IL NUOVO CIMENTO

VOL. VIII, N. 5

1o Giugno 195g

The Theory of Relativity, the Electromagnetic Theory and the Quantum Theory. H. T. FLINT Bed]ord College - University o/ JLondon E.

M.

WILLIAM:SON

St. Mary's College - D u r h a m

(ricevuto il 4 Marzo 1958)

I t is shown how it is possible, by means of geometry and the introduction of a principle of measurement, founded by analogy with the theory of g . Weyl, to discover a unity existing between gravitational, electromagnetic and quantum phenomena. Dirac's equation and an extension of it are derived from the principle of measurement, and an essential feature of the theory is that it incorporates a theory of the electron in which its mass appears as a geometrical quantity, entering into the equation as a consequence of the existence of a fundamental unit of length. The observed mass is given by this geometrical mass together with interaction terms which also enter naturally into the theory. Summary. --

The u n d e r l y i n g t h e m e of this work is the union which exists between t h e t h e o r y of relativity, the electromagnetic t h e o r y a n d the q u a n t u m theory, a n d its purpose is to p o r t r a y it b y means of g e o m e t r y and a t h e o r y of m e a s u r e m e n t . I t will a p p e a r t h a t the union includes the general t h e o r y of r e l a t i v i t y b u t t h a t in the applications in physics it is necessary to a d o p t the l i m i t a t i o n to the special theory. This applies to the t h e o r y of the electron which is an essential f e a t u r e of the union. I t has not y e t a p p e a r e d possible to progress in the direction of unification of these three domMns of physics b y the geometrieM w~y of thinking, t h a t is to say b y appeals to the m ~ t h e m a t i c M forms of geometry, ~s in the t h e o r y of relativity, unless an extension is m a d e b y the adoption of a c o n t i n u u m of

TtI]] THEOI~u OF R E L A T I V I T Y , T H E ELECTROMAGNETIC THEOI=tu ETC.

68]~

a greater n u m b e r of dimensions t h a n four. F o r the present purpose the n u m b e r of dimensions is five. B u t the requirement t h a t laws of nature must be expressed in a four-dimensional covariant form still remains an unfailing guide in the development of a physical theory. Thus, at present, whatever appeal is made to m a t h e m a t i c a l form in the search for new laws or for a new expression of known laws, the final form must satisfy this requirement. I t has been found possible to ensure t h a t in this appeal to geometry, the desired result is reached automatically. The equations obtained are either already in the correct form or are readily translated into it. Five-dimensional vectors and tensors are easily related to their four-dimensional counterparts and nothing is left which is of an artificial or unexplained character. The advantage of this m e t h o d of description is t h a t the results acquire a simple form in the analysis and the ideas leading to them readily suggest themselves. Einstein's genera] t h e o r y of relativity is essentially a theory of gravitation b u t other phenomena, especially those of electromagnetism, suggest the search for a wider t h e o r y in which t h e y would be united with those of gravitation. One of these attenlpts at unification was made b y WEYI. (1). I n it he regarded electromagnetic phenomena as the revelation of a system of measur e m e n t or of gauging appropriate to the physical world, gravitational phenomena revealing an appropriate geometry. B u t the concept of parallel displacement, which WEYI. introduced, does not appear to correspond to displacements in the physical world. Indeed, if WEu idea is assumed to be the basis of a physical theory, results follow which are not in agreement with experimental facts. Another a t t e m p t was suggested b y KALUZA (3), who proposed to a p p l y the principles of Einstein's t h e o r y to a l~iemannian c o n t i n u u m of five dimensions. The suggestion leads to a remarkable union of gravitation and electromagnetism b u t remains artificial since the fifth dimension is given n~ physical meaning. The line element of the continuum is (1)

da ~ = y,~ dx" dx" ,

where # and v can take the values 1 to 5. Values of the coefficient (y,,) are chosen which cause the geodesic of the continuum to represent the track of a charged particle in a gravitational and electromagnetic field. Kaluza's t h e o r y is a geometrical one and is a t h e o r y of relativity in five dimensions analogous to t h a t of Einstein in foul'. The unification it is n o w proposed to u n d e r t a k e is a t t e m p t e d b y the adoption of K aluza's ideas a n 4 (1) H. WEYL: Raum, Zeit, Matevie (Berlin, 1921), p. 110. (2) Tg. KALUZA: Sitzgsber. preuss. Akad. Wiss. (1921), p. 966.

