact that the eigenvalue of M 2 changes under d i la t ions , and that exponentiation of the inf in i tesimal 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 rat ios are observable. Then since every mass must undergo th© transformation indicated by (8.33) these mass rat ios w i l l remain unaltered by transformations (8.30). Notwithstanding the fact that some of the usual Minkowski space currents are not conserved, there s t i l l are f i f teen conserved quantities M^g, given by (7.35). For the Lagrangian density (8 .?b), i t is somewhat tedious but straightforward to evaluate these quantit ies. For example, one finds that the quantum f i e l d theory generator of translations in the y^-direct ion i s given by choosing the indices iA,Bl = f(6,4)+(5,4)}, on the surface t=t 0 ,L=L 0 , (8.34) P 4 S M 6 4 + M 5 / + = S ( t - t c ) S ( L - L 0 ) ( X L ) M V d * dL* •ii(^X) 2+i(vx)-(v X M L - 1 M 2 [ C X 2 + (~} KX) 2]} . By a s l ight change of f i e l d var iable, however, a l l the conservation laws (8.24c) can be brought into the form of conservation laws in  5-dlmensional space. This i s accomplished by writing the f i e l d equation 126 (8.7a) in the form of a f i e l d equation in 5-dimansional space governing  a massless spin-0 par t ic le . To this end, consider a f i e l d variable related to X. by (8.35a) H> = r" n X (n=some rea l number) so that written in terms of V the f i e l d equation (8.7a) becomes (8.35b) 0 = ^ k +r - 2 [ - ( r^ p)2-f4r^ r - K % " T = = r n [\ k )> k+r-2 [_r2 } y ^ r + 3 r ^ ^ Z n r } r . n 2 + 4 n + c ] } ^  . Clear ly , this can be brought into the form ( c . f . Wyler 1969) (8.36) = 0 w i t h z t = ( y k . r ) , t=1,2,3,4,7 simply by choosing the value (8.37a) n = 3/2 , provided the Casimir operator C is assumed to have the value obtained from (8.37b) 0 = -n2+4n+C = (15/4)+C ; i . e . C = -15 /4 . Then one does indeed have from (8.24c) the four conservation laws (8.38a) t ( 5 ) s u , u = 0 (8.38b) S ( 5 ) s t \ u * 0 (8.38c) D ( 5 ) u , u = 0 (8.38d) K ( 5 ) t u , u = 0 where the four types of current tensors are given by (8.39a) t ( 5 ) s u = -^Y> sY - S s u ( - n t Y *r> (8.39b) S ( 5 ) s u v = z s t ( 5 ) u v - * u t ( 5 ) s * (8.39c) D i 5 f = z * \ 5 ) s n - (3/2) f V»f (8.39d) K ( 5 ) s u = ( 2 z s z v - S s vz2)t ( 5 ) y u - 3 * s xp^uy +( 3 /2)Y 2 . 127 However, the approach of these authors is considerably di f ferent from that presented here, in that features such as relations resembling (8.40) which appear in their work do not arise from action principles developed in the manner of Section 7, in which stress is placed on the absence of translat ional invariance in order to determine the minimum requirements for an action principle based sole ly on rotational invariance [e .g . compare the divergence theorem constructed in Ingraham (19^0) with the rotat ional ly invariant Stokes' theorem constructed in section 7-l3. Neither i s an interpretation given i n their work in terms of eigenfunctions of the scale dimension in f ive dimensions, or in terms of massless f i e lds i n 5-dimensional space, as in (8.36). 8-2. Green's function for scalar f i e l d equation The condition (8.37), which yields a numerical value for the only non-vanishing Casimir operator C in the scalar representation, was motivated by the desire to be able to cast eq.(8.32) into a simple form. It i s also possible to derive this condition by examining the structure of a Green's function for the 6-dimensional scalar equation (8.7a). Covariant Green's functions for rotat ional ly covariant f i e l d equations have been developed by Adler (1972) for the case of f ie lds on the unit hypersphere i n 5-dimensional Euclidean space and on the n u l l hyperquadric in a 6-dimensional space with one time axis, by Fubini e t . a l . (1973) for the case of f i e lds on hyperspheres in 4-dimensional Euclidean space, and by Mack and Todorov (1973) for the case of manifestly conformally covariant f ie lds in 6-dimensional space on the hyperquadric L=0, Also, i n the 6-dimensional case dif ferent results have been advanced by Ingraham (1971) and Castell-(1966), who advocate the determination of Green's functions d i rect ly in terms of the coordinates y , r , L by 128 Thus one can maintain that fields associated with particles with mass  in Minkowski space can be looked upon as "projections", onto the  4-dimensional world, of massless particles in 5-dimensional space, spanned by the coordinates y k , r . As well, one can say that both the conservation laws and the broken conservation laws, associated with the transformation behaviour of particles with mass under the conformal group i n Minkowski space, are actually manifestations of the fifteen unbroken conservation laws associated with the transformation behaviour of massless particles in 5-dimensional space, under the transformations of the re s t r i c t i o n (5.10) of the 5-dimensional conformal group. Finally, i t remains to check that the commutation relations (7.63) i n 6-dimensional space actually correspond to meaningful commutation relations in Minkowski space. Using the definition (8.24a) of the conju-gate momenta, and assuming that bosons obey commutation relations and not anticommutation relations, the equal-time commutation relations (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 linear sum of creation and annihilation operators multiplying products (8.4U) X £ ( B - * (y ) r 2 Z y ( r m ) of solutions X(y) of the Klein-Gordon equation (8.15) and cylinder functions Z y (rm), where V i s given by (8.41b) V = +(4+C)i = = t\ • These results are to be compared with similar work by Castell (1966) and Ingraham (1971). This similarity i s due to the fact that for a scalar f i e l d , none of the techniques involving the operator U need be applied, and in particular 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 in 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. I f ^ A and ^ 2 A a r e the coor-dinates of two different points, each of which l i e s on v | 2 =L, then one looks for a formal solution of the wave equation which can be written in terras of ( " l i - " ^ ) 2 ' i » e « t for ^ -j ^ ^ £ o n e s«®ks a number q such that (8.42a) ( i m i ^ m , * 5 - C ) [ ( ^ 1 - . l 2 ) 2 ] q = o , where (8.42b) m l A B = i ^ l A 0 / ^ 1 B ) - i ' \ l B 0 / ~ > • Then for ^ j = * [ 2 there appear delta function contributions on the r ight hand side of (8.42a). To take these contributions into account, consider for the moment the special case L=0. For this case one has (8.43) y i i 2 = l \ 2 2 = 0 » V U 5 ~ i*1 K 2 ( y 1 - V 2 > 2 f o r L = 0 where y k , H are defined separately for *\ ^ and * J 2 . For a f i e l d ^ (y ) describing massless part icles in Minkowski space, define the Feynman function (8.44a) A ( ^ ( y r y 2 ) = - i <0| T ^(y, ) Y(y2)}|0> where T denotes the time-ordered product. This singular function is given by the simple expression (Mack and Todorov 1973) (8.44b) A m ( y ) = c (y 2 +i0) q = l im (y 2+i€ )<* , q rea l , where q=-1 for free f i e l d s , and (8.44c) c q = - i2~ 2 t l ( 4Tr ) - 2 T ( - q ) / r(q+2) . Then i f one defines the corresponding function in 6-dimensional space by mult ipl icat ion by powers of K , 130 (8.45a) ^ ( ^ , T 2 ) - = ( x , x 2 ) q A F 4 )(y ry 2) then comparison with (8.43) shows that (8.45b) AFC ^ 1 . ^ 2 ) = c q ( - 2 y l r * l 2 + i 0 K 1 v < 2 ) q • These results are generalized to the case L^ O by considering the general form of (8.43), v (8.46a) ( ^ 1 - ^ 2 ^ S M 1 X 2 { - LH 1 - 2 X 2 - 2 ( M 1 - H 2 ) 2 + ( y r y 2 ) 2 ^ for ^ j2=^\22=L. I f one defines the supplementary variables r^ 2 = L 2 / K J 2 , then one finds that (8.46b) (T)-rz)Z = LX j ~ 2 X 2 _ 2 (X 1 - X 2 ) 2 so that (8.46a) can be written (8.46c) (*l r/} 2 ) 2 = X L X 2 ( z 1 - z 2 ) 2 where (8.46d) z 2 = zt zt = y 2 ^ 2 , t=1,2.3,4,7 . Therefore, writing i n the general case (8.47a) A F<ni'n 2 ) = c q [ ( i 1 - 1 2 ) 2 + 1 0 * i * 2 ] q corresponding to the 5-dimensional propagator (8.47b) A * F 5 ) ( y 1 - y 2 ; K 1 - v t 2 ) H ( M i M 2 ) " q A F H T 1 2 > • then can be written simply as (8.47c) A* (5)( z1- Z2) = c q [ ( z 1 - z 2 ) 2 + i 0 ] q * This corresponds, up to a conventional factor 2 i , to the usual expression for the elementary solution i n f l a t 5-space with one time axis for the wave equation (8.36) describing massless particles (Gel'fand and Shilov 1964, p.280), provided q i s given by q=-3/2. Obviously, this 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 in which C i s given by (8 .37b) , Thus the Green's function corresponding to the f i e l d X for the case L^O i s given by (8.48) A [ 5 ) ( Z I . Z 2 ) £ ( r ^ ) ^ A ( 5)<V Z2> = L ~ q ^ 11 • yI-2) = = c q < r1 r2>" q [ ( *1-* 2) 2 +i<>] q IP so that when C= - 1 5 / ^ t q = -3/2, i s indeed a formal solution to the f i e l d equation (8 .32) for X when Z j ^ z 2 . To see that this value of C is the unique value which allows one to write a Green's function for 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 B [ o ^ - ^ ) 2 ] * 5 { - 8 ^ 2 q ( q - 1 ) - 4 q ( D - 1 ) ^ ^ ^ ( q - O ^ 2 ) 2 ^ ^ • If one assumes now that y / [ ^ 2 = ^  2 2 = L . this becomes (8.49b) m l A B m 1AB[ (v ? 1-/j 2)2]<l = = |-8q(q-1 )L-4q(D-1 )(*| j» ^  2)} [ 2 ( L - 1 * "| 2 ) ] q " ^ + + 8 q ( q - 1 ) ( L - ^ 1 ^ 2 ) 2 [ 2 ( L - ' | 1 ^ 2 ) ] q - 2 . Thus one can ensure that for ^  ^  ^ 2 the right hand side of this equation is proportional to ("| j - ^ by requiring (8 .50) -8q(q-1) = 4q(D-1 ) ; i.e. -q = (D-3)/2 . This choice has the significance that for Y{\^y\2' (8 .51a) i a u B * / 8 [( *| r •) Z ) 2 ] ^ ) I Z * -*(D-1 )<D-3>[(*1 r » | 2) 2}<D-3)/2 132 (8.51b) C = -£(D-1)(D-3) . For D=6, this yields C=-1 5/4 again* so that ins is t ing on the form (8.49a) for the Green's function automatically jus t i f i es the value of C arrived at heur is t ica l ly , by the requirements of s impl ic i ty , i n (8.37b). Since i t should be possible to recast f i e l d equations for part ic les with spin into a form resembling the scalar f i e l d equation (8.7a), e . g . by i te ra t ion , the constraint (8.51b) on the value of C can be used to advan-tage i n specifying the form of f i e l d equations for f i e lds having non-vanishing sp in . This procedure w i l l be employed below for th© f i e l d equation for s p i n - i pa r t i c les . 8-3. Spinor f i e l d In section 5-9 i t was shown that spinor f i e lds in 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 inear free f i e l d equation which i s to be employed for a description of s p i n - i part ic les i n 6-dimen-sional space. I f such an equation i s to be of not higher than f i r s t order in th© derivatives of the f i e l d , and sat is f ies the requirement of rotat ional covariance i n 6-dimensional space, thon according to the program proposed in Section 7. the most general such equation i s to be determined by considering th© most general rotat ional ly invariant Lagran-gian density jL which i s of th© form (8.52) i. = t ( ^ . X , m A B X ) and i s constructed out of expressions bi l inear i n X and expressions which are products of X and m A B X . *For D=5, one has C=-2 in 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 A B as defined i n (5.94), one can construct sixty-four linearly independent matrices (Drew 1972)* (8.53) I ; />A ; r * 8 ; f>kff U f a f c ) ; f ; fpk ; f1 . Calling these pi, i=1,2,...,64, the number of different of each type i s 1,6,15,20,15.6,1, respectively. Then one can form seven types of bilinear covariants X T^Tt. where X i s the a d j o i n t spinor defined by (5*115) and **** (5.124). These bilinear combinations of X. and K then have simple transforma-tion properties i n 6-dimensional space, e.g. (8.53) Scalar S = XX; Pseudoscalar Q = xfiX; Vector V A = xpAX . Keeping i n mind that the desired Lagrangian density can be an e x p l i c i t function of ^ since translational invariance i n 6-dimensional space i s not demanded, one can form four linearly independent expressions of the kind specified abovet* There are two scalar invariants (8.54) I] =^<r A Bm A BX ; I 2 = X X and two pseudoscalar invariants, 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 linear independence of these matrices follows from the observation that, except for I, they are traceless and that for each nijfr there exists a pj such that pi r ^ pi, and given pi and r j for i ^ j there exists a pn/i such that pi pJ= f n» Then, i f ^  c ^ i ^ , taking the trace yields ci=0 and taking the trace after multiplication with P ^ I yields Cj=0. The invariant Xv\k f^X i s omitted since i t can be replaced by a divergence plus excluded terms, by virtue of the identity 5 x^k^x 5 i »AB( xVp BX) + +L( xpk2kx + 2kxpk>i) --( x \ r k i*]BX + I ^ B ^ A P * * " ) • 134 considering conjugate momenta TT A 8 f given by (7.22), which multiplied into give rise to covariant expressions whose transformation behaviour i s i n part that of a tensor density. But i t i s precisely these conjugate momenta which appear i n the anticommutator on the l e f t hand side of (7.55)• Since the right hand side of (7.55) i s given, up to a factor, by the tensor quantity gAB(^.y) f defined by (7 .56), there i s a contradiction unless the c o e f f i c i -ent of the f i r s t pseudoscalar invariant i n X. i s zero. As well, since X. appears i n the expression (7.47b) for the angular momentum tensor SAgCDt i n order for the angular momentum M^R to transform as a tensor X. must consist of scalar invariants only, so that the coefficient of the second pseudo-scalar 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 Lagrangian density to be considered i s the linear combination (8.55) = alj+bl£ ( a»b complex, where the arbitrary coefficients a,b can be functions of L without affecting the rotational invariance of X. • The expression (8.55) i s not necessarily hermitian as i t stands, but since the Lagrangian density should be real up to a multiplicative constant, i n order for i t to be used i n the hermitian generator (7.46c) of the unitary operator (7.46b), £ i s brought into a real form by adding the hermitian conjugate £ + , From the r e a l i t y property (5*119) and the definition of m A B one has simply that (8.56) (X> Cm B CX) +=-m B C^ < rBC+ A+£= . ^ ^ A ^ . ^ % rBC% since from (5.124) A i s hermitian. Therefore the real part of / i s (8.55a) + ^ +)=4[a x V % B * - a* mAB ?V BX+(b+b*) % X.} . Now, the second term can be expressed i n terms of the f i r s t , since one has the identity (8.57) m^Xff-^X =m A B( X cr^Z ) - Z ^ ^ X . 