the final affine field laws. ii

The Final Affine Field Laws. IIAuthor(s): Erwin SchrödingerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 51 (1945 - 1948), pp. 205-216Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488482 .

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[ 205 ]

XVI.

THE FINAL AFFINE FIELD LAWS. 11.1

(From the Dublin Institute for-Advanced Studies.)

By ERWIN SCHRODINGER.

[Read 9 JUNE. Published 9 FEBRUARY, 1948.]

IN this paper I wislh first to indicate. the place of the theory e.xplained in

part I among a class that descends, from the theory of gravitation in empty

space by very natural anid straightforward generalization without any further artifice. We shall give a succinct survey of possibilities, some of which have been investigated earlier but without emphasis on their logical connection (sect. 1-4).

In the secolnd half (sect. 5) we examine, whether the equations (23b) and (23c) of part I might be identified with Maxwell's equations, of which they have lhe general structure.

L. General Survey.

All the theories to be surveyed here are dertived from the variational principle

81 = 8 Rkl -r = 0, (1,1)

where Bkl is the Ricci-tensor of an affinity rF i:

Bk; = _ ar:kz + a'kS + r'k, rP.l

- rJ, r sI, (1,2)

and, the contravariant density g kl is subject to the coiidition that its deter minant is negative, so that tensors g9ki and g k can and shall always be

uniquely definied thus:

Det. gk (= Det. g k0, g" = g gklm =g gmk= Sm (1,)

The theories differ in two relevant aspects, viz.:

i See Proc. R.I.A. 51 (A), 163,1947.

PROC. R.I.A., VOL LI, SECT. A. [28]



206 Proceedings of the Royal Irish Academy.

(i) In performing the variation either only the gik or only the I" ikl or

both can be regarded as independent variables. That gives three possibilities to which we shall refer as the metrical, the affine and the mixed cases

respectively. In the first two cases the relation between the two grolups

of variables must be furnished in another way. This introduces in principle

an unlimited variety, with regard to which we cannot, of course, atteinpt

anything like completeness. (ii) Either none, or one or both entities gik and P ikl may be assumed

to be syminetric from the outset, being thus reduced from 16 to 10 or from

61 to 40 independenit components respectively. That gives four possibilities in the mixed case, but only two each in the metrical and in the affilne cases.

Seven of these eight possibilities are dealt with in the following. It will be

pointed out why the non-symmetric metrical case is dismuissed.

2. The Theory of Gravitation.

In any case we have froin (1, 1):

SI = JRkl 885 dr + J& "aRkl dr = 0 (2,1)

Now if Ik1 is symmetric the second integral, with the help of the Palatini

expression for a1k? and partial integration, becomes

J 9kl ORki

dr J (ol ;a

- 8 gk%) arak? dr, (2, 2)

the semicolons referring to the affinity flak . It is known, that from this the field-equationis of gravitation follow in two ways, inamely

(a) Metrical theory (Einstein): One assumes gi = gki anid

flak? = {kl73} ; (2,3)

then the semicolon-derivatives in (2, 2) vanish identically, anid since the Ricci-tensor of the Christoffel-affinity is symmetric, the vanishing of the first integral in (2,1) requires

Rkl 0 . (2,4)

(b) Mixed theory (Palatini): One assumes againi gik = gk and (in order

that (2, 2) may hold) r'kl - f'lk . The two integrals have to vanish

separately. Hence the symmetric part of the bracket in (2, 2) must vanish,

from which one easily infers ?k,a = 0 and thus gkz; a = 0. The latter,

in virtue of the assumed symmetry of r, entails (2, 3) and we obtain (2,4) as

before. It will turn out presently, that the symmetry assumption about r

can be spared, without changing the result substantially.



SCHRODINGER- The Final Affine Field Laws. 207

3. Mixed Generalizations.

The attitude b is 'superior to a-thoulgh they both lead to the same

result-because it su(rgests very natural generalizations; whichl a does not,

since there is no simple and natural clue," by what the connectioin (2, 3)

should be replaced, if gik is non-symmetric. So we dismniss the case of a

purely metrical non-symmetric theory. For two of the three mixed generalizations we need the general form

wlhich (2, 2) takes for a non-symnmetric affinity. This decisive step has been

worked out in the appenidix of part I, viz.

