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Continuous symmetries and conserved currents
ANNALS
OF
PHYSICS:
Continuous
43, 445-466
(1967)
Symmetries TIMOTHY Harvard
University,
and H.
Conserved
Currents
BOYER*
Cambridge,
Massachtlsetts
The general relationship between one-parameter continuous symmetries and conserved currents in local Lagrangiatt field theories is reexamined. Defining a symmetry transformation of a system of fields as a one-to-one mapping of the fields carrying solutions of the equations of motion into solutions, it is noted t,hat not all symmetry transformations imply conserved currents, and conversely, not all conserved cltrrents can be derived immediately from symmetrics. However, we develop the one-to-one correspondence between those local cont inuotts symmetry transformations of the system which leave invariant the E:ltler-Lagrange eyrtations from the original and transformed Lagrangians, and local conserved crtrrents which can be written as linear ftttlctions of the Lagrangian derivatives. It is shown t,hat the theorem proved by Noether, which is customarily referred to in the const,ruction of local ronserved rttrretrts for some symmetry transformations, is in general related otrly tangentially to the problem in qltestion. The analysis presented for the corlst,rrtctiotls of conserved cltrretlts seetns of particular interest in the case that divergence terms enter the symmetry t,ransformatiott; also it illrrcidates the independence of conserved cttrrettts from the atnbiguityalwnyspresent itt Lagrangiatl formttlations dtte to divergence terms in the original system Lagratlgian. In the elemetltary cases of general interest in field theory, t,he procedrtres for the constrrtctiotl of conserved currents do not diverge significantly from the 1 raditional ones, altholtgh even here the difference itt att,itttde regardirtg the traIlsformation is Iloted. The converse construction of symmetries from cotlserved cltrrents has Itot been cotlsidered fttlly in t,he literat.ttrr. The getteral analysis is applied alld extended for systems of fields allowillg tlaive canonical qltantization or qttantiaation via the Schwinger action pritrciple. 111 particltlar, the consistelrry of the ittterpretation of integrals of the cltrretlts as qrtalllum generators of the original symmetry t ransformatiorrs is cspl~tretl in frtll getrerality. INTRODUCTION
There has been some continued interest ( I )-( 5) as t’o the relat,ionship behween one-parsmeter continuous symmetry transformwtions and conserved currenk in local Lagrangian field theories both classical and quantum. For some particulx symmetries, which might, bet,ter bc termed invarinnces of the Lagrsngian, one nssurnes the esistence of a continuous mapping of the fields which preserves the * Nat iotlal
Science
Foundation
Predoctoral
Fellow 415
446
I3OyEH.
value of the action integral, and then proceeds via Noether’s theorem to the construction of a conserved current (6’). Quite recently, Horn (1) and Dass (2) have considered in very limited quantum field theoretical contexts the converse of t,his construction; i.e., do all conserved current’s appear via Noether’s theorem from some continuous invariance transformation, and how does one recognize the symmetry associated wit’h a given current? Horn and Dass bot’h make explicit use of the usual spin-0 and/or spur-12 Lagrangians of quantum field theory. Their analyses proceed by use of quant,um commutation rules and stringent assumptions about the st’ructure of field equat,ions and of conserved current’s In this not’e, we wish to reanalyse the problem of continuous symmetries and conserved currents, not in t’he quantum domain but, in the classical context of differential equat’ions where the fundamental problem actually lies. The results carry over directly t#ot’he more specialized circumstances of quantum field theory. Section I gives the general analysis. Aft)er first clearly specifying the definition for a symmetry transformation which we shall adopt, we indicate a general construction of conserved currents for the special class of local continuous symmetries of the system which leave invariant the Euler equations from the original and transformed Lagrangians. In the following section, we not,e that every conserved current which can be writ,ten as a function linear in the Lagrangian derivatives is associated with a continuous symmetry of this specified class. Next we review the actual content of Noether’s theorem and conclude that the application of the theorem represents a detour from t’he question of interest. Finally, in Section II, we turn specifically to Lagrangian field theories which allow naive canonical quant’izat’ion or the application of the Schwinger a&ion principle. For these cases, it is always possible from any nontrivial conserved current to obtain a symmetry transformation which gives back a conserved current equal to the given current when the fields satisfy the Euler equat’ions of motion. Furthermore, the three-space integral of the zero component of tfhe current is shown, in general, to generate consistently as a quantum operatorthe same symmetry transformation on the fields. The arguments of the body of the paper are illustrated with sketches of examples relevant to quantum field theory. I. GENERAL
ANALYSIS SYMMETRIES
OF THE CONNECTION AND LOCAL CONSERVED
1. DEFINITION: A LOCAL ONE-I'ARAUETER FORMATION OF THE SYSTEM
BETWEEN CONTINUOUS CURRENTS
CONTINUOUS
SYMMETRY TRANS-
We consider a system of fields cpr(z), r’ = 1, 2, . . . , N whose time evolution is specified by a consistent and independent set of partial differential equations derived via Hamilt,on’s principle from the local Lagrangian density -C(z, (FJ .r), +f4~), ~&Pr(~), . . . ). Here, and in all functions considered in this paper where we indicate higher derivatives by dots, we are restricting the derivatives
SYMMETRIES
AND
CONSERVED
447
CURRENTS
arrlh2 . . . &‘pS to order n less than some n* finite and fixed. A local one-parameter continuous symmetry transformafiun well-defined for arbitrary functions
of the system will he defined as a mapping cp,(x),
= +d.q + of&:, d-4, add,
aravps(.4,. . ) + o(ey,
(1)
which for the continuous real parameter 0 maps in a one-to-one fashion those of t’he equatJions of motion into solution fields cp,(.c) which are solutions q,.“(z). Akhough we will continually consider equations only to first order in 8, it is clearly possible to regard these expressions as first-order differential equations in 0 which can be integrated to give expressions to all orders in 8. The analogous definitions and procedures involving symmetries are also available in the case of ordinary differential equat.ions. Thus a parallel analysis is possible for 1,agrangians of classical and quantum particle dynamics. We remark that in the above definition for a continuous synm~efry transforrrlaiion of the system,we have used only one single value of the argument .r. A traditional but much more limikd definit’ion occurs in what we shall call a local one-parameter cont’inuous inua~iance of the Lagrangian. This is a mapping I’ + 2% = x + 6x(x), 4-r
) 4 33 2”) = 4 x ) + 6*(~h)
= d.d
+ eddd)
+ 0(e2>,
(2)
which for the continuous real parameter 0, maps x and c,G.(2) in a one-to-one onto fashion such that ~(3, &(z), a,+,( z), . . . ) cZX/& when regarded as a function of .c, IP~X), a,dr), . , . is exactly &‘(r, Pi, a,,n,(r), . . . ). The symbols in all of the work presented in t’his paper st,and for mathematical functions, not for “physical quantities.” Not,ations such as aL?( .r, (mp(:l:j, . . . )/a++(r) stand for the partial derivative of the function d: wit,h respect to its second argument regarding all &her arguments as fixed. We assume all functions are analytic in their variables. In the recent work on symmetries and conserved currents, for example (1) and (a), t’he assumption seems t,o be present t,hat every “symmetry” leads to aconserved current and only the converse statement requires investigat,ion. Part of the difficult’y here is relat)ed to the variety of definitions for a “symmet,ry t,ransformation” which are abroad in the lit#erat)ure and folklore. Some writ,ers regard a “symmetry transformation” as what we have termed an irwariance of the Lagra?Lgian. We will find that indeed every ilwariance of the Lagrangian determines a syrrlmetry of the syste,ll whic*h preserves the Euler equat,ions from the original and transformed Lagrangians, and so indeed gives a conserved current. The proof can be obt,ained by rewriting the assumedequality
44s
BOYER
and applying the analysis of Se&ion 3. However, there are synmetGes of the syslelfz which give rise to conserved currents which are not associated with irlvariances of the Lagrangian. A very simple example is provided by the Lagrangian c = >@,cpd’p and the transformation cp(s) + +(.r:) = cp(z) + ox(x) + O( 8) for a fixed function X( 2) satisfying *X(.r) = 0. This is a symmetry of the syste/,l but not an invariance of th.e Lagrangian, which gives rise to the conserved current W” = a&( X) - (F&N zi. A precisely analogous situation arises for t,he Lagrangian density b: = - !.,$c?,A,c?“A”employed in the Gupta-Bleuler formulat,ion of quantum electrodynamics. I;urthermort, there are symmetries of the system which do not. give rise to conserved currents. In the system JZ= >$( &(P~(P- p’pp), the scale transformation p(s) + G(X) = exp [&I,( X) = cp(x) + a&(z) + 0( 19”)is a local continuous symmetry of the system by our definition, but it is not associated with any conserved current. Hill [see (4 ), p. 2573does not limit himself to invariances of the Lugrungiun but restricts the definition of symmetry transformat,ion to those mappings which satisfy Ey. ( 3 ) , i.e., precisely t,hose connected with conserved currents. On the other extreme, some aut,hors apparently are not sure t)hnt’ one should require t,hat a symmetry be a mapping of the functions pr , but might include under this heading merely changes of the Lagrangian such as addition of an arbitrary divergence term or multiplication by a constant. Also, the reader is reminded that the literature 011 the subject of symmet,ry t,ransformations contains several different, notations which are not rompat,ible. Using the not,ation of Eqs. ( 1) and (3)) the &variation and space-time derivatives commute, but’ the 8*-variat)ion and space-time derivatives do not. The precise opposite of this notat8ion appears in t,he article by Hill (4) and in the sect,ion 011 Noether’s t’heorem in the text by Roman (6). One st#andardt,est’ on quantum field theory within a single section seemsto mix together the two notations with apparent,ly erroneous procedures. See the present author’s note listed in Ref. (6) for an elementary analysis of this problem.
