On the nature of the electromagnetic field

Amsterdam Nuclear Physics 24 (1961) 388--399 ; C@ North-Holland Publishing Co ., Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON THE NATURE OF THE ELECTROMAGNETIC FIELD TETSUO GOTÖ Department o f Physics, College of Science and Engineering, Nihon University, Tokyo Received 5 December 1960 Abstract : Attempting to identify the electromagnetic field as an affine connexion for spinors, we investigate the nature of the electromagnetic interaction and the electric charge . All spinors interact with the electromagnetic field . Possibility of charge independence is studied . In our opinion, the violation of the charge independence by the electromagnetic field is caused by the violation of the flatness of space-time .

1 . Introduction Models of elementary particles have been proposed by many authors from various points of view. Especially, one of the most important aims of the theories and models already proposed is to derive the isobaric multiplets on the basis of simple or intuitive assumptions, for phenomenologically the isobaric multiplets of strongly interacting particles are the most characteristic . Though the isobaric multiplet structure is an important character of the strongly interacting particles, it is worth remembering that the isobaric spin seems to be closely connected with the electric charge. Why are there special relations between the electric charge and the third component of the isospin? And why is the symmetry of the charge independence violated by the electromagnetic interaction? Also other internal quantum numbers, the strangeness and the nucleon number, seem to have close connection with the electric charge . Thus, it may be considered that the internal structure of elementary particles is essentially related to the electric charge. Although the theories and models proposed by many authors have symmetries convenient for explaining the phenomenological symmetry of the charge independence and can derive the degeneracies of the mass eigenvalues, they cannot explain clearly the reason why the quantum numbers are related to the electric charge and the reason why the symmetry of the charge independence is violated and the mass levels are split by the existence of the electromagnetic interaction . For example, in the model of the rigid body or the model of the relativistic rotator 1 ) there exist internal rotational degrees of freedom which seem to correspond to the isospin. We cannot, however, understand the reason why the internal rotation of the rigid body is related to the electric charge. In Heisenbergs non-linear theory 2), the problem of the electric charge and the electromagnetic field will 388

ON THE NATURE OF THE ELECTROMAGNETIC FIELD

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not be clear until the dynamical problem is solved. Thus, on the basis of this theory, we cannot say anything about the relation between the electric charge and the isospin at the present stage. In the composite model and th,~- related theories a), it seems that the electromagnetic field has a special situat on among many other fields . Of course, it is easy to say that the electromagnetic field is also a composite one. If it is so, however, we cannot say anything about the relation between the electric charge and the isospin until the dynamical problem is solved. It seems to us that the symmetry of the charge independence and the violation of the charge independence by the presence of the electromagnetic interaction are the most important features of the internal structure of the strongly interacting particles. In our opinion, this fact has an intimate relation to the nature of the electromagnetic field and the electric charge. So we hope that the relation between the electric charge and the isospin and the reason for the violation of the symmetry in the presence of the electromagnetic interaction will be understood by the study of the nature of the electromagnetic field and the electric charge. After Einstein's gravitation theory, there have been many attempts to unify the electromagnetic field and the gravitational field. At that time, however, physicists did not know the variety of elementary particles or fields and then the unified theory was only an academic one and had no connection with the problem of the structure of elementary particles. Recently, Hara 4) attempted to amalgamate the so-called isospace and the usual four dimensional space-time by interpreting geometrically the electromagnetic field and the electric charge on the basis of Kaluza's five-dimensional unified theory 5) . In Kaluza's theory, the metric tensor of the five-dimensional space is composed of the electromagnetic field and the gravitational field. Thus the electromagnetic field is essentially related to the space-time structure. The charge is proportional to the momentum canonically conjugated to the fifth coordinate . Hara showed that Klein's coordinate transformation contained the rotation group in the three-dimensional Lorentz space (not Euclidean space!) . Anyhow, in his theory, the electromagnetic field occupies a special place among many other fields, as does the gravitational field. And this fact played an essential role in the amalgamation of the isospace and the ordinary space-time . In contrast with Hara's opinion, we attempt to interpret the electromagnetic field geometrically in the framework of the ordinary four-dimensional spacetime and to study the nature of the electric charge. In general, the structure of a space is prescribed by a metric tensor gig: and an affine connexion l'~1k . However, when we need to treat spinor quantities, an affine connexion l'k for spinor quantities, which is 4 x 4 matrix, must be also given. Thus, the structure of a space is perfectly determined by three quantities I'kj , it is go, l'fk and l',. . When we assume that l'i, is symmetric, i.e. r,, =