J 9k1 31k4 dr = J (? a k'a 31 "rmkl dr , (3, 1)

where

klu = kl,a

+ 8a> OrraL

+ gjkU *rJaP

- i (Ur?r a e) (3, 2)

and rikl - rikl + 23 itk r,,

['k = I - ['~~~~TTk) ~(3, 3)

rk I 1 (k, - re,,

from which identically

aP = rakk (3, 4)

If now Pkj is actually non-symmetric (64 components), the bracket in

(3, 1) nmust vanish and that means that the derivative (3,2) must vanish.

kl = - 0 (3, 5)

It is known from part 1, that this can equivalently be written in the

simpler form

gkl,a - gaz JP a - gku OrP l = 0, (3, 6)

and that it has the consequences ik

gVk - 9 (3,7) and

arMka a Or 1 = 0. (3, 8) a aXkr

2I admit that (3, 6) might be regarded as a natural assumption when written with r in place of

*r. But it is-reasonable only with fr explained by (3, 3). This seems to me too far fetched to be

adopted a priori.



208 Proceedings of the Royal Irish- Academy.

Now we have to distinguish the two cases.

(c) r1' non-symmetric, gik symmetric.

With a symmetric gz7 we get from (3, 6)

k= ii 't9)

while the vaniishing of the first integral in (2, 1) demands

A41 - (3,10)

Moreover, as we know from part I,,

- kl = R-1 + Fkl, (3,11)

where 1kl denotes the Ricci-tensor of the *1-affinity (which according to (3, 9) is in the present case symmetric); and

k= 3"2 &f' (3,12)

which is skew. Hence (3, 10) reads

- R = 0. - (3) 13)

So we get again only the pure gravitational field, nothing more. Fkz is a

gratuitous einbellishmnent, the vector F7, remaining entirely undetermined.

(d) Both r' t and gik non-symrhetric.

"k[ is luiquely determine4 by (3, 6), but not in the simple fashion (3, 9). (3, 11) with (3, 12) holds, but 11k? is no longer symmetric Instead of (3, 10) we get the full set

R kl - 0. (3,14)

We split it into its symmetric and skew parts. From the latter we

elimiinate Fkg by differentiationi. This yields

*ik - 0, (X3, 15a)

*-t ik,I +

.k1,i + 1 I,k ? * (3, 15b)

This is- with our *I'ikl playing the rOle of their r ih-precisely the theory of EihAtein and Straus.3 'For, our ??k is, aposteiori on account of (3, 4) and (3, 8), the same as their a priori manipulated (thQermnitized") Ricci tensor, which they call P:k. They give as. their field equations our (3, 4)

? Annals of Mathematics, fy6, 578,1945 ; ?7, 731,1946.



SCHRODINGER-The Final A/fine Field Laws. 209

and (3, 6)-of whichb

(3, 7) and (3, 8) are consequences-and our (3, 15).

But on accouint of the said "'manipulation" they could only obtain them

by imposing (3, 4) and (3, 7) as accessory conditions. 'illlis mars the sim

plicity of the foundation, compared with the poinit of view developed here.

We have still to examline the third possibility.

(e) rFikl synYMMetiic, gik noni-symmetric.

The vaniishing of (3, 1), whiel nowv takes again the simpler form (2, 2),

requires only that the symmetric part of the bracket vanish, thus,

g ;a - i2 a gk; - 4 8kg ip;

t = 0 (3, 16a)

while the first integral in (2, 1) furinishes

Rkl 0 . (3, 16b)

These are now our field equations, both the semi-colon and the B.-tensor

referrinig to the (symmnetrical) F ik,. For reasons of ecoiiomy we (defer the

furtlher investigation, because it canl be settled by a brief corollary to the

case g below.

4. Affine Generalizations.

If in performinig the variationi (1, 1) we wislh to allow only the F ikl to

vary independently, we must somehow give ouirselves the qi as functions of them. Now there is a simple and suggestive genieral manner of doing that,

and that is, to replace the variational principle ostensibly by a different onie

namely

I = a J(I kl)dr = 0, (4,1)

where ? is a scalar density, function only of the R Rkl, ortherwise arbitrary.

Then the array of its partial derivatives have the character of a contravariant

seconid rank density and we may supply the required relation between the gik

and the rFk: by defining

= aRkl (4,2)

Moreover, sinee a change of scale of all four coordinates in the ratio a (to

Wit, Xk =a Xk) wouild affect all the It kl by the factor a' and 1 by the

factor a4, i nust be a hoinogenieous function of the second degree of the tgkl .

