On a geometrical theory of the electromagnetic field

IL NUOVO CIMENTO VOL. VIII, N. 3 1 o Maggio 1958

On a Geometrical Theory of the Electromagnetic Field.

D. W. SCIA~A

Trinity College - Cambridge

(ricevuto il 24 Dicembre 1957)

S u m m a r y . - - Weyl's (1929) geometrical theory of gauge (phase) transformations for fermions and the electromagnetic field is generalized so as to apply to bosons as welt. This is done by introducing complex base vectors. The resulting theory is closely related to the Einstein-SehrS- dinger theory in its ttermitian form, and enables one to identify the electromagnetic field tensor. Two physical consequences are deduced from the theory: a) The charges of all bosons are integer multiples of a basic charge, b) The direct interaction between a particle and the electromagnetic field is invariant under space reflexions and time reflexions. A theoretical criterion is suggested (based on the argument leading to b)) for determining which interactions are parity conserving.

1 . - I n t r o d u c t i o n .

The aim of this paper is to propose a new geometrical theory of the electro-

magnet ic field (1).

Since m a n y geometrical theories have already been proposed, some justi-

fication is needed for introducing ye t another one. The justification lies in

the fact tha t the theory described here is not just a rewriting of the Einstein

and Maxwell t h e o r i e s - it leads to new physical consequences. These are t h a t

the charges o f all bosons must tie integer multiples of some basic charge (2),

~nd tha t the direct interaction between a particle and the electromagnetic

(1) A brief account of the theory has already been given, Phys. Rev., 107, 632 (1957). However, this reference contains errors, and is superseded by the present paper.

(2) The need for such a result has been emphasized by P. A. M. DIRAC: Phys. Rev., 74, 817 (1948), and E. P. WIG•ER: Proc. Amer. Phil. Soe., 93, 521 (1949), for both bosons and fermions,

418 i). w. SCIAMA

field must be invariant under space and t ime reflexions. Both these results are in agreement with observat ion (3).

The theory consists of a generalization of Weyl 's second a t t em p t (4) at a geometrical theory of electromagnetism, in which the gauge t ransformations of his original theory (5) became phase t ransformations (8).

Weyl 's formalism will be recalled in the nex t section, bu t it will be convenient ~0 ment ion here its main advantages and disadvantages. These re- marks also apply to other theories of the same type (7).

Advantages.

(i) The electromagnetic field is (( generated ~) by invariance arguments which specify the form of its coupling to ma t t e r (except possibly for terms depending on the field strength, which are usually exc luded- - the so-called principle of minimal electromagnetic coupling). This is in exact analogy with the gravi tat ional field (where the corresponding principle is called the principle of equivalence).

(ii) Despite its geometrical basis, the theory permits different particles to move in different orbits in the same electromagnetic field. This, of course,

is in direct contrast with purely gravi tat ional motion, and corresponds to the empirical fact t ha t the charge-mass ratio e/m is not the same for M1 particles.

Disadvantages.

(i) The electromagnetic potent ia l is par t of the spin connexion only, and not of the affine connexion of the Riemannian space. As a result, only fermions can in teract with the electromagnetic field, whereas we know empi- rically tha t some bosons in teract with it too. In the convent ional t r ea tmen t of the electromagnetic field (8) the boson interact ion is in t roduced by sup- posing tha t the boson field ~0 can undergo a phase t ransformat ion wi thout

altering the field equations. This supposition is based on the assumption tha t ~0 occurs in t h e Lagran-

(3) T. D. LEE and C. N. YA~G: Phys. Rev., 104, 254 (1956), discuss the evidence for parity conservation in electromagnetic interactions.

(4) H. WEYL: Proc. Nat. Acad. Sci., 15, 323 (1929); Zeits. ]. Phys.,56, 330 (1929); Phys. Rev., 77, 699 (1950).