q
2. CONSTIXJCTION
OF CONSERVEI)
CuHKEN’rs FROM CONTINUOUS
In the field system cp,(s), )’ = 1, 2, . . . , N,
SYMMEZRIES
-c(J:, Pr(d, dp’pl-CL),.
),
we assumethe existence of a local cont’inuous symmetry of the system $4~) -3 G%(x) = +(a) 1 S. S. Schweber, “Arl Illt.rodrlctiorl Harper alld Row, New York, 1962.
to Relativistic
+ &o,(x). Qrlantum
Field Theory”,
Sectiolr ‘ig.
SYMMETRIES
AND
CONSERVED
CURRENTS
449
Moreover, we assume that the symmetry is such that the Euler-Lagrange equations derived from .c( z, C&(Z), a,,&( .z), . . . ) when considered as a function of 98(Z), %t’Ps(Z), . . . are identical with those from .A?(2, (P?(X), a,,~,.( z), . . . ). This is a special class within the set of all symmetry transformations. In the system c = > 2 ( &$I”~ - p”p(p) mentioned above, the scale transformation ~4~1 -3 ~3x1 = exp Wl9(~>
= 9(x> + aOcp(x> + We’>
is a local continuous symmetry of the system by our definit’ion but’ is not’ of t,he required class, since t’he Euler equations of motion from ,c( 2, +( .r), a,,+(s) ) when csonsidered as a function of cp(.r), ?I,+( x ) are changed by a multiplicative constant from those of t,he original C. [See Strudel (7) for an analysis of such scale transformations.] The Euler~Lagrange equations of motion from the two Lagrangians arc [G(.r, $%(r),
. . .)Ja, = 0
and
where the Lagrangian
derivative
of
is t,he expression
Lqo, = g - d, I$$ + ’ . . + (9
s s
l)‘““i&,
. . . r&
aa
““. d -. Ir, . . P,,,(OS
Since t’he equations of mot’ion are identical, [,~(r, &(L), . . .) - Z( .r, ps(.c), . . .)Jrp, vanishes identically. By a theorem of tjhe calculus of variations, [see (6 1, Vol. I, p. 1931, this holds if and only if Z( .L’, &(x:)~ . . . ) - ,c( s, P.7(J:j, . . j is a divergence term. Thus weconclude that) two Lagrangiandensities give rise to identical Euler-Lagrange differential equations if and only if for arbitrary functions c+Q~, the difference of the Lagrangian densities is a divergence term. Our condition on the preservation of the Euler equations of motion is equivalent t#o
for some fixed functions ,!I” and arbitrary &I). Kate that the fun&)ns L>” are ambiguous because we can add t)o D” any t,erm of the form d,T”‘( .z, cp,(.r ), . . . ) where T”’ is antisymmetric in p and p and have d,d,T” s 0 as an identity in the fields. This ambiguity extends to the conserved currents and is what is used familiarly to obt’ain the symmetric energy-moment’um tensor from the canoni(a:ll
GO
BOYER
energy-momentum tensor. In general, assumptions on the tensorial character of the currents limit the terms in D' possible under this ambiguity. We are here thinking of functions s and jr which are analytic in all their variables. Thus we may expand S(s, C&(X), ...) - &(r, P,(Z), ...) as a power series in the fields dz), d&@.?(X), . . . . The fundamental functions D' will be those found by removing from every term in this expansion one appropriate derivative d, . Any term of the form d,8,YP( x, cpV(x), . . a) = 0 where 5”“’ is antisymmetrical in b and p, when expected in a power series in its variables, would, of course, vanish identically. aVow we can expand t,he left-hand side of (3) to first order in the parameter 8 as
where we have used the “commutativity” of t#he kvariat’ion and the derivatives d, which is evident from the definition of the 6 symbol given by ( 1) in which no change of the argument x occurs. When dealing with quantized fields, t,he position of t,hc fields in a product is of importance. We think of the functionf,. as being inserted precisely at, the place in the power series expansion from which the field (F?(S) was removed by different#iation. Kow using Eq. (-1)) we can rewrite (3) as
-[cQ9,S(o*= &WIL(x,As), d/d&(x.),. . . M, where [c],, is the Lagrangian
derivative
with respect t#ocpsand
TV% = (ac/aa,cp,)scp,+ e.. - D' where t,he terms in W’ other than
(5) (6)
D' arise from the identities
used to transfer the derivatives on 6q, in Ey. (4) over to the form required for the Lagrangian derivat#ive in (5). However, the Euler equations following from t,he application of Hamilton’s principle are precisely Bl,,
= 0,
I‘ = 1, 2, ...
, N.
(8‘)
Hence, when the functions cpssatisfy these Euler equations of motion, the identity ( 5) becomesthe conservation equation
0 = a,w”. Thus W’ given by Eq. ( 6) is a conserved current associated with the symmetry.
SY~I~IETHIES
ANI) CONSEHVEI)
CURRENTS
451
An elementary example of the c*onstru&ion is found for the case of a fixed space-t,ime t,ranslation in t’he free spin-0 1,agrangian 2 = ,!~(c?,,(FX’~- p’qq). Here the mapping is usually written in the form 2‘ + E’ = 2’ + 2’0, 4-r) + +,-(.?1 = p(s), with t,he definition of an invariance of the Lagrangian in mind. WC WII writ,e this in t,he form ( 1) preferred in this paper as P(.P) + cj3(x) = p( .P - ot ) = lp(x) - a,&?? + O( 8). Corresponding to ( 3), we obtain
either by direct substitution and computation or expanding the left-hand side 01 the identity C( +( c?), J,@(2) ) = C( p( s ), a,~( Y)) about I = .r. Thus we identify the divergencaet,erm of our analysis as D’ = JZE%,and the conserved current, of Eq. ( 0) is familiarly W” = dl;pdpq8 - 22’. This is identical with the usual result from use of Soether’s theorem. A secaondcx:m~ple for a symmetry of the system whicahis not an inv:ukm(.c of the Lagranginn is t’hah noted above for c = ,!~~,,qY’~. Here we cm (*ompute directly that, Z(d,+(s))
-
-C(ffl,p(.r))
= d,(pd”X(x)19).
We thus find the divergence term
and the conserved caurrent of Eq. (6) is w” = a$X(x)
- gPX(x).
