the general affine field laws

The General Affine Field LawsAuthor(s): Erwin SchrödingerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 51 (1945 - 1948), pp. 41-50Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488470 .

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( 41 l

IV..

THRE GENERAL AFFINE FIELD LAWS.

[From the Dublin Institute for Advanced Studies.]

BY-ERWIN SCHRODINGER.

[RBead 8 APRIL. Published 21 NOvER=., 1946.]

IN three . papers' I have examined the field laws which in an affinely

connected space-time, result from imposing the condition that stationary

be the space-time integral of a Lagrange- density which depends in an

unspecified manner on the tensor or tensors o'btained by contraction from

the. affile curvature tensor. But the previous work dealt with special

cases ouly. The affine connection was assumed to be either symmetric or of a particularly simple non-symmetric type, with 44 (instead. 64) inidependent components.

Thcangh already this so-called-weakly non-symmetric connection yields

all three fields in question, only the general case, dealt with here, is comn

pletely satisfactory and gives new iniformation. Only here the separation

of the two skew geometrical fields is unique and their identification with

the physical fields unequivocal, because one of them is governed by a

strictly linear set of equations of the Maxwell type (strictly linear, when

written with two six-vectors!) and must therefore be identified with electro

magnetism, while the field laws and the: gravitating effect of the other

one, the I' meson-field," are extremely involved. Whlen this field is weak,

they do appoach to the well-kcnown set suggested by Proca. But this is

of little mnoment, because the meson field in the niucleus and in collision

problems is probably far too stIronig to vindicate the linear approximation. The working out of the general case was delayed for two years by the

fact that a set of linear algebraic equations (viz. (1, 7) and (1, 9) below)

does not seem to admit of a simple solution. It has now turned out that

this Very fact entails the beautiful separationi of the two skew fields and

informns us of their intrinsically different nature. This is the reason why:

ill previotus, work (inc luding the author's bni earliest attempt2), which

accounted for one skew field only, this could not. without some unnatural

strain be identified with electromagnetis; For, from ouir present point of view, it had nothing to do with it.

ipr?c. Roy. I. A;49 (A), 43,1943 ; 237,275,1944;

2Procv Boy, I. AV 49 (A), 43,1943. PROC. R.IA., LI, SECT. -A. L 15]



42 Proceedings of the Royal. Ish Academy.

? 1. THIE VARIATIONAL EQUATIOxt.

It would perhaps be more consistent with our general attitude, if we started by using the general non-symietric affinity A'kz, and invariant derivatives with respect to it, without decomposing it. This method is quite feasible, injdeed it shortens some computations and yiel-ds the varia tional equations (1, 7), (1, 9) and (1,0) a&t one blow instead of piecemeal.

However, the slightly clumsier proceduite which we shall follow facilitates the comparison with previous work and avoids certain pitfalls involved in the inse of "non-symmnetric derivativep." Y&t I shall try to make the followinig presentation self-contained by indicatiig every step anew, so that the reader could check it up, albeit with much 'labour.

We split our general affinity Sikl into a symmetric one, P1kI, and a "skew" tensor of the third ran-k Uikz, thus:

A k-ki r ik* + U k:,

(Plk = r (kJi = ON U)(

By initroducing a notation for the trace vectoi of the latter :- . 0 =; 0 gt

= 3 7k,

we split U up into a trace-free tensor W-and aniother o ne, dependingt only on F:

Ult = rkI + i VI _ S'17kY (1,2)

Indeed, by contraction t: ~ ~ ~ ~ ~ 1' - 0 : Wlk??. (1,3)

We have to work out the two contracted curvature tensors in terms of the. , LVW and V's. The formulae (2, 12): and (2, 13) Roy. I. Acad. 49 -(A), p. 278, are usefuL We obtain

Rsz =IB3kz)r Vl; k - 2Y7k; I 3 WI'k4k 4 Aa + W,I++ 3 Vp J8kj

ari _k &Pak + 3 (V1 71- FAA; . (1,4) S.xk ax,

Here and throughout this p&per the semicolon (;) indicates invariant differentiation with respect to the syinmetrical r-affinity. By (Bi,) r iS

meant the Eii'steini tensor of the P-affinity, a tensor-Whose skew part is, by the way, just - I times the P-part of 54j.

