maxwell's and dirac's equations in the expanding universe

Maxwell's and Dirac's Equations in the Expanding UniverseAuthor(s): Erwin SchrödingerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 46 (1940/1941), pp. 25-47Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20490746 .

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[ 25 J

IV.

MAXWELL'S AND DIRAC'S1 EQUATIONS IN THE EXPANDING UNIVERSE.

By ERWIN SCHRODINGER.

[Read 12 FEBRUARY. Published 17 MAY, 1940.]

? 1. INTRODUCTION.

Previous workl on the behaviour of light and of particles in an homogeneous spherical Universe is completed in this paper. As regards light (or, more generally speaking, electromagnetism), I here supply the investigation of Maxwell's 1st order equations. In the earlier papers I had only dealt with the 2nd order wave equation (d'Alembert's) as a sort of substitute. As regards particle motion, the analysis of the Dirac equations, given in A.P. for the static Universe, is here extended to the non-static case. My " Physica " paper, which dealt with the latter, had also for particles substituted the corresponding second order equation, usually called Gordon's.

The space-functions (amplitude-functions) do not depend on whether B varies with the time or not. Therefore in the Dirae-case only the time

function remains to be determined. But with Maxwell's equations, which I had hitherto set aside altogether, the complete analysis has to be given here. I wish to acknowledge the valuable help of my friend, Wolfgang Pauli. Both the suggestion of dealing with the six-vector of the field itself (not with the potential) and the solution of the space-problem are his work. The time-problem of the Maxwell-case is of peculiar simplicity. It can be completely solved, up to a quadrature, withowt knowing the function R(t).

It so happens, that a few quite interesting items have been removed to the appendix. The spectral structure of the Maxwell-case (App. II), as compared with that of the Dirac- and of the scalar case, appears

to be characteristic of the spin or, on the other hand, of the tensorial

character of the wave-function (whether a scalar, a spinor, or a vector).

App. III incidentally answers the question: Is a homogeneous field possible in the hypersphere? (Answer: Yes. But if, e.g., you establish it as an

electric field, it alternates with very low frequency, changing periodically into a magnetic field, parallel to it.) App. IV shows that electromagnetic

"skin-waves," known from previous work, are circularly polarized.

1 Commentationes Pontificiae Academiae Scientiarum, 2, 321, 1938; referred to as

A.P.?"Physica," 6, 89?, 1939; referred to as "my Physica paper.''

PROC. R.I.A., VOL. XLVt, SECT. A. [4]



26 Proceedings of the Royal Irish Academy.

? 2. MAXWELL'S EQUATIONS IN THE EXPANDING UNIVERSE.

Let (kik be the covariant antisynmetric field-tensor in an arbitrary

metric. The set of Maxwell's equations, usually called the first, reads

a+Uk = , (2 1) axl]

where i $ k + 1 and the square birackets indicate that the one term

stands for three, obtained by cyclical permutation of ikl. (Vanishing of the cyclical divergence.) It is convenient to give the second set the same

form, with the dual tensor qik

__ik = 0. (2, 2)

We do not admit charge or current, which would otherwise form here the

right-hand side. The dual tensor is defined by

$ik - 1 Ejklm m = j EAkhN grgms 0's (2, 3)

E,kl is the well-known antisyinmetrical tensor of rank 4, whose

components vanish, unless ikim is a permutation of 1 2 3 4, and are

'I- g for an odd permutation, - v- g for an even one. (For the

contravariant components the sign would be the reversed.) The factor j just compensates that well-known repetition due to the summation convention. Since the qic and oik are real, equations (2, 2) and (2, 3) can be united by putting

F?* - (/)ik + s/ - 1 fi7cn (2, 4)

thus

ax,] o. (2, 5)

These are the equations to be solved. Of course there can be only three independent complex quantities Fik, Indeed there are these relations between them

Fik 2 1 4 Eizkm g gilns Frs, (2, 6)

three of which are independent. To describe space (i.e. the hypersphere) cylindrical coordinates are

particularly convenient for our purpose. What makes them so convenient is that the two families of planes, forming two out of the three sets of

coordinate surfaces in the flat case, acquire in the spherical case an

absolutely symmetrical standing. They turn into two families or bundles of great spheres, one bundle comprising all the great spheres that pass



SCHR6DINGER-Maxwell's and Dirac's Equations. 27

through a given great circle, the other one those which pass through the polar great cirele. (I call pole of a great circle a point having the same

distance from all the points of the great circle. There are infinitely many

of them and their locus is another great circle, which I call the polar one.

The relationship is reciprocal.) The family of coaxial cylinders, forming the third set of surfaces, acquires here the remarkable property of being

surfaces of revolution not only with respect to one, but also, and in a

perfectly symnetrical manner, with respect to the other one of these two

polar great circles. As regards connectivity, a cylinder is here of order 1,

like a torus. The two-dimensional geometry on it is Euclidean, as on an ordinary cylinder. It corresponds to an ordinary plane Euclidean rectangle of which opposite edges are put together (so that all four corners

meet in one point).2 The coordinates corresponding to the two sets of spheres are of the

character of azimuthal angles; they measure the angle between the spheres

(or rather half-spheres) of each set at the axis and run from zero to 27r.

We eall them 4 and & respectively. The third angle will be called 0). It

measures the radius of the cylinder, or rather one of its two radii, the

other one being . Thus w runs from zero to - only. 2

In these coordinates the spatial line element do2 is

dar = R2(d&02 + sin2w dc2 + cos2% df2), (2, 7)

where R is the radius of space, which will be considered a given funetion

of the time t. Taking for simplicity c 1, the four-dimensional line element is

ds2 = -R2(&(o2 + sin 2to d42 + cos2 d#2) + dtM. (2, 8)

So this is the tine element, for which we wish to solve the equ. (2, 5).