135 so that suppressing in the inconsequential divergence forming the f i r s t term on the r ight hand side of (8.57), one arrives at a Lagrangian density of the form (8.58) £ • = X ( T A B m A Bx -2 X U , X rea l . The f i e l d equation derived from the corresponding action pr inciple is found from (7.23), where the conjugate momenta (7.22) are (8.59) t f A B = "*2/ 0 (m A 3 x)= X«rAB . **** Hence, the spinor f i e l d equation for X reads (8.60a) - m A B j? <rA B-2X X = 0 and taking the adjoint, the f i e l d equation for X is (8.60b) (T^m^X -2A X* = 0 . As i t should, the second equation,(8.60b), also results from taking the variat ion of (8.58) with respect to X. Making use of the antisymmetry of o""*8, this equation can also be put into the form (8.60c) fi-^A 2 B X + i X X = 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 for spinors in 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 in 6-space to include 6- translat ions, along with the 15-parameter group of rotat ions, in order to write a spinor equation in a form d i rec t ly analogous to the Dirac equation in 4-dimensional space (8.61) ( i / i A J f A -M)¥ = 0 '. It i s then asserted that the parameter M can be looked upon as a "conformally invariant mass", giving r ise to a "conformally covariant mass" nM i n 136 Minkowski space. However, the above equation does not admit solut ions, for nonvanishing M, which belong to a speci f ic value of the scale dimension in 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 y l = ><0 i n 6-dimensional space (Barut and Haugen 1972). Moreover, X acts on solutions in Minkowski space of eq , (8,6l) as a d i f f e ren t i a l operator and not as a matrix, so that one cannot assert that the equation result ing from (8,61) i s covariant under the 4-dimensional conformal group. As wel l , since the operator W, and not the operator U defined by (5.23), was used i n considerations of eq , (8 ,6 l ) , i t cannot be maintained that Wt' transforms according to a representation of any conformal group, so that i t i s not meaningful to demand only that Wt belong to an eigenvalue of Furthermore, the alternative course of using the operator U instead of the operator W i n writ ing eq . (8.6l ) i n terms of Minkowski space variables results i n a complicated equation in 4-space which certainly has no simple relat ionship with the Dirac equation. Therefore, investigation of this 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 fact that (8.61) cannot be derived from a rea l Lagrangian density. The proof i s relegated to Appendix 3. where i t i s shown that only for the case M=0 can one construct a rea l Lagrangian density. 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 in rewriting the f i e l d equation in terms of y ^ . H . L the operators m g^ can be replaced by their equivalents in these coordinates, given by (5«5). The equation which results from this replacement i s (8.62) fcJVj V * « k \ - » T t T k [ ( H - 2 L - y 2 ) > k + y k y J > . - y k x> J + +/ 5 (yV>0>J + i AU =0 , 137 where the ident i f icat ions of o - 5 j c + w i t h K k and TT^, respectively, have been employed i n (8.62). Now, the operator U"^  i s given e x p l i c i t l y i n (5»127), and application of the s imi lar i ty transformation U( )U~^ to the matrix d i f fe ren t ia l operator in (8.62) y ields the equation for the f i e l d variable X =UX . Keeping in mind that i H i s not independent of y k , X , so that (8.63) V u ~ 1 > = i T r k ' ^ \ ( U ~ 1 ) = - i x - 1 L 2 T ? , one has (8.64) U ( < r A Bn A 3B+i A )U"1 = i K k \ + n - 2 L T T k ^ k +(i/2) K k TT + * \ -- K - 1 L ^ < r 7 k ^ k + w - l L 2 i r ? X ^ > < +(3/2)x- 1L^TT ?+iX . At th is stage i t i s convenient to go over to the coordinates ( y k , r , L ) defined by the relations (8.12). Then the solution of the f i e l d equation following from (8.60) and (8,64) proceeds exactly as i n the solut ion of the corresponding equation, (6.50), i n the 4-dimensional case. Again i t i s found that only for a part icular value of A , i n this case (8.65) A=-5/4 , does the iterated equation y ie ld solutions i n Minkowski space. The proof i s elaborated i n Appendix 4, and this value of A i s also suggested by the fact that i n the spinor representation the Casimir operator D, given by (8.5c), takes the form* (8.66) D = (9/2) ^{<rAB \ ^B-i(5/4)] . Using (8.65), i t i s shown i n Appendix 4 that in terras of the four 2-spinors defined i n (5«132), the spinor f i e l d equation has solutions *In the derivation of the re lat ion (8.66), use was made of the ident i ty 138 (8.67a) (8.67b) (8 .67c) (8.67d) X. = r 3 Z i ( r m ) X ^ y ) X 2 = r 2 Z_i ( rm) X ^ y ) = r 2 Z | ( r m ) X 3 ( y ) X 4 = - r 3 Z_ i ( rm) X^(y ) . where Z,Z are 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 func t ions of order f , and A. A i t i s not necessary to assume e q u a l i t y of ra and m. 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 k \ X ^ y J + i m X ^ y ) = 0 (8.68b) and (8.69a) ^ k ^ k X 4 ( y ) + i m X 1 ( y ) = 0 Z i ( r m ) { V k ^ k X 3 ( y ) + i S X 2 ( y ) } - i m Z i ( r m ) X ^ ( y ) = 0 (8.69b) Z ^ r m ) {^ k > b k X 2 (y)+3SiX3(y)}+iraZ - A(rm)X 1 (y) = 0 The 4-spinor f i e lds p (y ) , A (y ) are defined by (5 .133) . (8.70) ^(y) = X j t y ) \ ( y ) V(y) = X 3 ( y ) w4 In terms of these variables, eqs.(8.68) read (8.71) - i * k H ^ > (y)+m^(y) = 0 , so that (f>(y) sa t is f ies the Dirac equation with mass m. On the other hand A . A ~ <p(y) sa t is f ies the Dirac equation with mass m only i f X. i s constrained to obey the supplementary condition (5*137), so that the components <p(y) vanish, (8.72) - i * * > k ^ ( y ) + m p(y) = 0 i f y>(y)=0 . Hence there are three separate procedures, which one can employ at this point, that result in massive spinor equations. F i r s t l y , the f i e l d X can 139 be constrained to obey the supplementary condition (5•137)• so that the components (j) are eliminated. Then <p sa t is f ies the Dirac equation (8.72) and 9 =0. Secondly, the projectors E and E, which project out the compo-nents y> and <j> respectively, can be applied to X to eliminate either ^ or <p . Thirdly , the spinor f i e l d equations, (8.68) and (8.69), can be interpreted as equations for two di f ferent f i e lds whose behaviour i s governed by coupled equations and whose components are mixed by the* special conformal transformations. Of these three methods, the f i r s t seems the simplest, since i t results in a f i e l d which transforms according to an irreducible representation of the Lorentz group, and according to a representation of a conformal group, without having to declare certain components "unphysical" by the use of projection operators. However this interpretation rules out the poss ib i l i t y of encompassing both the electron and the muon at once in a single equation. For this reason, the view that projection operators should be employed w i l l not be ruled out in the fol lowing, [A s imilar analysis of the spinor equation (8.60) was carried out by Drew(1972), but instead of using U the operator W defined by (5.42) was employed in that work to rewrite the free f i e l d equation. Then eigenfunctions of Ji were deter-A mined by making use of projection operators E and E as outlined in section 5-5. These f i e lds then could be associated only with massless part ic les in Minkowski space.] For any of these interpretations, according to (5.131) the