the point charge in the unitary field theory

The Point Charge in the Unitary Field TheoryAuthor(s): Erwin SchrödingerReviewed work(s):Source: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 49 (1943/1944), pp. 225-235Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488459 .

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[ 225 ]

XI".

THE POINT CHARGE IN THE UNITARY FIELD THEOR1Y.

[From the Dtublin Institute for Advanced Studies.]

By ERWIN SCHRODINGER.

[Read 8 NovEMBER, 1943. Published 3 FBRUARY, 1944.]

? 1. THE NEW FEATURES.

Two new features of the electromagnetic field are entailed by the present theory,' the two being of an entirely different character. The first is the

incorporation of the Born-Infeld-theory, which sayQz that already the basic vacuum laws involve the two sixvectors, viz. P(B, E) and (H, D),4that coincide

in the limit of weak fields, but are connected by non-lihear algebraic

relations in strong fields. In our present task, which is to determilne the

fieldl of a point charge, this circuinstance only affects the " microscopic"

structure of the charge, for only in its immediate neighbourhood is the field

strolng enough to make the distinction relevant.

Inversely the secondfeature is only exhibited in extended fields. It says that the potentials (A, V) multiplied by a small negative constant, _ p 2, act

as sources. The possible consequences for the magnetic fields of the earth

and of the sun have been indicated.2

From the point of view of mathematical technique, these two rhodifica

tiolns of customary electrodynamics-for brevity call them the Born-effect

and(l the w-effect- are very neatly separated. Indeed, the Born-effect becomes

negligible as soon as the distance r from the centre is large compared with

the " radius of the electron" ' r0 (o0 3 X 10-13 cm), while the tA-effect only

sets, in when r becomes somewhat comparable with t-' (oc 3 x 10 cm). The

gap is wide eniough to permit complete separation.

Yet, precisely from our present investigation it will emerge, that the two new features are not entirely disconnected. For th-ey prove to be controlled

by virtually the same uniiversal constant, which at the same time accomplishes

the dlevoir of constant of gravitation. It is virtually the constant U2 itself I

%m speaking of.

1 Proc. R. I Acad., 49 (A), 43, 1943. 2 Proc. R I. Acad. 49 (A), 135, 1943.'

PROC. R.I.A., VOL. XLIX, SECT. A. [27]



226 Proceedings of the Royal Irish Academy.

The region of smiall r is dealt with in ? 2. The gravitational field turnis

out to have only an atmazingly weak singularity at the centre, comnparable to

that of anl extremely flat cone.

The region of large r, discussed in ? 3, would be entirely trivial in flat

space, where the problem is completely answered by the Yukawa potential,

eUlr. But when the point charge is embedded in spheric-al space, a new

trait emerges: we can present the solution corresponiding to aone sinigularity only-whiich cutrrent theory cannot. In other words, the ulniverse as a whole

need nio longer be electrically neutral.

? 2. THE STRUCTURE NEAR THE CENTRE.

(a) Mathematical Solutionz.

It is imperative to use the uniiversal constants in certaini lnew com

biniations, whieh I beg to explain comprehensively. The "n latural uInit " of electromagnetic field strength, expressed in

c-g-s- units, is Born's constant b Kin gramm-cm-units it is - ). b is

connected withi C and with the charge and mass of the electroni by'

b _ e 3 3957 x 10 ' e.s.u. , (2, la) r

2

e2

1 12361 m = 3-484 x 10-l3 cm , (2, ib)

200

12361 ... = - . . (2, ic)

r 0 is called " radius of the electron."

The letter f I introduce for a reciprocal length of the order of

l 0 cm -' , conniected with 7c the constant of gravitation, by

2 2kb2 1 f2 _ 2= k =_ 2-587 x 10-11 cm-'3 . (2, 2)

The constant f, or if you like, the constant

fJG (2, 3) a

3 The numerical values, hitherto disfigured by a ten years old mistake, are corrected here

according to Proc. R, I. Acad., 49 (A), 147, 1943,



SCHR6DINGEL-Thee Point Charge in the Unitary Field Theory. 227

(which is not very different from it) has in the new electrodynamics a fundamental meaining, which I lhave recently discussed in colnnection with geomagnetism.' From tllis discussion it would appear, that the numerical constant a is near to 12, with a fair guess that

- 2 27r e2 Ac ' ~~~~~~(2, 4)

the fine-structure-constant. This choice at any rate fits the geomagnetic data admirably.5 But we shall not commit ourselves to it in the following.