Referring to w, , ,& t (in this order) by the subscripts 1, 2, 3, 4, we make

a note of

a/-g-RI sin w cos w

1l- 1 22 =_ 1 1 } (2, 9) ii" Xi~~ ~ BR Sifl'w g1 1' Cos2to. "=l

In writing out (2, 5) explicitly, we wish to retain F14, F24, PF,, and there

fore use (2, 6) to express the other three by them

23 = i? sin w cos wF14, F3j = i cot w,E24, F1, -= iR tan wFn. (2, 1 0)

'The hyperspherical cylinder is also known under the name of Clifford's surface.

That its geometry is Euclidean can be confirmed from (2, 7), puttinig w = const.

114*1




Then (2, b) reads

(iR sin to cos wF,) - =0. at a

ai'.4 _ ar14 _ a 'iR cot w F24)

= 0. &w a~at

+ 2jiBta- (iitaFi

) + aF24 = O. (2, 11)

(RiB tan w FU4) + (iR sin w cos wFu4) (iR cot to F,4) = 0.

The unspecified function R(t) (depending on the law of expansion) can

be cancelled in the last of these four equations and can be made to

disappear from the others by admitting

a(BFkd= _VFk4 ; (k = 1, 2, 3); (2,12) at

with v a constant. (2, 12) is readily solved in the new variable

(2 13)

and gives

F4= e x function of t, 4, . (2,14)

We can obviously put

F (T e + tO fk4 ( ), (2,15)

B

with n and m constants.

Then (2, 11) reduces to - v sin w cos tJ fig + ilnfU

- irtfj 0

df 0

dat4 -nnf1e 4+ Vcot t f, =4 0

- infu4 - tan w f34 + a2? = 0 (2, 16) aw

dw imn tan tJot4 + p (sin at cos tof,) + in cot tofu4 ?-0

The possibility of static fields (v 0) is disposed of in Appendix I,

so we assume v + 0. Then the last equation follows from the other three

and ca be dropped, and the first one can be used to expel f,4 from the

second aid third, which then read:

v sinw cosw dfw4

+ mnf,4 - mf4 + V'coS%tf24 - 0

v sin cos cdw -mnf24 + n2 A4 - V Sin' Af34 = ? (2, 17)



SCHnaDmoNGrER-Maxwell's and Dirac's Equations. 29

So we have reached two equations for the two variables f24 and f34. Put

f f34 + f24 g - f.U -/f4 and x = cos' w. (2, 18) You get

df X (nA-_n' 2 V2 -2v z( )d

+ < v+ 2)C)2 f

- J(v^ i n m)(v - ff - m)y =

,, _2 + vM 2 V)f

+ (V + in M)(v -_n + n) f g=0

(2, 19) If here you put

m n zm n

f=x2(1-x)2 P, g=x2 (1 -a) Q) (2, 20)

you get by a rather laborious calculation the folowing second order

equations for P and Q respectively:

d2P dP x(1 - x)- * [in + 1 - (my + n + 2)x] dx

v + nt m v - n - In - 2 + 2 2 P =0 (2, 2 1)

x(1 - x) + [m + 1- (m + n + 2,x] dQ &2

)i'flfl

dx ?n? =

+In- v n m ), t +w +? 2 O

2 2

that is, the hypergeometric equation, in both cases. As usually it turns out that only polynomials are admissible, other solutions are too powerfully

siingular at x = 1, in spite of the additional factors in (2, 20). (Already

non-vanishing f and g are liable to produce infinite energy density on a

polar circle; see App. II, equ. (II, 3).) Thus, generally speaking, P and

Q are Jacobi polynomials, the ratio between them at x 0 to be deduced

from (2, 19). But the possibility of one of the two vanishing identically

(without the coefficients of the corresponding Gauss-equation satisfying the

polynomial condition!) has to receive special attention-otherwise one

would lose just the most interesting types of solutions!

m and n have to be integers. For, as W. Pauli has pointed out,3 branch points of the wave-function are excluded, if it is a tensor of integral order-in contradistinction to a spinor wave-function.

The unfortunately lengthy discussion, relegated to App. II, reveals that

the frequency v must also be an integer4 of absolute value > 2 and greater

* W. Pauli, Helvetica Physica Acta, 12, 147, 1938 ; p. 151, first paragraph.

4 The frequency in the usual sense is ?

27TXI




than i n i + I m 1, moreover -n + m (mod 2). For v > 2 the following

table gives the admitted values of i n i and I m and (third line) the multi

plicity, which we shall discuss immediately:

K0 1 2

2 1 0 In i 1 2 1v- 10 1 2 02 1 even

ml v-1 v-2 ... 1 v-2 v-3 v-44 0 / 4 2

multipl. 2 2 2 2 4 4 29 \ 0 1 ? o 1 odd.

2 2

The use of the table is that you choose an arbitrary integer, > 2, for v, and then find between the first two vertical double bars all those pairs

I n 1, I m I whose sum is equal to the chosen v, in the next set of columns those whose sum is less by. two, and so on. The end of the table on the

right looks, of course, different according to whether your v is even or odd.

That has been indicated by a bifurcation.

A duplicate of this table, for v < - 2, is obtained by copying it, but

writing v i instead of v.

The multiplicity arises from the fact that mn and 'a can be positive or

negative. Generally all combinations of signs are admitted, except that

when I v I = I m I + I n 1, the product mn has to have the same sign as r.

This case is still in another respect exceptional: as can be seen from the

table, zero is not admitted for either m or n.

The total multiplicity of the frequency v (considered apart from - v)

works out to be v2 -1. (2, 22)

That is interesting. The general form, postulated by group theory, is

Ni - N"2 (2, 23)

(see A.P., p. 377, last item in (4, 11)). N and N' have to be either both

integers, or both half odd integers. Now for a scalar function we got, in

A.P., N- = 0 for the Dirac electron N' = -i-- j. We here find, by comparing

(2, 22) and (2, 23), N' = -+ 1 for the photon. The intimate connection

with spin is fairly obvious. c

The ground frequency is 2, or, in ordinary counting ! h Its period

(with R = const.) is the time light takes to reach the opposite point of

the hypersphere. It is perhaps worth while mentioning that such low frequency vibrations as these, though they go on all right in spite of the

expansion, are yet quite likely never to perform even one complete cycle. For according to (2, 13), if R increa.ses more quickly than the first power

of t, the integral defining r converges. Infinity is then expressed by a

finite number in the time reckoning r that was forced upon us.