390

TETSUO GOTÔ

expressed by the metric tensor g=k and its derivatives . However, even in this case, l'k is not determined uniquely. The equations for determining .1'k are invariant under the transformation l'k --- rk+iAt, where A,,(x) is an arbitrary vector field. Thus, if we treat spinor quantities, it is necessary to prescribe a vector field A t (x) . We identify this vector field with the electromagnetic field. From this viewpoint, the electromagnetic field is closely related to the structure of space-time . A similar idea to the above concerning the electromagnetic field was already proposed by H. Weyl ?) . In his unified theory, Weyl attempted to introduce the electromagnetic field in an affine connexion I'i,. for tensorial quantities . At that time, the importance of spinor quantities was not recognized. After the discovery of the Dirac equation, the generalization of the equation in the presence of the gravitational field was made and then the theory of spinors in curved space was developed 8) . Infeld and van der Waerden 8) pointed out the arbitrariness of rk and suggested that we made use of this arbitrariness for introducing the electromagnetic field. In this note, we intend to develop this idea about the electromagnetic field . From the viewpoint stated above, the electromagnetic field acid the gravitational field are the fields which prescribe the structure of space-time and occupy a special position among many other fields . In contrast with the gravitational field, the electromagnetic field has a special relation to spinor quantities. In our opinion, the effect of the presence of the electromagnetic field appears only when there exist dynamical quantities described essentially by spinors. In section 2, we summarize some mathematical results and discuss the parallel displacement for spinorand tensorial quantities . We show in section 3 that the electric charge must be considered as a dynamical quantity described essentially by a spinor. As the electric charge and the electromagnetic field are intimately connected and from our viewpoint the electromagnetic field prescribes the character of space-time for spinor quantities, it is natural that the electric charge and spinors should be closely related to each other . After considering the dynamical nature of the electric charge, we discuss the possibility of introducing the isospin and the strangeness in sections 4 and 5. In section 6, we consider the gauge transformation and some miscellaneous problems. In this note, we do not intend to construct a realastic model for unified description of elementary particles, but to clarify the nature of the electromagnetic field and the electric charge andto discuss the possibility of introducing some internal variables (viz. the isospin, the strangeness and the nucleon number). 2. Spinors in Curved Space

In this section, let us survey some geometrical characters of spinors in a generally curved space. Throughout this note we consider four component

ON THE NATURE OF THE ELECTROMAGNETIC FIELD

39 1

spinors . The precise formulation of spinor theory in curved space was given by Infeld and van der Waerden ; we follow here, for the sake of simplicity, a paper by Schrödinger and Bargmann s). There are three fundamental quantities in a given space: a metric tensor gik, generalized Dirac matrices y~, k and an affine connexion r~, k for spinors. (For the suffices of world tensors we use Latin letters and for those of spinors Greek letters) . An affine connexion rfk is expressed in terms of gik and its derivatives on assumption of symmetric affinity, i .e. l;k-- r,', . The following relations hold between these fundamental quantities : Yi Yk + YkY=

ay= axt

2gik 1

r= Yf+rtyi--Yirt =

(2 .1)

0,

OktYi'Yioki ` R ktiY1 P

where Oki

art

ark

+ axk - ax

L

+rtrk_rkrt

(2.2) (2 .3)

(2 .4)

and Rxti is a Riemann's curvature tensor, i .e.