Usinig these notations the field equations read in gramm-cm-units,

(2 2__22 J b I bw @ka 4Ua - gkl -1 b

A k XU A2 ) (2 5a)

- a a_ aA k (2, 5b)

1 g a FJu7[ g (v! b - M2 + kl ) ] = 2 Ak, (2, 5c)

- = 71 + 2 kl - b 14 22 (2, 5d)

f kl is the primitive electromnagnetic field-tensor (B, E). I is its second invariant, always zero in the following. The asterisk indicates the dual tensor.-I beg to notice the arrangement of the constanlts. Every component

or A is virtually accompanied by the factor b , reducing it, as it were, to

Born's "natural measure." Only one futther conistant is involved, viz. ft playing the role of conetant of gravitation, but governing-under the niame

of y 2-also the new terms, recently discussed in geomagnetism. (In this sectioon we drop them.)

We adopt the spherically symmetrical line-elemnent in the form7

ds = - eAdrS - rt dO2 - r2 sin'Od#2 + evdt2 (2, 6)

where A and v are functions of r, to be determined, and t is the time,

4 Proc. R. I. Acad., 49 (A), 135,1943. 5 See I.e. Note added on proof.

c See (2, 3) above. 7 A. S. Eddington, The Mathematical Theory of Relativity, 2nd ed., Cambridge Press, 1930,

p. 83, equ. (38-2).



228 Proceedings of the Royal Ilrish Academy.

measured in light-path (g-c-units) The index 1 shall refer to r, 4 to t. The

only non-vanishing component of ckl is

41 4- 4jr), (2,7)

say. Thus

12kl1k = - e-A+v) # 2. (2, 8)

Since 12 = 0

W e- e( + V)42 (2, 9)

From (2, 5c) with k = 1, neglecting the p-term:

l d [r2e i ?x(A + v)(x #J O 0 (2, 10)

2d-r w (2 0

Integrated

2 c e -i (A + v) 2

c2/

E <+)# (2, 11)

whiere E is the integration constant. Thus

- e-i9A+v) # = e

b |?c2 + r4

_ I w - JJ r (2, 12) I o 2 _ _ _ _ __~2 rI

o kt #1 -U?r

2, b2 l e + r

Turning now to (2, 5a), only the equations with c I 1 survive. Moreover

o2 ~~~c 2 o,2 E2 e b72 q 4la

- = 9' # 14 e #-

2 2 ?+

4

(2, 13) o2o2 2~ () - 2e 2 - EeV

4 = 4 b24 - 12 e

1 2 + r 4

while the similar exDressions for index 2 and 3 vanish.



SCOHRODINGEI-The Point Charge in the Unitary Field Theory. 229

Using these results and the well-known values of the GCk, we get (neglecting the 1t-term)

C722 -= e 6-[1+ r(v'-X')] - 1 = tf2r2(lJ 6$2rt7) (2,14)

G4i = ev-AK i + 4X'v' - _

2~~~~~~ - 2 ev(l - J;?4)

The equatiol with C, has been onitted, for it only repeats the one with

G2,. From the first and the last one you infer

A' + V = 0 hence

A - - v (2, 15)

since both mnust vanish for r * oo . Then the second equation reads

ev(1 + rv') - 1 - f2(rz

- + r )

or

d

y (re') = 1 *.f2 ( JE +r r- r2) (2, 16)

while the first (ol the last) equ. (2, 14) cani be written

c_it (ev) f2 (2r - 2

d r,2 $E2- +

anld is thlus a consequence of (2, 16). From the lattter we get by integration

v ~~~~~f2 r

eV = e ?x = 1 - J ! (JeI + rW4 r2) dr; (2,17)

tihe constant is chosen so as to avoid a singularity of thefunction at the

origini.

To complete the mathematical solution, we talke from the first equ. (2,12), with regard to (2, 15),

b 2 o J + _ r_

8 A. S. Eddington, I.e., p. 85.




This is the radial field-strength in g-c-uinits. In g-c-s-uinits it is c + . We call that

B, = J -+ r 4b(2, 18) 2e

+ r

4

(b) Geometrical and Physical Discussion.

From the last equation, taking r large with respect to JJe l we infer,

that the effective charge in c-g-s-units is b . EEquatilng it to the electronic

charge e, we get from (2, la)

E r

T,his we introduce into the mzetrical coefficients (2, 17) and abbreviate

ii = *a' . (2, 19)

Then

ev = - 1 _ fro2j | (/1 + X - x2)dx (2,20)

It is not difficult to see, that the function of x decreases steadily from

x = 0 and approaches to zero at infinity. The strongest departure from

Galilean metric is thus at the centre, viz.