SCHRnODINGER-Maxwell's and Diracs Equations. al

Moreover it is noteworthy that the ground vibration is not simple, but

threefold degenerate, or sixfold, if you include v - 2. A further

discussion is found in App. III. The App. IV gives further information

about the exceptional case I v I = I n I + I m 1. It is the case where either

P or Q is zero, the other function being a constant so that no nodes in c

exist. For such large values of m and n, as correspond to waves of

"human"-as opposed to cosmical-dimensions, one obtains the interesting "skin "-waves.

? 3. DiRAc's EQUATION IN THE EXPANDING UNIVERSE.

For R = const. the investigation has been carried out completely in

A.P. The new point of interest is therefore only to cope with the time variation of R. In order not to commit errors, the whole process has to

be repeated from the outset. But I will be excused for giving only a

very succinct report. We use ordinary polar coordinates with the line element

ds2 ' R2

[clX' + sin'2 X (dO' + sin2 0 dp'2)] + dt2. (3, 1)

Not that they are handier in the Dirac-case, but they were used in A.P.

f is now an unspecified function of t, as in ? 2. With this line element

18 of the 64 Christoffel-symbols differ from zero. If X, 0k, t are labelled

1, 2, 3, 4, they are the following5

[X12 -= f'212 r1 = =l,3 =

cot x

EQ 1 4 41 = F124 = 142 = [343 = 143 =

r.221 = -Sill x COS x

r224 BR Sin' X (3, 2)

r23F = F32, = cot 0

rF3= - sini X cos x sin- O

133=2 _

Sin-lo Cos 0

F3,' =-R sin x sin2 0

5 The table in $ 100 of R. 0. Tolman 's book

'c Relativity, Thermodynamics and

Cosmology7' (Oxford, Clarendon Press, 1934) deserves gratitude.



32 Proceedings of --the Royal Irish Academy.

The metrical matrices 7yk, yk are

7y = l y _ _

yz= -ilisinX. a, 2 a

B sin X (3, 3) 7a

ilisinxysin O. a~ 9y = a

B sin X sin O

74 au = a4

with the ak an ordinary basis of Dirac matrices (constant, square, four rowed, anticommuting, square roots of unity).-The matrices controlling the parallel displacement of a spinor are obtained from the general equations, labelled (5, 4) in A.P., and are

r2 sin X.0 a2 a4 * 2COS X. a, a2

r8= ;2iR sin X Sin O.a3 a4 + 2 cos 0. a2 a342cos xsin 0. a3aal (3, 4)

From these data we set up the general Dirac equation, labelled (5, 1) in

A.P. Introducing, much as in (5, 14) A.P., instead of +t(X, 0, p, t) (the

Dirae function properly speaking), the function Q(X, 0, 4, t)

3

a(x, 0, 4, t) = B2 sin x jsin 4xJ 0X, ob, t), (3, 5) we obtain

.aa a4a aQ as C au aa a u\ at _B 5 7 + sin Xaf

+ R Y -pas&, (:3 6)

(y= 27rmc/h),

which is identical with equ. (8, 1) in A.P., except for R being now an

unspecified function of time. (A capital a is used here instead of the X in A.P., in order to avoid

the letter used in ? 2 for the radial cylindrical coordinate.-In A.P. the

variable x was temporarily abandoned for r = R sin X, which was a silly

manceuvre there and had to be omitted here.)

From equ. (3, 6) above follows the equation of continuity, labelled

(6, 4) in A.P., without change. But it is now more significant; on account

of the factor R1 in (3, 5) above, it states that the normalization of the

Dirac function yb is conserved, even with the expansion. That conforms

with the results obtained by scalar treatment in my " Physica "'paper. It

is worth mentioning that this conservation law has-at any rate in the



SCHR6DINGER-Maxwell's and Diracs Equations. 33

case of Dirac's spinor-a much more general standing. The equation of continuity holds for an arbitrary metric, and normalization will therefore be conserved in an arbitrary spatially closed Universe.6

The fact that R depends on t modifies only the last phase of the process

of solution of equ. (3, 6) above, as described in A.P. The assumption

"ce-2i"' must, of course, be discarded, replacing in equ. (8, 2) A.P.

by the general a

for which it stands, and trying to solve that equ. by

the following modification of (8, 12) A.P. = = [p (t) + f3, q (t)] i (X). (3, 7)

That means simply that what there (after splitting off the time-exponential) were two constants, to be determined, are now two ordinary functions of

time p(t), q(t), to be determined. t is the sante function of x which in

A.P. was called o. It depends on a double-valued subscript besides, and

p1 is the matrix (I ?) referring to this subscript. Now equ. (8, 2) A.P.

requires our two time functions to fulfil the relation

(P - 31q) 9b" - fLR ((%p + q) ilR (j) + 0q). (3, 8)

To satisfy it we have to put the factor of "1" and that of PB1 separately

zero, thus i1113 = n'p - sRPq

iR4 = - n"q -Ip. (3, 9)

There will be two independent pairs of solutions p(t), q(t). Either one or the other, or if you like a linear combination of them, has to be inserted

in (3, 7) above, which then gives the dependence on time, on the radial variable x and on one of the two subscripts into which the spin-index was

split. I feel I ought to copy out in full, for possible use, a complete

solution, and I will do it for the Dirac-function properly speaking. It

reads thus:

'[p(t) + q(t)] (cfj f, (Cos (Cos X) I [P(t) + q(t)]

eimf I R~~~~jn, (COS n)f,,,,j (COS X)

-Mi =R sin X Jsi A | t (C?Q 0) g7&tj (Cos X) (310 [P(O) - 2(t)]

tij/3 (Cos 0) g6ny (Cos X)

The meaning is that e.g. the third component is this:

ebn'4 B- sinI y sin7 9 [p (t) - q (t)] fpm (eos 0) gy"j (Cos X)@

The functions f and g are given explicitly in (7, 26) A.P., the admissible

combinations of quantum numbers n", j, mn by (8, 19) or (8, 20) A.P.

