Dirac matrices, teleparallelism and parity conservation

Nuclear Physics 7 (1958) 373--383; (~) North-Holland Publishing Co., ,4 msterdam Not

to be reproduced by photoprint or mierofihn without written permission from the publisher

DIRAC MATRICES, TELEPARALLELISM AND PARITY CONSERVATION H. S. G R E E N

University o/ Adelaide Received 13 F e b r u a r y 1958 A b s t r a c t : This p a p e r p u t s on a more f u n d a m e n t a l basis a proposal of Green and H u r s t for reconciling failures of p a r i t y conservation. I t is shown t h a t teleparallelism is a necessary consequence of t h e i n t r o d u c t i o n of Dirac matrices in general relativity, and t h a t these matrices require a reducible 8-dimensional representation. A suggestion is m a d e for t h e representation of all fermion fields b y a single spinor. The problem of particle-antiparticle conjugation is solved for general relativity. Finally it is shown how t h e formalism int r o d u c e d b y Green and Hurst, a n d W a t a n a b e , is contained in t h e theory.

1. I n t r o d u c t i o n One of the most promising attempts by Einstein 1) to construct a synthesis of general relativity and electrodynamics was his 'unitary' theory of 1928--30, with its concept of teleparallelism (Fernparallelismus). This is distinguished from the general theory of relativity in its best known form b y the existence, at every point of space-time, of a uniquely determined vector which is 'parallel' to a vector at any given point. It was soon observed, b y Schouten 2) and b y Pauli and Solomon ~) among others, that Einstein's theory had a special importance for the relativistic generalization of Dirac's equation, because it furnished a simple representation of a set o f four matrices 74, satisfying

{~, ~} ~

~ + y ~

= 2g~,

(1)

where gz~ is the metric tensor. The representation of the Dirac matrices obtained in this way is not, however, the most general one. After the similarity transformation ~ -~ y'~ :

V ~ V -1

(2)

is applied, the transformed matrices y'z still satisfy (1), whether the matrix V depends on the co-ordinates or not. For this reason, many authors have preferred to develop the theory of Dirac's equation in general relativity in a different way, following the method of Infeld and van der Waerden 4). One feature of this method is that the ordinary derivative y~,z of ~z with respect to x~ can be expressed in the form 373

374

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GREEN

r~,, = r L rv+ [ G , r~3,

(3)

where / ' ~ is the Christoffel affinity:

U 5 = ~g~ (g~. ~+g,~.~-g~,. p)

(4)

and Fa is a matrix which might be called the spinor affinity. There are several questions which might be asked about this approach. Does it still carry the implication of teleparallelism? Also, suppose one is given /'~ in an explicit form, how would one proceed to construct a representation for the 7~? These are purely mathematical questions, for which answers will be provided in this paper. But there as also some very interesting physical questions which deserve attention. Dirac's equation, assumed to be invariant with respect to general co-ordinate transformations and similarity transformations, has the form

r ~ (9, ~ - 5 9 )

= 0,

(5)

where 9 is a general spinor. Is there any evidence that this equation describes the behaviour of the elementary particles better than its counterpart in special relativity, i.e., that general relativity has anything useful to say about atomic or nuclear physics? If so, what is the interpretation of the apparently unique spinor field 9; how could it describe particles as dissimilar as the electron and the proton; and how can the invariance properties postulated b y general relativity be reconciled with the observed failures of parity conservation? These are questions to which at present no complete answer can be expected. The present paper will have fulfilled its purpose if it encourages the hope that satisfactory answers to the outstanding questions m a y still be found.

2. Dirac Matrices in General The main purpose of this section is to symmarize for the reader's convenience some relevant properties of the generalized Dirac matrices and spinors; a more extended account will be found in a review article b y Bade and Jehle 5). An attractive point of view is that the set of four Dirac matrices y~ is the fundamental geometrical invariant, and the metric tensor ga~ is defined b y (1). Then the geometry of space-time is properly described b y a linear matrix form ds = ya dx a. But as gaa is a multiple of the unit matrix, it is necessary to introduce as a postulate

[~%,, r~] = 0.

