IL NUOV0 CIMENTO VoL. 56 B, N. 1 11 Marzo 1980

Weyl-Dirac Theory with Torsion.

II: Foundations and Conservation Equations (*).

D. GREGORASH and G. PAPINI

Department o] Physics and Astronomy, University o/ Regina Regina, Sask., Canada $4S 0A2

(ricevuto il 15 Novembre 1979)

Summary. - - The Weyl-Dirac theory is extended to involve the complex Dirac field ft. The conservation equations of such a theory are obtained by using a co-covariant generalization of Rosenfeld's variational method. The electromagnetic current associated with the Dirae field is nonzero in general, and is highly nonlinear in the electromagnetic potential. Its form is reminiscent of the current of superconductors.

l , - Introduction.

In a previous paper (i), hereafter referred to as I , we reported on a gen-

eralization of the Weyl-Dirac unified theory of gravitat ion and electromag-

netism. The field equations admit nonzero torsion; thus the new theory is

suitable for s tudying problems involving intrinsic angular momen tum and

similar phenomena in curved space-times (2).

The purpose of this paper is to establish the mathemat ical foundations of

the theory and in particular to obtain a set o f conservation equations con-

sistent with Bianchi 's identities. For this, it is most convenient to use a co-

(*) Research supported in part by the Natural Sciences and ]~ngineering Research Council of Canada. (1) D. GRI~GOI~ASH and G. PAPI~I: II Nuovo Cimento B, 55, 37 (1980). (2) See, for example, F. W. H~HL: Gen. ReL Gray., 4, 333 (1973).

21

2 ~. D. GREGORASH and G. PAPINI

covariant modification of the method presented by BELINFANTE (a) and I~ OSENFELD (4).

Because of similarities between our theory and the special-relativistic for- mulat ion of the Ginzburg-Landau theory of superconduct ivi ty (~), we shall allow for the possibility of a complex Dirac field ft.

2. - C o . c o v a r i m a t c a l c u l u s .

We recall f rom I some of the results of the co-covariant calculus. We adop t

the following notation. Several tensorial indices may be designated by the nota t ion q~. For the

t ransformat ion x ' = x'(x) given by

(2.1) dx ' , = a~ dx~ ,

the corresponding t ransformat ion rule for q~ is denoted by

(2.2) ~ =- aa~(a) ~ .

The infinitesimal t ransformat ion rule for ~ is then (a.~ : ~ ~ a~)

~ --~ ~ - ~ = (a~)~" ~. ~ , , (2.3)

where

~a~'(a) ~ = ~ (2.4) (~2)." - ~a.~

are the infinitesimal generators of the t ransformat ion representat ion a (*). Using ~his notat ion, we write the eo-covariant derivative of a co-tensor

~ of power n (e) as (**)

(2.5)

(a) F. L. BELINFANTE: Physiea (The Hague), 6, 887 (1939). (4) L. ROSENFELD: ~lem. Acad. R. Belg. Sei., 18, 4 (1940). (~) H. B. NI~,LSEN and P. OLESEN: Nuel. 2)hys. B, 61, 45 (1973). (*) For example, for ~ = ~0a~, we have

a~B(a) ~ aa~o(a ) ~_ (a-1)~a~Q and

(6) The terminology used is that of P. A. M. DmAc: 2)roe. R. Soe. London Set. A, 333, 403 (1973). (**) The star and comma respectively denote co-covariant and partial differentiation.

W E Y L - D I R A C T H E O R Y W I T H T O R S I O N - I I

For a co-tensor density Q~ of power n, we have

For successive co-covariant differentiations, we obtain

(2.7) coa,,~o- o-~a,oa = - - (~ras)u~/~u,~o~% + ~oa.'`Qu~o + no~a/~.e,

where the in-tensors

q'`~ = - - D % + P%z

(2.8)

(2.9)

and

(2.~o)

23

are called the curvature, torsion and electromagnetic-field tensors, respectively. In particular, if the pair ( r , x) is metric compatible in the sense t h a t

g'`~,~ ~ O, then we obtain from (2.7)

(2.1]) R~o + /~ ' `~ = - - 2g'̀ ~b,~.

Thus we may state the following theorem: whenever ( f , u) is metric compatible, /~u~zo -?/l~'`ao vanishes if and only if/~o vanishes, t ha t is, if and only if u~ is integrable.

F rom (2.8) and (2.11), we have

(2.12) K ~ - - - P'`~,~ + 1%~,~ = - - 4t45.