8 Sitz. Ber. preuss. Akad. d. Wiss., 1932, p. 126, equ. (70),




As regards the time functions, it is readily seen from (3, 9) that if the

mass term were zero (A := 0) we would, just as in the Maxwell ease, obtain

exponentials in the variable introduced by (2, 13). It is the mass term

that complicates matters, not the fact that we are dealing with a spinor

instead of with a vector. The mass term seems to frustrate the desire for

a general solution. (What astonishes me more is that it is frustrated in

the case of a scalar with the second order d'Alembert equation. See my

"Physica" paper.) Since a general solution is impossible, the particular case of R varying

linearly with time, X = a + bt, is dealt with in App. VI. The solutions

are Bessel functions as in the scalar case. No point of particular interest

arises.

? 4. THE "ALARMI:NG" PHENOMENA.

Owing to the low time-rate of expansion, the time factors of the wave

functions have always very approximately the form of a combination of

ei&t and e-ivt. They assumne exactly this form whenerver f becomes

temporarily a constant. It has therefore quite a clear meaning to ask

oneself whether e.g. the form e* is permanent or may, in the course of

expansion, partly transform into e-i". For in case of doubt one can, in

any given moment, let f smoothly take on, a constant value and then

inspect the functions one has in hand. Physically the lack of permanence

would mean for light: reflexion in empty space; for particles: p-airi

production. These are what I called in my "Physica " paper "the alarming

phenomena."' It is very satisfactory that the treatment in ? 2 proves the rigorous

permanence in the case of light, for the slightest trace of the contrary

would have been all too paradoxical. But indeed the solutions can here

be obtained in the general form

ei- aid i witlh r J Xd

and clearly becomne et 't and erivt respectively (witlh ' whenever I(t)

becomes constant, - 1?.

In the case of particles no such simple proof of permanence is possible,

because equations (3, 9) cannot be solved in a general way. I have tried

in vain to find a proof on different lines. Unless an interesting general

theorem about equ. (3, 9) has escaped me (which is, of couirse, quite possible)

the permanence is not rigorous in the case of particles.

APPENmDX I.

Static fields, without charge and current, impossible.

It is perhaps not quite obvious that they are. Indeed the homo

geneous field of the fundamental mode (see App. III) is "nearly" static,



SCRH6DINGER-Maxwell's and Dirac-s Equations. 35

and might, on first guess, be suspected of being rigorously permanent. At any rate we are bound to take up equations (2, 16) with v =- 0, a

possibility, discarded in the text. The fourth equation is now no longer a consequence of the three others.

The four equations read -Q 4 df2

nf34 - T

=f2 4- _iifl4, inf4. (T,1)

d (sin wcos wfi4) + in cot w f24 + in tan co 34 = 0.

dw

First assume that not both n and in vanish, thus mn2 + nt $ 0, and put

if:,4 + nfj24 = U, (nt2 + n2) sinw cos o fi = V, log tan w = z. (, 2)

You get du du d2u + n2 int - }z

-=v, - = - (+ tanh z u. (I, 3)

4zZ2\ 2 + 2

z raniges from - oo to+ oo. Since - 1 < tanh z < + 1, the bracket is

positive. Thus any real solution (apart fronm u 0 , which is of no use)

is everywhere convex towards the z-axis and therefore cannot avoid becoming infinite at least at one of the limits. The coefficients being real,

the same holds for the real and imaginary part of a complex solution.

Hence no admissible solution in this case. If n m= m = 0, you get from

(I, 1) that 134, f2, and sin w cos 14 are constants. None of them can be # 0 without causing infinite field components "in natural measure"-see

equ. (II, 3) of the following appendix.

APPEMNDIX I.

The structural pattern of the electromagnetic proper modes of the

hypersphere.

By setting up, with the help of (2, 9), the inivariant

Oiko,; += giigkn4 mJk)

and comparing it with its familiar form f2 - E2, it can be seen that the

field components " in natural measuire" are

014 024 034

R ' Bsinw' Bcosw

Pf,23 _ _31 412

Rl sin w cos w' RK COS B' RI sin (T 1)

We shall indicate them, in this order, by the symbols

E,,, E Ed, HX, H4, H4. (I, 2)




In order to express them by the fk4 (k = 1, 2, 3), we have to compouand

the substitutions (2, 4), (2, 3), (2, 15), and then, in order to introduce f and g, use the first line of (2, 18) and the first line of (2, 16). The

following scheme is obtained:

ei(PT + no +} Jrm+) i;(vr + n4 + m+n))

B2 Cos w 2_ cos 3o (f g) AE ,-i14,

ei(zs + n4 + m+) ei(vT + n + m4)

f2 sin a,124 = 2S (g f ) = E - i14, (II, 3)

ei( t + NO ? ) iei(vr + 4- M4)

K fi1 = q 2 in -cos [n (f+ g) - m (f - 0)1 = la - tt,

which contains no concealed imaginaries. Further we recall from (2, 20)

f = cos%U sin"& . P (cos2w), g cos'nw sin"w . Q (cos2o). (II, 4)

Our pairs of solutions, (f, g) or (P, Q), have to be tried whether, following

(II, 3) and (II, 4), they produce any infinite field component E or H, the

7r daniger

zones being the polar great circles e = 0

anid w - 2

We first observe from (2, 19) that if (f, g) is a solution with v, mr, n,

then it is also one with v, - m, - n, whereas (g, f) is a solution with

-v, - m, n. Thus in general four solutions are derived from one, which

are either all four admissible or all four inadmissible, as a short reflection on (II, 3) proves. Hence it is sufficient to look out for all admissible

solutions with m and n not negative, but v both signs, and to deduce the

others by the aforementioned procedure. So we shall take m > 0 and

n > 0, until further notice." The factors cosmw sinla, in (II, 4) are now " favourable " anyhow, and

avoid the only remaining danger, that of the denominators in (II, 3), unless either n or m or both are zero. Hence it is these cases that require

attention. From (2, 21) we read the, Jacobi polynomial conditions,

for P:

either j(v-m-ni-2)=n1 or --k(im?n)=nt; (11, 5)

for Q:

either j(v--n ) =n'1 or - +m + n + 2) = n'; (11, 6)

(=n+1l)

where 'nl, n2, Ut1, n'2

T These considerations ought really to precede the statement in the main text

' ' that

2* and Q have to be polynomials. ' '

To look up the solutions for negative m and n

directly, without modifying the substitution (2, 20), would be inconvenient.