(5) H. W]~vL: Space-Time-Matter (New Yorl~, 1951), p. 282. (6) Unfortunately the phase transformations are usually skill called gauge ~rans-

formations. (7) V. FOCK: Zeits. ]. Phys., 57, 261 (1929); E. SCHg6DINGER: Berl. Bet., 105 (1932);

L. INFELD and B. L. v. I)ER WA]~RD~: Berl. Ber., 380 (1933); W. L. BADE andH. JEHL]~: Rev. Mod. Phys., 25, 714 (1953); P. G. B~RGMANN: Phys. Rev., 107, 624 (1957).

(s) W. PAVLI: Rev. Mod. Phys., 13, 203 (1941).

ON A G E O M E T R I C A L T H E O R Y OF T H E E L E C T R O M A G N E T I C F I E L D 419

gian only in the combination ?*~. However, this procedure destroys the geometrical significance which phase transformations have in Weyl's theory. Furthermore, in the absence of geometrical considerations, there is ~o a priori

reason why the theory should be invariant under phase t r ans fo rma t ions - the Lagrangian might, for instance, contain terms of the form ~s ~ § I t would be desirable, therefore, to have a geometrical theory in which bosons and ~ermions are placed on the same footing.

(ii) The conditions imposed on the spin connexion do not determine i t uniquely in terms of the metric. The electromagnetic potential is then. the part of the spin connexion that cannot be expressed in terms of the metric. While this is a self-consistent structure, it is very unnatural from the geo- metric point of view. The electromagnetic potential is, as it were, thrust into the space from the outside, rather than being part of its intrinsic structure. I t would be preferable if the electromagnetic potential could be expressed directly in terms of the metric (8).

(iii) The second advantage actually goes too far, for it places no res- triction on the possible values of the charge of different types of particle, whereas in fact it appears that all charges are integer multiples of one basic charge.

(iv) There is an arbitrary constant multiplying the Lagrangian of the electromagnetic field, so that the unit in which the potential is measured is undetermined.

The theory described in this paper retains the advantages of Weyl's theory, but eliminates disadvantages (i), (ii) and (for bosons) (iii). If the theory could be extended to eliminate (iv), it would lead to a value for the bare fine- ~structure constant.

The theory will be described in detail in Sect. 3, and its physical consequences in Sect. 4. I t can be simply summarized here by saying that it is essentially a complex version of Weyl's theory. Thus, where Weyl introduces a t each point of space four linearly independent real vectors e(a), these vectors are assumed to be complex in our theory. This implies that the metric tensor of the space is no longer the real symmetric tensor of Riemanni~n geometry, but it may be taken to be either complex and Hermiti~n or real and non- :symmetric. This is just the type of geometry used by EINSTEIN (10,~2) and SCItl~SDINGER (H,I~) in their unified field theories.

(9)~ Cf. H. WEYL: Proc. Nat. Acad. Sci., 15, 323 (1929), bottom of p. 329. (lo) A. EINSTEIN: Rev. Mod. Phys., 20, 35 (1948). (11) E. SCHRODINGER: Space-Time Structure (Cambridge, 1950). (13) M.-A. TONNELAT: La Thdorie du Champ Uni]id d'Einstein (Paris, 1955).

420 D . w . SCIAMA

However , the complex vec to r fields are needed in order b o t h to i d e n t i f y

the e lec t romagne t ic field and to i n t roduce its in te rac t ion wi th m~t te r .

2 . - W e y l ' s t h e o r y .

2'1. G r a v i t a t i o n . - The s t a r t ing -po in t of this t h e o r y is the i n t roduc t i on of

local Car tes ian axes (in) a t each po in t of a R i e m a n n i a n space (with Minkowsk i an

s ignature) . These axes consis t of four l inear ly i nde penden t real (14) vec to r fields e(~) ,

which I p ropose to call eons (15). W h e n an a r b i t r a r y co-ord ina te s y s t e m is

i n t roduced in to the space , t he eons will h a v e c o m p o n e n t s e~(~) (16). I t is con-

ven ien t ~ to choose the eons so t h a t t h e y are (Mfl/kowski-) o r t h o n o r m a l wi th

respec t to the met r ic g~j of the R iemann i~n space. W h e n this is done, we h a v ~

the re la t ion

(2.1)

where

g , = V(~fl) e~(~) ej(fl) ,

11 0 0 0 ~

=

0 1 0 0

0 0 1 0

0 0 0 - - 1

Tensor quant i t ies can t h e n be refer red to the i r eon-componen t s , e.g. fo r

a v e c t o r

A(o: ) = e i (o : )A i .