Hence = ji gk R kl . (4, 3)

This shows that, by adopting (4, 2), we fall back on the original form (1, 1)



2-10 Proceedings of the Royal Irish Academy.

of the variational principle whatever we may choose for- ?. On the, other hand, from (4, 1) and (4, 2)

aI -zJi!:dr = J&%UhzRsdI = 0 . (4,04)

The vanishing of this integral, that is of (3, 1), wlhich was a common featutre of all previous cases, is now the only demand. The consequenices are literally the same as before-depending' however, it will be remembered, on whether or not symmetry of rP,k is assumed. The other coindition, viz.

J BR dl 7-d 0 (4, 5)

must now not be demanded separately. (That it actually follows from (4, 3) and' (4, 4) is of no consequenice.), That part of the field equations which followed from. it, viz. B hl = 0 or 1?hi 0,_ as the case might be, is

now superseded by (4, 2). Only this part depends on the speciazl choice of the y .4 2. o.e

of ae

Lagrangian. We shall follow up here only-one particularly suggestive choice, namely

D =et. 1d h

(4,6)

The factor, 2 is put in for convenience. The equally irrelevant constant A -is to facilitate the adoption of 'a "human." scale for 7kz.-It has been explained in part I, that (4, 2) with (4, 6), takes the form

Rki = Agkl. g(4, 7)

which suggests to look upon A as the cosmological constanit. rThe set (4, 7) is common to the two cases we have to distinguish now.

(f) ri4k non-symmetric.

This is the theory put forward in part I, the bne this series' of papers is actually concerned with, so we can be brief about it. We see now why it is'

so closely -related to the Einstein-Straus-theory, explained sub d above. The relations (3, 2)-(3, 8), (3, il, (3, 12) are transferred without change; but since (3, 14) is now superseded by (4 7), the equia-tions (3, 15) are now replaced by

-Bl = Xgkl (4, Ba).

(B2k - Agk), I + (*Bk Agkl), i + (*Bl Xgi), k 0 (4,8b) Howe V V V V V V

However, as was ex~plaine'd -in part. 1,the relation,(4,7) affords a- more concise



SCUR6DINGER-The Final A/fine Field Laws. 211

way of expressing the field-laws, by inserting (4,7) in (3, 6), viz.

- B a rFka -Bkq ?el 0 (4X 9)

They are, if you keep ininind the definition (3, 3) of the r's, only about the original afflne field variables Ftwl, and they contain all the other informa tion, gkl having become just another name for Rag, expressed, if' you please, in a different scale.

(g) rikl symmetric.

Our variational condition (4, 4) demands, according to (2, 2), which is now competent, the vanishing of the symmetric part of the bracket in (2, 2). We had wtitten that out, sub e in (3, I6a), which was there to be supplemented

by (3, 16b), ?ow however''by (4, 7). (Remember that we had delayed further action because the relatively uninteresting case e will now be treated as a

corollary, by "putting A - 0'). With the abbreviation

0 v, a f (4, 10)

one easily obtains, by contracting (3, 16a) with respect to I and a:

g k = - -4k (4, 11) TJsing this in (.3, 16a)

dL ;J+ la i Ak + kk8ia t 0. (4, 12)

To solve these equations for the ri kl concealed in the semicolon, we have to supplement the symmetric density g kL in the uisual way by a metric,

which we must however not call gY,Al or gkl, because these symbols' have already been given an entirely different meaning, according to (1, 3). So we denote it by ski, its determinant by s and the contravariant tensor by S Al !thus

s _ Det.g A(= Det.Skl) 8kl = s_ ' / -8

kl f3 1 (4,18) 7IS Skrtn = m

The solution of (4, 12) works out. thus

rPA = hi L - jg,4i + i*8k it + * 8 ik (4 14)

The index s at the curly bracket means that the Christoffel-symbol is-to be formed of the skl. The lowering of the index in i and the " Latinization" is also performed with the s-metric.




By inserting this result into (4, 7) we obtain after some work

Bk: (FPo) = Bk: ? *(ik , I il, k) + ik il Agkl (4,15)

where the notation is, I think, unambiguous. We split this into its sym metric anid skew parts

-Rkl ( g )+ tk il Agkl (4, 16a)

(i, I - il, k) = Agkl (4, 16b) v

These are the fairly involved field-equations of the present case, heed being taken of the meaning of the s-metric aind the vector tk, see (4, 10). It is

the theory put forth by Einstein4 as early as 1923 and reviewed in great

detail by A. S. Eddington in Note 14 of the 2nd Edition of his Mathematical

Theory of Relativity. A very distinctive feature of the syimimetric case is, that not oinly is the