682

i{. T. F L I N T 3,Ild E . M. WILLIAMSOI~

Weyl's, with differences t h a t ~ppear to m u k e the resulting t h e o r y a physical one. Other suggestions for the d e v e l o p m e n t of a unified field t h e o r y huve been m a d e b u t those o~ WEYL and KALUZA give the b a c k g r o u n d of suggestions for uniting the theories of 8Tavitation and e l e c t r o m a g n e t i s m with %he q u a n t u m theory.

1. -

Geometrical

concepts.

The relations between the coefficients of the four-dimensional line element (2)

ds ~ = g ~ d x ~ d x ~ ,

w i t h m and n taking the w l u e s 1 to 4, and the coefficients (y,,) nre given in the following table

,(3)

/

'

'

= -- ~'

Y~

where (~,~) are the components of the electromagnetic potential. ~ a n d ~55 ~re independent of the co-ordinates. Certain values were given to these quantities in Kaluza~s theory, ~ being p r o p o r t i o n a l to the charge on the p~rtiele describing the geodesic ~nd ys~ depending on the constant of gravitation. The values ~dopted here differ in b o t h cases f r o m those of KALUZA. The constant is regarded as a universal constant equal to e/moC~ (3), where e is the ~und~m e n t a l unit of charge ~nd m0 is identified ~s the rest mass of the electron. An ide~ guiding the choice of v~lues of ~ and y55 in the present t h e o r y is t h a t the tracks of all particles moving in external g r a v i t a t i o n a l ~nd electrom a g n e t i c fields are null geodesics (4). E v e r y p h o t o n in the t h e o r y of r e l a t i v i t y describes such a p ~ t h a n d the generalization of this is t h a t e v e r y p~rticle does ,so in the new continuum. This m a y well be regarded ~s the expression of the analogy t h a t has been dr~wn between particles ~nd waves. To the question: W h y does ~n uncharged p~rticle m o v e along a geodesic while a charged p~rticles does not? ]~DDISTGTON has given the reply t h a t the charged p~rticle deviates f r o m ~ geodesic in order t h a t the t o t a l electromagnetic fource u p o n it m a y be zero (5). The question here is: t I o w (3) H. T. FLINT: ~:)roc. Phys. Soe., 29, 334 (1940). (4) j. W. FISH~R: Proe. Roy. Soc., A 123, 489 (1929); H. T. FLINT and E. M. WILLIAMSO~: NUOVO Cimento, 3, 4 (1956). (5) A. S. EDDINGTON" Math. Theory o/Relativity, 2nd Edition (Cambridge 1930), t). 191.

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does it come a b o u t t h a t a particle moves along a null geodesic? The answer is t h a t it modifies the g e o m e t r y in such a w a y as to do so. This modification is v e r y simply expressed b y m e a n s of the value of y55 which depends on the n u m b e r n' of unit charges e, a n d the multiple n of the f u n d a m e n t a l mass m ~ssociated with the particle, the relation being (6)

{4)

7~5 = n'21 n~ ,

n' is an integer b u t n is not necessarily integral. The reason for this choice for y~5 will a p p e a r later on. I t is evident t h a t y55 is independent of the sign of the charge. The line element da is related to ds b y the equation (5)

da ~ = d s ~ + - ,

~'55

where dx 5 is the fifth covariant c o m p o n e n t of the displacement, dx5 = ys~dx '~. Writing ds 2 = - c~d~ ~, where dT is an element of proper time, it appears t h a t in order to m a k e d a 2 = 0, dx~ m u s t be identified as (6)

dx~ --~ ~ n' c d v , n

which can be w r i t t e n simply as (7)

dx5 = - - c d v , n

since n' can h a v e b o t h positive and negative integral values. This is the physical i n t e r p r e t a t i o n of the fifth co-ordinate. W h e r e v e r a particle exists in the c o n t i n u u m dx5 = (n'/n)cd~: the g e o m e t r y of the space is d e t e r m i n e d b y the gravitational and electromagnetic fields external to the particle. I f the particle itself is regarded as a generator of a field, this field does not affect the values of the coefficients (y,,). The fields of' the particles h a v e a n o t h e r p a r t to p l a y ; t h e y are regarded as determining the s y s t e m of gauging.

2. - A note on the relation b e t w e e n four- and f i v e - d i m e n s i o n a l quantities (7).

I f the co-ordinates (x ~) are subject to general transformations to new coordinates (x ''~) independent of the fifth co-ordinate, and if this co-ordinate is t r a n s f o r m e d according to x '~ - - x 5 4 - f ( x ~ ) , it follows t h a t the four c o m p o n e n t s (e) H . T. FLINT a n d E . M. WILLIAMSON: 2~UOVO Cimento, 3, 4 (1956). (7) H . T . FLINT; PFoc. Phys. Sot., 29, 417 (1940).