The e lements of in te rva l ds ~ can t h e n be wr i t t en

ds ~ ---- gi~ dx~ dx~,

= ~(~)r dx(~) ,

(18) L. P. EISEN~_aT: R i e m a n n i a n G e o m e t r y (Princeton, 1926), Chap. I I I . (14) WwYL "takes e(4) to be pure imaginary, but as a preparation for our generalization

it is more convenient to take e(4) real, and %o introduce the matrix ~(:r in (2.1). (1~) They are variously known as vierbeine, tetrads, tetrapods, orthopods, qua-

druplets, four-legs and r~p~res mobiles. (16) In what follows, Greek indices always number the eons, Latin indices refer

%o the co-ordinate system. The summation convention is used for both sets of indices.

ON A G:EOMETRICAL T t t E O R Y OF THE ]~L]~CTROMAGNETIC F I E L D 4 2 I

which shows tha t the dx(~) are co-ordinate differences in the local tangent flat space. They are wri t ten with a line through the d to emphasize tha t they are not perfect differentials unless the !~iemunnian space is flat.

In a space with a given metric, the eon field is not uniquely de te rmine4 by (2.1). Any solution of (2.1) remains ~ solution under a Lorentz transformation L(~fi) defined by (~v)

where L ' is the transpose of L. Fur thermore , the parameters of L need not be the same at different points

of the Riemannian sPaCe, but can v a r y arbi t rar i ly with position. The laws of physics are ~ssumed to be inv~riant under these ~rbi t rary

Lorentz t ransformations (is) as well as under a rb i t ra ry -co-ord ina te t ransformations.

This new invariance proper ty leads as Usual to an identi ty. To obta ia this identi ty, consider for simplicity the Lugrangian density B of a single material field, ~. For ~n infinitesimal IJorentz t ransformat ion we have the va- riational equat ion

+f e

where ~ ( a ) ( = ~/~e~(~)) is the energy-momentum tensor density of the m~terial field. Assuming the material field equations 8~/~y == 0 to hold, we get

f ~ , ( a ) 3e~(a) d~ 0 I

:Now for an infinitesimal Lorentz t ransformat ion we have

3e~(a) --~ do(~fl)~(/~y)e~(y),

where do(aft) is an infinitesimM skew-rhatrix depending arbitrari ly on position. I t fol lows tha t %~(~)~(fly)e~(7), tha t is, ~(gfi), is symmetric in a, ft. Since we have

i t als0 follows tha t ~: , is symmetr ic in i, j. In the limit of special relat ivi ty,

(1:) F. D. MURNAGnA~: The Theory o/ Group Representations (Baltimore, 1938), p. 352.

(is) For our present purpose we need consider only proper Lorentz transformations.

422 D.w. SCIAMA

this symmetr ic ~ , coincides with the Belinfante-l~osenfeld definition of the energy-momentum tensor density of a mater ia l field (xg).

We now introduce an affine connexion for the cons. To do this we paral-

lel ly t ransfer the cons at one point P to a neighbouring point P ' b y means

of the affine connexion/ '~k of the Riemannian space. These t ransferred cons will in general differ infinitesimally from the local cons at P ' . We shall make

the simplest possible assumption about this difference, namely tha t it con- :sists of an infinitesimal Lorentz t ransformat ion L(atS), where

L(a/~) - ~(a~) + d o ( ~ ) ,

do(aft) being an infinitesimal skew-matrix (and not a perfect differential, unless the space is flat). We fur ther assume tha t do(aft) depends l inearly on the displacement P P ' ( = dx~); t ha t is,

do(at3) = %(aft) dx~ .