I\‘ot,e bhat in this derivation of the conserved current, we use the requirement, of the invarinnc*e of the Euler equations from the original and t,ransformed Lagrangians, but we never make explicit use of t,he fact that the mapping pr( x) + &(r) is a symmet,ry of the system. In Se&on 3, we show that, every mapping which exactly preserves the Euler equations of mot,ion, or equivalent,ly, for which ( 3 ) holds, is indeed a symmetry Of the syat,em. We summarize t,he construction of a conserved current from a continuous symmetry of the system as follows. Zj’ a rrqpz’ng p,.(:r ) + &(s) = P,.(X) +&,c~(.r,) (?/ the forw ( 1) gives ISq. (3) for some jixerl junctions D’ and arbitrary qv , then the mapping is a symmetry t~ansfomation of’ the stlstsw oJ’ the t!ype giaing served curwnf W’, which cwrenf is given by Eq. (6). 3. DEMVATION CONSERVEI)
We
now
OF -4 COXTINUOUS CURRENT
SYI\I;\IETRY TRANSFORMATION
vise to a
con-
FROM A I,o~AI,
turn t,o the converse of the previous sec+tion.WC will show that from
4:i2
BOYEH
some local conserved currents, one can deduce continuous symmetry transformations which in turn give rise to the original local current by the construction indicated in t,he previous section. Considering the same system of fields pl- and Lagrangian ax,
4%(z), 4fPr(2),
. . . ),
we now assume the existence of a local current W”( 5, C&.X), dppT(x), . . ) c011served by virtue of the Euler equations of motion satisfied by the fields p,. . Then W is immediately associated with a symmet,ry of the system of the form indicated in the previous section, if and only if for some non-vanishing functions SAT PAJJt:), 4&z), . .) -[~l,J?
= d,W”
(9)
holds as an identity for all fields + . However, in general we are given only the equation 0 = a,W ‘, valid when the equations of motion hold, and we can not conclude that we can write an identity of the form (9). The obvious suggestion by comparison of Eq. (9) with (5) is to try to define the symmet,ry transformation as the mapping cpr(x) + (p7(X) = cpV(t) + &,( .r j with 6~~ = Ofr + O(IF?“) where the f are the functions appearing in t,he identity (9). The procedure of proof consists in reversing the steps in the construction of W ‘. The Lagrangian density &‘(X, &(x), a,,&-(x), * . . ) can again be expanded to first) order in e in the manner of Eq. (4). Then using Eq. ( 9) and the identities (71, we have
which, with the appropriate designation of the functions D', is precisely the Eq. (3 ) required for the derivation of the conserved current W’. We note that therestill remains someambiguity. If W” contains any terms which vanish identically when we take t’he divergence d,W’, then t,hesewill not be reflected in the functions jr . Hence we arrive back from the mapping cp,(I) + C++(X) + @4x, CPA JJ:), . . . ) + 0( 0’) only if we depart from the choice of the fundamental functions D ’ t#oinclude these further terms in 191’. We thus have proved that if a conserved current W” can appear in the identity (9)) wit,11not. all the functions f, vanishing, then there is a mapping (PJXf + &CC) which gives t,he conserved current in the manner of the previous section. It remains to be shown that this mapping doesindeed preserve identically t)he form of tQe Euler equations from the initial and transformed Lagrangians, and also that it is a symmetry transformation by our definition.
The fact, that t’he mapping leaves invariantS the Euler equations follows from the theorem of the caalculus of variations quot,ed above. The proof follows immediately from t,he property i-hat when regarded as functions of ~.(.r ), a,(~,( .r), . . t,he two Lagrangians differ only by :r divergence term. This can be cSonverted to a surface integral in t)he action, and so makes no (~~JIit~rihutiOtl in t,hc application (Jf Hamilton’s principle which requires that the variation of qFrand all suihably high derivatives should vanish on the boundary surface. Finally, we must, prove t,hat, tho mapping is indeed a symmetry traIlsfornl:ttioll of the syst,em in the sense of the clcfinit~ion given above; i.e., whenever qr sat isties the Euler equations
[C(s, ys(.rj, ‘. . ,I,, = 0, then zF satisfies
where
This is the statement that +,( .r) is a solution of the original Euler equations whenever ppr(.r) is a solution of these same equations. We form the action int8egral (Jf 13~. ( ?I) with the second term tfransposed to the right-hand side over the full region B of physical interest, and apply Hamiltjon’s principle for a variation ++( .r) ---f qra( .r:) = p?(s) + NT,(X) where q( .r) and a suitable number of derivatives vanish on the boundary for 2, but is arbit’rary inside S. When qF1(.r ) satisfies the Euler equations of motion, t,he right-hand side of the expression has a stationary value and hence so does t,he left side. We have thus when qc,( x) satisfies the equat~ions of motion
where +5”(s) 3 qFra(.r) + a~,.~( P). Carrying out the derivative in CLand then integrating by part,s in .r where we can discard the surface terms since qr( .r‘) and it,s (first several) derivatives vanish on the boundary, Eq. ( 10) becomes
We now consider Eq. ( 11 ) as a power series in the paran&er 0. By separating the contributions to the terms which are zero and first order in 0 and using the assumption that the Euler equations of motion are satisfied for cpSwit,h respect to
454
BOTER
a,.c, (-PI(X),. . . ), wc
see that we must have
0 = s, f14.dC(X, d-L.), . . ~)l&h,%(X)
(12)
at, least to first order in 0. Analysis for higher orders of 0 gives the result for all orders. Since Q(X) was assumed arbitrary in the interior of Z, we deduce that the quantity multiplying Q(Z) must vanish. Thus we have, as required, that, Mz, dd, . . .)I&, = 0 holds whenever (p7satisfies the equations of motion. The invertibilit,y of the transformation follows from the assumed expansion to first order in 0 and the continuation of the transformation by iteration. Note that we have made the tacit assumption that the functions 6~~ = 0f7 + 0( 8) are welldefined when t#he fields satisfy the equations of motion. We conclude that any transformation (p7(zr) 3 &(z) = qr(z) + 6~, satisfying (3), and in particular t.he transformation derived from the conserved current 157” in (9)) is a local oneparameter continuous symmet’ry transformation of the system. We noted that it is not generally true that every current W’ conserved when t,he equat,ions of motion are valid is of this form. Since W” is conserved when t,he fields satisfy the equations of motion, it might seemnatural to allow changes in the functiona for~z of W’ to an “equivalent current,” 151’”which still maintains the samevalue asW” when the fields obey the Euler equations. (This is an entirely different question from that involving the ambiguity of the functions D’. In general, a change of W” by a function dpTBP(z, p,(z), . . . ) would change the zlalueof t’he current..) One example would be to change t,he fun&)nal form of W” over to 191’”= w” + [E],,hr + a,[C],,h,P + ‘. . . In spin 3; electBrodynamicswith
R = $8( $‘a”#
- m#j - J&F”( d&A, - &A,)
+ xF”“F,y
+ e&$‘$Ap
the current W’ = -a,( 8‘A” - d”A”) is indeed conserved but a,,W” = -8# (#A” - #A”) = 0 holds as an identity and we can not find any nonzero functions jr for Eq. (8). However, we can writ,e 151” = wp - 2&[C],“’ + [Z],, = e$-f”l), and so obtain
-[S]+(
-ie#)
-
(if$)[~]~
= a,w’“.
In this manner, we associate the current with the gauge symmetry t)(x) + exp [-ieel+(
&cc> * G(x) exp [+iee].
SYMMETRIES
AND CONSERVED
43.5
CURRENTS
We remark also that if we are able to write d,,W’ as -[J3],,gr - a,[cl,,gr”
-
. . . = drW’(x, R(X), . . j
(13)
for some nonvanishing functions gr( .c, ‘py(z), . . ’ ), gTP(Ic,pFr(CC),. . . ), . . ., theu we could rewrite this in the form -[s],,fr
= d,(W”
+ [C],,g,p - . . . ) = drw”qx,
$&(.I-), . . )
(14)
where jV = gT - d,,gPP+ . . . Here again W NPis a new function which has the samevalue as W’ when the fields satisfy t,he equations of m&ion. The symmetry corresponding t,o ( 14) would give a (*onserved current in the fun&ma1 form W”‘.