SCHRnDINGEw-Maxwell's and Dirac's Equations. 37

are non-negative integers, viz., the orders of the polynomials. At least one of these four conditions must be satisfied. We conclude

v Mn + (mod 2) and v j > n + i. (It, 7)

We deal with the case v = m + n first. Then n' (being zero) is a

non-negative integer, but none of the others (n1, n2, n'2) can be. Hence Q must be a polynomial of zero order, thus a constant, while P can only be

zero, because its polynomial condition is not fulfilled. Indeed, with these assumptions for P and Q in (2, 20), (2, 19) is satisfied. Inspecting the field components it is seen that the aid of the sin-cos-factors in (II, 4) to overcome the denominators in (II, 3) can in this case not be spared. So zero has to be excluded for mn and n.

The same (P, Q) gives a vibration with v, - - n, whereas on exchange

of P and Q (thus with P = const., Q 0 O) you get one with - v, - m, n

and another one with - v, i, - n. These four solutions exhaust the case

I v I = m + n. For if we now turn to investigate the case v m - n, we

would only find them over again.8 The case we have now settled absolute value of the frequeney equal to the sum of the absolute values of the azimuthal quantum numbers-is exceptional, in that zero is excluded for the latter, and their product has to have the same sign as the

frequency. It is also of exceptional interest (see Appendix IV). There remains the case I v I > m + n. Then either of the conditions

(II, 5) entails one of (II, 6), and vice versa. Hence both P and Q are

Jacobi polynomials. They are of consecutive order, Q following if v > 0,

P if v < 0. With this in mind, we insert (2, 20) in (2, 19) and put x = 0.

We get unequivocally

P(o) , n (II, 8) Q (o) - +ifl-f (it8

(to apply only for non-negative m and n; see footnote, p. 36). This incidentally indicates the factor to apply to the polynomial obtained for P, in order to make the solution complete. With this factor applied you certainly get a solution for every set of values v, mi, n, chosen in accordance

with (II, 7), with the agreement to investigate directly only the non negative m and n, an,d with the condition of the present case, viz., I v I > m + n. And you equally get a solution for - v, m, n. From these

two six others follow, covering between them the eight possible com binations of sign, all eight solutions different already on account of the exponential occurring in (II, 3) (except for the obvious reductions when m or n vanishes). So what remains to investigate is the finiteness of the field components in the present case. This is secured by the sin-cos-factor in (II, 4), unless m or n vanishes. Now just for these critical cases all

combinations of sign follow from the one with positive v in the way

8 To explain why : because e.g. with P

= const., Q

= 0, and m, n positive equ. (2, 19)

are not satisfied.




mentioned at the outset, which does not interfere with finiteness of the components. So we can restrict this investigation to v positive, m, n not negative. Then in both (II, 5) and (II, 6) the first altemative prevails, and we get, paying attention to (II, 8),

= 1 - MT - 9U

p= v + n GnC1(t + gn + 1, in + 1, cos2w),

Q G,1+ (n1 +n 1, m + 1, coswd), tII, 9) with

nl i 2 ( m - A - n -

2).

(By G we mean the Jacobi polynomial, defined as usual, G(O) = 1.)

We need the ratio of P/Q also for x = 1. From a well-known formula

,

(?

? + m? + 1,

n+ 1 1)( 1)n

( + 1) (nt + 2) (n 3). (n + n1)

'(m +1)Qnm-i-2) (m +3) ... .m+n)

and similarly for the other, Ol + 1

From these data we get

P (0) v- - n

Q (O) v + m - n

P (I) -w-n

Q(1) - TZin? (II, 10)

These ratios are never zero or infinite. If m = 0, the first one takes the

value -1, whereas n = 0 gives to the second one the value + 1. This

circumstance compensates in the critical cases for the absence of the sin

or cos-factor, or of both in (II, 4), and makes all field components finite, as will be recognized by careful consideration of (II, 3). For it means

that P ? Q, respectively, vanishes at the limit in, question, and since they

are polynomials in cos%i, that means that they must contain a factor

Cos2 or sin 2( respectively. The comprehensive result is that in the case

when the absolute value of the frequency is greater than the sum of the

absolute values of the azimuthal quantum numbers zero is not excluded for the latter and all combinations of sign are allowed.

That exhausts the discussion. The results are summarized in the text. It may be that strictly speaking the continuity of the components on the circles where the coordinate system is singular ought to be tested. But I shall omit that.

APPENDix III.

The electromagnetic fundamental mode.

From the table on p. 30 it is seen that v 1 = 1 is not admitted, no system of in, n being compatible with it, because the only odd value for

I m I + I n 1, not exceeding 1, is 1 itself, and in this case m and n must

not be zero. Thus the lowest frequency is I v I - 2 (in our reckoning;



SCHRmDINGERn-Maxwell's and Dirac's Equations. 39

C in ordinary units). v = 2 is obtained with m = n = O, with m = n = 1,

and with m = n -1. We shall look into the first case. The other two

represent essentially the same vibration only in different orientations, a little less convenient to analyse. The frequency v - 2 is obtained with n = 0, = - withm tn-ta ndwith in - - 1, n l. This type of

vibration is the electromagnetic mirror image of the first with respect to space.