(6)

Defining

~

= ~-[y~, ~ ;

r~v -- ½{y~, r~};

(7)

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one finds that these are anti-symmetrical in all pairs of suffixes, so that ~a~,p = ea~p ys,

(8)

where lea~,el = 1, defining Vs. Further, ½[Yx, Y~,] = ga~Y,--ga, Y~; ½{ra, r~,e} = ga~,r~p+ga~7,,t,+gao r~v; Yas ~ Tag5 : y52 : g,

--Y~Ya :

(9)

1/6ga~,emPC'Y~p~;

in which g means the determinant of the ga,, which is empirically negative. One defines ga~,, ya and y5 b y

ga,t,g~ : ~a;

ya : gat, y~;

y5y 5 :

1.

(10)

The ring generated b y the Ya has only 16 independent elements, e.g. the unit matrix ~, the Ya, Ys, Ya~ and Ya~. It will be postulated that the derivative Ya, ~ is an element of this ring and can therefore be expressed in the form p~ p5 Yp~. Ya,~, = Eat,6+Eat,v Y+E~,,ys+Ex~,Yp~,+Ea~,

(11)

But from (10) it follows on differentiation that ½{Ta.v, Yg}q-½{ra, rj,,v} =gag, r

(12)

and b y substituting (11) into (12) one finds Ea~ = O;

E~,,gp~,+E~,,gpa = ga,,~;

E ~ 7 ~ v ~ + E ~PCYap~ = 0;

Ep5 , .2_ 1~o5 A~,gppta..,pv~Ap

=

(13)

O.

From the third and fourth of these equations it follows that E ~ and E ~ have the form

= E 2 aaP, defining E ~ and E~ 5, so that (11) reduces to

7a, t, = E~,, 7~q- ESa,,75-+-E S yavq- E ,? 7as-

(15)

This result can also be written in the form of (3), where 1 5 a P~ = - - ½ ( ~ C a v~ r a y~q-Ea~y 7s-+-E~,v y , q - e , S y s )

(16)

and

c L _=

L.

(17)

To establish the equivalence of (15) and (3), it is necessary to use the fact that the tensor C~, gp~ is anti-symmetric in ;t and #, which follows from (4) and the second of equations (13). When (I6) is substituted into the generalized form (5) oi Dirac's equation,

376

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S,

GREEN

this reduces to ~,a V,,a_½{ga. C L r.+ga,. E~, r s + E / ' + (El, ~6,,~'P- - 2 L '1a p ir75 oal~vp~., ~ IYpv

+ (E~,~ g~'"--½gq g.,. CL ea~'"~)7,es}~o = O.

(18)

The physical interpretation of this result will be considered in more detail later (§ 6). But it m a y be noticed immediately that the imaginary part of ga~'C~, has the character of an electromagnetic field, that ga~'E~/, has the character of a pseudo-scalar meson field, and that the imaginary part of E ~ ' acts as a mass field. The coefficients of 7~, and ~p~ in (18)havenot yet any firm physical interpretation, but could represent fields involved in such phenomena as /3-decay. 3. T h e Field I d e n t i t i e s

. Eat,, 5 E~" and Et,5 which appear in (15) cannot be The coefficients Eat,, assigned arbitrarily, but must satisfy a set of identities in order that (15) be integrable. These identities are obtained from (Ta,~ ) , . : (7~, ~),a by working out each side of this equation with the help of (15). But first one requires the following results: p p 5 5 6 7at,.. = (E~., 7%-- E~. 7't,p) + (Et,~)'as-- Ea, Y~,5)+ (Ei,~ Ya-- E]~ ),~,) ; ~,. = (E~ ~p--EL ~ p ) + Es. 5 Y~ + (Esv ~ Ya-- E ~ Ys),. (19) 75. v : E ~ , v p + E ~ y 5 + E f Vsp. where, to point the symmetry, the notation

E L = g,,~ E~p,

E~5 v = --gg-- ~ L ,

E L = E~p v

,

E L : ~E?,

(~0)

has been introduced. These results follow easily from (7), (8) and (15). On working out (Ya, a),. with the help of (15) and (19), one finds that the conditions that the result should be symmetric in # and v are cr 5 ~ (r E~,, .+Eatp , E ~rpv@ Eaa Es.+ E ~6 , Eo. o" p o" 5 o" 6 o" . : Ea~" l,+Ea. Ept,+Ea. Es~,+Ea. Eel,, E 5•~t', "-~'EAI~ p 5 5 5 o Ep"+EA~ Es~+Eat' E~ __ E 5 t,+E~. 5 s 5 e 5 . -- ~, E m , + E ~ E s t , + E ~ v Eel,, o" p o" 5 o" o" o" . Ej,..+Ej, Ep~+E~ Es~ : E~,~,+E. p E po" ~ + E v5 Es~ ,, E~. , 5 5

(21)

where

E~. = E;; EL = E2.