_ arkt + arkm + Riktm - ^ rißt r3km- rifmrßt,t , ax axi

(2 .5)

From eq. (2.1), generalized matrices Yk's are determined by a gives metric tensor gik . And using eqs . (2.2)-(2.5), we obtain where

,~,kt

4RktijS'J +~iefkt 1,

-`

S" -- 4 CYiy'--yJY'~, yi = g`kYk~

(2.6) (2 .7)

1 denotes the 4 x 4 unit matrix and fkt is an arbitrary antisymrnetric tensor . Then, if we assume that there is no gravitational field, or that the space-time is flat, rk is almost determined and the result is as follows :

r~. k = Z ieAk(x)ô~a,

(2 .s)

where A k (x) is an arbitrary real vector field and e is a constant . (The reason why rk must be pure imaginary in (2 .8) is not clear only from eqs. (2.1)- (2.7) . A precise discussion is given by Infeld and van der Waerden.) The tensor fik is expressed by A k (x) 2 ; follows: aAk _ aA i

axk ! axi . ik

TETSUO GOTÔ

392

The covariant derivative of a spinor field e(x) is defined by eik(x) = ÔL-Va (x)_r,',kV (x)1 and in the case of no gravitational field becomes

(2.10)

(2.11) == (ôk--zzeAk(x))Va W ~ Now, we consider the parallel displacement of tensorial and spinor quantities. For example, we consider the parallel displacement of a vector Pß (not a vector Vjk(x) = â,L yr2 (x)--ZieA k (x)Va (x)

Fig . 1 .

field) given at a point A from a point A to a point B along a given curve C(C : xf = xz (s)) ; see fig. 1. In this case, we must treat the following equation as an initial value problem : dp dxk = 0. (2 .l2) + rßkpf ds ds Similarly, if we consider the parallel displacement of a spinor (not a spinor field but a spinor given at a point A), the following equation must be studied: ci a

......

eft ==o. (2.13) ds If there is no gravitational field, the equations (2.12) and (2.13) become t +

FI, k ds

, o'

(2.12')

dx k + 1 ieA k (x) e = 0. z ds ds

(2.13')

ds d'

Then, as far as we treat tensorial quantities in the case of no gravitational field, we can consider that the space-time is flat . However, if we must treat spinors, we cannot consider the space-time to be flat even if there is no gravitational field. In order to consider the space-time to be completely flat, the electromagnetic field At(,T) must also be zero.

t With regard to eq. (2.13') we remark, that, as will be shown by eq. (2.14), we of course can obtain the equation of parallel displacement for ~ as Q/ds = 0 by a suitable choice of the coordinate system in spin space. However, the choice of the coordinate system in spin space should depend on the curve or integral path C.

ON THE NATURE OF THE ELECTROMAGNETIC FIELD

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From now, since we confine ourselves to the usual Minkowski space, it is sufficient to take rk as given by the eq. (2.8) . And then for the parallel displacement we need to consider eqs. (2.12') and (2 .13'). Integrating (2.13') we obtain that k(X)] 1 ie exp f dxA e,"

J

where Jo is an initial value given at a point A. If Ak(x) = akA (x), i.e. f{k = 0, the phase factor in (2.14) is independent of the path C and expressed by --12# [A (xB) -A (oA) ] . If A k (x) is time independent and the curve C is closed in the three-dimensional sense, i .e . there is only a static magnetic field, the phase factor becomes Z ie fc Adx =-lie fs not AdS and this is just the magnetic flux in the closed circuit C. It may be worth-while to remark that this fact is similar to that of Bohm's Gedanken-Experiment 9) which illustrates a peculiar function of the electromagnetic potential A k(x) in quantum mechanics . From (2.13'), we can see easily that d d 0, (ee) = ds ds

(~Yke)

= 0, etc.

(2 .15)

These facts can be easily understood by remembering that (~$), *k $) etc. are ordinary tensorial quantities and there is no effect of the electromagnetic field on ordinary tensorial quantities. (See eq. (2.12')) . 3. The Electric Charge as a Dynamical Variable It is well known that the electric charge of a strongly interacting particle and the third component of its isospin are linearly dependent on each other. Then, if the isospin results from quantization of an internal dynamical quantity, the electric charge must also be considered as a dynamical quantity. Therefore, in pre-quantum theory or classical mechanics, the electric charge is susceptible to any arbitrary magnitude. The discrete character of the charge of the existing elementary particles can be interpreted as the result of the quantum condition for an internal dynamical variable . Since we suppose the electromagnetic field to be an affine connexion for spinors, it is natural to consider that a dynamical variable related to the electric charge is described essentially by a spinor quantity. Now, let us study the motion of a particle in a completely flat space (e .g. Ax.(x) = 0) . And let us suppose that the particle has an internal dynamical variable described by a spinor $. If the quantity described by $ is not affected by any force, i.e. ~ does not change with time, the equations of motion for such a particle are given by de _ d2x= _ 0, 0' ds2 ds