684

tI. T. F L I N T 3,nd 5E. M. WILLIAMSON

(A "~) of a five-vector (A ") f o r m a four-vector, while the c o v a r i a n t c o m p o n e n t A~ is a scalar in four-dimensionM analysis. I n order to distinguish between the four- and five-dimensionM quantities, whenever possible, the f o r m e r will be denoted b y a small letter and the fivedimensional c o u n t e r p a r t b y a capital letter. Thus (8)

A m=

a~.

Reference to the relations (5) a n d (6) shows t h a t the scalar edv is equM to dxJ~/~,~ and the notation. &IV~

(9)

= c~.

is a d o p t e d for the scalar a. corresponding to As. B y means of the relations (3) it is possible to relate other components of vectors and tensors. Thus

I A~

(10)

A5

c~q~mA~

a.

Similar relations exist for tensor components. Thus (11)

=

,

=

,

where am. is a four vector, and (12)

Amn = amn -~- ~r

@ ~/~5~o~q)~a~. @ y55~cf,,~q)na.. ,

where a . . a n d a,,. are c o v a r i a n t vectors and a.. is a scalar. (13)

A~5 _ A~

a~onA~~ _

o~m~

A useful relation in the case when A ~ and B ( A I~ = - - A ~t', B~,~ = - - B ) is

c~n am~.

are a n t i s y m m e t r i c tensors,

A~"BI,~ = am~bmn -c 2a m. b~. .

(14)

3. -

Also

Dependence

on the fifth co-ordinate.

W h e n the results of this analysis are considered, it a p p e a r s t h a t , in order to m a k e it the basis of a physical theory, the dependence u p o n x 5 is a simple one. Some quantities like gm~ a n d ~om are independent of it, and w h e n e v e r

THE

THEORY

OF RELATIVITY~

THE

ELECTROMAGNETIC

THEORY

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685

it occurs it does so in the factor exp [2zin'xS/lo], where n' is a positive or neg a t i v e integer a n d lo is a length. The integer n' is introduced into the t h e o r y in this w a y and is the same integer t h a t enters into ys~. Thus n' is identified as the n u m b e r of f u n d a m e n t a l charges on the particle u n d e r consideration. The length lo is regarded as a f u n d a m e n t a l length a n d will later be identiffed as the C o m p t o n w a v e l e n g t h h/moC. This, it has been suggested, has the p r o p e r t y of a m i n i m u m length. The occurrence of x 5 in this f o r m is reminiscent of certain circuital problems in other branches of physics where there is periodicity in t i m e and where the interest is in the fact of the periodicity and not t h e actual values at intervening times. I n all eases considered, x 5 occurs only as an integral multiple of lo/n'. I n t h e ease of the electron or positron, x 5 occurs only as a multiple of 10. ~ 0 o t h e r values h a v e physical meaning. I t is suggested t h a t this represents somet h i n g f u n d a m e n t a l in the t h e o r y and is the expression of a law of nature.

4. - A t h e o r y of m e a s u r e m e n t (8).

The unification of the q u a n t u m t h e o r y with the theories of g r a v i t a t i o n and e l e c t r o m a g n e t i s m is b a s e d on the idea t h a t this t h e o r y is essentially a theory of m e a s u r e m e n t . I t is the expression of a principle of gauging analogous in its f o r m to the t h e o r y of Weyl. I n order to explain this point of view and to derive the q u a n t u m equation f r o m it, it is necessary to express the line element of the c o n t i n u u m in the f o r m of a matrix. This is done in the f o r m : (15)

da ~ 7, dx l~ .

T h e coefficients (7,) are matrices a n d m a y , in general, be functions of the co-ordinates. I f t h e y satisfy the relations (16)

it follows t h a t da2 = 7,, dx" dx" in a g r e e m e n t with the f o r m for da 2 already introduced. A n y vector (A ~) will be said to h a v e a m a t r i x length (17)

A = ~,A ~

(s) H. T. FLI~T: Prec. Roy. See., A 870, 150, 432 (1935); H. T. FLINT and E. M. WILLIAiSOn: Zeits. ]. Phys., 135, 260 (1953).

686

n.T.

FLINT

& n d E. g .

WILLIAMSON

and its length L is determined b y two gauging factors 0 and ~o, such t h a t /i = OAyJ.