Therefore, it follows from (2.6) tha t

(2.13) ~a*,~--Qa.~a = --(aa")u"l~'`,zQ g'-2, -}- ~2~.'`Q'`z~ -+- (-R'`'`~o + nJze)~2a ~-

= - - (aa")u"Kavaof2, + Qa,,Q"~.a + (n - - 4) Qa]~o

for a co-tensor density of power n.

Of considerable importance in a variat ional problem in physics is the re- lation for the co-covariant divergence of a co-vector density. For a co-density ~u of power n, (2.6) implies

(2.]4) 9,' , , , = ~",, - - ~ , (Q%~ + nz'`).

We conclude this section with Bianchi 's identities, which are the s t ructure

equations of a space. Being algebraic consequences of the definitions of the part icular tensors, these identities show tha t any co-covariant geometry is

24 D. GREGORASH and G. PAPINI

completely determined by its curvature, torsion and electromagnetic-field tensors. We have

i) Bianchi's first identity,

(2.15) R--%aQ,,, ~ R%.,,~. e + R%.oo.a -q- /7~,~,~Q%,, ~ /7u,c,~Q~'~e q-/7~,c~Q%~ -- o ;

ii) Bianchi's second identity,

(2.16) Qx.Q + Q~q... q_ QZq.. Jr Qa~,~,Q% e q_ QX~,eQ%. _~ QXa.Q%~, =

iii) Bianchi's third identity (homogeneous Maxwell equations),

or, equivalently,

1..,~ + 1~.,. + 1.~,. = 0 .

The contracted Bianchi identities are of greater relevance in a theory involving gravitation. With the Ricci tensor and scalar curvature defined by

(2.18)

and

(2.19)

respectively, the contracted identities are

(2.20) (/7%-- �89 5%/7).. ---- �89

and

(2.21) Q~,,~ q- Q~,~,~- Q ~ , , q- Q~,~Q%, = / 7 , ~ , - / T u , - 2]u~.

3. - The formal ism of the variational principle.

In this section, we describe the mathematical formalism of the variational principle used to obtain the field equations and conservation equations of our theory. With suitable modification, this extension of the method of Belinfante (3) and Rosenfeld (4) can be applied to a wide class of classical physical systems interacting with gravitation.

For our purposes, it is sufficient to describe a variational principle as the hypothesis that there exists a function f3, dependent upon some specified

W E Y L - D I I C A C T H E O R Y W I T I I TOICSION - I I 2 ~

geometr ic objects and their der ivat ives (*), for which the ac t ion in tegra l

(3 .1) I ( V ) - f ) 2 d ' x V

is i nva r i an t unde r one or more t r a n s f o r m a t i o n groups.

I n par t icular , in a Wey l -D i r ac t h e o r y wi th source, ~ will in general depend

on the var iables (**) (fl,/~, g~ , x~, F ~ , , ~A), where ~0~ is a (real) co- tensor of

power n, t oge the r wi th the first par t ia l der iva t ive of each of these. Since such

a t h e o r y is, b y hypothes is , a eo-covar ian t one, it follows t h a t we can wri te

in te rms of co -covar i an t objects ,

(3.'.,)

Let us consider the invar iance of I ( V ) u n d e r a co-ord ina te t r an s fo rma t ion

x ' = x '(x) . F o r any tensor ia l objec t o~ of t r a n s f o r m a t i o n t y p e a~ b, we shall have

(3.3) ' ,' . o o ( x ) = a 2 0 ~ ) o ~ ( x ) , ~ - ~x~

I n par t icular , for an infini tesimal t r a n s fo rm a t ion x '" = x~ + ~(x) , we ob ta in

(3.~) (O/a(X ') : (Oa(X) ~_ AO)a r~ (1)a(X) ~_ ((~ab)lv(l)b(X)~l:v.

As is well k n o w n (7), the q u a n t i t y Ao~ in t roduced in (3.4), be ing the dif- ference of two tensors a t separa te points , is no t itself a tensor . Thus, while

we could do so, it is no t conven ien t to work wi th the A-var ia t ion.

We m a y , however , define ~ t y p e of va r i a t ion which yields a tensor for any

geometr ic object . F o r example , for a co- tensor o~ of power n, we define 8o~ accord ing to

(3 .5 ) ~,~o = ~ ,~ ' o (x ' ) l~ .~ - ~o~,(x) ~- ~'a(X) - (oo(x) .

The p rocedure invo lved here is seen to be analogous to t h a t of dif ferent ia t ion

in t h a t the t r ans fo rmed componen t s (o' a are (( pulled back ,) to the po in t x (***).