Lv - e -^ 1 -_ fir0' (for r - 0). (2, 21)

Even this departure is exceedingly small, since from (2, lb) anid (2, 2)

the product f2r02 lies between 10-" and 10-44 . Yet we must not

call -the metric regular at the centre. The suirface of an infinitely snmall

sphere is not,47r times, but only 4r (1 - f2 r02) times the square of its

radius. The singularity is of the kind that in two dimensions is presenited

by the vertex of an extremely flat cone.

Let us compute the invariant curvature from (2, 14). Using (2, 19)

G _ g"lG, + 2g G22 + 9 44 G44

(f 2 2x' + I 2)

(2,22)

It is seen that, although the gik differ so little from their Galilean

values, G approaches to infinity for r -*- 0; as, of course, it must, since

the invariant inass-density of Born's electron is known to do so.

Moreover a new physical meaning transpires of the constant f which

we have already come to know as "1 constant of gravitation " and, under



SCnRdDmcGER-The Point Charge in the Unitary Field Theory. 231

thie form of yO [see (2, 3)], as controllilng the behaviour of extended

electromagnietic fields. It is now seen to determnine also the curvature

inside Born's electroni, I mean. to say for moderate valuies of r. E.g. for

x -1, i.e. r = r,, you have

G = f2 (3 /29 - 4).

To compare this with a f our-diineisional hypersphere, we eqtuate it to 12/R 2

and obtain R = 43,724 km,

which is to be held, as to order of magnitude, against

au-Ioc 30,000km,

determined from geomagnetism. -

It remains to be shown that the metrical coefficients (2, 20) for x >> I

ilndicate a gravitational mnass equal to the inertial and energetic mass rn

The integral can be transformed thus

x Jx dx x [ x dx

w~ ~~ d VI + XI d - V/- + X- ix 1 +zd -2 1+ '

- 2

By combining these two formulae:

/i + x4d =2 x dx

4.

-V/ 1

+ Xr d.v = - - 4

+ --V/+

+ X4

Using this in (2, 20) we get:

41 x~~~~~~~~2\ e= eA 1 - r02(17 +jr + 3 11+ ox4

(2, 23) F'or x >> 1, with (2, 19):

ev- e- 1 f'2rO 27 d 4 ; (r >> Io).

F'rom (2, 2) and (2, la)

42 X o3 = 2e- 2/ (2fW dx +' 4 2 1c1, 4




the latter from (2, lb) and (2, lc). Hence finally

ev _ c-A = 1 _ w r m ; t (r ?> r0) . (2, 24)

This agrees with the well-known Schwartzschild solution for a point

mass mi.

Since Born has shown, that Mn0 c2 is the field energy of his electron, we have here, for the first time, the model of a point-source whose

gravitational field is accounted for by its electric field energy. The

singularity itself contributes nothing. This is not so in the case of the trtaditional solution, which (in

simplifying units) reads9

2m 4vr e2 e v e ^ -r + 12 (2, 25)

Here m and e2 are independent constants of integration, to which any i alues could be assigned. However, with e + 0 the electric field

uaergy is infinite anyhow. How is nevertheless a finite gravitational mass

produced? Obviously by incorporating an infinite negative mass in the singularity itself. - That is clearly seen by comparing the behaviour of

(2, 25) for r ->- 0 with that of 1 I

? 3.-THE COSMIC STRUCTURE OF TIHE POINT-CHARGCE.

In the region

r0 << r < '

we have, from (2, 18), approximately

=bE

When this region is exceeded, the xu-terms in (2, 5) must be taken into

account, while the metric can now safely pass for (lalilean, X = v - 0

in (2, 6). Then we easily get from (2, 5b) and (2, Sc)

V=a gl s r G e e_ , (3,I)

where V stands for c A. I apologize for the clash of the two e's.

9 See A. S, Eddington, I.e., p. 185.



SCHRSDINGER-The Point Charge in the Unitary Field 7Theory. 233

No particular relevance, as far as I can see, is attached to the pt-teria in' (2, 5a), although it reaches the same order as the rest, as soon as r

becomes comparable with u - 1p.

The physical meaning of space becoming flat, is obviously that it

becomes what it would be without the point charge in qucation. It is

xvorth while to examine the latter when embedded in spherical space.

In other words: what is the Yukawla potential (3, 1) in spherical space?