(2~)

These relations m a y also be obtained by a similar process from (3). If Ra. is the spinor curvature, defined by

R~. = C . , + P , r , - r , , a - r . G .

(23)

and Rx~ . ; is the Riemann Christoffel tensor, constructed entirely from the

DIRAC MATRICES, TELEPARALLELISM AND PARITY CONSERVATION

377

Christoffel affinity, one has [~,~, R#~] = R ~ y p .

(24)

On decomposing this identity, one obtains

~ -(CL

Cp.+Eat ` Es~+Ea# Eel)

o, 6 o. ,~ . C p,,,+ E ,~, Es,,,+ Ea,, E~,/,) = R~,,,,,, 5 b+C,~l, p Ep~+E:L# 5 e E5~ = ~ 5 - - C pa. Ep#+Ea~ s e Es . Eat, "~//,~e~,

-e

,o.

p

6

~

5

e

~6

-- C p

e

5

15-aa/.+C;~t, Ep.+E:~I, Es. : r'a.5,-~ av Ep,a+Eav 5 p 5 E~/~ --E~/# = E~P E,j,--E# E,v,

E6

(25)

.

5~,

where /v means the covariant derivative with respect to %, in the usual sense of general relativity. It is readily verified that (25) and (24) are equivalent.

4. Teleparallelism Consider the simultaneous differential equations

ha, ~ = E~, p hp + Ea~ 5 , h5+E~he; hs, #

E 5# p h p ~a_~'e ~5# ~ '~6. ' he, t, = E~t, hp+ E 56~ h 5.

(26)

The conditions that they are integrable are obtained b y working out (hx, t,),~, (hs.a),v, and (hs,#),v and expressing the fact that these must be symmetric in # and v. But these conditions are found to be satisfied identically b y virtue of (21). Thus equations (26) are integrable, b y virtue of (15). There is one integral of (26) which does not depend on the E's, so long as they satisfy the integrability conditions; this is

h ~ h:~+g-1 h52--he 2 = constant.

(27)

This enables one to prove the theorem: corresponding to any vector h~(O) at the point 0 o/space-time, there exists a uniquely defined 'parallel' vector

at any other point P o] space-time. The proof is as follows. Let h be any fixed constant, and at 0 let h~ = ha(0 ), h e ----h, and h~ = {(--g)(h~h~--h2)}½; then the constant on the right-hand side of (27) is zero. Furthermore, the solution of (27) is uniquely determined throughout space-time. Hence the vector hha/h 6 is uniquely determined everywhere, in particular at the point P. It will be noticed that the "uniqueness" of the parallel vector at a distance depends on a special choice for the value (h) of h 6 at 0, which thereafter plays a fundamental role in the metrical properties of the space-time manifold. In this respect the present concept of teleparallelism is more general than Einstein's where no such parameter was involved. Einstein's theory is obtained b y imposing the restrictions g-½h 5 = constant and h e : constant,

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which have the effect of reducing E~,, Eg ~ and Eg 5 to zero. The author e) has recently discussed the application of this theory to electrodynamics, concluding that it is still tenable. Generalizations of Einstein's theory in s were discussed by Einstein which only Eg ~ and Eg 5 vanish, and not Exg, himself ~), by Pauli and Solomon a), and by Schouten and van Dantzig s). 5. R e p r e s e n t a t i o n

In this section, it will be shown how one can construct an explicit representation of the Ya, given the components E]g, Ex~,, 5 El,~ and Eg 5 of the spinor affinity. As a by-product of this work, a general solution will be obtained of the set of simultaneous equations (21). The first step is to determine a set of six independent solutions h,x, hrs, hr6 (r = 1, 2, 3, 4, 5, 6) of the simultaneous equations (26). This is done by taking any solution of the equations ~,h~x h~, = gag;