In the last, section of this paper (Subsection II.4), we will prove that for the casesof interest in quantum field theory, it is always possible to find an equivalent, current W” and an associated symmetry such t,hat W” is generated by the symmetry in t,he manner of Subsection 1.2. 4. INVARIANCE NOETHER's
OF THE ACTION THEOREM
INTEGRAL
AND (:ONSERVEI)
CURRENTS
FROAL
At this point it would seem helpful to review briefly the content’ of Noether’s theorem (9) in order to clarify its relevance to t,he problem of relating symmet,ries and conserved rurrents. We arc concerned here with transformations dependent, upon a single real parameter 0. Hence the part of Noether’s paper which concerns us is the proof that the action integral is invariant under a oneparameter continuous transformation if and orlly if an identity of t’he form (5) holds for suitable functions W’. Specifically, the hypotheses assume a function e(.r, G(X), 4dfd~), GLd.r~, . . ) and a continuous mapping .r + 2 = h( x, (FT(x), d,p,( x), . . . ) = x + 6x,
pJx) ----t&.s(fO)= RSx, P.(T), d,cpx(x),...I = dx)
+ ~*PAx),
( 15)
where 6~ and 8*pprcontain terms that are first, order in e, and O(e”), such that
for an arbitrary region !J and image region fi for all values of the continuous real parameter e. Sote in part,icular that .Z may be a function of the fields cFr(zj, 4i’Dr(~!, . . . This is not the casefor what we have defined asan invariance of the Lagrangian. The analysis proceeds as follows. We combine the integrals over the single arbitrary region Q to first order in 0 by approximating the integral over the differ-
1.5Ci
BOPEII.
01’ tht two regions a - I! by s a,1do,,&r”~ where C?Qis the boundary surface of !?, and then using Gauss’ theorem to convert, t,o :L volume integral over fi. (These elementary manipulations are carried out in det,ail in the paper mentioned in refcrencae ( R ).) We obtain to first order in 0
enw
0=
t/“r[GLr, s 11
$&(.c), dp&(“r),
. . . 1 - se(.r, (FJ.I.1, dpppr(.r), . ” ) + a,(cwd:(.r,
q%(r), d,qdJT),
(17) . . . I,).
i\‘ow the bracket may be simplified by the same procedure given above in Subsection 1.2, t,hrough the expansion (4) and ident,ities (7), to give 0 = s, ri4.r(,b.,.6c,
-
d, (g-“,
a r
+ ‘. . -
again to first, order in 0. Since Eq. ( 1s) holds for arbitrary must vanish and we derive the ident,it,y -[[C]&LYr
C&r”
)I
)
(1s)
regions a, the integrand
= d,W”8
(19)
where ‘W’O = (d6/ddpcp,)6q3, + . . . -
62s.
(20)
This result is precisely of the form of the identity (5) when we identify 62’~ wit’h D'. As of yet, we have made no reskictions on t,he character of the functions P~( x). If, however, we now assume that the fields (p7(s) satisfy the Euler equaCons from the applicat8ion of Hamilt80n’s principle relative to s 1 t14”rC(“r, @4x), dr(Fr(.r.),
. . ),
where S is the entire physical region, then we have [,c],, = 0, 1‘ = 1, 2, . , N so t,hat t,he left-hand side of ( 19) now vanishes, and we derive that the local current ‘157’ is conserved, d,W’ = 0. The queet,ion of the converse problem for _\oet,her’s theorem (somesto t
(22)
With these definitions, it is easy to reverse the procedures of Noether’s derivation out,lined above to obtain the equality of the action integrals in Eq. ( 16).
Sc)tc
that’
Noether’s
c*ormerved This only
is the
the
identity
REI,ATIONS
the
arc
BE’I’\VEEN
:md invariant
c~hangcts
and
terms
Eq. real
out
Euler
equntious
other
for
one
terms because
differeut
symtnetrp for
The
itlterest
q 4
lies
selves,
the
from
the ;tntl
are
futictiotis
when 9J.c)
the
from
arise
forth a&ion
the
divergence theory,
derivation,
it
terms
have
tcrnl
it1 t,he
c~onserved of
divergence
in no
no
c+urretltJ the
identit,y term
I)“.
( 5) always
sirlcae
+,( .c)
one
We
that
this
would the
of
make
linear
the
iderttic~ally.
vanishes
identically.
Any
contribution
tto
( The
of such I,agrangim
All change,
question to
t,he is a
current. that
in
generators, cahartgcs
Lagrangian
currerlt.
c*ontribution
vanishes
rise
of
mapping
of coordinate.
mlserved
secondary
cshauges
operator
), gives
cahattge
alteratiotts
differential
a coordirtat,e this
crucial
OIltO t,hem-
a caouserved on
ex-
example the
Euler t,he
from
as an aside
paper
to my
earlier
this
a c~ontinuous of
whkh
an
il’o
from
of time-independent of
change
is that
of the
appear
ttotc
for
Lagrangiatts.
whet,hcr
divergetlc*e
of motion
separate
CJf
case
recaonsider
form
cmstructiotl
associated
C itself W”,
+
of coordinates
equations the
Forrnulatiott
II.3
on t’he
Lsgrattgi:u~
the
referet1c.e clear
effect
it1
test
applic*atiotl
Subsection
hec*otnes
&( .r )
use of coordinate Hence
author
transformed
changes
t,ertns
immediate
(V)II-
+
appropriate
SOIlle
will
present’
preserves and
Xoethcr
the
the
allow
work
integral. by
We
the
of the
it1 the or
in t.he
from
make
and
transformed
P, pFr( x)
The
hetwecn
absorbed
t ~d,qY‘q. by
initial
relevant
field
illustrated
whicah
=
mordiuatc
ever
which,
x
mapping
the
to
and
.r +
Lagrangim.
of the
a/mz,?/.s
of solutions
tr:ttisforni:ttioti, S.rP,
quantum
side
preferred
mapping
qFr are
tcrnms
(as
X in
involving
synintetry here
(o +
it1 the
c:onsiderutions
if and
D,“. invarinttc~e
view
of
for
:t
IA~v~4m4n-c~~s
pr( x)
origittal
transfornmtiot~
h:tc*k
they
point’
cspecinlly
equations
ANI)
:tpproac~hes
6x, even if it’ is necessary to make n change on the vnlues of the field. q qF7( .I’), as is indeed the
below.
holds
symmetries
ttm1sfortn:~tion
qr from
:I Noet)her
couvcrt
is it1 the
appear
itivolvitig
( 16)
C:UIZ~~ENTS,
two
for
integral
cm
divergeme
ample
whet1
interest.
of caoordinates depends
as to
( 19). about
of Soether’s
w’c’ presented
current,
the
ittterest
rterle~
question
identity
theorem:
CONSERVEI)
the
that
Soether’s
c~ruckl
of the statements
scope
SYMAIE~~UES,
t,he a(+ion
to apprcciat,e
I~urther
the
sec*ticms,
iuvariant
leaving
the
on
form
INTEGIZAL
previous
I,ugr:utginns,
hint
of Soet.her’s
holds.
beyond
of :t c*otlserved
lcavitlg
no in the
c~otltent
( 19)
:~(‘TtON
structiott
gives
be writktl
entire
czurrettts
OF THE Iti
K~II
then if
conserved 5.
theorem
caurrent
terms
.r
with
by
divcrgettc*e
pure
to the
&(
Also,
j
this
divergence fundutnetttal on
derivat.ive
the
left of :t
)?
L Olte furt,her advantage in omitting ally reference to :t coordinate change 6.z might be that it worJd reorder less likely remarks whose talltological nature is hidden under the variety of symbols. The note by Romnll (5) 011 ihe effect of divergence terms in 2 seems entirelv misdirected.