Then %(aft) is the (skew) connexion we are seeking. This connexion can be expressed explicit ly in terms of the con field as

follows. F rom the definition of %(aft), we have

~x~ + F$S(a ) + o.(a#)e"(#) = O,

t h a t is, e~(a) has vanishing covariant derivative. This equat ion can be re- garded as defining both F ~ and oq(o:fl), for %(o~fl) can be eliminated by using its skewness in a, fl, a n d / ~ can be el iminated by using its symmet ry in r, q. The first elimination leads just to the Christoffel relations for / ~ (in terms .of e~(a), of course, ra ther than gsq), while the second elimination leads to

Oct(a) {e"(a)oo(~r) + co(#) o0(Ta)} = e,(~) co(p) ~xo e~(~) ~xo ] "

We can calculate o~(~zfl) from this by cyclically interchanging the Greek indices

and combining the resulting equations. The gravi tat ional field equations can now be expressed in terms of e~(g)

ins tead of g~q. In order to do this we define a curvature tensor in the usual way, t ha t is, f rom the change in an a rb i t ra ry vector A(~) when i t is parallelly

(19) L. ROSEN~'ELD: Aead..Roy. Belg., 18, No. 6 (1940).

ON A GEOMETRICAL THEORY OF TH]~ ELECTROMAGN]~TIC FIELD 423

t ransferred around an infinitesimal closed circuit. The resul t is

~o~(~) ~%(~) R~o(~) - - ~x; ~x~ + %(~) oo(~) - - oo(~y) %(r~),

which is skew in p~ q and ~, ft. By comparing the changes in the vectors A t,

A ( ~ ) , we get the relat ion

,(2.~) R~(~/~) = R~o e,(~)e~(~),

where R ~ is the Riemann-Christoffel curva tu re tensor. We may define a Curvature scalar R b y the relat ion

R =- e~(~)e~(f l )R,~(~f i ) .

I t follows from (2.2) tha t R is equal to the I~iemannian curvature scalar. The Lagrangian density of the gravi tat ional field is now taken to be

sR

where s is the de te rminant of the mat r ix ei(~) (and is equal to v/~gg). The vanishing of the var ia t ion of f (eR~-B)dT, with respect to e~(g) then leads to

the gravi tat ional field equations.

2"2. E l e c t r o m a g n e t i s m . - So far our account of WeyFs theory has consisted .simply of a rewriting of Einstein 's theory of gravi ta t ion in terms of eons. We must now see how WEYL introduces the electromagnetic field. This is ,done in terms of the sp in connexion which will nex t be defined.

The first step is to introduce ~t each point of the Riemannian space Dirac matr ices which are vectors with respect to the Cartesian f rame e(a). These matr ices y(~) satisfy the commuta t ion relations

~(~)~(fl) + ~(fl)7(~) = 2~(~/~)1.

:Now there exists (so) a mat r ix S such tha t

(2.3) L(~f l ) y( f l ) -=- S r ( g ) S - ~ .

This matr ix S (spin t ransformation) is uniquely defined in terms of L up to

(so) R. BRAUEIr and H. WEYL: A m . Journ . Math., 57, 425 (1935); W. PAULI: Ann . Ins t . t l . Poincard, 6, 109 (1936).

424 D.W. SC~A~A

an arbitrary complex factor k. that

This factor can be normalized by requiring

de tS ~ 1 ,

which restricts k to ~= 1, ~= i. The latter two values can then be excluded: for proper Lorentz transformations J5 by requiring that these L should give rise to a connected group of spin transformations S.

If (2.3) is solved for an infinitesimal Lorentz transformation 8(aft)~-do(afl) r

we get (~) s = 1 + -~ ao(a~) s (a~ ) ,

where

Now in order to differentiate a spinor field with respect to position, the spinor at one point must be parallelly transferred to a neighbouring point and then subtracted from the spinor actually at that point. However, these two spinors will be referred to cons that differ by the infinitesimal Lorentz transformation ~(afl)-~ do(aft), where do(aft) is o~(afi)dx~. The spinors themselves will then differ by the corresponding infinitesimal spin transformation, in addition to their basic dependence on position. Hence the covariant derivative of a spinor field ~ is given by

~ 1

The spin connexion o~ is thus given by

o~ = �89 ~(a~)

If the Dirae Lagrangian is rewritten with eovariant derivatives instead: of ordinary derivatives, then the extra terms involving o, describe the coupling between the ~ field and the gravitational field.