~,~h~ h~5 = 0;

X, h,5 h,5 = e; 2~ h,5 h~s = 0;

= 0; ~ , hr6 h,6 = h 2

(27)

at the point 0; the solution, which relates only to this point, is a set of numbers, not field variables. But, by means of the integrable equations (26), the values so obtained can be continued to any point of space-time. Moreover, since gag, 0, g, 0, 0, h ~, satisfy the same simultaneous ditferential equations [by virtue of the second of eqs. (13)] as ~ h~xh,g, etc., the relations (27) hold not merely at 0 but throughout space-time. Let h~ = gas h~, h~5 = g-1 h~5, he = h-2 hre; (28) then from (27) it follows that h~:t h ,a + h~5 h ,5 + hr 6 h,e = (~,.

(29)

Also, by multiplying (29) by hsx' g, h,5' g and h,~, g, one finds that the simultaneous equations (26) are satisfied identically, provided Zlg= E7 :

X, h : h,a, ~, ~,, h~" hr6" #,

E L = ~ , h/h,x, ~, E~,a : ~ , h, a hr6" g.

(30)

It can also be verified that, by virtue of (26), the relations (30) furnish a solution of the simultaneous differential equations (21). The most general solution of (21) is therefore provided by (30), with hrx, h,5 and h~ unrestricted except by (27). To achieve a representation of the 7x, let ~, (r = 1, 2, 3, 4, 5, 6) be a set of 6 constant mutually anti-commuting matrices: {~r, ~,} = 26,.,;

(31)

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TELEPARALLELISM AND PARITY CONSERVATION

379

these generate a Clifford algebra and as there are 2 X 3 of them, they have well-known representations in 2s ----- 8 dimensions. Let

& = Z~ h~a ~..

ft, = L h~e ~

fl~ = } ; h~ ~..

(32)

so that. by (27) one has

(33) and by (26) one has /~,/~

v

5

6

= E;tt, fl~+Ex~,fls+E~t~,fl6; =

/~6. # =

E~ & + ~ ; E.~~ + E e t , ¢ ~55 .

(34)

Let (37)

it satisfies

{~, ~:} = {~, ~) = (~., ~) f172 = -- 1;

=

o,

fiT,F, = O.

(as)

Then, according to (33), (34) and (38), ½fl~fl,(1-4-ifl7 ) satisfy the same equations as 7a; and one can make the identification

7x = ½fla fla(l ! ifl7) •

(39)

The representation thus obtained is 8-dimensional, but reducible to two four-dimensional representations of which ½(l+ifl7 ) and ½(1--if17) are the unit elements. There appears to be no way of obtaining the 4-dimensional representations without introducing the dichotomic variable f17,which appears in a perfectly natural way in the theory. One is therefore impelled to regard in (5) as an eight-component spinor, and to consider the generalization fla fie (~v,Z--B:t)V

(40)

_ 1 ~ ~ B~--~-(~c~. ~+E~. ~ ~5+~L ~ ~.+E~5~.)

(4~)

of (5), in which

is the spinor affinity. One concludes that the natural generalization of the 4-component spinor equation of special relativity is an 8-component spinor equation. The significance of the extra components will be discussed in § 8. It should be mentioned that the above results could be expected on arguments derived from group theory. The group of similarity transformations under which the theory is invariant is isomorphic with the group of rotations in an abstract 6-dimensional space. This group is well-known to require a 2 3 = 8-dimensional representation.

380

H.S.