(18)

0 and ~0 are not independent, as will be seen later on, in fact 0 is the p r o d u c t of y/, the complex conjugate of ~o and a m a t r i x chosen to make L a scalar quantity. The vector with components (A ~) is said to undergo a parallel displacement when t h e y change to A ~ - ) d A ", where d A ' - - - - A~" A'dxa, the coefficients (/J~) being the bracket expressions of the continuum of five dimensions. The result of this displacement is, in general, a change AL in the value of L. If it is assumed t h a t AL is linear in the displacement components (dx ~) and the components (A ~) for all displacements and vectors a result of the form: (19)

O; ~

= O~,"H.~ ,

follows as the equation for F, it being borne in mind t h a t ,9 is dependent, upon y~. In Weyl's t h e o r y the place of H~ was taken b y a component of the electromagnetic potential. In the present case it is a m a t r i x operator depending linearly upon field components. The fields are those regarded as associated with the particle itself. T h e y m a y be, as in the case of the electron, electromagnetic in character, or, in the ease of a nucleon, t h e y m a y be nuclear fields, The general form for H is

(20)

HI, = sl B t, + s~y~B ~ q- say~y~B,,o + e~y ~'y~ B~o~, .

The components B , , B , , etc., are interpreted as field intensities. The quantities e are introduced as adjustable constants. I t is necessary to introduce t h e m so t h a t t h e y can be chosen to cause the expression on the right of equation (19) to satisfy the requirements of eovariance and further to satisfy dimensional and numerical requirements. T h e y are thus matrices multiplied by some factor. If there is no change in length with parallel displacement the equation to~ be satisfied is ~W (21) 07" ~ = 0 , which leads to the consideration of (22)

y~ ~z ~

as the equation which determines v2.

=o,

THE THEORY

OF R E L A T I V I T Y ~ T H E E L E C T R O M A G N E T I C

THEORY ETC.

687

This is the form of Dirac's equation and suggests t h a t F be regarded as identical with the ~p function of the q u a n t u m theory. The matrices (y') have so far been regarded as depending upon the co-ordinates, b u t at present no progress can be made without the limitation t h a t they must be related to. the Dirac matrices. This amounts to limiting the considerution to the case when there is no gravitational field. The electromagnetic field still remains so t h a t the situation is t h a t which confronts the q u a n t u m t h e o r y where the gravitational field does not influence the problem. The matrices (y,) and their associated components (7~'), where y~' = 7~y~, are related to their four-dimensional counterparts denoted b y (fi~) and ft. b y means of equations similar to (s) to (13). Thus (23)

The function F will be assumed to depend upon x 5 in the usual way so t h a t

~y~ _ 2nin' ~f ~X 5

1o

After substitution of this and of the matrices given in equation (23) into (22} the following equation in four-dimensional quantities is obtained: (24)

flm(h ~ ~ ~

n'~h

~o ~

)

~ + ~

n'h

~~' = o .

F r o m equation (16) and the corresponding contravariant relation

(25) it follows t h a t (26) and when a gravitational field is absent the matrices (tim) have constant com~ ponents and (27)

~-

+ ~

-- 2 ~ .

In this case, b y multiplication of equation (24) b y (2s)

~, - i ~ 4 ~ ,

~ = i,

ifi 4 and writing

~ = i~,~.,

4)88

H.T.

FLINT

~nd

E. ~ . , WILLIAMSOI~

e q u a t i o n (24) becomes

(29)

m(h 8 ~ ~27d 8x"

n'c~h

)

n'h

Zo vm

a n d the ~m and fl are Dirac matrices. I n classical mechanics ~ewton~s second law of motion provides the definition of mass. In q u a n t u m mechanics the rest muss is defined b y Dirac's equation. Mass is the q u a n t i t y 1]/o occurring in the last t e r m of this equation in the expression Mocfl~f. Thus the mass is to be identified with the geometrical quantities introduced in this theory and (30)

Mo = n ' h / V ~

clo.

Mo is placed equal to nmo, so t h a t ,(31)

nmo = n'h/v/7-~ elo .

/In addition, the factor of ~m is to be identified with n'e/c, so t h a t (32)

e = c~h/lo.

F r o m (32) (33)

lo/e~ -~ hele ~

a n d since hc/e ~ is a dimensionless q u a n t i t y e~ is a length ro, such t h a t (34)

lo/ro = hc/e ~ 9

F r o m the assumption t h a t all particles travel along the null geodesics of t h e continuum, it appears t h a t a must be identified with e/mo c~, so t h a t it follows from equation (33) t h a t

lo = h/moO a n d from (30) t h a t %/75~----n'/n. The n u m b e r n is thus to be regarded as a geometrical quantity.

5. - The significance of the lengths lo and ro.

If it is accepted that, in all cases where the t h e o r y can be applied, changes Jn the co-ordinate x 5 less t h a n loin' have no physical significance in the s t u d y

THE

THEORY

OF

R]ELATIVITY~

THE

ELECTROMAG~]~TIC

TI-I]~ORY E T C .