WEYL introduced the electromagnetic field into his theory by generalizing the normalization condition for the spin transformation S. Instead of re- stricting k by taking S to be unimodular, he allowed the phase of k to be an arbitrary function of position. This introduces an extra term into the covariant derivative 6f a spinor field depending on the difference dk between the values of k at the two neighbouring points. This difference is assumed to depen4 linearly on the displacement, so we can write

dk --~ ik~ dx~ .

(31) E. M. CORSON: Introduction to Tensors, Spinors and Relativistic Wave Equation~ (London, 1953), p. 40.

ON A GEOMETRICAL THEORY OF THE ELECTROMAGNETIC F I E L D 4 2 5

Since dk is assumed to be an imperfec t differential, ks is a new non-integrable vec tor field, t ha t is, i t is not in general the gradient of a scalar field. The covar iant der ivat ive is now given by

so i t is na tura l to identify k~ with the electromagnetic potential. I f this new covar iant derivat ive is inserted into the Dirac Lagrangian,

there arises the familiar coupling te rm

where

In this form the theory is invar iant under the t ransformation

~ r = eiO ~p ,

8O k '=l~ ~x~"

The ident i ty derived from this invariance is, of course, the conservation of charge

8J~ O. ~ x v

This is a covariant equation because i ~ is a vector density. The scheme is completed by adding a ~axwel l i an Lagr~ngian (with an

a rb i t r a ry coefficient) to the previous Lagrangian. The advantages and disadvantages of this scheme have been discussed in the introduction, where it was concluded tha t a new theory is needed. An a t t em p t a t such a theory

is described in the nex t section.

3 . - T h e o r y o f c o m p l e x e o n s .

We begin by assuming tha t it is possible to introduce at each point of space four l inearly independent complex Vectors e(~). The metric tensor of

the space (g~j) cannot now be real and symmetric . :However, it can, be either complex and Hermi t ian or real and non-symmetric . In the former': case we

can suppose tha t the complex cons are (( or thonormal )) in the sense tha t

(3.1)

where the star means complex conjugate. The la t te r case can then be derived

426 D.w. $CIAI~IA

by taking the symmetr ic and skew parts of the metr ic to be respectively the real and imaginary par t s of g , . t towever , we shall confine ourselves to the complex t t e rmi t i an ease.

We see then t ha t the space of the complex con fields is also the space of the Einstein-Sehr5dinger theory (lo-1~). ~Now this theory is usually t h o u g h t not to contain the electromagnetic field, since its equations of mot ion do no t yield the Lorentz force (2~).

This led the au thor (~) (following P A P A ~ R O ~ and RoBInSOn) to suggest t ha t the theory described just the gravi ta t ional field, whieh would be t ier - mi t ian (or non-symmetric) if its sources h~d spin.

However~ we shall see tha t the in t roduct ion of complex cons enables one to adapt the Einstein-SchrSdinger formalism so tha t it includes electromagnetism, wi thout affecting the in terpre ta t ion in terms of spin.

Le t us first consider the physical significance of complex cons in relat ion to the real cons which a n observer can introduce as his Cartesian reference system. In the previously quoted paper (~) it was shown tha t in the Einstein-

SchrSdinger theory the orbits of neut ra l test-particles are the geodesics of a t~iemannian space whose cont ravar ian t metr ic tensor is gf~ (or, in terms of the complex metric, the real pa r t of giJ). Hence the physical ~pace in which the observer will map the gravi ta t ional motions of bodies is this l~iemannian space. The original space has no direct physical significance, bu t it makes its presence felt indirect ly by v i r tue of the fact t ha t Einstein 's original (1916) field equations will not hold exact ly in the Riemannian space. A similar si- tua t ion arises with the complex cons. They have no direct physical meaning, bu t real eons which do can be constructed from them. This construction is demonst ra ted in the relation

where ~R means the real pa r t of, and E~(e) are the real, physical eons. Of course, the complex cons cannot be completely el iminated in this way since the real eons will not satisfy the Weyl equations described in the last section.