394

TETSUO GOTÔ

and the Lagrange function is then given as &

-- 'lm

dxk dxk ds ds

+

1i 9

le

de de e . ds ds

(3.2)

However, in the case of 1''k = Zi Ak (x) (e.g. the presence of the electromagnetic field), the fact that $ is not affected by any force or $ does not change with time is not expressed by de/ds = 0, but by k

de -}- 2 ieAk (x) dx e = 0. ds

ds

(3.3)

This means, that we must change d/ds to d/ds+ 2 ie A k dxk/ds for spinors. For tensorial quantities, we need no change for any operation. Then, the Lagrange function (3.2) becomes m de dxk + d i 2 ds ds ds

eA k dxk a ds

__ 1

and Euler's equations derived from (3.2') are d 2 xk e dxi , d$ _ ]. dxk -2 k Z fki ' eA ds ' ds2 2m ds ds where ôA i OA k f ki = ôxk i ôx

(3.3)

represents the field tensor. As is easily shown by (3.3), the following conservation law is valid: d = 0. (3.4) ds Ce) Because of the first equation of (3 .3), 2$ expresses the total charge of the particle and the charge conservation law is expressed by an equation like (3.4) . The magnitude of the total charge 2R may have any value, chosen as initial value. Eq. (3.3) is similar to the ordinary equations for a charged particle . There is only the one difference that in our case the total charge must be considered not as a parameter but as a dynamical variable . The interpretation of the electromagnetic field as an affine connexion for spinor quantities is consistent with the interaction between the electromagnetic field and a charged particle . If Ak(x) = akA(x), the field strength fik = 0, the path of the particle is straight and e = ei"A(x) e0f therefore, if we transform $ to 011'1(x) ~, the equation (3.3) becomes identical with (3.1) . Thus, in the case fik --_ 0, the space-time can be considered to be perfectly flat.

ON THE NATURE OF THE ELECTROMAGNETIC FIELD

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Now, it is clear that the internal variables described by spinors have an intimate connection with the electric charge and the dynamical quantities described by tensorial quantities have no relation to the electric charge. Thus it is hoped that the internal quantum numbers related to the electric charge are expressed by dynamical variables of spinor quantities. 4. On Charge Independence The dynamical vaiables discussed in the previous section do not contribute to the energy of the particle, they stick to the particle. However, if the internal variable e varies, it contributes to the energy of the particle . For example, let us consider the following Lagrangian (in the absence of the electromagnetic field) _

dxk dx m° -asds

+

Z

dM z (Z ' )' ds

where M is a function of f and $ only. From this Lagrangian, we obtain the Hamiltonian _ 1 (4.2) H 2nm kl% + M2 (~~ ) 0

and then M (f, ~) gives the rest energy of the particle. Now, since M is dependent on ~ and f, the mass of the particle is also a dynamical variable. And as $ is a Lorentz spinor, the motion of $ contributes to the angular momentum of the particle . Even if the particle is at rest, i .e. its momentum is zero, it has in general an angular momentum which is caused by a motion of the internal variable $. Thus, it is clear that the three quantities, e.g. the spin, the mass and the electric charge of the particle are correlated with each other through the dynamical variable $. In quantum theory, M( , ) plays a role of a mass operator. Hence, if M( , ) has a symmetry property or is invariant under a transformation, the mass eigenvalue of the particle, in general, degenerates. Thus, if the transformation is represented by a rotation group in a three-dimensional Euclidean space, we can consider that the particle forms the so-called iso-multiplet and the transformation can be identified with a rotation in iso-space . Actually, there exists such a transformation . It is well-known that the linear transformation given by Pauli and Gürsey 10) is isomorphic to the rotation group in the three-dimensional Euclidean space, where

~ -} ~' = a~+by5 C-1 ~,

~ -->~ ~' =

(aj2 +jbj 2 = 1 .

a*~ -f- b* $CY5 ,

(4.3)

396

TETSUO

GOTÔ

This transformation is independent of the Lorentz transformation for $. The infinitesimal generators of this transformation are given by Ti =

C YsC-1

-

CYs ~

T2

= 2 ZC

YsC-1

-f-

CYs l

Ts

=2 .