f i e l d s <f> and y belong, respectively, to the eigenvalues +§ and -•§• of the scale dimen-sion ^ s o that the postulate that f i e lds with mass in Minkowski space must be eigenfunctions of -^(5) has been ver i f ied in this case. With these resu l ts , 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 def in i t ion (8.59) of the conjugate momenta, the angular momentum tensor in 6-dimensional space (7.47b) is given by 140 (8.73) = -* ? ^ D ( " A B + *AB>* H S A C S B D Z and for solutions £ of the f i e l d equation, the Lagrangian density vanishes ident ica l ly . To write this tensor in terras of X_, i t i s necessary to perform the s imi lar i ty transformation U( )tP 1 on the operators (mA3+0~AB^* and the relations (5.22) to (5.39) can conveniently be used for this purpose. CD The conserved quantities M A B are given in terms of S A B u by (8.74) M A B = f A | S A B C D d S C D = J ^ d f y 2 S( K-1 *f-t) S( ^ 2 - L ) -• ^ - ^ c S f A B l ^ - ^ ^ c C s ^ j 0 6 - s [ A B f 5 ) ] Substituting the exp l ic i t form (8.73), with £=0 for solutions X , one finds that (8.75a) ^ ( S g ^ 0 6 - * ! ^ ^ = = -x ( ->^(r - 6 5 -L^T 7 ) U(mAB+<rAB)U-1 X and (8.75b) 1CS[AB]C^ = ' ^ [yjr jMK 44(^" 2L.yV 4]("i A BVAB ) ? S = - i x X [n^-2r ^ 7 4 +2r 2 TT ^-2y^ (T 6 5-2y 4 rtr 7 ] U C m ^ <TAB)U"1 X . by using the ident i t ies (8.76a) Ucr j kU- 1 = ( r ^ T T ^ T T j (8.76b) U T T V 1 = TT k (8.76c) UK klT 1 = K k - 2 y k ( r 6 5 + 2 y j 0 - k J + 2 r o - k 7 + 2 y k ( y j T r J - r T r 7 ) - ( y 2 - r 2 ) T T k (8.76d) U (y^V 1 = <r^.^TT k+r n 7 For example, choosing the set of indices {A,B}= i(6,k)+(5.k)] , one finds that the generator of translations in the y k -d i rec t ion i s given on the surface t=t Q , L=L0 i n terms of the 4-component f ie lds <p and <f> by 141 (8.7?) P k 5 M 6 k+M 5 k = f t = t dn d L d V S ( t - t Q ) S ( L - L Q ) . L=L° . { K U £ ^ k $ +* * 3 i jVok? + i K 2 L 2 i ( f + f )} • Now i f the t r a n s v e r s a l i t y condition (5*137a) i s imposed on the f i e l d % , then according to (5.137c), the f i e l d components (f> vanish. In the expression (8,77), t h i s means that a l l the terms i n the braces vanish save the f i r s t term, which i s proportional to the fourth component of the energy-momentum 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 vanish, then the second term i s proportional to the fourth component of (8.78b) t ' k ^ = i J> V^k(p . The l a t t e r s i t u a t i o n occurs i f instead of the condition (5.137) one makes use of the projection operators E and E which separate the spaces belonging to the components y and >^, as explained i n section 5-5• As i n the case of the scalar f i e l d , one can reconcile the existence of f i f t e e n conservation laws with the presence of mass by rewriting 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 variable (8.79) y s r(5/2)vy # t h i s equation reads (8.80) Ukak+S7}7)f=0 *^7~\ » k=1t2,3.^ w i t h • If on the other hand the condition (5.137) i s used to eliminate f, then with (8.81) (f> = r ( 3 / 2 ) y , the f i e l d $ s a t i s f i e s 142 (8.82a) U ^ k + t f 7 ^ ) ^ = 0 • i f r 5 ° • with (8.82b) £ 7 s + ) $ 5 . Thus one can say that 4-component spinor fi e l d s with mass are the manifesta- tions i n Minkowski space of 4-component massless spinor fields i n 5-dimensional space. In addition, there are precisely two such types of  f i e l d possible, with masses m and m, as i n (8.71) and (8 .72). As a f i n a l step, i t remains to evaluate the commutation relations (7»62) for 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* appearing on the l e f t hand side of (7«63)« If one assumes that fermion f i e l d s obey anticommutation relations, then applying the operator U, these compon-ents yield the anticommutation relations satisfied by the f i e l d components ^ + with their conjugates, (8.83a) j£(y.r) , $Xy« . r ' ) } t = t , = $ e.<M') r 3 L " 2 S(r-r') . On the other hand, combining the relations found for A^,B} = f(6,4)+(5,4)} in (7.62) with the results 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 ' ) } t = t , = 0 (8.83c) {^(y.r) , ^ +(y« ,r« ) } t = t , = 2 J g ( y - £ » ) r V 2 $ (r-r«) . This discussion has shown that viewing spinor f i e l d s describing particles with mass in 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 anti-143 commutation relations of f i e l d components. Therefore, i t i s not without interest to consider further a covariant Green's function for the spinor f i e l d equation, 8-4, Green's function for spinor f i e l d equation The problem of constructing a Green's function for the scalar wave equation has already been treated in section 8-2, In order to f ind a corresponding Green's function for spinors, i t i s only necessary to perform an i terat ion of the spinor f i e l d equation, in order to bring i t into the scalar wave equation form. In order to be able to make comparison with the 4-dimensional model constructed in Section 6, this i terat ion w i l l be carried out for spinors in any D-dimensional f l a t space with |3 -matrices sat isfying (8.84) {|3 A , ; B $ = 2 S A B for any f l a t metric, A,B=1,2,,..,D, and a l l other symbols being defined in complete analogy with the 6-dimensional case. In the D-dimensional space, one then finds (8 . 85a) mj&P = -2 v^ 2 ^ * A +(2D-4) \ k * A +2 ^ A * A ( ^ B JJB) (8.85b) o ^ a ^ 5 = ±D(D-1) (8 . 85c) (TupP ''^Jk)^B W + ^ A *A so that (8.85d) ( <rA Bm A B)(<rC DmC D) = | m A B m A B - ( 0 - 2 ) 0 - ^ ^ . Now the f i e l d equation analogous to (8,60) can be i terated, with the resul t (8 .86) (<T ^^V1/^ ) = - C ^ m ^ H f i C D ^ V + A ] A^B"*8-144 valid for any number y " 0 In particular, for (8.87a) y> = - \ - K D - 2 ) the iterated equation is of the form of the scalar wave equation, (8.87b) [ ( r A B v l A V i X - ( i / 2)(D . 2 ) J [<rC\ViX] = = -(l / 8)m A Bm A B + \ [>4(D-2)] Now, according to (8.51) the above equation has a Green's function of the form (8.88a) [(l,-,2)2]-(D-3)/2 only for (8 .88b) 4 A [A+KD-2)] = -KD -1)(D -3) ; i.e. \ = -KD-2)+(i> For example, for D=4, one has > = -3/4, and the l a t t e r number agrees with ( 6 .69 ) , which was arrived at by an entirely different route. For D=6, one has X = -3/4, -5/4, and again the l a t t e r value is the same as that i n (8.65) and ( 8 . 66 ) , For D=5, the possible values of X are - 1 . Then according to ( 8 . 87 ) and ( 8 .51 ) , one has ( 8 . 8 9 ) ( « - A B l u 2 , B - i){p% c ) l i D - i i | [(H-V2]-'} • 0 for 1 i ^ l 2 » 8 0 that the spinor Green's function i n 5-dimensional space i s given formally by ( 8 . 9 0 ) s ^ v ^ . v ^ ) = [r C D v l l c^ l D-(i / 2 ) ] [0\i-*]2>-2] • [Allowing for t i = l 2 corresponds to replacing (*[ 1-I2) 2 w ^ t h a l i m i t , as i n (8.4?a).] To evaluate this expression, one needs the relation (8.9D < T A B a A ^ B ^ l l - , l 2 ) 2 q = - i q(M 2) 2 ( q- 1 ) n / f A W B-VW so that with v t 1 2 = v^ 2 2 =L, one finds 145 (8.92) S ( 5 ) < l j . V If one chooses L=1, then this formal solution is equivalent to the form given by Adler for f ie lds on the unit hyperquadric in 5-dimensional Euclidean space (Adler 1972). An ident ica l procedure can now be carried out for the case D=6, with the result (8.93) S(«| H a ) = * , B -(3/4)] [ (ni-t2) 2 ]" 3 / 2 = _ ( 3 i / 2 ) h , A f A t 2 B f B - L ] hrv 2] 5 / 2 This expression i s to be compared with the two alternative guesses proposed by Adler for the special case L=0 for the spinor f i e l d equation with X= -1 (Adler 1972), (8o 94a) S-CMz) - 1,AfA \ z B f B ("h C 1 2 c r 3 (8.94b) S ^ , , ^ ) = ( ^ l , A l 2 A r 2 • The f i r s t of these is rejected by Adler since i t provides no contribu-tions to amplitudes. This resul t follows from defining a Feynman rule for a vertex where a current with polarization 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 cal led a "pseudopropagator" because i t i s not a formal solut ion to the f i e l d equation. Since the propagator S(>^ ,^2) defined by (8 . 