GREEN

6. Physical Identifications The physical interpretation of this theory is naturally somewhat tentative because no field equations -- apart from the identities (21) -- have yet been postulated, and no attempt to do so will be made in the present paper. However, certain identifications can be made with confidence even in the absence of a set of field equations, assuming of course that there is some physical sense in the theory at all. If e is the 'bare' electonic charge and ~v~ the electro-magnetic four-vector potential, one sees from (18) that t

A similar identification was made b y Einstein, though, as he was unaware of the connection of his theory with Dirac's equation, he did not introduce imaginaries. The pseudo-scalar meson potential ~v5 can be identified in a similar way: ¢ = (43) where e' is the appropriate coupling constant, which, in an 'unrenormalized' theory, might well be the same as e. From (18) one sees also that //,

=

J(E

t')

defines an unrenormalized mass field p, which ought to be constant over regions comparable with the de Brogiie wave-length, even at low energies. The real parts of Cx~, E ~ and E ~ do not play any part in the account given of the elementary particles b y special relativity, and are therefore connected with gravitational phenomena. This is confirmed in part b y (25), where the Riemann-Christoffel tensor R ~ appears linearly with the real part of C ~ , / ~ - - C ~ / t ,. So far there is no direct discrepancy with observation. However, one question demands immediate consideration. H o w is it that, with a single spinor equation (40), the several different kinds of fermions found in nature can be represented? To answer this question, it must be stated how ~o is related, for example, to the wave-spinor of the electron, which has no interaction with the meson field q~, though it interacts with the electromagnetic field. It will be shown that there are unique, relativistically invariant, similarity transformations V 0, Vs, V6 which 'switch off' the interaction of the electromagnetic field, the meson field, and the mass field, respectively, with the spinor field ~o. Then, if ~0 represented the proton field, VoW might represent the neutron field, V 5 V 6 ~omight represent the positon field, and V 6 V 5 V 6 ~ might represent the neutrino field. There are, of course, other possibilities such t The symbol J

denotes i times the imaginary part.

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as V 5 ~o, V e ~o, V o V5 ~o and V o Ve ~o which might be needed to represent the /,-meson and the hyperons. To construct Vo explicitly, let k~~, for u ---- 1, 2, 3, 4, be eigenvectors of g ~ , regarded as a four-dimensional matrix, so that ga~, k ~ ' =- g'at, k~',

where g'~, is diagonal. Let the k~a be normalized so that ~uku ~ k ~ = g ~ . Then if a ~ is defined by ax = 24 k~a a~,

(44)

where the u-summation is from 1 to 4, one has Vo

-1 = l+a fl +½aa Rv~a-t-ga P ~ T±~ x V a = lq-aafls; Vs = l q - a e f l s ,

~',v~,

(45)

where a aa and fl,z are formed from a x and flz in the same way as ya* and ~za are formed from ya and yz (sec eq. (7)), a s = as, and a s = a s. The reciplocals of Vo, V5 and V s are given by ;ts(1 +/Sa aZ+ 1R

Vo-1 G

=

a~X±lR G

= Zs(l+&a%

where ~s, 2~ and '~e are constants which can be determined in a straightforward way. If the above suggestion is correct, an interaction which involves, for example, the emission of a positively charged meson, has an amplitude proportional to V0 Y5% when a similar interaction involving the emission of a neutral meson has an amplitude proportional to Y5yd. It is worth noticing that V0, V5 and V s commute, so that it is unnecessary to distinguish between V s Vs, for example, and Va Vo. But V0 fl~ V5 is distinct from Vsfl~,V s, because Foil a = a;tVo;

V~fl 5 = % V~;

Vet5 s = % V s.

(47)

These formulae enable one to bring Vo, V5 and Ve to the left- or right-hand side of any amplitude in which they occur. 7. P a r t i c l e - A n t i p a r t i c l e Conjugation In spinor field theory, the view is generally held that a Dirac spinor represents simultaneously two rather different things: particles and 'holes', or, in the quantized theory, the annihilation of particles and the creation of anti-particles. Another spinor, obtained from the first by particle-antiparticle conjugation, represents the creation of particles and the annihilation of antiparticles. This operation of particle-antiparticle conjugation, is per-

382

H.S. GREEN

formed by taking the complex conjugate and multiplying by the transpose of a matrix ~7. In the general theory of relativity this matrix cannot be identified with 74 as in the special theory, and the object of this section is to show how it m a y be constructed. If 7~* represents the hermitean conjugate of ~,~, ~?must have the property }'x = 7~* r/.