45s
UOYEH
It, is significant that although we have considered when cpr(s) and &(x) satisfy the same Lagrangian equations of motion, nothing has been said about the equations satisfied by the functions &( 3). Indeed &(z) considered as a function of L is in general not a solution of the same equation of mot(ion as (C,(X). Nor do we obtain any easily ident8ifinble equation of mot,ion if we apply t,he full Noether mapping (15) to the different8ial equation of motion for &z). If we consider the Lagrangian 6: = J,$,,&‘(p for one of our examples, then t,he symmetry of the syst,em P(X) + 1;1( x:) = q(e) + 0x(r) + 0( 0’) where a%( X) = 0 is associated with the very cumbersome Soether mapping
which is an invariance of the action integral and gives rise to the sameconserved current W” = d”(oX - (~8% obtained earlier. However, ~(3) considered as a function of z is not a solution of the original equation of motion; here a,a”(p(%) # 0. Also, the full Noether mapping above applied to $J”p(s) = 0 gives a complicated equation of motion for +( 2). We may summarize the difference in att#itudes by viewing t#hemas diverging from the identit,y (5) or [ 19) which is the pnint, common to both. Invariance of 1,agrangian Present, Analysis Invariance of Euler Equations
IdenMy
Noct,her’s Theorem Invariance of (5) or (19) s Action Integral Yields satisfy Euler Equations
I Symmetry of System
I Conserved Current’
To the knowledge of the present writer, virtually all authors, [seefor example Hill (4) or Roman (S)], have tacitly adopted an attitude intermediate between the two above-mentioned positions; they allow coordinate changes 6x(.x) independent of the fields cpr(x) but also include divergence terms separately into the new Lagrangian. Our feeling is that in a general analysis, any reference t,o a coordinate change is superfluous in the actual derivation of t’he local conserved current.
SYMMETRIES II. RESULTS 1.
SOME
AN11 CONSERVED
AN11 EXTENSIONS QUANTIJM
LIMITATIONS
RELEVANT FOR FIELD THEORY
OF THE STRWTURE
459
CURREBTS LOCAL
OF QUANTIZED
LAGRANGIAN
FIELD
THEORIES
ln the previous sect,ions, we have tried to present with some generality the problem of connecting symmetries and conserved current’s. Here we wish to review and extend t,hese results for the c*aseof local Lagrangian quantum field theories as formulated by n:iive cxronical quantiza.tion and by the Schwinger action principle. WC restrict, ourselves to the following t’wo syst,ems of classiral or quantum 13,&c)). fields I&X‘), i’ = 1, 2, . . . , N with Lagrangian density 2(x, opt, Case ( i) (Naive Canonical Quantixation). The Euler equations of motions are second order partial differential equat,ions in which the second order t,ime derivatives of the fields are regarded as dependent, and all fields, space derivatives and first time derivat’ives are independent at a point. Because of the form of t,he Euler equations from C( .z, qr( 21, a,,~.( x)), if &&P, appears in [ZIPr , then &&~l- appears in [J-?],~; consistency requires t,hat the identities
can be solved for the second time derivatives in the form dodo~r- ‘I’,( x, pFs,d,~ , d,d,d
= Ark .I’, pr , a,(~s)[c],, ,
I’ = 1, 2, . . . , N
(21)
for all functions p’r . An immediate example is the free spin-0 I,agrangian c = !,*~(a,,&~ - p’~pF). The identity (24) is here
Case (ii) (Schloinge, Action Principle). All t,he Euler equations are first order partial differential equations from the Lagrangian d: = f i(‘pr.@:,&cp,~ d,,+ .@$,c6) - 6( cp&) . If any field pr must be regarded as dependent, or if any first derivative of an independent field, (this includes all first time derivatives of independent fields), is dependent, when t,he equations of motion hold, then we can write as an identit)y holding for all fields qr (dependent quantJity) -3 = a,[~],,
+ &‘i3k[&J,,
(25)
where 3, a, , and atk are functions of only those quantities which can be regarded as independent at a point z, even when t,he fields satisfy the equations of motion. Quantixation of t,he field theories, of course, involves many further restrictions. [See Schwinger (IO), p. 919 (1951); p. 716 ( 1953).] However, the limitations not,ed here are those which will play a role in our analysis, especially in deriving symmetries from conserved currents.
4tio
HOYEH
The previous analysis of Subsection I.? for t#heconst~ru~tion of corlservcd NI'rents may be taken over in its entirety; limitat~ions WI the strurt,ure of t,hetheory here add 110 new information. Simply, given my continuous mapping pr(.r) -
&(K)
= pr(.r) + &p,(s) = SJS) + B.&-r, p?.(.xC,a,p,(s‘,) + fh02),
we construct C( .L’, +,( .r), a,,;,(x) ) but, now regard it, as a function of @Js 1, ap(F?4 d) d,&d .r) , . . ; i.e., everywhere +,.(.r ) appears, we replace it by qr( s ) + 0fr( s, pr( s), a,~,.( .I:)) to first’ order in 0. Kow subtract C( s, pr( .I’ ), 8,1pr(x) ). If we can find some funct’ions D' such t’hat, to first, order in 0, C(r, &(x1, d&(-c))
- C(r, pArI,
ap4d.r) 1 = a&“(s,
p,(s), d,p,(r))
t “6)
holds as an identity for all fields (F~(.I’J, then the mapping is a cont8inuoussymmet,ry transformat,ion of the class assocktt~edwith conserved currents, and the conserved L current W’ is W”% = ( a~:jaa,Lc,)jyo - 1)“.
( “7 )
We note from the construction of W”, using t,he fundamental functions 11” of Subsection 1.3, t,hat#because both t,he functions c and jr involve no derivatives of order higher t,han first, the same must be true of W”. 3. INTEGHALH SYMMETRIES
OF CONSERVED
C~RIZENTS
AS QUANTUM
GENERATORS
FOR
If the current, ‘W’ is conserved when the fields satisfy the equations of motion, then for fields which can be regarded as vanishing at spatial infinity,
is a time-independent quantity. One proof makes use of a four-dimensional Gauss’ theorem
taking t,wo of the bounding surfaces as rn = cl , xc = c2 for cl aud c2 conskmts, and allowing the remaining surfaces to receed to spatial infinity. In the case of quantum fields, the condition on the behavior of the fields at spatial infinity should be viewed as a restriction on the allowed domain of states. [See Schwinger (IO), footnote p. 917 ( 1951).] It is usual in quantum field theory to regard the operators W arising from symmetries as the generators of the original symmet’ries. Hence it is of interest here to ascertain the circumstances under which the int,egrals of the local conserved currents from these local cont,inuous sym-
SY3lMETIZIES
AND
CONHEHVEU
l(il
CUHKENTS
metries do indeed serve consistently as t#he operator generators of these same The analysis presented is more general than that symmetry transformat,ions. available for invariances of the Lagrangian---which invariances require that 6~ is independent) of the fields. k~ analysis of quantum operators for symmetry transformation both local and nori-local strictly from t,he point of view of canonically quantized field theories has been given elsewhere” by the present author. In the case of quant,um field theories resulting from naive canonical quarttization, the answer to this question of consistenc~y is that for symmetries of the form
involving no higher than first derivatives of the fields, the integrals of the currents always generate the original symmetry. We saw in Subsection I.2 that the functional form of d,W’ associat~ed with the symmetry was given by the ident,ity (5) which here takes the form
- _cas - aP$1 ( aa
fX. c,pa(x),a, cps(;c))= a, w’(.r, $%(:I:),a, q,<(x)1.