..The Lagranae density is t.o hea ana unspecified function of the 22 quan tities Bki, Ski, thus .3 (B ki, Ski) - But since 'we have-- introduced four redundant variables,'we havei to add a suitable term, in order to be allowed to v-ary the 68 quantities F, W, J[ independenty and to fulfil equ. (1, 3)



SCHRODINGER-TJte General Affine F"eld Laws. 43

afterwards. Thus ouir variational principle reads

0 =8J[(Rkz, SkA) + 2pk W'k.j dCXdX2 CX3dx4

f | ( kl BRkl + 0kl Skl + 2pk 8 Jawk) dr i dr[2 ds3 dx 4. (1, 5)

The meaning of ? k, 0kl is obvious, while v k is the Lagcrange multiplier.

Moreover we put

2kl _ gkl + fk, ?3k - 9k1 - fkl (1, 6)

so that 8kl and fki are the ordinary derivatives of ? with respect to the

symmetric and the skew part of Rkl, respectively.

(a) Varying the r's.

The well-kInown relation

a (B kg) r - (3P zr );a + (3r a)

is of great help. One inust not forget to vary the P's concealed in the

semicolons in (1, 4) and be aware that e.g. a term ?IW a rfl1, yields two

terms in the factor of 3 r'mkl, one for P. = k, ip ; I and another one for

p - I, v = k. The rougfh variational equation must be contracted and the

result-used to simplify it, paying attention to (1, 3). Then you obtain

A + 8' (Aik _ kft fJp _ jrk) + 3k (pl - 7 2jrt) +

+ 3 3k1 V + ffl Wk V5+ fPkwl3 = 0. (1,7)

Here we have introduced abbreviations for the two "current densities," viz.

f kP ;_ fk,p =tc kgi ; P 3k $, - rk (1, 8)

(6) Var ying the grs.

Here the procedure is quite straightforward. Only the last tlhree

terms in Rk,, eqn. (1,4), contribute, anid, of course, the termi in (1, 5) that

contains the Lagrange multiplier pt. Thus the variational equationi contains

the latter. One expresses it by other quantiules by contracting anid

demanding (1, 3). After eliminating p in this manner you get

fkt;'l k15(4j + fk8V ) + Ska (4t + f PV)

- 3 fkg Va + 9P1 wkpx + gkwvl 5a = 0. (1, 9)

This equation has a structure very similar to (1, 7).

Let me mention by the way, that if you add (1, 7) anid (1, 9), then, owing

to their symmnetry character, their full content is preserved and takes a

remarkably simple formi, because the terms (except those with i and r) all



44 Proceedings of the Royat Irish Academy.

combine to form a peculiar kind of invariant derivatives of V kt viz.

a a 1 + _P?Ak + 2kP&-p - A 2 t( *flp +

where A'it, _Atka - ik(Aflp -

and it is easy to verify that A%ea = Aai. This shows inciden.tally tthat equls. (1, 7) and (1, 9) leave V, undetermined.-However, we shall not use these conceptions in the following, we have menRtioned themn only to comnpare

with Einstein, l.c.

(c) Varying the V-s.

This is also quite straightforward.. Bt the ro'ugh result cointains the contraction g1 which can be obtained from (1, 7). If the result, drawn fromn there, is substituted, the variationale quation with respect to the V's amounts to the simple statement

4rk _ tk - (1, 10)

According to (1, 8) this says that a certain linear comnbintiona of the skew

tensors 0 and f has a- vanishing divergence.

? 2. THE SiKEW FiELDS. Putting

k= k? + # ' Bik = k? - #k (2, 1)

we rewrite (1, 4) thus4

ViY (Bkl4r . i(F71 + VI; k) 3VZVk + Wk4 W%4

=l 4. - j(Vk - k,l) W kl;a + 3VpWPkl au ark

SkI aX ax1 + 3(V?,k - Vk,t). (2, 2)

The field equations are obtained by inserting into these 22 equations the values of the r's and the W's drawil from (1, 7) and (1, 9) (which are linear algeb raic equations with respect -to these variables) andi by 'adjoining the definitions (1, 8)- and the statement (lt 1). /

Because thle .Lagrangian is left undete'rmin-e'd for the time .being, each' of

the three fields will be represented by two 5' conjugate" tensorial entities in the field, equations, gravitation by g and y, the skew fields by f and 0 auld by

0 and S respectively. One out of each pair is to be regarded as the primitive field variable, the other on-e is the derivative of the Lagrange funiction with

3 AvEinstein, Annals of Mathematics 46?578j 194?5? A second pa?er on the sam??; subject bjr

A. Einstein and E, G. Straus was availaible to me in typescript by the k|nd,aessof the authors. -* The underlining of the subscripts in (Mm)t is short for" symmetrical pa?t of."