9 5 ) is certainly a formal solution of the f ie ld equation, the argument against eq.(8.94b) cannot be raised here. As for the argument against (8.94a), using (8.96) N * A ^ Bf B = ^ 2 - L . one has (8.97) rA(^) S ( ^ 2 ) r B ( l 2 ) = - 6 i ( 1 l i 2 r 5 [ L 2 fk f ^ ^ -- L ^ l f ) ( l 2 p ) f B + ^ A 7 2 B ^ l f - l 2 f ) -so that this propagator-vertex chain does not give zero contributions A A to currents since the f i rs t two terms do not contain or ^ . At this point, one could go on to define (8.98) S ( y l t H l S y 2 . x 2 ) = U ( ^ ) S O f , , ^ ) V~\*lz) and construct Minkowski space scattering amplitudes in terms of the formulation in 6-dimensional space, but further developments in this direction wi 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 ield equation (8.60) is given by (8.58), and by inspection X~ is invariant under phase transformations of the f ield X\ , (8.99) X * * « p ( i A g ) , for constant A • However, as i t stands £ is not invariant under the vy -dependent transformation (8.100) x + VX , X * X T 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 Ag(^), which 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} = Xo-^AB+gF^)*: -2X X X is 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 Ag^), which undergoes a gauge transformation (8.103) A B * A B +? B A0\) 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<rAB(mAB +g ^ y x . 2 A X X The entire Lagrangian density describing the mutually interacting \ fields and A fields must also include a term X^ which w i l l y i eld upon variation of the action the f i e l d equations satisfied 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 for 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 R of this scalar operator equal to a c-number C means that A B w i l l 148 satisfy a second-order linear partial d i f f e r e n t i a l equation, and this equation w i l l not be very different from the form ( 8 . 7a ) . For the vector representation (5 .66a) , one has e x p l i c i t l y (8.106a) | ( s A B ) C D ( s A B ) D E • 5 % \ (8.106b) (sA B)C D (m A B) SDE = 2im C E , so that the operator C i s given identically by (8.107) ( O C E = j e ^ . ^ c j [ M A B S D E + ( S * B ) D E J . = ( i m A B m A B +5) %CS +2im C E . Thus the vector f i e l d equation may be written (8.108) (|mAB m A B +5 -C) A D +2im D £ A 8 * 0 As i t stands, this f i e l d equation i s not gauge invariant under the f u l l gauge group (8 .103) . However, one can extract from i t s structure the family of permissible gauges A under which i t i s invariant. To this purpose, one can make use of the commutators (8.1 09a) [m A B, Jc] = - i S A c )B*± $ C B ^ A (8.109b) [m A B mAB, = - 4 ^ f -10 ^ Then replacing Afi by Ag+ )f A » as i n (8 .103) , one finds that i f Afi is a solution of (8 .108) , then so i s Ag+ j5gA provided (8.110a) 0 = ( i m ^ ^ - C ^ A + a i i ^ g ^ A 2 ^ { ( i m ^ B - O A ^ ) } i.e. provided A is a solution of the scalar wave equation (8 . 7 a ) , (8.110b) ( i m A B m A B -c) A("p = 0 . From the analysis of this equation, i t is known that A(Y{) must be formed 149 from a product of functions not involving y k and a function A(y) (8.110c) A = f(L) r 2 Z + ( i + + c ) i (rm) A ( y ) where A(y) satisfies (8.1l0d) O ^ W ) A(y) = 0 , m real . At this point, one can remark that eq.(8.110d) is reminiscent of the gauge condition which i s imposed on the function A(y) i n the theory of the, Stueckelberg f i e l d B(y) (Stueckelberg 1938). In this 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 additional compensating f i e l d which renders the Lagrangian for a massive vector boson formally invariant under gauge transformations (8.111a) B k * Bk+ } k A ( y ) for functions /\(y) constrained by (8.1I0d). This i s accomplished by the requirement that B(y) undergo the transformation (8.111b) B(y) -» B(y)+mA(y) whenever B k transforms as i n (8,111a). The remaining relations governing the free fields are given by (8.112a) O k l » k - H n 2 ) B k ( y ) = 0 (8.112b) (} k^ k+m 2)B ( y ) = 0 (8.112c) " i ^ (yJ+ m B C y ) = 0 where the auxiliary condition (8.112c) is akin to the Lorentz condition. Then the equation i n Minkowski space for the spinor f i e l d Y(y) interacting with Bj^y) is written (8.113) -i^ k(^ k-igB k ) r(y)+m F v V(7) = 0 This equation i s then covariant under spinor f i e l d transformations (8.114) ^(y) -t» t(y) exp[ig A(y)] provided such replacements are accompanied by the gauge transformation (8.111). Since adoption of the f i e l d equation (8.108) for the vector f i e l d i n 6-dimensional space leads one automatically to the family of gauges 150 A ( y ) satisfying ( 8 . 1 1 0 d ) , one may attempt to determine under what conditions the remaining relations ( 8 , 1 1 1 ) and ( 8 , 1 1 2 ) follow from the 6-dimensional free f i e l d equation. In particular one seeks an auxiliary condition on A R which reproduces i n Minkowski space the Lorentz-like condition ( 8 , 1 1 2 c ) , An obvious choice for such a gauge condition i s that which brings ( 8 , 1 0 8 ) into the form of a vector analogue of the scalar wave equation ( 8 , 7 a ) , v i z , ( 8 . 1 1 5 a ) A B ^ A' B = A B +^ B A'(n) ( 8 . 1 1 5 b ) 2 i m D E A ' E + 5A»D = 0 . It i s shown in Appendix 5 that this condition can indeed be implemented by a gauge transformation, and that ( 8 . 1 1 5 ) is preserved under subsequent gauge transformations by restricting the gauge group ( 8 . 1 1 0 b ) further to functions A"(*\) which are homogeneous of degree ( 2 C / 5 ) » i«e. ( 8 . 1 1 6 ) 5*ID & A " - 2 0 A" = 0 . Therefore, since L i s homogeneous of degree 2, for A" (^ ) the function f(L) i n ( 8 . 1 1 0 c ) i s ( 8 . 1 1 7 ) f(L) = L ( C / 5 ) . With the condition ( 8 . 1 1 5 ) , eq.(8.108) reduces to the simple form (8.118) ( i m A B mAB-OA0 = 0 . Now the prescription ( 5 « 7 ^ ) can be used to write (8.118) as a set of equations for A k, keeping i n mind the fact that the operator (U)^ Q. given i n ( 5 * 7 0 ) , brings into consideration the extra f i e l d variable A 7 , where the component A7 is defined i n terms of ^j^A 8 by ( 5 . 8 4 ) , ( 8 . 1 1 9 a ) A ? = -L"2 I R A 8 . According to ( 5 . 8 8 ) this definition leads to ( 8 . 