689

of the motioI~ of a partic]e associated with n' units of charge, it follows t h a t

dx5

--

dx5

~T~ dx'~ ~: lo n' '

or

(35)

lo

c dT ~/~

~T~dx~ r n-7 .

I f there is no field of force, this equation gives d~ ~ lo/nc. This means t h a t a n i n t e r v a l of proper t i m e less t h a n lo/nC or h/Mo c~, where Mo is the mass of the particle, cannot be associated with its motion. Thus l o can be regarded as a m i n i m u m proper length, h/moc. This is a m i n i m u m length measured along the world-line of the electron. I n the case of an e x t e r n a l static field, T~ = T~ = Ta = 0 and T4 = iT, where is the electrostatic potential. I n this case, substituting d x 4 = icdt, e dT lo ~ d ~ + a c t dt ~: n--7 . I f the velocity of the particle is small, dt can be placed equal to the proper t i m e dT and (36)

d'v -~

h/nmo c~

1 + n'~T/n"

T h u s the existence of T modifies the m i n i m u m interval a n d tends to diminish it if n' T is positive, as in the ease where a positive charge m o v e s in the field of a n o t h e r positive charge. B u t if n' T is negative, the m i n i m u m interval increases a n d it would be meaningless to speak of location in t i m e in a case in which the m a g n i t u d e of n'~T/n approaches unity. I n this ease another law of limitation

ln'cp/nl< 1/~

(37)

is suggested. I t could be said t h a t w i t h o u t this restriction it would be impossible to locate the particle on its worlddine. An e x a m p l e is p r o v i d e d b y an electron in the field of a positive charge, such as t h a t of a nucleus. I n the ease of an electron, T m u s t be less t h a n 1/~ or moe~/e in order to satisfy the condition. If a positron and electron a p p r o a c h one a n o t h e r u n d e r a Coulomb field T = e/r, r m u s t be greater t h a n e2/moc ~ or to. This gives a meaning to ro as a limiting length. The t w o limiting lengths 10 and r0 are related b y lo/ro = hc/e 2 in a g r e e m e n t with equation (3~). 45 - I1 2Vuovo Cimento.

690

6. -

H.T.

The mass

FLINT

3,1~d E . M. W I L L I A M S O N

of a p a r t i c l e .

When a change of length occurs with a parallel displacement the q u a n t u m equation (3s)

y ~

- ~.~.w

= o,

in accordance with equation (19), so t h a t it can be written in the f o r m

ifl~y" ~

(39)

,

in order to relate it to Dirac's equation. If the gravitational field be neglected the form t a k e n is

(40)

h ~ am 2~i~x'*

n'e ) h , c q~'~ w - ~ n m o c f l ~ p - ~ f l ' y H~y~= O.

Thus the mass t e r m of Dirac's equation is replaced b y the sum of two terms, and if M'o becomes the mass defined b y Dirac's equation, writing Mo = nm0, t

(41)

§

h

The last t e r m in this equation arises from the field due to the particle itself and, in practice, the observed mass would approximate to the mass, Mo, t h a t has appeared from geometrical considerations, when the interaction terms are small. These terms represent an interaction of the particle's field with quantities depending on the function F and the matrices, which can be regarded as measuring some p r o p e r t y of the particle. This inclusion of the interaction terms as p a r t of the mass of the particle is in agreement with the process of mass renormMization. In applying these ideas the case of the electron is of particular interest, b u t similar considerations would apply in other cases, as in t h a t of a p r o t o n and its nuclear field.

7. -

The theory

of t h e e l e c t r o n .

The field associated with the electron is derived from a vector p o t e n t i a l with four components (Am), the fifth component A~ being zero. I t is assumed t h a t t h e y depend upon x 5 in the usual way. The four-dimensional counter-

THE

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THE

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p a r t s (a,~) are the potentials usually t a k e n to be vector potentials of the electron theory. Since A5 = 0, Am =- a~ (10). The field c o m p o n e n t s are (B,,) where

(42)

B,~-

~x,

~x~ .

F o r convenience k is placed equal to 2:~moc/h und for the electron n ' = ~ = 1. F o r the case # =-5, ~ = m, it follows f r o m equation (42) t h u t (43)

Bs,~ -~ i k A , ~ .

I n this case according to equation (20) (44)

H u --~ e~7~B~,

a n d the additional t e r m in the q u a n t u m equation is h

_

/t

v

Thus f r o m equation (39) it follows t h u t

(45)

h ~F cF+ifidY" 2 ~ i ~x t'

hc 2 ~ F+fi~V%27~FB'~ = 0 ,

a n d this will be w r i t t e n in the f o r m

(46)

cy~+ifidY" 2 ~ i ~x ~

4 '-

--~ + I

B I~,) - - 0 .