SCJR6DINGER-The 6General A/fine Field Laws. 45

respect to the first. The choice is in prin;ciple arbitrary, --and that

indepenidently for each of the tlhree fields. The exchange of r4les of a pair of variables cai 'be brought about by a

contact transformation with respect to it, in a manner that is familiar from

thermodynamics and classical mechanics. With the original Lagrangian,

,y, 0, S are the primitive field variables, , f, 0 are defined via the Lagrangian. For the time being we keep to this point of view. Only in the eventual

field-equiations we shall reverse it mntally in the case of the gravitational

pair 7, 8:

Turning now more particularly to the skew fields, which are described by the last two equations (2, 2) and by (1, 8) and (1, 10), we have to settle another vital poinlt. The secondary field variables are defined by

= t - tt a kg + fl + Oklas,11 (2, 3)

But since + and S are of the same tensorial character, any linear

transformation of them with constanit numinerical coefficients, acconmpanied by the induced transformatloni of f and 0, leads to variables of mathematically equal right. And if one such trainsformation is uniquely determined by the fact that it brings about an enormous simplification of the field equations by separatiing them with rospect to the two fields and making one of the sets exactly linear, there is reason to believe that those aggregates rather than * and S themnselves correspond to the- physical fields of observation. We

shall see' that this state of affairs obtains.

For, contemnplate such a transformat:ion

-*. = all + a 12S (Det. a,i 1) (2,4)

aind the one it induces, viz'

,~~~~~~~~~a + -, -.f - a622 f - a2. 0 : (2, 5) *

- s 0 = -al2f+ al10.

Now, from (2, 2) one and only one linear aggregate of * and n , viz, just S itself, is the -rotor of a potential. This induces s' to put

a ! , (\2,6)

Moreover, from (1, 10) and (1, 8) the' tensor density 40. - f' has a vanishing

divergence.: It- would be utterly ludicro-us to have- the divergences of two

fundamental physical fields proportional to each other! From P2, 5) and

(2, f) this can: 'now' only be aveivded by taking a ,i and' a .2 in the ratio



46 Proceedings of the Royal Irish Academy.

4: 1. So we put, with no further essential arbitrariness,

II a - 1, a0 = - , a21 0. (2,7)

The result is., that the field

kl k_ i f fkl (2 ,8)

is governed by an exactly linear set of equiations of the Maxwell type (of course, with two six-vectors !), while the second field is singled out, uniquely as

- - ki 011 Skl :

f= f l,- (2,9)

with tlhe presumption that it corresponds to the meson-field. Its field equations are. the first equation (1, 8) and, from (2, 2),

4 i . = Ea? - ap, 4 . + r

+ kl = _ _ a r ? +4( Z,k-Vk,4)-: Wa^kt;a tf +3PV WV1.

(2,10)

Th.he Maxwellian field S, S , (2, 8), except for its gravitating effect,

whieh will tuirn out "correctly," is kept entirely aloof fromn -the rest by the

remarkable fact, that the V-vector drops out rigorously front all the other equations except the last eqn. (2, 2). that is to say, it will be shown to cancel rigorohsly in the first eqn. (2,j/2) and in (2, 10), which-has superseded the second equation (2, 2). By this freedom it mtakes the [r-termns in the eqnS for S/C/ illulsory, they just add to the potntial which is undetermined anyhow. So the Maxwellian field is settled, we. have nothing more to do twih it. The field q, f with its current density i are termed the meson field. On f and it we put nio bars since they are not needed. We proceed to determilne the approximate equations of this Geld.