We can now carry th rough the complex generalization of the Weyl theory in the obvious way. The complex cons are not uniquely defined by (3.1);

any solution remains a solution under a (~ quasi-unital T ~> t ransformat ion U(~fi), defined by

where U + is the conjugate transpose of U.

(22) J. CALLAWAY: Phys. t~ev., 92, 1567 (1953); W. B. BON~OR: Ann. Inst. H. Poincar~, 15, 133 (1957).

(2~) D. W. SClA~A: Proc. Camb. Phil. Soc., 54, 72 (1958); cf. also 0. COSTA PlY, B]~AUREGARD: Journ. de Math., 22, 85 (1943) esloeciMly footnote 1 on p. 129.

ON A G E O M E T R I C A L T H E O R Y OF THE E L E C T R O M A G N E T I C FI]~LD 427

Hence the relationship (3.2) between ei(oc) and E~(~) is unal tered by in-

dependent un i ta ry t ransformations of the ei(~) and Lorentz transformations.

of the E~(e). B y analogy with the real theory of Sect. 2, we now assume tha t the con-

nexion /~q is Hermi t ian and tha t the con connexion is skew-hermitian. W e

write this la t ter connexion u~(afi) since it has to do with quasi un i ta ry transformations ra ther than Lorentz ones. The two connexions are determined b y the equations

(3.3) ~e,(~) axe- + r fS (~ ) + u~(~fi)e~(~) = o .

As before, the con connexion can be eliminated, this t ime using its skew- hermit ian proper ty . The result is

3g~Z ~x~ + F ~ g ~ 4- F~,g ~ - O,

where (3.1) has been used to eliminate the cons. This eqnation, with its cha- racterist ic ~ - - differentiation, is well-known from the Einstein-Schr6dinger theory (10-12).

We can also eliminate J7~ by using its Hermi t ian property. This leads t a

e~(y)e*q(oOu~(fl7) + e*~(7)eq(fl)u~(ya ) -- eq(fi) ~e*~(~) ~e~(/~)

~x~ e*~(~) ~x~;-

which gives uq(o:fl) in terms of the con field. The curva ture tensor derived from uq(~fi) is given by

~uo(~) (3.4) R~(~fi) = ~ x ~ ~x~ + u~(~y)u~(yfl) - - u~(o~7)u~(yfl) ,

which is skew in p, q and skew-hermitian in ~, ft. I t is related to the curva ture tensor formed from / ' ~ as follows

(3.5) R~q(gfl) = R:~q e~(~)e~(/~) . .

I t has two interest ing contractions, the scalar curvature R given by

R ---- e'(~)e*q(fl)R~(~fi)

and its trace R~q(~) (which was zero in the real case). From (3.5) we have

428 I). W. SeIA~A

the relation

r ~ q ,

which we shall need later. We now consider the physical meaning tha t can be given to this geometrical

s t ructure ; in part icular we shall a t t emp t to identify the electromagnetic field. As was pointed out earlier, physical space ~is determined by the cont ravar ian t metr ic tensor g~(. In order to describe mat te r , we introduce a field ~ into physical space, which may t ransform like ~ scalar, spinor, vector , etc., under Lorentz t ransformations of the real physical eons E~(a). Now comes the crucial s tep: we suppose in addit ion tha t ~ transforms like a scalar density of weight

under quasi-unitary t ransformations of the complex eons. More complicated t ransformat ions laws can be envisaged, which m ay have to do with families of e lementary particles, bu t for our present purpose the discussion will be ~estricted to the simplest possibility.

The covariant derivat ive of our field quant i ty ~0 will contain an ext ra t e rm arising f rom its density character. This is so because after parallel t ransfer to a neighbouring point, i t will be referred to an eon f rame differing from the local one by an infinitesimal quasi-unitary t ransformation. This t ransformat ion will al ter the value of ? and so contr ibute to its derivative. The magni tude

of this contribution is

~u~(~a)~.