1 1 9 b ) A 6 + A 5 = 0 . In Appendix 5 i t i s shown that one can also find an appropriate gauge 151 transformation with gauge function A n ( ^ ) such that (8.120) A 6 - A 5 = 0 , 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 scalar wave equation i n section 8-1 • The gauge transformation induced on the f i e l d A k can be brought into the form (8.111a) by identifying the f i e l d B k with (8.122a) B k = x A k so that (8.122b) B 5 *A7 • In Appendix 5 i t i s shown that the f i e l d s Bj^y) and B(y) satisfy the f i e l d equations (8.112a,b), and that the meaning of the auxiliary condition (8.115) i s that the Lorentz-like condition (8.112c) i s satisfied i n Minkowski space. That B i s a scalar f i e l d under Poincare transformations follows from (5«82b) and the fact that X- i s a Poincare scalar, since Ag i s assumed to transform as a vector. Alternatively, 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. Finally, one can return to the minimally coupled spinor f i e l d equation specified by To determine what alterations occur i n the form of the equation satisfied by the spinor f i e l d X=Ux when interaction with Ag is included, one must perform the transformation U( )U~1 on the matrix <rBCYl$c, with the result (8.123) U t c r ^ g A ^ i n 1 = (K-1LTTk+iMKK+L2<rk7)Ak+ +(x tr 6 5+li TT 7)rA 7-K M <r 6 WTT 7)( A 6 +A 5 ) . 152 In the gauge i n which A^=A^=0, the la s t term vanishes. Hence the equation which X satisfies follows from this relation and from (8.64), (8.124) { i K k(o k-igBkJ+i-2 TTk( ^ - i g B ^ - r <r ? k ( h <r65(-rh r+igrB)+ +r F ?(-r^ r+igrB)4K kTr k +(3/2)rTT 7+i>}X =0 . Just as i n eq .(8 0113). the replacement ^ - i g B ^ occurs i n the f i e l d equation, and the function of the corresponding replacement r i j ^ r ^ r - i g r B i s to ensure the gauge invariance of the relevant terms. The extra terms i n X. which result from including B are (8.125) -2gr%(<r 6 5+rTr 7)BX = gr( ppp+ ^ 5 ^ ) B + 2 g r 2 ( ^ 5 ^ ) B . where the 4-component spinors (p and <f> are defined i n (5.133) • This shows that i n the situation i n which one imposes the transversality 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 interacts with the pseudoscalar f i e l d B with the customary meson-nucleon interaction (Schweber 1961). Alternatively, one can replace (8.100) by the phase transformation exp ( i jS?Ag)« In this case the replacement ^ k - i g occurs i n the f i e l d equation, and the extra term i n X. which results from including B a. A i s of the form <p pB. As well, any combination of these two phase transformations can be considered. The complete system of equations describing the X- and B-fields i n mutual interaction consists of (8.124) together with the equations for Bjj and B including terms for the fermion current. Clearly, exact solutions of this system of equations w i l l be exceedingly d i f f i c u l t to obtain, as is the case for coupled spinors and vectors i n the usual formulation in Minkowski space. 153 Therefore i t would be appropriate at this point to embark upon a treatment of a perturbational approach to these equations, using the covariant Green's functions developed in sections 8-2 and 8-4 to construct a set of Feynman rules for scattering amplitudes due to the minimal coupling interact ion. This course w i l l not be followed here. However, i n conclusion i t should be noted that while (8.100) is the simplest gauge group which one can consider, by no means need attention be restr icted to this case. If one were to allow more general groups of gauge transformations, then as a resul 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 principles has been carried out for the general case in Section 5. and for the part icular cases of spin -0, spin--§-, and spin-1 part ic les in Section 8. Not enly does the formalism used here accommodate the mass in 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 for the study of the perturbation theory associated with a minimally coupled spinor f i e l d i s developed in this invest igat ion. Making use of the results presented here, such a study has potential for providing interesting insights into quantum electrodynamics and the problems associated with the theory of the intermediate vector boson, which i s believed to mediate the weak interact ion. As wel l , the intrusion of the strong interact ion into ordinary gauge theory i s adum-brated by the automatic appearance i n Minkowski space of the pseudoscalar meson-nucleon coupling as a resul t of minimal coupling in 6-dimensional space. I t also seems useful here to l i s t some other of the as yet incompletely explored directions in this f i e l d . F i r s t l y , i t may be plausible 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 poss ib i l i t y would be to use a Petiau-Duffin-Kemmer formalism for a unif ied description of boson f ie lds (Petiau 1936, Duffin 1938, Kemmer 1939). In fac t , Kemmer has shown that there are precisely three non-t r i v i a l real izat ions of his matrix algebra in a space with six dimensions (Kemmer 1943), so that in 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 direct product of the 8X8 y3-matrices introduced in Section 5, so that using the result ing 64x64 matrices in a Dirac-Hepner type equation, as in section 8-3, one could conceivably develop f i rs t -order equations for spins 0,1, and 2 d i rec t ly from 6-dimensional space. Also, since the rotat ional ly invariant action principle developed i n Section 7 i s not dependent in i t s essentials on the dimensionality of the space concerned, i t could be used in the 5-dimensional DeSitter space harbouring rotat ional ly invariant actions whose projections into a suitably chosen 4-dimensional space can be interpreted as those corresponding to part ic les in a Riemannian space of constant curvature R. Then in the l i m i t of large R Poincare invariant actions could be recovered. In the same vein, 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 rotat ional symmetry as the guiding pr inciple in 6-dimensional space. In th is case the method of Utiyama (1956) only prescribes how to augment a t ranslat ional ly invariant Lagrangian density by the introduction of compensating f i e lds so as to render the action integral manifestly invariant under transformations whose parameters depend upon the coordinates of some underlying (possibly curved) reference manifold. Since only rotat ional and not translat ional invariance i s required as a prerequisite for candidates for any action function in 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. F ina l ly , i t should be noted that the phenomenon of the "anomalous scale dimension" (Wilson 1969) assumed by f ie lds engaging in interactions could be more readi ly dealt with from the 6-dimensional standpoint. 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In order to study the transformation properties of the weak f i e l d approximation of the Einstein equations, one must evaluate the commutator (3.27a), which consists of four terms. F i r s t l y , one has (A1.1) ^ n ^ S k r 9 s ( k p ) r t s u - ( k p ) k r J s S r t S S u V n = =Vn(kp>Vu " (k p >Vu^ n " -4i{-(1 +i>yy pV«} £t^u + ^ ° V k j t u ^ m • Secondly, (A1.2) ^\S k r$ n s(VYu " ( k P ) k r J sVV rtS n u " = A ( k P>Vu - ( kp)VnAi " = { 2i(1 -i)Sp V 2 i V A u V 2 i < 1 ^ ^ p . V ^ i S p ^ M i y ^ ^ J -^ K \ ^ y % \ ^ u \ ^ } s V 2 ^ p ^ n t u V 2 ^ P n ) k n t u ^ j + + 2 ^ V ) k n t ^ \ - 2 y m ^ p m ) k J t n \ ^ • The th i rd term is (Al .3) > \ S 3 r ^ ( V 7 u - < V V s ^ A ^ u " • vvv V. - <vVtA, • where the las t equality follows from the identi ty <*.»