(*) Specifically, we t'~ke ~ to be a polynomial in the ilrst derivative o fany object in order that the resultant field (~quations be second-order differential equations (albeit possibly nonlinear). (**) We denote the complex conjugate ot' A by .~. (7) J . L . ANDERSON: Principles o~ Relativity Physics (New York, N.Y. , 1967), p. 39. (***) hr fact, the reader will notice, in the final expression, that --~oJ~ is the co-covariant aualogue of the Lie derivative (ff ~o~ with respect to ~ .

26 D. GR~GORASH ~ I l d G. I ' A P I N I

We may write

~o~o = ~ : ( x ) - ~o(x) + ~ ' ( x ' ) - ~:(x') __ 5 ~ o - (~o(=')- ~.(x)) .

Using the co-eovariant generalization of Taylor's expansion,

a).(x') = co~(x) 3- [e).,~,-- nu.o~.] (x'" - - xl') 3- O[(x'~-- x")~] ,

we finally obtain

or, in manifest ly eo-covariant form,

( 3 . 7 ) ~r a = ( g a b ) , , f, Ob [~ , ,~ , _ _ Q /Z ) . ~) . ] _ _ f.Oa,/z ~iz.

In particular, for the metric tensor we have

(3.8) ~g,. = - - ~ /z . . - - ~../z + (Q,.~ 3- Q.,~) ~ .

while, for any co-scalar field Y2, we have

(3.9) 8~ = -- tg, e~e �9

The above prescription may be applied to both F~/z, and =,. However, it is easier to obtain the variations of these in the following manner: we first note tha t

(3.10)

so tha t

~(~o,,) - - ( ~ ) , , ,

By using (3.7) and its analogue for ~%., and then applying (2.7), we obtain

(3.11)

and

(3.12)

In addition, we do not apply (3.7) in order to evaluate ~/~u,~; rather, using the definition of the eurvatm'e tensor, we immediately obtain

(3.13) ~/~v~ = -- (~r~-),~ 3. (~F~u~),, 3. Q%Q ~/~z~,r.

W ~ E Y L - D I I ~ A C T t I i E O R u W I T t t T O R S I O N " - I I

If the action integral I(V) is to be invariant, then we must have (*)

f~'(x')d4x'-- f~(x)d'x = o . V V

For an infinitesimal transformation ~x~ = ~(x), however, we have

d4x '= det (6~ + ~,~) d~x ~ (1 + ~,~) d4x,

so that , to first order in ~,, we have

f{s -- ~(x) ~- s = 0 . V

The integrand contains the expression

~ ' ( x ' ) - - ~ ( x ) = A ~ = ~s § s

so that , finally, the variational principle becomes

(3.14) [ { ~ + (~) ,~} d~x = 0 o

V

In order to most easily demonstrate the usefulness of this method, let us consider a Lagrangian s dependent upon g~ and some other unspecified fields, but independent o f / ~ Q . Then we have, for some coefficients s s and s

(3.~5) ~2 + ( ~ ) . ~ = ~g~ ~g~ + ~ - - s ~ . ~ . . ~ + ( ~ ) , ~ =

- ( ~ . ~ + s + { ( ~ s s + ~ , , , + s ~ , ~ , , } . ~ .

Because of (3.8), we may write

-

where

(3.16) ~ v ~ ~ ~s ~ 1 - +

27

(*) The notation ~(x) is purely symbolic and does not imply that s is explicitly a function of x.

~ 8 D. GREGORASH and G. PAPINI

which is a co-tensor of power --2, is the symmetric (total) energy-momentum tensor associated with ~.

Thus the invariance of the action integral may be expressed as

(3.17) f{[-- ~ , ~ ~ Q ~ -- ~#~Q~% -}- 9~ § 9~v,~ + 9~e~Q~-

If we consider variations ~e such that the divergence term in the integrand vanishes (via Stokes' theorem), then we have

(3.18)

In general, then, the total energy-momentum of a physical system interacting with geometry will not be a conserved quantity.

Having obtained (3.18), we may now consider variations ~q which do not cause the divergence term to vanish. If (3.18) is to be still valid, then we must have

that is

(3.19) [(~e, -+- 5~9~ -- ~ " § ~,~,+. § s ~e-- ~g'"~%~.],. -?