(48)

By differentiation with respect to x, and multiplication from the left by ~-1, one finds

~1 7:t t,~--Ta, t,- --2(Ial, Y~+I,t~,Ts+Rt,~7:t~+Ri, ST,ts) (49) from (15), if I ~ and 1 ~ represent i times the imaginary parts of C~a and E~z, and Rz ~and Rz 5represent the real parts of E~ ~and Ez s Since g~p C~a, and hence g~pl~t~, are a n t i s y m m e t r i c a l in ~ and p, (49) is satisfied by

(50) ~?-z~,~ = ~1 I ~~ 7~ 7 ~--- i ~5 7 ~ 75 + R~' 7 . + R t , 5 75. One can verify by a straightforward though tedious calculation that (~. z),, computed with the help of this formula is symmetric in # and v, by virtue of the identities (21). So (50) is integrable, and requires only a boundary condition to make ~ uniquely determinate. This boundary condition is supplied by assuming the Galilean value 74 for ~ in regions remote from matter. Under similarity transformation, ~ transforms somewhat differently from the Dirac matrices. Corresponding to (2), one must have 7] --->"n ' :

(V--l)*7] V - l ,

(51)

in order that the property (48) should be invariant with respect to such transformations.

8. Conservation of Parity As general relativity usually presupposes the invariance of physical laws with respect to all co-ordinate transformations, including reflections of the spatial co-ordinate system, it is in apparent contradiction with recent experimental evidence 9) that parity is not conserved in the weak interactions affecting the decay of certain elementary particles. A crucial test of the theory is therefore to examine whether it is consistent with the failure of parity conservation. In a recent paper, Green and Hurst 10) have suggested that the condition of invariance with respect to spacial reflections might be relaxed, by requiring only invariance with respect to parity conjugation in the sense of Lee and Yang 11). In Watanabe's terminology 1~), this implies that chirality, rather than parity, is conserved. Green and Hurst 10) and Watanabe 12) independently introduced 8-component spinors to allow the

DIRAC MATRICES, TELEPARALLELISM AND PARITY CONSERVATION

383

representation of the chirality operator, and Green and Hurst defined a set of operators Pl, P2 and Pa operating on the 8-component spinors. The matrix Pa always commuted with the Dirac matrices; Pl and P2 commuted with the Dirac matrices in representations for particles of 'pure' parity and anticommuted with the Dirac matrices in representations for particles of 'mixed' parity. Failures of parity conservation could be attributed to the interaction of particles of pure and mixed parity. All these considerations are automatically validated by the theory of the previous sections. One can make the following identifications: pl =

(-g)-~85,

p~ =

(g)-~8587,

p~ = - i 8 7

(52)

and the conservation of chirality is ensured by the requirement p3w = ~

(53)

which is equivalent to the choice of ½(1+i87 ) as the unit element. There are two possible and equivalent representations of the Dirac matrices, furnished by the Yx = 8ix 86 and the 8x 85 of this paper. The Yx commute with Pl and P2, as well as P3 and are therefore appropriate to particles of pure parity. The 8x r5 on the other hand, commute with P3 but anticommute with pl and p~, and are appropriate to particles of mixed parity. The following m a y therefore be asserted: the failure of parity conservation is not merely consistent with general relativity, but is an almost necessary consequence of the theory. References I) A. Einstein, Sitz. Preuss. Akad. Wiss., P.-M. Klasse (1928) 217, 224; (1928) 2, 156; (1930) 18, 401 2) J. A. Schouten, J. of Math. and Phys. (Mass.) 10 (1932) 239 3) W. Pauli and J. Solomon, J. Physique 3 (1932) 452, 582 4) L. Infeld and B. L. van der Waerden, Sitz. Preuss. Akad. Wiss., P.-M. Klasse (1933) 380 5) W. L. Bade and H. Jehle, Revs. Mod. Phys. 25 (1953) 714 6) H. S. Green, Proc. Roy. Soc. A 7) A. Einstein and W. Mayer, Sitz. Preuss. Akad. Wiss., P.-M. Klasse (1931) 541 8) J. A. Schouten and D. van Dantzig, K. Akad. Amst., 35 (1932) 642, 843 9) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105 (1957) 1413 and subsequent publications by these and others 10) H. S. Green and C. A. Hurst, Nuclear Physics 4 (1957) 589 11) T. D. Lee and C. N. Yang, Phys. Rev. 102 (1956) 290 12) S. Watanabe, Nuovo Cimento 6 (1957) 187