With v = 2 and m = n = 0 we have from (TI, 4), and (II, 9) (where we choose to reverse the sign)

f P= 1; gQ -( - 2 cos'co), (IIT l)

(G1(1, 1, x) being 1 - 2x). Hence

f24- 2 2= COS (; f3 = = Sin2w. (111, 2)

Hence from (II, 3)

-E, 0 ~~~~11(00, sincoi' O14=^- B,

sn _sin COS t sin l sin lt

COS (t) COS vt COS wt)

sin it

BR2 = B2

We state at first glance: the absolute values of the field-vectors are

constant in space; H is an exact replica of E, but that it advances in

phase by a quarter of a period; the energy density E2 + H2 is constant in space and time, if R is constant (otherwise, following a general rule, it is inversely proportional to the fourth power of R., see my "Physica"

paper). Since the w-components are zero, the field lines are on the cylinders

= const. We can produce at all points of space line elements, all of

equal size, every one in the direction of the vector E in that point, following (2, 7), if we make

d= 0, do = - = const., (III, 3)

throughout space. This is a rigid movement of entire space, a rotation through the same angle in both azimuths. It can be continued until every point returns to its original position after completing a closed trajectory of length 2rR. The trajectories give a complete picture of the field at large. It is nearly evident that the trajectories are great circles, but doubts can be removed by forming the differential equations of the geodesics of space from (2, 7). You easily obtain

da (Slldt =0 d dew; ' 2 LK 2_)

K H

3II1, 4) which can be satisfied by (III, 3).




Both polar eircles which form the scaffolding of our cylindrical coordinates are among the trajectories, thus are lines of force. Obviously they determine the entire system uniquely, for the only thing that is left open-from where you begin to count the angles i and &-can make no

difference. What is remarkable, though, is that the reverse is not true:

our two polar circles are not at all distinguished among the trajectories, they have no prerogative whatever, you obtain the same system of

trajectories in the same way by starting from an arbitrary one and its

polar. That is proved by observing that if you pick out an arbitrary

trajectory of the system, its polar circle is sure to be also one of them,

because it is the locus of those points which have the same distance from

all the points of the first trajectory-and this locus, in a rigid movement,

must move into itself. This polar pair determines the rest of the system

uniquely. So this type of field is determined uniquely, if you pick out one point

of the hypersphere and give the electric vector in that point. There is

thus a three-dimensional manifold of orientations and that is just covered by the three proper vibrations we got for v - 2 (and by their linear

combinations). Returning to that orientation which we have chosen for investigating

it (v = 22 m = n = 0), we observe that with v -2, m ;- n = 0 the

functions f and g have to be exchanged, which changes the sign of the

+-components (see (II, 3)), whilst the change of v changes, in addition, the

sign of the H-components. The first has the effect of changing the

trajectories into those of the rigid movement

dw = 0, doq = -d- = coinst., (III, 5)

which is an exact counterpart to (III, 3), but differs from it not merely

in orientation-it is its mirror image. The second change of sign, that

of the H-components, merely has the effect that the magnetic field now

retards by a quarter of a period instead of advancing. I confess that this

circumstance puzzled me for a while, for there is actually no possibility of getting a solution with H advancing in the second case, nor one with

H retarding in the first case. But that does not mean that the right screw

is, in spherical space, physically distinguished from the left screw, it only

means that we are at liberty to give the magnetic pole in North-America

the, negative or the positive sign, as we please.

Every cylinder e0 const. is entirely covered by either kind of the trajectories (III, 3) and (III, 5), respectively, so that there' is one of each

kind through every point, with the angle 2,u between them. The one

system winds in a right screw, the other in a left screw round the cylinder,

which, for the purpose of visualising the state of affairs, you best replace

by a torus in flat space. (It so happens that a torus in flat space also

carries two families of circles, all of them congruent, isogonal trajectories

of each other; they are less popular than the parallels and meridians).



SCHorDINGER-Maxwell's and Diracs Equations. 41

Properly speaking we are faced with just the two families of "straight"

generators, to which every quadric is entitled, only that with a cylinder in flat space they happen to coincide. Since the generators are real, our

cylinders are ruled surfaces, much like a hyperboloid of one sheet in flat space.

Incidentally these solutions answer the question that someone might put: Is a homogeneous (say electric) field possible in spherical space? The answer is: Yes. But it cannot be a strictly parallel field, because a

bundle of strictly parallel great circles does not exist, not even a narrow one. It has to be given a slight twist either to the right or to the left,

such that neighbouring lines of force, viewed at large, are interlocked (like successive links of a chain). In other words, if you would extend a small ribbon-like surface between them, this has not the plain form of a

driving strap but is twisted round like a Moebius band, only not once but twice, so that every edge (line of force) runs back into itself, not into the

other one. Now this twist means a curl of the electric field in its own

direction and, in the course of time, arouses a magnetic field in the same direction, having the same kind of twist and curl and thereby gradually consuming the electric field. When the latter has disappeared, the

magnetic field is at its maximum. It begins to be weakened by the electric

field in the opposite direction, which now begins to develop; when the latter has reached a maximum the magnetic field has become zero and so on.

All that is precisely as with an ordinary standing wave; the peculiar thing about it is only that the two fields, instead of being orthogonal, have the same direction. The period is of course exceedingly large.

Whilst left screw vibrations by themselves and right screw vibrations by them'selves produce nothing new on composition, but only the same kind of thing in a different orientation, you get new types by composing them with each other. There is a one-to-one correspondence between these

more complicated types of the ground vibration and the general rigd movement (rotation) of the hypersphere into itself-just as between the special types we have studied and the special rotations (III, 3) and (III, 5).

APPENDIX IV.

The cylindrical skin-waves (electromnagnetic).

In spherical space there are types of proper vibration which depart

from the usual by being different from zero practically only in a. small

region. One type, which can be called "skin-like," is virtually restricted

to the immediate neighbourhood of any one of our cylindrical surfaces.