This m e a n s t h a t the real p a r t of the additionM t e r m in (45) has been added to the mass t e r m moc2~+fi~f. The factor of Bsv in (45) m u s t be a c o n t r a v a r i a n t tensor of the second r a n k and this m e a n s t h a t the m a t r i x in s~ m u s t be t a k e n to be ft.. I n this case the t e r m contains tile p r o d u c t fid~fi.7,, a n d for the case # = m, ~ = n, this becomes ifia~o: ~, so t h a t y~+iflo:,~p becomes a factor of B ~ : This t e r m is a c o n t r a v a r i a n t tensor of the second r a n k and is thus w h a t is required. es m u s t also be multiplied b y a numerical a n d dimensional constant to give the t e r m the correct m a g n i t u d e a n d dimensions. The expression #y~+iflo:~*y~, where # is the B o h r m a g n e t o n , is a well k n o w n expression for polarization density, so t h a t it is n a t u r a l to write I *~' in the

692

I t . T. F L I N T t~nd E. M. W I L L I A M S O N

form :

allowing g~# to a b s o r b t h e f a c t o r 2 h c / = .

(47)

F r o m this it follows t h a t

I * .... = Iron _ g~#~+iflc~mC~ny j .

T h e c o m p o n e n t I *'n is i m p o r t u n t becuuse it is a four vector. uecordin~ to (47) I~ ......

Its

v a l u e is

igl~yj+~my ~

b u t since this is a v e c t o r q u a n t i t y a n d I *~" is a tensor, t h e r e q u i r e m e n t s of cov~riance are satisfied w i t h ~ n o t h e r c o n s t a n t in this c~se. M a k i n g use of this f r e e d o m , t h e v a l u e m ~ y be t ~ k e n to b e : (~8)

'

I : ~'~ =

i m = __ ig~#F+Ct~n~.

-

]~-rom e q u a t i o n (14) = i*-~nbm~ + 2i*% b~,.

I*l'B

a n d since I*mn = Iron ,

i *~nn = i'~n .

Moreover i*m. = __ i,~. 9 Thus ~ (I*"'B/,~ + P ' B * , . ) =. ki'*n(b,~n . . . . + b* ) + ~-i% (5*% - - b'~.) , b * , . = ika,~* ~

b% =: - - i k a ~ .

Writing

/m

=1

~( a m + a * m )

t h e expression b e c o m e s : (49)

89

],,~ -

k g ~ # ~ f + ~ f ]~ .

T h u s t h e mass of t h e electron m~ is given b y

(50)

m ' o c ~ + ~ -= m o c ~ + ~ -

~ g ~ + i ~ n ~ / m ~ + g~e~§176 ....

The electron can t h u s be said to possess a p o l a r i z a t i o n d e n s i t y # ~ + i f l ~ " ~ ' y , ~ w h i c h i n t e r a c t s w i t h its field w i t h a s t r e n g t h m e a s u r e d b y l g , a n d also t o possess a c u r r e n t d e n s i t y w h i c h i n t e r a c t s w i t h a s t r e n g t h m e a s u r e d b y g2.

THE

THEORY

OF RELATIVITY,

THE

ELECTROMAGNETIC

THEORY

ETC.

693

Dimensionless constants have already been introduced into the theory of the electron (9) in this way and similarly into theories of the nuclear field. In the case of the electron, values have been calculated for them.

8. - The e q u a t i o n s of the field of the electron.

It will be assumed that the field of the electron is described by two tensors (B~,) and (] 7 ,J. A polarization tensor (I~,,) is introduced with (51)

V~ ~-- B , - - I , , .

This assumption is in accord with the theory of G. Mie and is in contrast to that of H. A. Lorentz, who based his work on the assumption that the field is to be described by a single tensor (B,,) or, as he described it, by a single electric and a single mugnetic intensity. The theory of Born and Infeld also follows that of ~ie. The theory of Lorentz emphasizes the importance of the charged particle which is, in his theory, the generator of the field, so that the field is not the primary aspect of the theory. In contrast, the field is primary in the theory of Mie and the charge is a manifestation of it. In this way it may be hoped to avoid reference to the structure of the particle. In the present theory this is avoided by regarding the m~ss as a geometrical quantity with the addition of contributions from interaction terms, derived from the function F, certain matrices and constants of the theory. The tensor (Bz,) is derived from components of potential as already stated (42). One set of field equations is thus (52)

~B,~ ~x z §

3B~a ~ +

~B~, ~x , - 0 .