We may treat ~ , ~Q,. and ~,<~:) as independent quantities. By expanding (3.19), we obtain

(3.20)

-4- [(~" + ~ 's - - ~ : + s § s +

Thus we h~ve

(3.21)

and

(3.22) ~e ~ = s 6 ~ 2 + �89

together with the equation implied by the vanishing of the coefficient of ~e, which is equivalent to (3.18).

W E Y L - D I R A C T H E O R Y W I T H T O R S I O N - I I ~ 9

We wish to app ly this formal ism to the Lagrangian of a Weyl-Dirac theory which can be wri t ten as the sum of a curva ture part, ~,(fl, fi, g~, l~u~) , a (~ conformal pa r t ~), s fl, fl,~, fi,,, g~, ]~), and a source par t , ~( f l , /~ , g~, ~a, Va,,). We assume these to be in-densities, so t ha t our formal ism can be

applied separa te ly to each one.

4 . - T h e e q u a t i o n s o f c u r v a t u r e .

Under an infinitesimal eo-ordin~te t rans format ion 8x~ * = ~u(x), the in-

t eg rand of the var ia t ional principle of an in-densi ty ~.(fl,/~, gu~,/~,~e) is

(4.1) ~ a q- ( s ~

where

(4.2) ~ _ ~fl ,

(4.3) ~ = ~ f i ,

(4.4) ~ , ~ = - ~ \ ~ + ~ , /

and

(4.5) ~ . ~ -- + ~R~u~ ~R~,~ "

F r o m (2.14) and (3.13) we m a y write

(4.6) �89 !D2 '~v ~/~x~ = - - (!Dxuvq,e + ~ v ~ Q % ~ + 1 !l$~.u,~ Q%o) ~1 ~v +

Thus (4.1) m a y be wri t ten as

(4.7) $Y~R § (~,~).~ : ~ ,~f i + !32~fi--~u~$g~, + !D~.u~$Pxu~ +

+ ( s ~ + ~3x~ ~F%).~,

where, for convenience, we define

(4.8) ~ z ~ = --(~3~.,~% + ~"~(2%, + ~ ~3~,~Q~,~) .

Upon subst i tu t ing (3.8), (3.9) and (3.11) into (4.7), we find t ha t

~ 0 D. GI~EGORASH and G. PAPINI

I f we write

(4.9)

then we obtain

(4.~0)

F r o m the coefficient of ~e we obtain the conservation equation

= - - !3~fl ,~- !3~/~,~ + !3a"~R-~,,~ .

We shall see below that , whenever s ~ / ~ this is equivalent to the contracted

first Bianchi identi ty. I f (4.11) is to be valid for a rb i t rary infinitesimal 65, then we require

In this expression we find the terms

(4.~3) {[~.,~Q~.~- ~ , ~ ] ~ . , - ~,-~. ,~}. . + + { [ ~ , ~ Q ~ - ~, . ] ~ . ~ - ~,-~. ,~} Q~ =

= [ (~g ' " . . § ~x"'Q%~.)QX.e- ~e" + !~.,.Q%..]~%~. +

Because ~o~c~)= 0, this includes the terms

Thus the r ight-hand side of (4.13) equals

W E Y L - D I R A C T H E O R Y W I T H T O R S I O N - I I 31

This implies t ha t (4.12) m a y be writ ten as

[2 (~ + G~'~. + ~#.~'/7%.~ + �89 ( ~ ' / ~ . - - ~ d ~ q ~ . ) ] ~ , ~ +

+ a t e r m l inear in ~ = 0 ;

consequently we obta in

(4.14)

where

(4.15)

Equa t ion (2.13) and Bianehi 's second ident i ty m a y be used to rewrite (4.11) and (4.14). We have

~ = --(~d,~,~ + ~ ' . 9 ~ + � 8 9

= + { ~ , ~ % ~ - ~ ~

Therefore, (4.14) becomes

(4.16) 2~eu = 2~ou- - ~e~

thus we m a y write (4.H) as

Let us now briefly consider the invariance of the action integral for ~ , under conformal t ransformat ions . Under an infinitesimal t ransformat ion ,

~gu, = 2ogu~ , (4.18)

we have

(4.19) ~fl = - (off a n d ~f l = - ( o f f ,

while ]~x~,e is an in-tensor.

The integrand of the var ia t ional principle for such a t rans format ion is, therefore, given by

Thus the invarianee of the act ion integral under conformal t ransformat ions

32

requires

(4.20) ( ~ , = _ 1{/~ +/~!~2},

regardless of the specific form of ~ . The simplest real in-density involving fl and the scalar curvature is

(4.21) ~a = <=-# I # l ~ .