> ( ^ n t ^ V ^ t n • Thus the third term is given in terms of the second by interchanging j and k. 171 For the fourth terra one has (AI.5) A w y Y u - <vVs^\u = = A ( V n t n u - ( V V s ^ « t u = 2i{( 1 - j > ) s p j M p V ^ - K 1 - i J S p ^ p V ^ J + z y ^ J « k - S J ^ p --y V k - r V J ) ^tu + 2 ( «V ) n n t u ^ ( cr p k)n n t ub^ ^ p m ^ t u ^ - 2 ^ < V k J m £ t u ^ n * Several different types of terms occur i n the commutator (3»2?a). Col lect ing terms of one type, (A. .6) « 6 ^ ) « t a ^ ( « r - p J ) t a ^ n J ( r - I _ ) t a t u . 3 - 2 ( < r p k ) * , t ^ 1 1 . The remaining terms involving the spin matrices are (AI .?) 2 y n { - ( ^ p n ) ^ u V 3 n H ^ p a ) k J t n V M ^ p j ^ u A n + ( V ) ^ u > " + • s \ U p ^ V V . - s V . ^ p + 5 i t 5 p k ^ ' - s J ' I ' k > « i p t F ina l l y , col lect ing a l l the contributions to the commutator, one arrives at the resul t (3.27b), so that the d i f fe rent ia l operator which acts on the tensor f i e l d in the weak f i e l d approximation of the Einstein equations does not commute with the generator of speci l c nformal transformations in Minkowski space.172 Appendix 2. Ver i f ica t ion of area elements and an integral in 6-space The def in i t ion (7-12) of d S A , (A2.1) dS A = - d 6 ^ U ^ 2 - L ) 2*^ , 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 o n ^he surface ( f+L )=0 to be (A2.2) \ j = y 1 , \ 2 = y 2 , X 3 = y3, = y \ > 5 = K. . Then from (7.4c) one has e x p l i c i t l y (A2.2a) d S A l = tk]B]BzB3BkB5^^/^\0''^v[B5/'b^5)d^'''dX5 • and defining the sixth independent parameter by (A2.3) A 6 = L , then one can produce the Jacobian J ( * ^ , X ) i n (A2.2a) by forming (A2.3a) ("b^A/>X6)dsA = -£ B 1 # #. B 6(>1 B 1 /^A 1)...0 '9 B 6 /^A 6)-•d ) j > ( » d = = - J ( Y ^ , A )d A,j . . . d A ^ with J ( ^ , X ) determined by (A2.4) d\ = J ( v ^ , X ) dky d*. dL . Using the relations (5*1) which connect the sets of coordinates (v^A) a n ( j ( y ^ . H t L ) , one f inds for the determinant of the 6x6 matrix "i(v^A)A(y»*»L) (A2.5) J ( v ^ . A ) = - H 3 / 2 . Now checking the resul t (A2.3a) with the def in i t ion (A2.1) of d S A , one uses (A2.6) Mkm6 = ° . H 5 / * x 6 = M 6 / n 6 • 1/< 2 vO to obtain (A2.7) ( ^ A / U 6 ) d S A = - X ( » l 2 - L ) d 6 r l = -Jty.X) S ( ^ 2 - L ) d V dH dL so that with the def in i t ion (A2.2) this agrees with (A2.3a) provided integration over the area element (A2.1) i s carried out over • r L ( > » In the case of dS^g, given by (7.15), the def in i t ion (7.4b) reads (A2.8) d S A l A 2 = ^ A 2 B 1 B 2 B 3 B 5 O r l B l / ^ ^ ) C M B 2 / i X 2 ) ' '(>>*\Blh\3)0\B5/>>\5)d ^ dX 2 dA 3 d \ 5 . Then one has (A2.9) a r \ A M 4 ) (M B /> V d S A B = - J ( r \ ' X ) d > 1 d A 2 d > 3 d X 5 • On the other hand, one finds by straightforward computation that since (A2.10) ^ A / * A i * = SV- H y 4 (^ A 5 +S A 6 ) . the def in i t ion (7.15) of d S A B y ields (A2.11) (^^ A / o> 4)(>>1 B/U 6 )dS A B = - d 6 1 S(-^2+L) S(-K - 1 / | ^ t ) and according to (A2.4) this agrees with (A2.9) provided integration over the area element (7.15) i s carried out over -H.^. To see how to transform the f i n a l two terms i n the integrand of (7.43) v ia Stokes' theorem in the form (7.16), one must evaluate (A2.12) ^ D{(i/2) t( f J S | {) t [#«lD J D +1) J " f l -Now from (7.42) one has (A2.13) ^ ^ = 0 so that the las t term in (A2.12) vanishes. Also, for l inear transformations, (A2.14) v | D g r ^ = ivf , so that (A2.12) becomes (A2.15) ^ C D \ ( V 2 ) / ( f ?"\c)\ = \U ) • Therefore eq,(7.43) can be written as the sura of the two terms given in (7.44). 174 Appendix 3. Inadmissibil i ty of 6-translation invariant spinor Lagrangian Consider the Lagrangian density Xpostulated by Barut and Haugen (1973b) for use in an action principle associated with eq . (8 .6 l ) , (A3.D 2 B H = i t f * A V - M y f . Since these authors include translations among the permissible motions in 6-diraensional space, their var iat ional equation takes the form of an ordinary Euler-Lagrange equation (A3.2) ^lmM - J A ^ B a / a c *AT> = 0 • Now, using the rea l i t y property (A3.3) (?p A V ^ - ^ f AY • which follows from (5.120), the rea l part of (A3.1) can be written (A3.*»> i , B H " * < ^ B H + ^ B H ) = -M Vf+ ( i / 2 ) f f A ^ + ( i / 2 ) ^ qj^AV = = -M9f+(i/2) « f A (9p A Y) and i t s imaginary part i s (A3.*b> 2! B H = - ( i / 2 ) ( ^ B H - ^ H ) = i ( 9 f A ^ - ^ 9 ^ ) • Since the second term on the r ight hand side of (A3.4a) i s a divergence, i t cannot affect f i e l d equations of the form (A3.2), so that by inspection one has that the rea l and imaginary parts of ^ B B - y ie ld separately the equations (A3.5a) M t = 0 (A3.5b) fk ^ = 0 respectively, in agreement with (8,61) only for the case M=0. The converse statement i s also true, that the f i e l d equation (8,61) cannot be derived from a real Lagrangian density, since i f (A3.6a) / ( i i i ) B H = a S ^ A ^ b 9 f 175 then the real part is (A3.6b) ^ i v ) B H = K ^ ( i ^ ^ ( i B j ) + ) = = i( a 9 JBA -a* \ Y f A * )+K b+b*) H7 Y = = K a + a ^ Y f A ^ ^ b + b ^ Y r - l ^ a ^ p A ^ ) # leading to field equations of the form (A3.6c) | 3 A 3At+NM* = 0 , 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 solution 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 spin matrices, eq.(8.62) can be written as the set of four equations (A4.1a) r 2 ^ k X 2 + ( i / 2 ) r ^ r X l + i > X 1 H r ^ k X i + + i r 2 ^ r X 3 - ( 3 i / 2 ) r X 3 * 0 (A4.1b) i ^ k X i - ( i / 2 ) r ^ r X 2 + i ( 2 + > ) ^ 2 ^ r ^ k X 3 = 0 (A4.1c) - i ^ k X i + - ( i / 2 ) r i r X 3 + i ( 2 + > )Xy$n%X2 = 0 (A4.d) - r 2 X k i k y 3 + ( i / 2 ) r ^ r X ^ i > X 4 - | r X \ X 1 + i r ^ r X 2 - ( 3 i / 2 ) r ^ 2 = 0 As was the case in the comparable si tuat ion which occurred in section 6-6, this set of equations can be greatly simpli f ied by recombining the four equations into equations for combinations of components of X: (A4.2a) i r 2 ^ k X 2 + ( i / 2 ) r ^ r ( X 1 + r X 3 ) + i > ( Xj+rX-j) = 0 (A4.2b) i ^ k ( y 1 + r X 3 ) . ( i / 2 ) r ^ r X 2 + i ( 2 + X ) X 2 = 0 (A4.2c) - i ^ k ( X 4 + r ^ 2 ) - ( i / 2 ) r i r X 3 + i ( 2 + \ ) Xj = 0 (A4.2d) - i r 2 ^ k X 3 + ( i / 2 ) r ^ r ( X i + + r X 2 ) + i > ( X 4 + r X 2 ) = 0 . This arrangement greatly f a c i l i t a t e s performing the i terat ion of the f i e l d equation for X , since operating from the l e f t on each of these equations with tfk>ik has the simple results (A4.3a) [r 2 & > k - ( r> r ) 2 +6r ^ r+4 X ( X +3)] ( + r Q>) =0 (A4.3b) [ ^ ^ ^ - ( r ^ ^ r ^ X O ^ ) ] <p = 0 1 7 7 where are the 4-component spinors defined in (5»133) • (A4.4) (f = * 3 .*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) simplifies to (A4.5) { r 2^ k-(ri r ) 2 + 6 r^ r-f4>( A +3)}^+(4X+5)y = 0 . Hence, X must have the value (A4.6) > = -5/4 i n order that the equations for y> and y? can be solved. In this case, the iterated equations can be written as (A4.7a) {r 2 ^ k - ( r ^ r ) 2 + 6 r i r - ( 3 5/4)}y> =0 (A4.7b) ^ ^ - ( r ^ Z ^ r - O S A O l f " 0 . Since the variables separate, ip and f are products (A4.8) F(r) ^ ( y f A tn — >(r) X 3 ( y ) " F'(r) ^ ( y ) • 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 m 2 ) * 1 , 4 ( y > B 0 (A4.9b) ( f V m 2 ) X 3 f 2 ( y ) = 0 • Since there i s no compelling reason for equating m and ra, this assumption w i l l not be made. The pos 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 for F,F',F,F' can be written i n the form (8,17) for cylinder functions Z y 178 (A4.10a) F(r) = r3 Z v(rm) , F»(r) = r 3 Zj,(rm) (A4.lOb) F(r) = r 2 Z y(rm) , F»(r) = r 2 Z'(rm) A A where Z . Z ' . Z . Z * are four possibly different cylinder functions and v i s given by (A4o10c) vZ - \ ; i . e . v = +_• or ^ = - i . However, since J _ i = - Y i , Y_ A . = J . | . , any linear combination Z „ A of J _ A . and Y_i. can be written as a different linear combination Z A of J I and Y i , so that 2 Z 2 considering only ^-j yields a l l solutions. E x p l i c i t l y , the components of X are (A4.Ha) X, = r3zi(rra)X 1(y) (A4.11b) X 2 = i^Z'^rni) * 2 ( y ) (A4.11c) X 3 = r ^ r m ) X 3(y) (A4.11d) X 4 = r3z«i(rm)X 4(y) . Now rewriting eqs.