Another type, which occurs as a limiting case, when the cylinder

degenerates into a great circle, is restricted to the neighbourhood of that circle, and can be named "thread-like" or "tube-like" (but not a hollow

tube; the greatest intensity is along the middle line). A detailed account PROC. R.I.A., VOL. XLVI, SECT. A. [5]




of_ the latter type was given in A.P. for sealar wave functions. The skin

waves were just mentioned there; my collaborators9 studied this general case in more detail. In the present solution they are obtained with I v I = I m I + I n I and very large values of the azimuthal quantum numbers

m and n. If both are very large, you get a skin-wave. If only -one is

large, the other one unity, you get a thread-wave. It is quite interesting

to look into them, on account of their peculiar state of polarisation. From App. II we have in this case, if we take v, m, n positive, P = 0

and Q = a constant, which we take to be 2. The field-components are

readily obtained from (II, 4) and (II, 3). In writing them out we will

use the following abbreviations for the phase and for a common amplitude factor

(P= n + mn4 + vr

A - B-2 cosm-lw sinn1W. (IV, 1)

Then we have

E.= A sin I1 El=,-A cos ?

1'= - A cos w cos 1 = A cos w sin 4)

E, = A sinwcos4 ) Hp- A sinwt,sin 4). (IV, 2)

It is seen that E2 = H2 = A2, so A is the amplitude, constant with

respect to time (apart from the possible variation of R). It is also constant on every cylinder w = const., but has a very sharp maximum with

respect to X at --= w0o where

tan woj' - (IV, 3)

That is what restricts the whole wave phenomenon to the immediate neighbourhood of this cylinder; it begins to fade away very rapidly,

virtually as an exponential, at a distance of the order of VAR fromn the surface of the cylinder, on either side (A is the wave-length). We may, therefore, practically take w = w. in the equations (IV, 2).

The phase qp remains constant in the direction of increasing W, i.e., in the normal of the cylinder surface. To find the direction of fastest increase in the cylinder surface (which will be that of the ray) we have to make

niA# + mA4, a maximum for a constant small increment Ac

A-c-2 = B2 (sintw AO" + costw -A42).

You readily find

- cottw. A tm

9 P. O. M?ller, Phys. Zeitschr. 40, 366, 1939 (owing to a misprint on the cover of

my copy I gave a wrong quotation in my "Physical paper). W. Hepner, Edinburgh theses, to appear presently.



ScmiH@DNmam-Maxwell's and lirac's tquations. 43

Since , is practically w0 and m and zn are supposed to be very large, this

means AO = At. Thus the rays are in this case the one set of generators,

which we know from (III, 3). That was to be anticipated. What else

should I he rays be but great circles!

If you keep in mind that

RAw, B sin w AO, R cos w)A4

are local Cartesian coordinates, you find (IV, 2) to represent locally just

an ordinary circularly polarized plane wave. A local observer on the

cylinder X = &, would call it so. And if he were able to explore regions

at a distanee of five or ten times V\AR he would be astonished to find

that his system of plane waves extended, for all that he knew, to infinity

in the directioni of propagation and in one of the directions in the wave

plane, but faded away rapidly in the second direction in the wave plane,

orthogonal to these. The "lines of constant phase"' on = - 0 are not,

in general, the other system of generators (for the latter are not, in

general, orthogonal). Starting from a fixed ray and walking along a line

of constant phase, you will cross the same ray again, but the first crossing

will always be at a different point. Olnly after several crossings the same

point is reached. That is a rather remarkable connectivity.

The inquiry into the sense of circular polarisation has to be handled

with care. With given absolute values of v, m, n there are only four solutions of this type, just one for waves in either direction along either system of generators. So the sense of the polarisation is determined; the

question is, with what it is correlated and how. For this correlation it

must be unessential whether the directions of increasing , qb 4 form a right-hand or a left-hand frame (one of which would make our formulae

comply with the customary convention about the North American magnetic pole-but that is uninteresting, because a change of this convention does not change the kind of polarisation of a given light wave).

I beg the reader to make a sketch of the local (4, 4)) frame. The wave

(IV, 2) proceeds in the direction of decreasing 4 and A. Indicate it by

an arrow, pointing to the third quadrant. Now take the moment when P is zero. Then

Et, = O E, <O, E > 0.

Thus the electric vector is in this moment in the plane of the paper, and points to the quadrant 4) < 0, + > 0. In the following moment, since B

becomes negative, the vector E will rotate towards that normirul of the paper along which w decreases. That is the direction towards the contre of the 4-axis and away from the centre of the ,)-axis (in the large they are circles; from (2, 7) the centres of the 4-circles are at = 0, those of

the 4-ecircles at -

You may choose this direction to be the inner or the outer normal of



44 Proceedings of the Boyal Irish Academy.

the -paper. In one case you get left-hand polarisation, in the other case

right-hand. I cannot tell you in which ease one or the other without seeing

your sketch. But I can tell that when you get left-hand polarisation you

find the ray winds in a right screw around the cylinder, and vice versa.

To ascertain the sense of winding, try to imagine the 4- and + -axis of

your sketch slightly vaulted in the correct way, inward and outward

respectively, so that the sketch acquires the appearance of a saddle, and

then imagine it applied to the neck of a one-sheet-hyperboloid. That will show you the sense of winding, and convince you that it actually changes

with the direction of the normal you choose for decreasing o and,

therefore, with the polarisation of the wave.

The correlation can also be expressed thus: the light vector rotates in

the same sense in which the ray winds around the cylinder.

The other cases (v, -)m, -fn; -v, m, - n; -v, m, n) give nothing

new. They furnish, as I said, the waves in the opposite direction and

those in either direction along the other system of generators. The waves

in opposite direction along the same system have, of course, the same

polarisation. So the state of affairs is like that in an active crystal, not

like that in the Faraday-effect.

With in ?) 1, n = 1 or n >> 1, 2n = I we get fronm (IV, 3) 0 = 0 or

-r -o - respectively. These are the thread-formed waves. The carrying

2 cylinder with all its generators degenerates into one great circle. The

distinction of circular polarisation disappears, because the two opposite states can be superposed to give an arbitrary state of polarisation. I

consider these waves the adequate generalisation of the plane wave in flat

space. It can be shown that the ensemble of thread waves with arbitrary

axis, wave-length and polarisation is sufficient to build up an arbitrary

state of radiation in the hypersphere. They do not form a complete

orthogonal set, though. They are not, in general, strictly orthogonal, but

very nearly so, if their directions deviate appreciably. And they are, on

the other hand, superabundant: a finite number of them for every

frequency [viz., 2(v2 - 1)] is already sufficient.