Inisolving (1, 9) and (1, 7) (where, by the way, IL k 2lr is to be

replaced by i' according to (1I 10)) -approximately for the unknown P andq W;, we must, takef care neverr to drop terms with F, in, order to make good that they rigorously cancel. The procedure amouiits8 to a

devQlopment in powers of the f s as follows. From (1, 9) the IF's are of the first order in the f's. Hence the. W-terms

in (1, 7) are of the second o.rder, 'and we may dropthem a first and solve'..

for XthgP's Ucon,cealed in the semicolon. By -familar methods we get t

K -= 4t,im ? aCQJwtl + F,) V+ am+ %/C +t VC) + em. ;(2, 11)ai



SCaRDN6Mn-a-ho Ge neral' Affine Field Laus. 47

Here one has defined " Latin " p's in the-. usual way

- X /tkZ 4 g g k Z gkI , - 5 w ka(2,12)

and "appointed" thlem for ratsiig. and lowering of indices. The g is appointed to turn any de nsity into the correspondinjg ten'sor and vice

versa. The } are the Christoffel symsbols of the g's.

The tensor e, symnmetric in i, I, is of the second order and is determined

thus. First you carry (2, 11) into* the sEemicolon of (1, 9) and find that all the terms' vith V cancel. At the same time you carry (2, 11) into the sendi colon of (1 7), where of necessity all ternms cancel except those containing either e or W. So you have for the exact determination of these two tensors again 64 linear algebraic equations none of which contains V. HEence 'and

V are rigorously independent of 'V. The only place where this vector

appears in our, solution is where it openly shows up in (2, 11).

it is klear how WV and e could, in principle, be obtainek to any approxi

mation by alternatirng mtutual substitution from one set of equations to tle

other. For the momenit we are conitent with the first order approximation (2, 11), without 0, anid with the corresponiding one for fT, obtained fromn' (1, 9) by using in the semicolon for Jr the (2, 11), without 9. By familiar

methods one finds

WkiMn = -g`i (fmiI I + fii I in fim I i) + a3klim i-c3 iz. (2,13)

The vertical bar ( ) means here and in all the following invariant differ

entiation with reference to the ( }

After contracting (2, I1),.

r, a- i1 + 5bV + 0'a (2,14) we carry (2, 11), (2, 18), and (2,14) into (2,10) and, dropping quiadratic terms iz f, we obtain as the field equations of the meson field sn linear approximation

I1 (\ ? - + j (f fIfamt I bi*)a

-r fkzlnl) (

tog'ether with

from (1, 8> These would be Proca's equations except for the term which contains

eplicitly second derivatiVes. In pri&icple this means an' enormous complication, because it makes the cyclical divergence of jk non zero, expresses it by third derivatives. But in actual fact the additional term only amounts to a slight direct influence of gravitation on the meson-field and is hardly contradicted by the current view about the latter, which never tad

occasion to contemplate such iufluence, direct or indirect.



48 Proceedings of the BoyZ Irtish Academy.

Indeed if there is no gravitational field5 the order of derivatives can be

exchanged and the first two terms in the bracket merge with the rotor-term

modifying its coefficient j. The last term is the wave operator on f i,

wbich according to the very Proca equation is proportional to f,. and this,

in the linear approximation, must be virtually the same as *ht' say C#&Z.

Hence the last term merges with thie first membier, modifying the coefficielnt 4- 1 of f h1* Jf you work this out you 'get ;

= (1S 2j) (a . akD?g x (2,16)

where /' stands for terms wieh- vanish when there is no gravitational field. In this sense a weak f-field is governed by the Proca equations. By this

I will nQtj say that the present theory, if accepted, confirms the Proca

equations for the meson from the classical poin1t of view. For there is reason

to believe that the enormous mesonic field strengtl within the nucleus does

not cdrrespond to a weak /-field.

? 3 GRAVITATION.

The first eqn. (2, 2) reads more elaborately

a h ka Jr lak a rFkai'

axhk

J,(Vk; + V;tk) -.3 VI Vk + W V W7fia' j (3,1)

Here we insert (2, 11) and (2, 14), not forgetting the Ps concealed in the

semicolons. The lengthy labour, which is mostly impended on terms that

eventually cancel, is facilitated by using transitorily a frame in which the

first derivatives of the g's vanish. 'Thfie nal result reads

h = 9h-G

e i jkl

+ i(eakI

+ e + Wak

p W'k

a 4 J + itht *

''( 2)