CCcurrent " i' niot necessarily zero-as against (3, 7), which holds in the

non-symnmetric theories-but the occurrenice of any skew field at all is tied up witlh this non-vanishing ik. Indeed for ik = 0, we get gk: = 0 froni

(4, 16b), and Ski coincides then with gkl, so that (4, 16a) become the

equations of the pure gravitational field with cosmological term. Yet in

view of the exceedingr smallness of the constant A very small values of ik

may suffice to produce a noticeable skew part of gk.l

In interpreting this theory, one seeins to have the option whether to

regard gki or Sk1 as the gravitational potential. I believe that only the

second alternative has been enivisaged, which is simpler. The g-ravitating

effect of the skew field must then be squeezed out of the A-term in (4, 16a)

by developing gk: = Sk1 + ... , (4,17)

where the dots indicate a series of ascending even powers of the skew field.

However, to ancount in this way for the comnparatively strong gravitatiing

effect which from general principles one is used to and feels bound to

attribute to the electromagnetic field, seems to require-in view of the

exceeding smallness of A-too appreciable values of the go, inadmissible

if this skew tensor is to represent electromnagnetism.

It is ilot our task here to deal with these questions, but we must give

the promised corollary, settling the case e that we left open. With A = 0

4 Sitz. Ber. d. Preuss. Akad., pp. 32, 76, 137 (1923).



SCHR6DNGEU-TheO Final A/fine Fteld Laws. 213

we are left with

Rg :zts)+ Pk& =s it-0

k I I,Z4 k 0. (4,18)

These are, with the metrical tensor skZ, the equations of pure gravitation, enhanced only by an7odd energy-tensor, which depends pon a vector-field i k of v'anishing curl and, from (4, 10), vanishing divergence

tk= 0 . (4, 19)

This seems to be an absurdly trivial generalization. The curl condition alone allows us to choose a frame in which the -kS are: constait, simply

by choosing the, scalar, of which ik, must be the gradient, as- one of the- coordinates. We have mentioned the case only for comploteness, and also to show that the so-called cosmological term can- matter a lot, in.

spite of its -smallness.

5. The Skew Fields.

We return to the non-symmetric affine theory (which is the subject proper of these papers), more particularly to the equations (3, 7) and (4, 8b), which from their build could be Maxwell's equations. Can they?

A preliminary task is, to express the densities ti and gY by ga V and gik. The latter we regard as the metric. A more convenient notation is noW desirable. We put

'ffm=kik , -/u = h (5,1)

thus gqik - t + ?ft.k

The determinant of hik we call 4, its normalized minors- 1k. First we compute the determinant g. A suitable (complex) transformation reduces hik to uiity and Ak to the components f', and A4. In this frame

g (I + f2) ( + fA ') =I + f . f + fi2f

We rewrite this; still iii the special frame, thus

g = A + j A A" AkrnA4f,m + t2, (5,2)

whete Zi =i fjk/fm. (6, )

But (5, 2) is obviously invariant. To obtain the V and gt* we behold that

dg = ggikdgik - ggik(dh i + dfik) = ggdkdAk + ggj dfij,.

PROC. R.I.A., VOL. LI, SECT. A. 1-9j




Heiice

g ik - _ 1 agh i v g )fik

To carry ouit the differentiationl, we miust remember that

a h = , _ ik _ 9 hik a (hit hkm + hiin hkj. (5,5)

(The latter follows from-i the well-known relation

dhik i- h hkm dhim,

in which however the h-product nmust be symmetrized, in order to be the derivative.) In writing out the equations (5, 4), we introduce the convention

that the inidices of fXk shall be raised with the help of hi'. This can cause

no confusion, because no other conventioni for raising and lowering has yet

been introduced in the expositioni of this theory. So from (5, 2)-(5, 5) we

easily obtain g - h + If 1tmffM + 32Y (5, 6a)

r 5 [hhr (1 + if -mfim frmfs ] (5, 6b)

gv= f g (fr+ ? 32 tErslmfm) (5, 6c)

If the hi7, are of the order of unity, and the /ik are small, g and gs8 differ

fromn h and v'-h hra respectively only by quantities of the second order;

gv is of the first order and differs from /hfre only by quantities of

the third order. Moreover the field relationship between the contravariant

density gv and the eovariant tensor fr, (= gra) is precisely the one

that in Born's non-linear electrodynamics, written in the general metric hik, joins the contravariant deilsity and the convariant tensor of the electro

magnetic field. Turning Inow to the suspected Maxwell equations, (3, 7) and (4, 8b),

which we re-write for convenience, 9va - - (3, 7)