The other set is (53)

~V'~

-~x~ = o.

These are Mie's equations except that they are understood in the sense of five-dimensional analysis. If # = m (1, 2, 3, 4) Y m

(54)

~ x ~ + i k v ~,~o = o ,

(9) H. A. BET~E and S. SAL•ETER: Handb. d. Phys., 35-I, 178 (1954).

694

H.T.

~. WILLIAMSON

F L I N T a n d E.

a n d t h u s b e c o m e s on s u b s t i t u t i o n of V "5

(55)

~x~ --

ik

- - ~ F ~ V "~ .

F o r t h e case of t h e electron n ' = n - - - - 1 so t h a t 7~5 = 1. d i m e n s i o n a l quantities this e q u a t i o n b e c o m e s

In

terms

of four-

~V mn

(56)

~x ~ --ik~v~--ikv%.

T h e first t e r m on t h e r i g h t vanishes if t h e r e is no e x t e r n a l field, a n d w h e n v "*~ is real it c a n be o m i t t e d as an i m a g i n a r y q u a n t i t y . I n general, it is to be reg a r d e d as a c u r r e n t d e n s i t y ~nd it does, in fact, a p p e u r as a c o m p o n e n t of t h e e n e r g y - m o m e n t u m - c u r r e n t tensor. I t will be o m i t t e d here. Substituting v~

9

= b%

-

i% =

-

-

-

ika "~-

i "~.

the e q u a t i o n b e c o m e s iki't

-- k~a ~

~x ~

t h e f o r m of w h i c h suggests w r i t i n g i n. = (57)

ikzG

whence

~v'~'~ ~x ~ -- k~(X.~_ a ~) .

T h e expression on t h e r i g h t h a n d side t h u s takes t h e place a n d is d e n o t e d b y ]~. I f v e c t o r s H ~nd D be i n t r o d u c e d w i t h

H x=v23

H v

D~ = i v 1~

D

'

y

v 31

= i v ~4

'

of

H z =

v12

D~ -

i v 34

a current density

t o g e t h e r w i t h a c u r r e n t d e n s i t y J a n d a charge d e n s i t y ~, w i t h j~ = ]1

,

j,j

--

]5

j~ = ]8

,

t h e field e q u a t i o n s t a k e t h e f o r m :

(58)

curlH--1_D

= J ,

c

div D = ~.

,

~ =

__ ij~,

THE

THEORY

OF RELATIVITY,

THE

ELECTROMAGNETIC

THEORY

ETC.

695

I n the absence of an external field, the first set of equations (52) become

I

(59)

curlE§

[

=0, div B = 0 ,

where B~ = b~ ,

B~ ---- ba~ ,

B~ -= b~ ,

E ~- ibl, ,

E~ = ib~ ,

E~ = iba~ .

I n Mie's t h e o r y there is no specific relation given between the vectors (D, H) and (B, E) although one is assumed to exist. I n the present case

serves as ~ definition of v~ . If vectors I and P are introduced with I~ --

i ~3

Iy

P~ = -

ii~

P~

i 3~ ~

ii~

P , = _ ii34

the relations B =H+I,

(60)

D=E+P

follow.

9.-

The Lagrangian f o r m of the theory.

I t is possible to derive the q u a n t u m equation and the field equations from the Lagvange function (61) L = ---41 ( B * ~ n B ~ - - I * ~ B ~ - - I ~ B * { ~ ) - - 4 ~ ihc{,+iR~ \ ~ ~ ~ ~ 7~x~

~+ifi4y~ ) ' ~x ~

the vMues of the polarization components being, us before, given b y (62)

I *~ = g~u~+fi4y~fl. 7~p,

I ~ = gllz~o+y~fi.y~fl4~o.

L is thus a lunction of % ~+, ~ l ~ x ~, ~ + l ~ x ~, ~A~/~x ~ and ~A*~I~xL The q u a n t u m equation (63) 5,i

__ /

_8L _

]

~L

0

696

H.T. FLINT ~nd E. ~. WILLIAMSON

is readily seen to reduce to the f o r m (46), and the field equation __

[

leads to equation (53). The equations conjugate to (46) and (53) result f r o m similar equations in ~p and ~A,/~x'.

10. - The field tensor of energy, m o m e n t u m and current.

The energy tensor (OQ) is f o r m e d in function:

(65)

O5

-

~(~A~/~x ~') ~x"

the usual w a y

~(~W~x,) ~x~

from Lagrange's

c o n j u g a t e t e r m s + ~q,L.