This suggests t ha t we in terpre t u~(~) as the electromagnetic potential , and ~ as the charge-density of the ~ field (in arbi t rary units). I f the ~ field is quantized, ~t would then be a measure of the charge of the resulting particles. W e can support this in terpre ta t ion by an explicit calculation of u~(~) in terms of the eon field. F rom (3.3) we have

:Now the real pa r t of u~(otfl) is skew in ~, fl, and so contr ibutes nothing to u~(~) .

Hence

f . , ~e~(~) } - - u o ( ~ ) = i Im ~ e ~ ~ - + / ~ ,

Ins = i Im ~- iF~ .

v

I t follows tha t a quasi-unitary t ransformat ion of the eons of de te rminant


j0 induces the transformations

q9r= e'a~ ,

r

% ( ~ ) = % ( ~ ) - - i - ~0

~X ~ '

which is just the phase t ransformat ion characteristic of electromagnetic theory. This makes our in terpre ta t ion of %(:r a reasonable one. Fur thermore , by giving phase t ransformations this geometrical interpretat ion, we ensure tha t the ~ field can in teract with the electromagnetic field whether ~ is a tensor or a spinor quant i ty in physical space. The only basic assumption we have made is tha t the laws of physics are invariant under quasi-unitary transformations of the complex cons. From�9 the geometrical point of view this is a ve ry na tura l assumption.

The electromagnetic field F,~ is defined b y

F rom (3.4) we have

where

~ u ~ ( ~ ) ~ u . ( ~ )

F .~ = R . o ( ~ ) ,

- - ~ / 0 q '

~x~ ~x~ '

�9 p

V

Hence the electromagnetic field in the Einstein (lo) theory is R:,~. Now if one adopts Einstein 's va r i a t i ona l principle this quant i ty vanishes (~o), so tha t one would then expect the equations of motion to refer to neutral particles only. However , if one adds to the Einstein Lagrangian a Maxwellian Lagrangian for E,~ and a mater ia l Lagrangian (containing covariant derivatives), one obtains non-vanishing electromagnetic effects. This will be discussed in detail elsewhere. We end this paper with a brief account of some

physical consequences of the theory.

4. - Phys ica l consequences of the theory.

(a) F rom the group theoret ical point of view the weight ~ can be any

constant. However , one can restr ict its possible values by physical considerations. Suppose ~ describes an assembly of bosons. This system has a

[~ 28 - I1 N u o v o Cimento. c a

430 D.w. SC~A.~A

well-defined classicul limit when u large number of particles are in the same state. In this limit, ~s is u elussicul field-strength (or ra the r potentiul), which curt be meusured by meuns of u (complex) ~0-charge. I f this ehurge is g, suy, then the (reul) force acting on it due to ~ will be (~) the gradient of the quunt i ty gq~*+g*~s. Now if ~ is not un integer, the phuse of ~s will no t be completely determined by the field equations: ~ will be of the form ~oe ~"~, where n is uny positive or negutive integer (or zero). Similarly, g will be of the form go e2~'~, where m is unrela ted to n. Hence the force on g will not be single-valued. However, this is impossible since the force is measurable. I t

follows thu t ~ must be a positive or negutive integer (or zero). Hence the churges of all bosons huve (Positive or negative) integer ratios.

Unfor~unutely, this urgument does no t upply to fermions, which ure pre- ven ted by the ~'auli exclusion principle f rom having u clussical limit. In addition, of course, u spinor field is not measuruble becuuse it does not t runsform

us u single-vulued representut ion of the Lorentz group. The quantizat ion of churge for bosons suggests u me thod of culculuting

the (bure) fine-structure constant ~. Fo r p resumably the charge e corresponds to tuking ~ equal to I. However, this does not de te rmine ~ directly, because ut this stuge of the theory the vector potent iul is meusured in arb i t rury units. These units will be determined by the Lagrangiun of the theory, unless i t contuins un arbi t rury constunt. This question will be discussed in detufl in

unother paper.