($R ik - Afik), I + ('-R k - f4kl), i + (%R Bi -

Afii),h = 0 , (4,8b)

we see, that they are not Maxwell's empty space equations for the two

tensors that are in the Born-relationships, because from (3, 11) and (4, 7)

B1ik + F k = Xfi k



SCHEODINGER-Tke Final, Affine Field Laws. 215

There is, however, one :distinctive possibility of interpreting thema as Maxwell Born equations including sources. If you -write (4, 8b) thus-we now use square brackets to indicate the sum over the cyclical permutations, as is often done

/f[2*, ]='X *[B,], (5, 7)

you may look upon ;the second member as representing the sources of the flk-field, or what is usually called the four-vector current.

Analytically the equations (5, 7) are very complicated, not of the first, but

of the third order in Ak. Third derivatives are concealed in the second member, if you consider that RBg is the Ricci-tensor of 'the s ix, and>

that the latter are by (3, 6) extremely involved functions of the g. and their -first derivatives. IHowever. one must not expect a very simple connection if it is, a question of accounting by first principles for charge and current, which in a non-unitary theory are looked upon as primitive data.

I do not say that this is necessarily the competent interpretation, but it is a possible one. It has one feature that is very attractive, if fL (- g k)

V is to represent electro-magnetism. The source function forcibly turns up in the "cyclical " set, not in the other one, (3, 7) where by inveterate habit we

would expect it. This entails, 'contrary to common usage, that f&4 is to he associated with the magnetic field and the space-spatial components of 4i

with- the electric field. It has been pointed out long ago5 that this association

provides a more reasonable description of the reversibility of Maxwell's equations ; for then the components of the magnetic field and of the current ehange sign automatically when the sig n of t: is reversed, since they are just

those which contain the index 4 once.

Formally one can, of course, always exchange the r6les of the two sets with the help of the 6-density, and i"n a non-unitary theory it is a matter

of taste whether you introduce any skew tensor or its dual as the primitive entity. With -us, however, the equations turn up naturally 1in the above

form, and e.g. gis is naturally associated with g,4 and not with g23. And

the latter two components cannlot -be exchanged; they do have a very

distinctive meaning. Is there symmetry with respect to the sign of the charge?

Given a tensor gAl, let me call the tensor whose (k1)-component is

,g 1k its mirror image; similarly the mirror image of an affinity rtwk shall be the affinity rPi1- r[k'. Then if gAl and P'k, satisfy (3, 6), their V

5 A. Einstein, Sitz. Ber. d. Preuss. Akad., 1925, p. 414. See my report in "Nature,"

153, 572,1944.




mirror images also do. By this, let me call it, reflexion the Ricci tensor 1R4 is also turned into its mirror image, on account of (3,4) and (3, 8).

Hence the equation (4, 7), which from (3, 11) reads

'Rkl + Fkl Xglk,

if it was originally satisfied will also hold for the mirror images, provided

Pk, is also reflected, which means changing its sign. Thus with every solution of the field equations is associated another one, obtained from it by changing the sign of the skew tensors. Of course the proposed charge current density thenl also changes sign. Let me remark by the way that the symmetric part of the starred, but not that of the non-starred affinity

is the same in the two associated solutions. The ostensible lack of symmetry between the charges of opposite sign

has often been aommented on. To-day it seems to have reduced to the fact that the negative counterpart of the proton is not known. However, a careful theoretical scrutiny by Heitler and J. McConnell" has shown the chances for discovering it to be so small, that the question of its existence is hitherto undecided.

In part I I made the point, that the magnetic field surrounding a celestial

body is probably a direct consequence of its mechanical rotation and that

the unified theory ought -therefore to account for it. The case has since

been strongly advocated by P. M. S. Blackett,7 who collected valuable new

evidence. The symmetry established above does not preclude the possibility of a general explaniation-the kind advocated by Blackett-along the lines

of the present theory, but it shows that it is not a very simple thing.

For it must necessarily be based on the intermediary of at least a general

comprehension of the structure of matter. The sign of the field must be

bound up with the fact, that the matter producing it is composed of positive

protons and negative electrons. We would have to expect rotating matter

of the inverse constitution, if it exists, to produce a magnetic field of the

opposite sign.

8 Proc. Roy. ?r. Acad. 50 A12 and another in press. 7 P. M. S. Blackett, Nature, Vol. 159, May 17, p. 658, 1947.



the final affine field laws. ii

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