(4.4)

It is a remarkable fact that T3 = . 1 $E$ is identical with the total charge discussed in the previous section. Thus, if M(f, e) is invariant under the P-G transformation (4.3), we can introduce a dynamical variable corresponding to the isospin . From our viewpoint, it is easy to understand that the charge independence is violated by the presence of the electromagnetic interaction : the electric charge is not invariant but transforms as T3 and hence the electromagnetic interaction is not invariant under the transformation (4 .3) . In our opinion, the presence of the electromagnetic field means that the flatness of space-time is violated for spinor quantities. Then, it. is considered that the violation of the charge independence reflects that of the flatness of space-time . As already pointed out, e has an angular momentum . Thus, the magnitude of the isospin is restricted by the magnitude of an ordinary spin .The infinitesimal generators of the Lorentz transformation for $ is given by

Lxz - 2~f [Yxt Yz]$. By comparing T3 with the third component of the angular momentum Ll2 , it is easy to see that the states possessing integral spin have an integral isospin and the states possessing half-integral spin have half-integrai isospia t. 5. Strangeness and Fermion Number Now, let us consider the strangeness and the nucleon number. The strangeness is also the quantum number which is related to the electric charge. Then, as in the case of the isospin, we introduce another dynamical variable 'I which is described essentially by a spinor. Such a dynamical quantity defines an electric charge 2Fq. The angular momentum of il is also similar to that of e. Then, the possible total angular momenta of the particle which has two internal variables y and tj are given by the following table : g spin

spin integer integer half integer half integer

i

integer half integer integer half integer

total spin integer half integer half integer integer

The $ spin is the same as the isospin (Cf. section 4) .

As sho-.%-n in the previous section, the integral and the half integral character of angular momentum is the same as that of the isospin . Then, identifying qj? t This discussion is made on the assumption of the quantization rule (J,

~t} + a

1.

ON THE NATURE OF THE ELECTROMAGNETIC FIELD

397

with strangess is, we hope, sufficient to explain the existing experimental facts by means of two internal dynamical variables ~ and rl . Finally, in the case of half-integral total angular momentum, the wave functions resulting from the quantization procedure have to be describerl by spinor fields. Then, electromagnetic interaction with the spinor fields is necessarily introduced by virtue of the space-time structure : the derivative ak for spinor fields must be replaced by the covariant derivative ok-rk = ak- 2ieA k (x) for the sake of covariance. Hence, the total charge of spinor fields is given by The last term, 1, can be called the fermion number, because it must be added in the case of fermions. The fermion number just corresponds to the nucleon number introduced phenomenologically . 6. Concluding Remarks 6.1. ON GAUGE TRANSFORMATIONS

From our viewpoint, the theory is invariant under the transformation --> V = ell ieA$ ' I ->,q' = e1i,Aq' V' (x) = e } ieA V (x) -> f (x) = ~(x)

for spinor fields,

v (x) ->

e(x) ~k

k -

Pk -}- 2 e(~~ + ~~

for tensor fields,

-}-1) akA for spinor fields,

Pk + 2 e (~$ -I-- W M

for tensor fields,

(6.1)

Ak(x) -> Ap k(x) = Ak(x)-0kA(x) .