For such a Lagrangian, we find the following:

(4.22)

and

(4.23)

(4.24)

while

(4.25)

D. G R E G O R A S H and G. P A P I N I

Equat ion (4.22) means that

From (4.22) and (4.24), we see that (4.17) is identical to the contracted first Bianchi identity.

5. - The equations of con_formal structure.

We assume tha t the Lagrangian describing the conformal structure of space- time is an in-density ~(fl, ~,fl.~,, ~.~,, g~,., f~,,). Then with the help of the definitions

(~.~) ~ - ~--~ - ~ f l - ~ , ~ , j , , - ~ , ~ Q ~ "

(5.2) ~ = - ~ ,

(5.3) 9.1., = eg,, q- ~g , , '

(5.4) i~ = #~-;Z,,, + fi ~,~

and

(5.5) ~ " = 2 ~ ; ~ _~T, , vlv~/

- - ~ 1=~o~t~=

W E Y L - D I R A C T I I E O R Y W I T I I T O R S I O N - II ~

W~ hg~TO

~,~ = ~ ~I~ + ~ ~fi + i ~ ~x~ + '5 ,~"

Xow

_ _ 0 v/~ - - = = 2 uv ~ ~ ~t~ 2~,,~(~+),~ ~ ,~ ~ , + {~ ~s,~,

so tha t , wi th respec t to an inf in i tes imal co-ordin~te t r ans fo rma t ion , ~he in-

t e g r and of the v~r ia t iona l pr inc ip le is

+ ~ + ~/-~.~ ~ ..~

Because (5.6) conta ins no higher t h a n f i rs t -order de r iva t ives o~ 8~', we see i m m e d i a t e l y that

(5.7) %~,_- ~ + ~/~-~ ~,o-~-~,~ s

s~tisfies the conservation equation

Under an inf in i tes imal conformal t r ans fo rma t ion (4.18), we have (4.19) a n d

(5.9) a(fi , ,) = - - o)f l . , , 5(fi,~) = - - ~fl-.,,,

while fu, is i nva r i an t . Thus the in tegr~nd of the va r i a t i ona l pr inc ip le t akes the form

from which we ob ta in

(~.~) w , , = - / ~/~ +~*,M.- , + / ~ + L , ~ . , j �9

We set now

3 - I1 N u o v o C i m e n t o ]3.

~ D . G R E G O R A S H and O. P A P I N I

With this particular choice and that given in (4.21) for ~ , the vacuum La- grangian, ~ . + ~ , very strongly reminds us of the special-relativistic formu- lation of Ginzburg-Landau theory (~). However, we expect important dif- ferences to exist between that theory and the present on% because, in the former, fl and the electromagnetic field have not the geometrical significance they have in the latter.

With ~ given by (5.12) we find

(5.13) ! = / t . + +

. i

- - - ~ t~'*~ol --

A detailed study of (5.13) goes beyond the scope of this paper. However, it is interesting to note the form of (5.13) when the Einstein gauge (]fl] ~ 1) is chosen: with fl ~-- exp [i9] we obtain

(5.14)

where

and

k V'----g {r (A s + B 2) - - 2 A ( ~ o - 9 , ~ , o ) - 2 B ( ~ 9 o + ~Qg"~)}, + (A s + B~) �89

A ~ ~ - - ~'~,~

B ~ 2u"~,~.

The second term on the r.h.s, indicates the presence of a scalar field ~ inter- acting with electromagnetism.

The vacuum current i~ exhibits a quite unusual behaviour. For the present choice of ~o, it takes the form

(5.15)

In general, this is a rather complicated expression, but its form in the Einstein gauge, f l= exp [i~], is quite interesting.

We, in fact, have

(5.16) i" = 2k V ~ (Auu + B~,,} (A s + B~) ~

The first term here is reminiscent of the Meissner current of super- conductors, while the second allows for possible fluxoid quantization.

V r T t t E O I ~ Y W I T I I T O R S I O N - I I 35

This s imple resu l t ind ica tes t h a t a t t e m p t s to r educe the ~ r e y l - D i r a c t h e o r y

to t he E ins t e in -Maxwel l t h e o r y (s) arise in genera l f r o m a fa i lure to under -

s t and the t rue phys i ca l s ignif icance of W e y l ' s geomet r i ca l hypo theses .

6. - The equations for a min imal ly coupled source.