(29’1
Since, as we noted above, W’ is a function only of t.he fields and first derivatives we can determine the functional form of dW’/d&cp, by t’aking the deriva.tive with respect to d&~,,. of both sides of (29). We t’hus find
again as an identity theory, we have
However,
holding for all funct,ions pr . In the quantized
from the original canonical
3 T. H. Boyer, Nzcovo Ci’mento 44A,
quantization,
613 (1966).
we require
version of the
42
BOTEH
bl-(~),
WI
=
+c-6
e(2),
(33)
Gf4~))
precisely as required. Since W is invariant under time and space translations, it also generates the appropriate transformations for the derivatives of the fields. In the case of quantum field theories from the Schwinger action principle, t,he int’egral of the current acts as the generator of the original symmetry only in the case that all of the fields are given in t’erms of the canonical variables. In simple Loreritz transformations are particular, for quantum elect’rodynamics, not reproduced by the quantum generators in the radiat’ion gauge quantization but rather further terms are int’roduced; there are also gauge symmetries giving rise to conserved currents whose integrals do not act as generators for the symmetry. For spin+ qua&urn electrodynamics specifically, the symmetry of the system 1J4.s) -3 exp [-;eN,.r)0]+(z), A,(x)
-j A,(.c)
-
$(.r) &A(x:)e,
4 $(.c) exp [+;eh(X)e], P”“(x)
+ F’“(z)
produces the conserved current W’ = e$-f#(x)X(x) - F”‘(.r)d,X(x). However, the space integral of the zero component will not generate t#he required quantum symmetry. The failure of this generator W is symptomatic of the difficulties in quarkizing the electromagnetic field. In the Schwinger radiation gauge formalism only t,he t,hree-transverse parts of the fields are canonical variables. We remark that the integrals of the quantum currents from gauge transformations in the Gupta-Bleuler formalism, where na’ive canonical quantization applies, do indeed generate the required gauge transformations. Again we take W“ from the fundamental functions D” and so conclude that if c and fr contain derivatives of at most first order, t’hen so does TV”. But now from the identity (5)) or ( 29) here, which again holds for t.he construction, we seethat the left-hand side involves only first derivatives, since all the equations of motion are first order. But then (,dw”/d&zcor)drax’Fr = 0
(34)
for all a,~~ . This can hold only if dW”/&3,p, is antisymmetric in p and 0~.In particular, this implies dW”/&30pp = 0. In the Schwinger action principle, the conjugate variables are pairs of fields pr and (P+such that in d: all time derivat,ives of these fields appear as >$(Q+.d~cp,~- I~~P~.c,+‘).But then we can consider the partial derivative of the identity ( 29) with respect to &cp,t and so derive (35)
SYMMETRIES
AND
COSSERVED
CURRESTS
463
In deriving the terms on the right-hand side, we have made use of the nntisymmetry noted above, and also the fact that W” involves derivnt,ives of at most first order, to find t’he relations
The effect of the generator
W on p,. is computed as
where we have made use of the equal t’ime commutation
rules including
[cpr(x),cpd~jl* IL,I=L”r= ;63(x - $7,
(37)
of an integration by parts in three-space, and of t,he identity ( 35) now employed when the fields satisfy the equations of motjion so that [s], = 0. (The analysis can be followed easily in t’he example of space translations of the free spin-J+j field,
The time and space-independence of W means that, the generator acts appropriately on the derivatives of the canonical fields. -Vow the dependent fields arc connected by the equations of motion to the independent fields and derivatives of these fields as indicated in the identities (,2.5). Thus W also generates the required transformation for all t,hose dependent fields cormected to the canonical fields. The generator fails only for those unquantjized fields which are not, connected to the canonical fields. In considering t,he consistency of the quantum operator generators arising from symmetries through conserved currents, we have taken the current arising from the fundamental divergence terms. However, the current could be modified by terms aP”(.~, (O,(X), . . . ) whose divergence vanishes identically due to the nntisymn&ry of T”” in p and V. When considered in the integral ( ‘28) for IV
40-l
HOTER
such terms contribute J” d3x8~7’“” which may be replaced by a spatial surface integral. In the case that hhis surface integral can be regarded as vanishing, the t,ime-iildependellt quantity W is t,he same as that obtained by omitting such terms. We note also that if we are not concerned with the tensor aspects of the currents, then from a conserved current W” we can obtain other conserved currents merely by taking derivatives W:,) = d,W”, etc. However, on taking the space integral of the zero component,, the new quant’itfy becomes a spaGal surface integral or vanishes. 4.
THE
ASSOCIATION
OF
C~ONHERVEU
CURRENTS
WITH
SYMMETRIES
The second half of the problem considered in this paper, t,hat of finding symmetries associat,ed with conserved currents, allows substantial comment when restricted to the relatively simple forms required for quantum field theory. Here we can decide that any nontrivial local conserved current has associated a symmetry transformation producing a conserved current of magnitude equal t’o t,he original current when the equations of motion are satisfied. The analysis bears a basic resemblance to a theorem of Fletcher [see (51, p. 741 which the author discovered after carrying out the very natural procedures described here. Also, during the revision of the present paper to move the examples from footnotes into the body of the text, an article by Dass* appeared in print. This paper purports t’o carry out the converse construction of symmetries from conserved currents for arbitrary Lagrangian systems. The crucial st,ep in Dass’ general analysis turns out to be the first, where it is asserted that every conserved current W’ WII be written as an identity d,W” = f([61p,). In our analysis to follow, we will note that for some current’s W’, for example all those associated with what Fletcher terms “strong” conservat’ion laws and also for some trivial cases, we have d,W” = 0. The restrictive assumptions we introduced earlier were made so as to allow more than a heurist’ic argument’ in the converse construction. In the case of quantum field theory, it seems ent’irely natural to change the functional form of the current 157’ while maintaining its value. The quantized version of the theory contains meaning only when the equations of motion hold, and indeed t,he action of operators on the fields is defined only through the equation of mot,ion by the action on the canonical variables, which are again determined by the form of the equations of motion. Thus a natural procedure for finding a corresponding symmetry transformntion is as follows. First we separate out all the distinct conserved currents WI;,,. ,4 power-series expansion of W” in the fields p, , arcpr , . 1. would allow one t’o 4 T. Dass,
Phys.
Rev.
160, 1251 (1966).
SYMMETRIES
AND
(‘ONSERVED
C’URRENTS
405
select a monomial term and then t,o choose a minimum number of further terms necessary t’o obt,ain a conserved current, WY;, . Then we remove all the derivutives which allow W’;O to be written as the derivative of a conserved current %‘;;I. This last ?6’;!, is t,he current W’ which we now analyze for :t relationship t(J a symmetry transformation. We proceed by removing all dependent fields and derivatives by use of the equations of motion. Then on application of t’he conservation condition 0 = d,W”, the only dependent’ quantities arc the further derivatives of t,he fields. The need for somesuch consistent procedure in eliminating the dependent variables is illustrated in the example of spirl-,l,2 quaut,um electrodynamics. The current, \Q“ = -~3,(#‘~4” - i3”.4’) gives d,W” = 0. However, when reduced t,o an expression in terms of the independent fields, this is immediately seen to be the electric current associated with gauge transformat,ions of the first kind. In t,he case of field t,heories allowing na’ivc canonical yuantization. we can remove the dependent derivatives by the use of the identities ( a4j and so find
as an identity for all fields qI . We can rewrite thk as noted in Suhsect,ion I.3 as
(39) where W”’ has t.he same value as W’ when the fields satisfy tjhe equations of motion. We remark that here W”’ and 191” have also t’he same funct,ional form. Then Eq. ( 39 ) coincides with t)he identit,y ( 5) and so if the right-hand side does not vanish identically, we can find a symmet#ry gcnerat’ing a conserved current W”” which differs from W”’ only in terms of the form d,T’”where 7’“” is antisymmetric in /* and Y. We have noted t#hat such t)erms give spatial surface integrals when we try to form an operator generat,or II’. In the casethat t)hebracket on t,he right-hand side of (39) vanishes, we conclude that t,he original current] was kivial in the sense that even when expressed in terms of t,he canonical variables, the conservation of t,he current does not, depend upon the equations of motion for the dynamical variables. For example, the csurrent W” = 131~ W1 = --~?~q”,\Q2 = &q, W” = -&+c is trivial in this sense.The possibility that -w’ was the spatial derivative of a conserved current, in which case our transformation from W’” to WN’ would give a vanishing identity ( 39)) n-as spec3ically avoided by the initial procedures. An analogous result holds in the case of the systems allowing applicat,ion of the Schwinger action principle. In some casesof fields in the Schwinger formulation, the procedure described here is unnec*essarily circuitous, although the symmetry found is unaltered. For csample, for the free spin-0 field c = $q”.d,q
-
! gLp”.p
-
,I yp”U” - ~‘q(p, the current cp+ P + a&,
corresponding
to space translations
'pp + p* + apgtP,
w@ = ,!,p*.ap(FtP - f~dppP.EPp - (>$pY.dvp - ;&(F".p
- ?yp"pv - &p)tW
allows immediate present,ation in the form of the identit,y ( 5). However, the procedure prescribed in the test would remove the dependent fields (ok, &q and convert to W’” involving second spatial derivatives. Partial integrations of t,he form mentioned above are then necessary to arrive at the identity (39). If we had used the Lngrangian of naive canonical quantizntion A? = .1;(i)PPDd”P - M’~(F), then for CF+ CF+ d,,&’ we have
w" = avpaPpEp- j 2(a~c"as~v - p'?p'cF)tY, which does not involve any dependent quantities so that our rule gives immediately the identity (5). We conclude that nontrivial currents can be traced to symmetry transformations for these very restrictive field theories which allow quantum interpretations. ACKNOWLEDGMENT I would
like
to thank
L. DeRaad
for reading
the manuscript.