This corresponds to the usual gauge transformation for strongly interacting particles. If A k (x) = akA (x), it is, of course, possible to eliminate the affine connexion A k (x) for spinors by the so-called gauge transformation . Hence there is no physical effect of Ak(x) . A similar state of affairs exists for the gravitational field. If the affine connexion for tensorial quantities satisfies the integrability condition (i.e. if the parallel displacement does not depend on the path), we can make rfk zero by a suitable coordinate transformation . In such a case, Riemann's curvature tensor is zero, i.e. Rifka -= 0, and we call the space flat. Similarly, we may call the space flat for spinors if the field tensor file vanishes . The gauge transformation by which we can make Ak = 0 just corresponds to the suitable coordinate transformation by which we can make ß rfk= 0.

6.2. PRESENCE OF GRAVITATIONAL FIELD

If the gravitational field is present, the dynamical properties of the electric charge become important . In such a case, the motion of a particle is influenced

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TETSUO GOTO

by the interaction between the charge variable and the gravitational field. For example, we show the generalization of the equations of motion (3.1) in the case of a general curved space d2x; +h dxf dxk i ~ ~ dxk I'k k _dis- ds m ds ' ds2

dea +

7s

ra

dxk

'#' k ds

'

(6.2)

where rt's are in general 4 x 4 matrices. Even if the electromagnetic potential At = 0, the I',, do not vanish and are determined by the gravitational field . Thus, we can distinguish in principle between motions of charged or g{7&) uncharged particles by the use of the gravitational field even if there is no electromagnetic field. 6.3. SUMMARY

In this note, we discuss the possibility of identifying the electromagnetic field with an affine connexion for spinors in ordinary four dimensional space time. In this way the electric charge has intimate relations to the dynamical quantities described essentially by spinors . In our opinion, the isospin and the strangeness correspond to internal variables represented by spinors jlt and 'I. We have not examined the physical meaning of these variables. The variables $ and,j may correspond to degrees of internal rotation of two rigid bodies t or to internal variables of a composite system of two particles having Zitterbewegung. The discussion given in this note is not influenced essentially by any assumption about the nature of $ and 7 It is essential in our discussion that these quantities must be described by spinors. From our viewpoint, the effect of the electromagnetic field appears only when there exist spinors as dynamical variables. All spinors, no matter whether they are fields or not, necessarily have an electric charge and thus interact with the electromagnetic field. The violation of the charge independence by the electromagnetic field is the result of a curvature of space-time . The conservations of the third component of the isospin of the strangeness are naturally interpreted as the conservation of ~$ and #J, even if the electromagnetic interaction exists . The author wishes to thank Professors T. Toyoda, T. Okabayashi and H. Suura for their stimulating discussions. t A composite system of two rotators is discussed by H. Fukutome 11 ).

References 1) T. Nakano, Prog. Theor. Phys. 15 (1958) 333; T. Takabayashi, Prog. Theor . Phys. 23 (1960) 915 (see also the refs. of this paper) 2) H. P. Dürr., W. Heisenberg, H. Mitter, S. Schlieder and K. Yamazaki, Zs. f. Naturf. 14a (1959) 441

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3) Y. Katayama and M. Taketani, (preprint) ; Z. Maki, M. Nakagawa, Y. Ohnuki and S. Sakata, Prog. Theor. Phys. 23 (1960) 1174 4) O. Hara, Prog . Theor. Phys . 21 (1959) 919 ; see also O. Klein, Nuclear Physics 4 (1957) 667 5) P. G. Bergmann, Introduction to the Theory of Relativity (Prentice-Hall, New York, 1955) 6) E. Schrödinger, Space-Time Structure (Cambridge University Press, 1954) 7) H. Weyl, Space, Time and Matter (translated into English by H. L. Brose) (Dover Publ ., New York, 1950) 8) E. Schrödinger, S. B. Preutz Akad . Wiss . (1943) 105; V. Bargmann, S. B. Preutz Akad. Wiss . (1932) 346 ; L. Infeld and B. L. van der Weerden, S. B. Preutz Akad . bliss. (1939) 380 9) Y. Aharonov and D. Bohme, Phys. Rev. 115 (1959) 485 10) W. Pauli, N. C. 6 (1957) 204 F. Gürsey, 1V'. C. 7 (1958) 411 11) H. Fukutome, Soryushiron-kenkyu (mimeographed circular in Japanese) 20 (1959) 311