The L a g r a n g i a n for a m i n i m a l l y coupled source field ~a of p o w e r n will

be an in -dens i ty ~( f l , fl, g~,,, ~f~, q~,~,). F o r convenience , we shall use She de-

f init ions (*) (**)

( 6 . 2 ) z , - z : - eft,

(6.3) i~ ~ = ~ = - - n ~ c~0a, ,

(6,4) ~u~. : - - 2 ~T,au~ - - ~ , ~

and

(o'a~)~ u ~

-

U n d e r an inf in i tes imal co-ord ina te t r a n s f o r m a t i o n , the i n t e g r a n d of the va r i a t i ona l pr inciple , sf~md*x = 0, is

(6.6) 1

j~

W e use (3.8) a n d (3.11) to ob t a in

(s) J . L . PIETENPOL, R. INCOUL and D. SP]~ISER: Phys. Rev. Lett., 33, 387 (1974). (*) In any relativistic theory, the group generators (aAB)u~ satisfy (aa~)u~ = - (aaB)t u. We use this without further mentioning it. (**) If ~m is the Lagrangian for the source field ~a, then ~A= 0, the validity of which we henceforth assume.

~6 D. GREGORASH and o. PAPINI

and

with

(6.7) ~'~ = ~"~,~ + ~,~Q%~. :

~urthermore,

Thus the final expression for the integrand (6.6) is

( l ~, /Q . , . _ ~ f l , _ ~:2fl , . + im i v . - 5 +

1 ,

&?~,, ~*~

From this expression we find that

(6.9) ~:e~ + � 8 9 = t d ,

where

(6.10) te ~ = &q ~,~ 8 ~ , ~ ,~

is the c a n o n i c a l energy-momentum tensor. I t is~ therefore, apparent that ~ e is the energy-momentum density associated with intrinsic angular mo- mentum.

Furthermore, the canonical energy-momentum tensor satisfies the con- servation equation

(6.11) iq~,~ + t~'q*,. + t.'Q",~ : ~,#,~ + ~#*~-- i~%~ + 1~,.~%~.

To conclude this section, we shall discuss the invariance of 2~ under an infinitesimal conformal transformation. Under such a transformation, the integrand of the variational principle is

= (~ . . - - #~:1-- fi~:2 +

3~A$/~ ~-

+ cg~,t, j j "

~ , V E Y L - D l l g A C T t l E O I % Y ~ V I T t I T O R S I O N - I I

Therefore, we mus t have

[w + w,,, aw , , j "

37

7. - C o n c l u d i n g r e m a r k s .

Although this pape r is intended pr imar i ly to establish the ma thema t i ca l foundat ion for a Vteyl-Dirac theory with torsion, the foregoing analysis yields

several interesting results which could have impor t an t physical consequences.

The most str iking feature of our theory with complex Dirac field fl is the

fo rm of the geomet ry current , given b y eq. (5.16), when the Einstein gauge

is chosen. Despi te the high nonl inear i ty of this current , its form establishes an interest ing paral lel wi th the theory of superconductors.

Secondly, the conformal invariance of the theory leads to a constra int on the t race of the e n e r g y - m o m e n t u m tensor associated with any Lagrangian. I t is impor tan t to r emember (cf. I) t h a t this equat ion can be regarded as a con- sequence of the generalized Einstein field equat ion of a Weyl-Dirac theory, and thus not all the field equat ions are independent .

Finally, the var ia t ional method leads to the conservat ion equations (4.11),

(5.8) and (6.11). We have shown these to be consistent with Bianchi 's con-

t rac ted identities, eqs. (2.20) and (2.21), thus the ma themat i ca l formal ism does not yield any internal contradictions.

I t is our goal to utilize the formal ism contained in this pape r to present , in future publications, an analysis of the weak-field approx imat ion (part icularly for a conformally flat space), a der ivat ion of the equations of mot ion and a s tudy of certain physical problems involving par t icular choices of the source Lagrangian, ~m.

�9 R I A S S U N T 0 (*)

Si estende la teoria di Weyl e Dirae a eomprendere il eampo di Dirae eomplesso ft. Le equazioni di eonservazione di una tale teoria sono ottenute usando una generaliz- z~zione eoeov~riante del metodo variazionale di Rosenfeld. La eorrente elet~romagnetiea associata col campo di Dirac ~ diversa da zero, in generale, ed 8 fortemente non li- neare nel potenziale eleetromagnetieo. La sua forma ricorda |a eorrente nei super- eondu~tori.

(*) Traduz io~e a cura della Redazione.

38 D. GR~GORASH and G. pAPINI

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