RECEIVED: October 27, 1966 REFERENCES I. 1). HORN, kn. Phys. (S. Y.) 32, 444 (1965). 8. T. D.\ss, Phys. &c,. 146, 1011 (1966). 8. P. ROMAK, X7covo Cimenlo 10,5% (1958); H. STEUDEL, Z. Sa/wforsch. l?A, 129(1962); 1). B. FAIRLIE, Suouo Cimenfo 37, 897 (1965); H. STEUUEL, X71ovo Cimento 39, 395 (1965); C. A. LOPEZ, N~tovo Cirnenfo 40, 19 (1965); T. D.\ss, Muovo Cimento 42A, 730 (196(i). 4. E. L. HILL, lieu. Mod. Phys. 23, 253 (1960). 5. J. C. FLETCHER, Rev. Mod. Phys. 32, 65 (1960). 6. P. ROMAX, “Theory of Elementary Particles.” North-Holland, Amst,erdam, 1961; T. H. BOYER, Att,. b. Phys. 34, 475 (1966). 'Y. H. STEUDEI,, 2. Natwforsch. 17a, 133 (1962). 8. R. COUR.YNT AND 1). HILBERT, “Methods of Mathematical Physics.” Interscience, New York, 1953. 9. E. NOETHER, 1Vachr. Akud. W&x. Gottingen, Kl. II, Math.-Phyik. 234 (1918). 10. J. SCHWINGER, Phys. Rev. 82, 914 (1951); Ibid. 91. 713 (1953).
OF
PHYSICS:
Continuous
43, 445-466
(1967)
Symmetries TIMOTHY Harvard
University,
and H.
Conserved
Currents
BOYER*
Cambridge,
Massachtlsetts
The general relationship between one-parameter continuous symmetries and conserved currents in local Lagrangiatt field theories is reexamined. Defining a symmetry transformation of a system of fields as a one-to-one mapping of the fields carrying solutions of the equations of motion into solutions, it is noted t,hat not all symmetry transformations imply conserved currents, and conversely, not all conserved cltrrents can be derived immediately from symmetrics. However, we develop the one-to-one correspondence between those local cont inuotts symmetry transformations of the system which leave invariant the E:ltler-Lagrange eyrtations from the original and transformed Lagrangians, and local conserved crtrrents which can be written as linear ftttlctions of the Lagrangian derivatives. It is shown t,hat the theorem proved by Noether, which is customarily referred to in the const,ruction of local ronserved rttrretrts for some symmetry transformations, is in general related otrly tangentially to the problem in qltestion. The analysis presented for the corlst,rrtctiotls of conserved cltrretlts seetns of particular interest in the case that divergence terms enter the symmetry t,ransformatiott; also it illrrcidates the independence of conserved cttrrettts from the atnbiguityalwnyspresent itt Lagrangiatl formttlations dtte to divergence terms in the original system Lagratlgian. In the elemetltary cases of general interest in field theory, t,he procedrtres for the constrrtctiotl of conserved currents do not diverge significantly from the 1 raditional ones, altholtgh even here the difference itt att,itttde regardirtg the traIlsformation is Iloted. The converse construction of symmetries from cotlserved cltrrents has Itot been cotlsidered fttlly in t,he literat.ttrr. The getteral analysis is applied alld extended for systems of fields allowillg tlaive canonical qltantization or qttantiaation via the Schwinger action pritrciple. 111 particltlar, the consistelrry of the ittterpretation of integrals of the cltrretlts as qrtalllum generators of the original symmetry t ransformatiorrs is cspl~tretl in frtll getrerality. INTRODUCTION
There has been some continued interest ( I )-( 5) as t’o the relat,ionship behween one-parsmeter continuous symmetry transformwtions and conserved currenk in local Lagrangian field theories both classical and quantum. For some particulx symmetries, which might, bet,ter bc termed invarinnces of the Lagrsngian, one nssurnes the esistence of a continuous mapping of the fields which preserves the * Nat iotlal
Science
Foundation
Predoctoral
Fellow 415
446
I3OyEH.
value of the action integral, and then proceeds via Noether’s theorem to the construction of a conserved current (6’). Quite recently, Horn (1) and Dass (2) have considered in very limited quantum field theoretical contexts the converse of t,his construction; i.e., do all conserved current’s appear via Noether’s theorem from some continuous invariance transformation, and how does one recognize the symmetry associated wit’h a given current? Horn and Dass bot’h make explicit use of the usual spin-0 and/or spur-12 Lagrangians of quantum field theory. Their analyses proceed by use of quant,um commutation rules and stringent assumptions about the st’ructure of field equat,ions and of conserved current’s In this not’e, we wish to reanalyse the problem of continuous symmetries and conserved currents, not in t’he quantum domain but, in the classical context of differential equat’ions where the fundamental problem actually lies. The results carry over directly t#ot’he more specialized circumstances of quantum field theory. Section I gives the general analysis. Aft)er first clearly specifying the definition for a symmetry transformation which we shall adopt, we indicate a general construction of conserved currents for the special class of local continuous symmetries of the system which leave invariant the Euler equations from the original and transformed Lagrangians. In the following section, we not,e that every conserved current which can be writ,ten as a function linear in the Lagrangian derivatives is associated with a continuous symmetry of this specified class. Next we review the actual content of Noether’s theorem and conclude that the application of the theorem represents a detour from t’he question of interest. Finally, in Section II, we turn specifically to Lagrangian field theories which allow naive canonical quant’izat’ion or the application of the Schwinger a&ion principle. For these cases, it is always possible from any nontrivial conserved current to obtain a symmetry transformation which gives back a conserved current equal to the given current when the fields satisfy the Euler equat’ions of motion. Furthermore, the three-space integral of the zero component of tfhe current is shown, in general, to generate consistently as a quantum operatorthe same symmetry transformation on the fields. The arguments of the body of the paper are illustrated with sketches of examples relevant to quantum field theory. I. GENERAL
ANALYSIS SYMMETRIES
OF THE CONNECTION AND LOCAL CONSERVED
1. DEFINITION: A LOCAL ONE-I'ARAUETER FORMATION OF THE SYSTEM
BETWEEN CONTINUOUS CURRENTS
CONTINUOUS
SYMMETRY TRANS-
We consider a system of fields cpr(z), r’ = 1, 2, . . . , N whose time evolution is specified by a consistent and independent set of partial differential equations derived via Hamilt,on’s principle from the local Lagrangian density -C(z, (FJ .r), +f4~), ~&Pr(~), . . . ). Here, and in all functions considered in this paper where we indicate higher derivatives by dots, we are restricting the derivatives
SYMMETRIES
AND
CONSERVED
447
CURRENTS
arrlh2 . . . &‘pS to order n less than some n* finite and fixed. A local one-parameter continuous symmetry transformafiun well-defined for arbitrary functions
of the system will he defined as a mapping cp,(x),
= +d.q + of&:, d-4, add,
aravps(.4,. . ) + o(ey,
(1)
which for the continuous real parameter 0 maps in a one-to-one fashion those of t’he equatJions of motion into solution fields cp,(.c) which are solutions q,.“(z). Akhough we will continually consider equations only to first order in 8, it is clearly possible to regard these expressions as first-order differential equations in 0 which can be integrated to give expressions to all orders in 8. The analogous definitions and procedures involving symmetries are also available in the case of ordinary differential equat.ions. Thus a parallel analysis is possible for 1,agrangians of classical and quantum particle dynamics. We remark that in the above definition for a continuous synm~efry transforrrlaiion of the system,we have used only one single value of the argument .r. A traditional but much more limikd definit’ion occurs in what we shall call a local one-parameter cont’inuous inua~iance of the Lagrangian. This is a mapping I’ + 2% = x + 6x(x), 4-r