I n this f o r m the tensor is not s y m m e t r i c b u t it can be m a d e so b y a f a m i l i a r procedure. The t e r m s added for s y m m e t r y do not affect the conservation of the tensor and do not a d d to the t o t a l energy. Thus, since these are the t w o points with which the present t h e o r y is concerned, t h e y m a y be a d d e d or o m i t t e d as is convenient. The p a r t due to the field t e r m s will be t a k e n in its s y m m e t r i c f o r m and the r e m a i n d e r will be left as it stands. The energy tensor t h e n becomes:

(66)

T~

= ~ (V'aB~ + V*VaB~a)- - 6~ 7~

-- *

"0~ - - * "~

~x ~

+

Ux~ ~7~

9

F r o m the q u a n t u m equation (46) the t e r m s in I *~ and I ~ together with the last t e r m in b r a c k e t s vanish. I t appears f r o m the structure of T~, t h a t it does not contain the co-ordinates x 5, so t h a t the equation (67)

OT~

~x; -- 0 ,

which the components satisfy, becomes for the various values of v

~T; ~X m

_

o.

TIll5 T H E O R Y OF R E L A T I V I T Y , T H E E L E C T R O M A G N E T I C T H E O R Y ETC.

697

Thus the energy tensor can be regarded as (T~) and the energy density is. - - T ~ , the total energy being

W = --fT~, dx dy dz. F r o m equation (66)

(6s) If h ~ ul~--2~iOx/~ 2~i

e 091~ and

u~+ --

h ~ 2~i~x ~

-~ = ~p+(c~%~ + e~ + moC~fi)~p- - - ~

e c 9~' ,_

(k = 1 , 2, 3),

_.~

+ I B~) ,

k being summed over 1, 2, 3. This is the energy density of the particle and can be regarded as made u p of energy W~ consisting of kinetic and potential energy and of interaction energy W~. Thus

rVI~B* The terms can be rearranged for convenience, bearing in mind the value of Wz, so t h a t T' 4 =

+ V * ' a B ~ z ) __ ~1 (. v. . . . ,8B ~ 2. - V _ ~ Z B *~r ~ +

89

88

~ : ~ - W,z .

F r o m the field equation it follows t h a t

f V*~r

dv : - -

A~

~x4 + V *'n4 ~

] dv ,

and 1 ~x~

V,m4 OA~1] dv

(dr -= d x d y d z ) .

dv = f A m ~V*~4 Ox4 dv , and it is possible to write: (69)

- 1 ~ B *~'~ T~ = ~1 ( Am ~V*m4 ~x4 + A*m ~V*'~4~ ~-] --~--We.

698

H.T. FLINT

and

:E. M. WILLIAMSON

Tile energy density is (70)

W=

. . .'~4x + A ~ - ~ ] . W~--~ B*~B~,~--~ (A "~.~V*'

I n the case when V "4 does not depend on the time and there is no magnetic field, the conditions are static and

W::

89

The energy consists of the energy of the particle and the electrostatic energy of the field. There is no suggestion of energy of interaction a n d this would not be expected in a n y case, for if the particle gains b y interaction the field m u s t lose and the total energy is not affected b y the interchange. This point m a y well have a bearing on the difficulties t h a t have arisen over the interaction terms in regard to their infinite value. The expression (70) for W must be finite, if it is to have a n y physical meaning. I f p a r t of it is regarded as energy of the particle and p a r t as energy of the field, it m a y be t h a t in m a k i n g the division a term tending to infinity has been t a k e n on the one h a n d which m u s t be balanced b y an equal t e r m on the other. B u t there is no significance in this division. The interaction terms applied to the particle or to t h e field m u s t be finite and s if a rule agreeing with the requirements of experiment can be found, which shows h o w these infinities can be avoided, it can be applied w i t h o u t upsetting the covariant prop e r t y of the energy tensor. Such a rule would appear to be as justified as the original choice of the form of the interaction terms (lO). (10) Reference should be made to the article quoted under (9).

RIASSUNTO

(*)

Si dimostra come sia possibile, per mezzo della geometria e dell'introduzione di un principio di misura basato, per analogia, sulla teoria di H. Weyl di scoprire un'unitariet~ esistente tra i fenomeni gravitazionali, elettromagnetici e quantistici. L'equazione di Dirae e una sua estensione si dcrivano dal prineipio di misura e un aspetto essenziale della tcoria g di incorporare una teoria dell'elettrone nella quale la massa .di qucsto appare come una grandezza geometriea che entra nell'equazione come conseguenza dell'esistenza di un'unit~ fondamentale di lunghezza. La massa osservata data da questa grandezza geometrica assieme a termini d'interazione che a loro volta entrano naturalmente nella teoria.

(*) Traduzlone a cura della Redazio~e.