Ckh is the Einstein-tensor of the curly brackets of the Ykz. To throw the

matter tensor into relief we subtract from this equation ha-lf its contraction

times ghz. Moreover we use ani importanit identity, which spring from the demand that the Lagrangian t must -be a scalar density'

ka + ka S f 4 t ka = -, (,3) 14L~~~. a .... .D

;,? In tuli rigour this is an qnallowed abstraction. By its very existence the/-field produces

gravitation! :,._ .' ;. ,_':.. 6 Profit B.I.A., 49, 237,1944. There we had only one skew field, now we Have two. Moreover

one sign was different there owing to a clumsy definition bite



ScmRIDiNGm;-The General Afne Field Laws. 49

Thus we get

( t _i SI + -Skj - +

+ f {h -b jIkif"P?68p + Yki, (3,4)

where G is the invariant curvature of the g-metric and Yki is a symmetric tensor depending only on the g's and f's and of quadratic order in the latter, viz.

t T (4ig - jSIi^i) - e klia + I (akil +eIlk) - i g kl + ePaj) ,p + W L 18 el - igkj War SW' (3,5)

The explicit expression up to quadratic terms of the f's could easily be found, as indicated in the preceding section, by carrying (2, 11) into (1, 7), solving for the e's and replacing the W's in the result (and also il the last

two terms of (3, 5)) by their linear approximation (2, 13). However (3, 5) is exact, and the appoximation of the e's and W's can in principle be

continued up to any order, as indicated above. In fact, since they are

determined by & set of linear algebraic equations which are bound to be compatible thie solution can be formally written down, but I could not find a surveyable form of it.

Disculssing the gravitational equations (3, 4), we first observe that the first term on the right may include a cosmological term if S becomes a

multiple of v/- g when the skew fields vanish. The s-terms, together with

a contribution from the first terin, describe the gravitating effect of the electromagnetic field. The remaining terms, again with a contribution from the ?-term, ought to render the gravitating effect of the mneson-field. We do

not wonder to find the description of this effect rather complicated, even in

the quadratic approximation, since in actual fact also the meson field

equiations (2, 1-5) were rather complicated, even in the linear approximation,

in the presence of a gravitational field.

No useful purpose would be served at the moment by examining this

effect more closely, I mean to say by carrying out the evaluation of Yk1 in

terms of the f-field in the manner described above. A closed, compact solutioni for the W's and e's in terms of the g's and fs would, of course,

be most welcome, but I have not yet succeeded in constructing it.

? 4. SUMMARY AND CONCLUSION.

In this paper 1 have at last completed a geometrical field theory on which I began work more than two years ago.7 Whether it Is physieally

7 For its earlier history see my report in ?? Nature," 153, 572, May 13,1944.

PROC. R.I.A., LI, SECT. A. [6]



50 Proceedings. of the' Roya Irish Academy.

right or wrong, that is to say whether it has a direct bearinlg on the physical

fields it purports to account for geometrically, or not, it must I think' be

called the affine field theory, since it rests almost entirely on the assumption

that the fundamental connection of space-timne is purely affine. This includes,

of course, that no metric is envisaged a priori.

The only restriction is, that in the sel-ection of the special field laws only the curvature tensor is involved, and this, only by the second rank tensors it engenders by contraction.

Three fields re-sult, two of which 'and their interaction conform with our ideas, based on experiment, about gravitation and eectro-niagnetism and their interaction. The laws governing the third field ax much more complicated. But in linear approximation t4iey do conform with a' simpl linear classical set of equations, proposed for the meson field by Al Proca,. conform with it if we disregard a certain direct interaction between gravi7 tation and meson field, an interaction Which the -present theory asserts.

while the proposed set had neither the occasion nor the intention to include anything of the sort,

This encourages one to regard an affine connection of space-t'ime as the competent geometrical interpretation (from the classical point of view)- of the three physical texnsor fields we know.

'Let us accept this for the sake of argu -ent. The most important concleusion then concerns the meson field. The linear approximation thal I mentioned just before is, for- reasons which I' do not detail here, almost

certain to be entirely insufficient for the enormous men field strength' inside the nucleus and in collisions. If this is So, if the classical field laws of the meson are violently non-linear-, we can hardly hope that they, will be of much help in' guessing the true quantumi laws- of the meson. This

appears, to be a despondent negative result, But it -would give us an

understanding why the attempts, based on linear classical field laws, are not vindic%ted by anything like a, strikinig order they bring into the knownl facts.

The present theory offers a less. despondent, indeed ;rather. bold, :alter native.. If we believe affine connection to engender the "classical analogue"

of the true laws of Nature; this -might induce us to try and modify, the

classical notion of affine connection itself in such a manner as to produce the true laws. But this requires a new: basic idea of a very futndamental kind.



the general affine field laws

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