(b) The quasi-unitary group contains the operators thu t inver t the con ~xes e*(~). Fur thermore , these operators are continuous with the identity, for

the quusi-unitury group is connected (~). t tenee, the geometricul properties of the buse space (and so electromagnetic

effects) cannot detect the chirMity of the E~(~) frume, since there is no covar iant relution between the chirulities of the e~(~) f rume und the E~(~) f r a m e - the relation curt be chunged in u continuous wuy by quusi-uni tary t runsformat ions of the e~(~). I t follows from this thu t the direct interact ion between u pa r t i c l e und the electromagnetic field is invariant under u spuce refiexion and u t ime

reflexion of the E~(~).

This consequence of the theory is known to be t rue exper imental ly to con-

sideruble uceurucy (3). Until recent ly it would huve seemed u ra ther tr ivial

result, bu t the discovery of par i ty non-conservat ion in weuk interact ions (~')

(24) G. W~N~Z~L: Quantum Theory of Fields (New York, 1949), p. 60. (25) H. WE~L: The Classical Groups (Princeton, 1946), p. 194. ~u proves this

for the unitary group, but the proof carries through for the quasi-unitary group. (26) C. S. Wu, E. AMBLER, R. W. HAYWARD, D. D. ~{OPPE$ ~nd R. P. HUDSON:

Phys. Rev., 105, 1413 (1957); R. L. GARWI~, L. 1~. L~EDEI~I~AN and M. W]~INRIC]Et: Phys. ]~ev., 105, 1415 (1957); J. I. F]CI]~D~AX and V. L. TELEGDI: Phys. Rev., 105, 1681 (1957),


has r a i s e d t h e g e n e r a l q u e s t i o n of w h y p a r i t y c o n s e r v a t i o n ho lds for some

i n t e r a c t i o n s a n d n o t for o the rs .

Our t h e o r y sugges t s t h a t for an i n t e r a c t i o n to b e p a r i t y c onse rv ing i t s

u n d e r l y i n g g r o u p m u s t in some w a y c o u n t e r a c t t h e d i s c o n t i n u o u s n a t u r e of

s p a c e - t i m e re f lex ions (27).

W h e t h e r th i s is i n d e e d t r u e for s t r o n g ( n o n - e l e c t r o m a g n e t i c ) i n t e r a c t i o n s

r e m a i n s to be seen.

I a m g r a t e f u l to t h e H a r v a r d College O b s e r v a t o r y , to T r i n i t y College,

C a m b r i d g e , a n d to I%IAS, for t h e i r s u p p o r t a n d e n c o u r a g e m e n t whi le th i s

w o r k was b e i n g p e r f o r m e d .

I shou ld l ike t o t h a n k P r o f e s s o r P . A. M. DllCAC, Dr . D. }012~KELSTEIN~

Dr . C. W . MISNER~ Dr . O. PEN~0SE, Dr . R. PENR,0SE~ Pro fe s so r A. SALA~r

a n d Dr . J . C. TAYLo~ for h e l p f u l d i scuss ions .

(eT) Cf. P. A. M. DIRAC: Rev. Mod. Phys., 21, 392 (1949); J. M. JAuc~I and F. I~OHRLICII: The Theory o] Photons and Electrons (Cambridge, Mass., 1955), p. 86.

R I A S S U N T 0 (*)

Si generalizza la teoria geometriea di Weyl (1929) delle trasformazioni di gauge (fuse) per fermioni e per il Gampo e]ettromagnetico in modo da renderla applicabi]e anche ai bosoni. Ci6 si ott iene introducendo ve~tori complessi. La ~eoria che ne r isul ta

s t re t t amente connessa ella teoria di Einstein-Sehr5dinger helle sue forma hermit iana e permet te di identificare il tensorc elet tromagnetico. Del la teoria si deducono due aonseguenze fisiche: a) le eariche di t u t t i i bosoni sono mult ipl i interi di una carica base; b) l ' in terazione d i re t ta t ra una par t ieel la e il campo elet tromagnetico ~ inva- r iante per le riflessioni dello spazio e del tempo. Si propone un eriterio teorico (basato su un argomento cite conduce a b)) per determinare quail interazioni conservino la pari th.

(*) Traduz~one a c u r e de l la R e d a z i o n e .

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On a geometrical theory of the electromagnetic field