APPrrDix. V.

More general type of solu tion (electromagnetic).

From the eigensolutions with I v I = I m I + I n 1, which we investigated

in the preceding Appendix, a more general type of solutions can be derived, which contain arbitrary functions and are no longer eigensolutions. To fix the ideas take the case with v, m, n all positive. We then have P -, Q = 2. From (II, 3) and (II, 4) it will be seen that the six field

components are obtained from the real and imaginary part of

cos", sin" w ei (VT n 4 J m ) (V L



SCHnDINGrat-Maxwell's and Dirac's Equations. 45

with individual factors which depend on the field component in question and also on c, but not on rn and n. Now (V, 1) can be written

a1n 4(V, 2) with

C1 = Cilt ( eil (T + )

By superposing all the solutions of this type (itn 1, 2, 3, 4 . . ., and

n = 1, 2, 37 4 . independently) with arbitrary constants you evidently

get a solution whose field components are formed-with the same individual factors mentioned above-from

1WFi,} t)

where F is an arbitrary analytic function of the two complex variables

i? and ;. Since the latter are restricted to the interior of their unitary

circles respectively, the field need have no singularities. It is amazing that solutions of such a general type can be set up

without specifying the time law of expansion, which enters only in the

definition (2, 13) of the variable T. Apart from this, the state of affairs

is analogous to the well-known use of analytical functions of a complex

variable for setting up solutions in flat space. For simplicity sake let us

envisage d'Alembert's equation in flat space, using, like here, cylindrical

coordinates:

Dj lae I a2 _ af O -

+ 1 a 1 f, + a f

alA2 al. 'p 2 bz at24

A solution is furnished e.g. by the real or by the imaginary part of an

analytical function of the complex variable C,

f - rei(t t+z) (vY 5)

For the first three terms in (V, 4) will cancel, because they constitute

Laplace 's operator, expressed in plane polar coordinates r, 0; the last

two terms cancel, because t and z occur only in the combination t + z.

APPmrix VI.

Dirac's equation with R a linear function of timie.

If in equations (3, 9) we make the special assumption

R = a + bt (VI, 1)

and introduce a new independent variable z,

P ? ?(= b ) (VI, 2)

instead of t and instead of p, q a new pair of dependent variables

t(z) = z- p, v(z) _= z- q (VI, 3)




we obtain the equations in" I

di b 2 dz z -

+ v (VJ ,4)

in" 1 dv b 2 _ _ v - U.

d-z z

They coincide with dC

_A-CA + CA-, dz z

dx-l1 X1 - -d---z= - CAl- CA. (VI) 5)

These are well-known relations, connecting two Bessel-functions CA and CA1 which differ in parameter by 1 (the real unity). More precisely, they hold for every one of the familiar functions-as J-function, first kind

Hankel-function, second kind Hankel-function, Neumann-function-when it is defined in the usual way as a function of two variables, argument and

parameter. Taking for C any one of these functions,

' = Cin,t 1 (z)? v =

Ci,,, 1 (z) (VI, 6) b + i b2

is a solution of (VI, 4). The general solution is obtained by superposing (with arbitrary constants) two independent solutions like (VI, 6), e.g. the

one obtained with the first kind Hankel-function and the one obtailned

with the second kind Hankel-function.

From the bracketed part of (VI, 2) it can be seen that z is extremely

large anyhow; for ,u is of-the order of 1013 cm.-, and the "velocity" b is

certainly not of higher order than the velocity of light (taken as unity

throughout this paper). As regards the parameter, it is also very large,

if the wave-length is of "human" (as opposed to cosmical) dimension, for

n" is the number of wave-lengths on the circumference of space. But the

ratio z/n" can be of moderate order (wave-length expressed in the unit

10-13 eml.). Under these circumstances the Hankel-functions are very

nearly imaginary exponentials and a very good approximation is afforded

by Debye's method. With the abbreviations

k - and Sin a = b6 (VJ, 7)

I find (adjusting an arbitrary common factor in u and v conveniently)

2i I u(z) = (z Cos a + Tuano a) eik(Cota a) _

2

vQc) = i(z Cos a - Tanga)eikota-)?(V,8



SoHRnDINGER-Masxwell's and Dirac's Equations. 47

The modifications (as compared with the scalar case, treated in my

"Physica" paper) are (1) the imaginary terms under the square roots,

(2) the real terms in the exponents. The first have a trifling influence on

the phase, but I dare say of the order of the neglected quantities only. The real terms in the exponent have the important task of making

I ~~12+1 V 2 Cosa2 z s2 -ig a + 1 Tang2 a z (VI, 9)

so that (see VI, 3)

1 p 12 + q J2 = independent of time. (VI, 10)

And this is an exact and general relation, which follows directly from (3, 9), independent of (VI, 1).

The conclusions with respeet to frequency, phase velocity, group velocity are thus essentially the same as in the scalar case. The influence

of the expansion on these quantities proves to be extremely small, far

beyond all practical interest. Yet I feel obliged to reveal a criticism

applying to both the former and the present treatment. I fed I cannot

warrant more than an upper limit for the order of magnitude of the

corrections. For they are so small that errors of the same order may be

introduced by cutting Debye's semiconvergent expansion short at the first term. The second term would have to be included, if anything more than

the order of magnitude were of interest. But that is hardly the case.

In order to complete the solution (3, 10) for the special assumption

(VI, 1) of the present section, we make a note, that

q =b- 2 ) (VI, 11)

give the complete solution, if we take first (in both p and q) the Hankel

function of the first kind, second (in both p and q) the Hankel-function

of the second kind, and combine the two solutions by two arbitrary

constants.



maxwell's and dirac's equations in the expanding universe

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