(A4.2) with X= -5/4, one has (A4.12a) i x 2 < ^ k X 2 + ( i / 2 ) ( r r - 5 / 2 ) X 1 + ( i r / 2 ) ( r i p - 3 / 2 ) X 3 = 0 (A4.12b) | X ^ k X 1 + i r V^ kX 3+(i / 2)(-ri r + 3 / 2)X 2 = 0 (A4.12c) - i^ k> kX 4 - i r^ kd kX 2+(i / 2 ) ( - r i r +3/2)X 3 = 0 (A4.12d) - i r 2 X k ^ k X 3 + ( i / 2 ) ( r ^ r - 5 / 2 ) X ' ^ ( i r / 2 ) ( r ^ - 3 / 2 ) X 2 = 0 . Then multiplying (A4.12c,b) by r, and adding to (A4.12a,d) respectively, one arrives at the simpler coupled equations (A4.13a) i r ^ k ^ k X 1 + ( i / 2 ) ( r i r - 5 / 2 ) ^ = 0 (A4.l3b) - i r ^ k a k X 4 + ( i / 2 ) ( r i r - 5 / 2 ) X 1 =0 . Making use of the formula for the derivative of a cylinder function, one has 179 (A4.14) r^r[r3Zi(rm)j = 3r3Zi(rm)+Ai j-(ljferm)Z^(rm)+Z_i(rm)] , so that eqs.(A4.13) are simply (A4.15a) 0 = r 4{Z|(rm)Y ka k*i (y)+imZ '_i(rm) X ^ y f l +i(3-1-5/2)r3z ' i(rm) ^ ( y ) (A4.15b) 0 = r^-Z'i(rm)^ kA k^( y)+imZ -i(rm)X 1(y)Vi(34-5/2)r3Zi(rm)X 1(y) . Therefore, i f this 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 satisfied by ^ ( y ) , (A4.1?) (-i* k^ k+m) p(y) = 0 . Similarly, for the remaining components X ^ , o n e finds (A4.18a) 0 = ^(rm)y^ k X 3(y)-imZ' -A(rm)X 2(y)+ Z|(rm)y^ k X 1(y) (A4.l8b) 0 =-zi(rfi)rt kX 2(y)-imZ_A(rm)X 3(y)-Z'i(rm)y kX kX^y) 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 ^(y) satisfies the Dirac equation only i f the f i e l d y(y) i s simultaneously constrained to vanish. To establish the connection with the 5-dimensional Minkowski space, one must change the f i e l d variables to 4-spinors and M* defined by (A4.20) p s r < 5 / 2 ) y . £s r < 3 / 2 > $ . Then i n terms of 2-spinors Y^ 2 a n d Y-j (A4.21) * = A. v. J the f i e l d equations (A4.13) and (A4.12b,c) read (A4.22a) r * k b k ^ \ +ir >> r Y 2 = 0 (A4.22b) - r Y K > K Y 2 + i r * r 7 ] =0 and (A4.23a) r k fy-irb r ^ 2 + r V k > k Y i =0 (A4.23b) - r ^ ^ z - i r - b ^ i - r ^ ^ z - 0 respect ively. These equations immediately y ie ld the 5-dimensional equations (8.80) and (8.82). 181 Appendix 5» Gauge properties of the vector f i e l d To show that the condition (8,115b), (A5.1) (2im°E+5 S D E ) [ A E + / A » ( ^ ) ] s o , can be implemented by a gauge function A 1 ( ^  ) belonging to the family (8,110b), one must f i n d a A 1 s a t i s f y i n g both (A5«1) and the wave equation (A5.2) (imAB mAB . c) A«(n> = 0 . Contracting (A5«1) with ^ J J , (A5.3) n D(2H»DE+5 S D E)AE = - 7 D(2imD E + 5 0 D E ) ( J E A ' ) -S-(nD E m^ E . 5« ] D ^ )A« . and comparing with (A5.2), i t i s seen that i t i s indeed possible to f i n d such a gauge function A* by equating (A5.4) [c-(5/2) »|D jp] A ' = -1^(211^2+5 S D B)^ • Then any subsequent gauge transformation with gauge function A" preserves the condition (A5»1) provided (A5.5) 2 i m D E ^ E A " + 5 ^ D A M s 0 • From the second equality i n (A5»3), t h i s means that i n order f o r subsequent gauge functions to s a t i s f y the d e f i n i t i o n (8,110b) of the gauge group of the f i e l d equation f o r A B, (A5.6) (imAB mAB-c) A"("\) = 0 . while also preserving the condition (A5»1)» they must be r e s t r i c t e d further to the set of functions homogeneous i n v| of degree (2c/5)» (A5.7) 5 "{D ^ D A " - 2 C A n = 0 . Consider now the behaviour of the components Ag and A r , under a gauge transformation. F i r s t l y , using the formulae (5«3) f o r the derivatives the gauge transformation f o r the f i e l d A k i s (A5.8a) A k = A k-y k(A 6+A 5) -» A k-y k(A 6+A 5)+ ^ k A - 2 y k H * L A = • A k + * - 1 o k A I) 182 and that for A^+A^ i s simply (A5.8b) A6+A5 = A5+A3 A6+A 5+2K^ L A . For the f i e l d A 7 one has (A5.8c) A 7 = A ?+^- 1L^(A6+A 5 ) = -IT* ^  B A B + v ^ L ^ A ^ ) -» A 7 - L - i ( K ^ ^ + 2 L i L ) A + H - 1 L i ( 2 K ^ L A ) = = A 7 - K L - T ^ K A . and the combination of components A5-A3 vanishes by virtue of the definition of A 7 . In fact, this i s a gauge invariant statement since (A5.8d) 0 = A6-A5 = A6-A 5+2y kA k+2 L a A ? - (y 2- H.-2L) (A6+A5) A 6 - A 5 = 0 . From (A5.8b), (A^+Ai j ) can be made to vanish by a gauge transformation provided one can find a function A"( K\) satisfying (A5«6) and (A5.7), as well as (A5.9) 2M* LA" 5 - ( A 6 + A 5 ) = - ( A 6 + A 5 ) . To show that the gauge condition (A5«9) i s not at variance with (A5«6), multiply (A5«9) by the wave operator, (A5.10) -KimrjE m^-OU^) = (_»_)_ H P ^ « C ) ( H ^ _ A n) = s M ^ L ^ ( i n i D E m D E - C ) A B } + H - 1 > K > K A , , + 5 M o L A , , + 2 V < 2 ^ > , i L A ' » . Now, the l e f t hand side vanishes by the f i e l d equation (8,118), so that A " satisfies (A5,6) only i f (A5.11) VT 1 ^ K > K A " + 5 ^ L A " + 2 K 2 o K * L A N = 0 . But (A5«6) reads e x p l i c i t l y (A5.12) ( - K - 2 L o k o K + H 2 ^ > K + 5 H > M - C ) A " = 0 , so that (A5.11) is satisfied i f (A5J3) [ K \ ( R ^ I k + 2 L ^ l ) + 5 L O l + 4 H ^ - C ] A " = 0 . 183 Now, (A5«7) reads e x p l i c i t l y (A5.1<0 ( H V+2L « L ) A " = (2C/5)A" . Therefore a gauge function A" satisfying 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 verified identically. The f i e l d equation (8.118) written i n terms of the coordinates y^, L, and r = M - 1 l i " yields (A5J6) \ - r 2 ^ k ^ k J l r - i r + ( r ^ r ) 2 . c } A k - 2 r 2 > k ( A 6 + A 5 ) = 0 , so that i n the gauge A6=A5=0, with C given by (A5«15). one has (A5.17) { - r 2 > k > k ^ r > r + ( r ) , r ) 2 + l 5 / ^ } A k - o # Solutions A k are products (A5d8a) A k = r 2 Zi(rm) A k(y) with (A5.18b) (> ko k+m 2)A k(y) = 0 . The la t t e r equation entails (8.112a) i f one identifies BK with (A5.19) \ S H A k . To see what f i e l d equation ky s a t i s f i e s , consider the identity (A5.20a) 0 =v(D(imABmAB.c)AD = (KB ^ B - 0 ^ DA D ) - 3^-2^ D(V | B AB) +2I^ BA B so that (A5.20b) 0 =%L^( im A Bm AB-C)A D = faufP-C){u\p)--3L"i ^BAB-2L-^ D ^D( *[ gAB) . Now the condition (8.115b), when multiplied by L~2" and contracted with *Vp, reads (A5.21 ) 3L-* ^ - 2 1 * L - i 1 DcF( V B ) = 0 so that from the definition of A^, i n the gauge A5=A^=0 eq.(A5.20b) states that A 7 satisfies the f i e l d equation 184 (A5.22) {-r 2 i k > k - 4 r^ r+(r ^ r ) 2 + 1 5 / 4}A 7 = 0 with solutions (A5.23a) A ? = r 2 Z'i(rm) A ?(y) where (A5.23b) ( o k H + m 2 ) M y ) = 0 ' The latt e r equation i s just (8.112b) i f (A5.24) B = H A ? . As a la s t step i n the study of the free B-fields, 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 result (A5.25) 0 = -3A 7-2r I ^ ^ r } r A 7 = = (-3+4-1 J r 2 Z'i(rm) A7(y)+2mr3 Z'^rm) A ?(y) -- 2r3 Zi(rm) * kA*(y) . This simplifies to the form (8.112c) i f the cylinder functions Z,Z' are related by (A5.26a) Z».i(rm) = -Zi(rm) , i n which case (A5.26b) 0 = -2r3 Zi(rm) j^ kA k(y)+mA 7(y)} . 

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