I eonsider it quite relevant (though perhaps not very astonishing) that a solution with only one singularity can be indicated, With the classical

potential (4 0) that is not so, for obvious reasons: the lines of force

issuing from the charge must end somewhere, they cannot go off to infinity,

because in spherical space there is no infinity. Let me recall the classical features, beginning with the two-dimensional

sphere. Here the only solution, apart from the constant, of the Laplace

equation

1S ~ ~ ~ .~ ~~~~~~~ v 2V R-2sn YysinO0 ~- )=0 (3, 2) is

9 V = log tan , (38,3)

which has two sinlgularities, for 0 = O and for 09 i.

On the three-dimensional hypersphere, using X for the angular distance from the centre,

v2 = _ tsi ; (sin2 X f) = 0 (3,4)

has, apart from the constant, the only solution

V = cotan x (3,5)

again with two singularities, at x 0 and at x - 7r.

Of course, the two singularities need not be antipodic. TShe structure

of (3, 3) and (3, 5) is best grasped by attributing to a single point charge

thle potentials

loc 2 sin0 or X cotan y (3, 6)

respectively. They hiave not vanishing, but constant Laplacians

(viz. - 1 and 2 respectively) 'hat is to say, they -2

B 2 and ye

represent single point-chariges (at 9 = 0 and at X - 7

respectively),

rROI. R.I.A., VOL. XLIX, SECT. A. [281




with a neutralising charge spread uniformly over the sphere (or hyper

sphere). - By superposing 10 two or more of themn,

- e k log sin- or Y = - h ekX 7 cotan X k, (3,7)

witil

7 0, (3, 8)

you get more general solutions, of which (3, 3) and (3, 5) are simple

special cases. (The 07 and w - x 7 are the angular distances from the

different poinat charges.) But (3, 8) must be observed, the "uniiverse"

as a whole must be electrically neutral.

Dropping from now on the two-dimensional case, which only served Cas a more familiar illustration, let us include the ,u-term 1in (3, 4):

V2 V= I a I -v 2V. A2sin2x x ksin x - (3,9)

Put for brevity

a (3, 10)

(which is a very large number, roughly 1020 ) and introduce, the variabtle

z = sin (3,11)

Trhen (3, 9) turns into the hypergeometrie equation

z (1 - z) V" + (3 - 3z) V' - a' V 0 . (3, 12)

To obtain the solution we need put

V = z- C] , z = 1 - x (3, 13)

which gives

x( - x) U"" + (4 - 2x) U'- (a2- DU 0. (3, 14)

Comparing this with Gauss' standard form

1 a ( [y (a + (3 + 1) x] y' - aJy = 0, (3, 15)

-e have to take

2 2 (3, 16)

10 The factors - 1 or - -, respectively, in (3, 7) serve to give to the ek the exact

meaning of charges.



ScERWiNGEr-The Point Charge in the Unitary Field Theory. 235

Then

U = P (a, 13; y; x)

is a solution of (3, 14) and, from (3, 13),

V= z-F(1 + ,/1 - a', 4 - - a; 4; 1-c) (3,17)

is a solution of (3, 12). It is the one we need.

The factor z - embodies, according to (3, 11), the required Coulomb

singularity at X - 0, i.e. at z - ; while F here approaches to a

constant. Indeed, since

-y - a - j3 = j > 0 ,

-we have from well-known formulae

F (a, /3; y; 1) - (7)rY - a- 13 ) [(,y l a) Py-()

r( Qr () __ _ sinh7r>Ja --l r (I - /1 - a2)fr(I + /1 -a2) 2 /a2- I

This is an enormously large number. To remove it and obtain the potential of unity charge, let us supply a suitable multiplier in (3, 17) and, at the same time, express z by X and drop the irrelevant 1 under

the square root (V (1) means "V for unit charge"):

YV(1) <- I + iB 4 - i ; Cos 2X) sinh (Pp r) sin(318

At the antipodic point, x -r , F approaches to 1 and V ( 1) becomes

-anishingly small, viz. as

-e 2

In the neighbourhood of X = 0, thht is of the value 1 of its 4th

argument, our Gauss series, as it stands, is the poorest tlhinkable inistrument (though it converges uniformly, all right). ' It is bound to behave there as e- R X. But it is hardly worth

while to go to the trouble of proving this fact. Any amount of solutions (3, 18) can be superposed with arbitrary

factors e kA the various X ,i being the angular distances from the various point-charges. Condition (3, 8) is not imposed. The universe need not be electrically neutral.



the point charge in the unitary field theory

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