) 4 33 2”) = 4 x ) + 6*(~h)
= d.d
+ eddd)
+ 0(e2>,
(2)
which for the continuous real parameter 0, maps x and c,G.(2) in a one-to-one onto fashion such that ~(3, &(z), a,+,( z), . . . ) cZX/& when regarded as a function of .c, IP~X), a,dr), . , . is exactly &‘(r, Pi, a,,n,(r), . . . ). The symbols in all of the work presented in t’his paper st,and for mathematical functions, not for “physical quantities.” Not,ations such as aL?( .r, (mp(:l:j, . . . )/a++(r) stand for the partial derivative of the function d: wit,h respect to its second argument regarding all &her arguments as fixed. We assume all functions are analytic in their variables. In the recent work on symmetries and conserved currents, for example (1) and (a), t’he assumption seems t,o be present t,hat every “symmetry” leads to aconserved current and only the converse statement requires investigat,ion. Part of the difficult’y here is relat)ed to the variety of definitions for a “symmet,ry t,ransformation” which are abroad in the lit#erat)ure and folklore. Some writ,ers regard a “symmetry transformation” as what we have termed an irwariance of the Lagra?Lgian. We will find that indeed every ilwariance of the Lagrangian determines a syrrlmetry of the syste,ll whic*h preserves the Euler equat,ions from the original and transformed Lagrangians, and so indeed gives a conserved current. The proof can be obt,ained by rewriting the assumedequality
44s
BOYER
and applying the analysis of Se&ion 3. However, there are synmetGes of the syslelfz which give rise to conserved currents which are not associated with irlvariances of the Lagrangian. A very simple example is provided by the Lagrangian c = >@,cpd’p and the transformation cp(s) + +(.r:) = cp(z) + ox(x) + O( 8) for a fixed function X( 2) satisfying *X(.r) = 0. This is a symmetry of the syste/,l but not an invariance of th.e Lagrangian, which gives rise to the conserved current W” = a&( X) - (F&N zi. A precisely analogous situation arises for t,he Lagrangian density b: = - !.,$c?,A,c?“A”employed in the Gupta-Bleuler formulat,ion of quantum electrodynamics. I;urthermort, there are symmetries of the system which do not. give rise to conserved currents. In the system JZ= >$( &(P~(P- p’pp), the scale transformation p(s) + G(X) = exp [&I,( X) = cp(x) + a&(z) + 0( 19”)is a local continuous symmetry of the system by our definition, but it is not associated with any conserved current. Hill [see (4 ), p. 2573does not limit himself to invariances of the Lugrungiun but restricts the definition of symmetry transformat,ion to those mappings which satisfy Ey. ( 3 ) , i.e., precisely t,hose connected with conserved currents. On the other extreme, some aut,hors apparently are not sure t)hnt’ one should require t,hat a symmetry be a mapping of the functions pr , but might include under this heading merely changes of the Lagrangian such as addition of an arbitrary divergence term or multiplication by a constant. Also, the reader is reminded that the literature 011 the subject of symmet,ry t,ransformations contains several different, notations which are not rompat,ible. Using the not,ation of Eqs. ( 1) and (3)) the &variation and space-time derivatives commute, but’ the 8*-variat)ion and space-time derivatives do not. The precise opposite of this notat8ion appears in t,he article by Hill (4) and in the sect,ion 011 Noether’s t’heorem in the text by Roman (6). One st#andardt,est’ on quantum field theory within a single section seemsto mix together the two notations with apparent,ly erroneous procedures. See the present author’s note listed in Ref. (6) for an elementary analysis of this problem.
q
2. CONSTIXJCTION
OF CONSERVEI)
CuHKEN’rs FROM CONTINUOUS
In the field system cp,(s), )’ = 1, 2, . . . , N,
SYMMEZRIES
-c(J:, Pr(d, dp’pl-CL),.
),
we assumethe existence of a local cont’inuous symmetry of the system $4~) -3 G%(x) = +(a) 1 S. S. Schweber, “Arl Illt.rodrlctiorl Harper alld Row, New York, 1962.
to Relativistic
+ &o,(x). Qrlantum
Field Theory”,
Sectiolr ‘ig.
SYMMETRIES
AND
CONSERVED
CURRENTS
449
Moreover, we assume that the symmetry is such that the Euler-Lagrange equations derived from .c( z, C&(Z), a,,&( .z), . . . ) when considered as a function of 98(Z), %t’Ps(Z), . . . are identical with those from .A?(2, (P?(X), a,,~,.( z), . . . ). This is a special class within the set of all symmetry transformations. In the system c = > 2 ( &$I”~ - p”p(p) mentioned above, the scale transformation ~4~1 -3 ~3x1 = exp Wl9(~>
= 9(x> + aOcp(x> + We’>
is a local continuous symmetry of the system by our definit’ion but’ is not’ of t,he required class, since t’he Euler equations of motion from ,c( 2, +( .r), a,,+(s) ) when csonsidered as a function of cp(.r), ?I,+( x ) are changed by a multiplicative constant from those of t,he original C. [See Strudel (7) for an analysis of such scale transformations.] The Euler~Lagrange equations of motion from the two Lagrangians arc [G(.r, $%(r),
. . .)Ja, = 0
and
where the Lagrangian
derivative
of
is t,he expression
Lqo, = g - d, I$$ + ’ . . + (9
s s
l)‘““i&,
. . . r&
aa
““. d -. Ir, . . P,,,(OS
Since t’he equations of mot’ion are identical, [,~(r, &(L), . . .) - Z( .r, ps(.c), . . .)Jrp, vanishes identically. By a theorem of tjhe calculus of variations, [see (6 1, Vol. I, p. 1931, this holds if and only if Z( .L’, &(x:)~ . . . ) - ,c( s, P.7(J:j, . . j is a divergence term. Thus weconclude that) two Lagrangiandensities give rise to identical Euler-Lagrange differential equations if and only if for arbitrary functions c+Q~, the difference of the Lagrangian densities is a divergence term. Our condition on the preservation of the Euler equations of motion is equivalent t#o
for some fixed functions ,!I” and arbitrary &I). Kate that the fun&)ns L>” are ambiguous because we can add t)o D” any t,erm of the form d,T”‘( .z, cp,(.r ), . . . ) where T”’ is antisymmetric in p and p and have d,d,T” s 0 as an identity in the fields. This ambiguity extends to the conserved currents and is what is used familiarly to obt’ain the symmetric energy-moment’um tensor from the canoni(a:ll
GO
BOYER
energy-momentum tensor. In general, assumptions on the tensorial character of the currents limit the terms in D' possible under this ambiguity. We are here thinking of functions s and jr which are analytic in all their variables. Thus we may expand S(s, C&(X), ...) - &(r, P,(Z), ...) as a power series in the fields dz), d&@.?(X), . . . . The fundamental functions D' will be those found by removing from every term in this expansion one appropriate derivative d, . Any term of the form d,8,YP( x, cpV(x), . . a) = 0 where 5”“’ is antisymmetrical in b and p, when expected in a power series in its variables, would, of course, vanish identically. aVow we can expand t,he left-hand side of (3) to first order in the parameter 8 as
where we have used the “commutativity” of t#he kvariat’ion and the derivatives d, which is evident from the definition of the 6 symbol given by ( 1) in which no change of the argument x occurs. When dealing with quantized fields, t,he position of t,hc fields in a product is of importance. We think of the functionf,. as being inserted precisely at, the place in the power series expansion from which the field (F?(S) was removed by different#iation. Kow using Eq. (-1)) we can rewrite (3) as
-[cQ9,S(o*= &WIL(x,As), d/d&(x.),. . . M, where [c],, is the Lagrangian
derivative
with respect t#ocpsand
TV% = (ac/aa,cp,)scp,+ e.. - D' where t,he terms in W’ other than
(5) (6)
D' arise from the identities
used to transfer the derivatives on 6q, in Ey. (4) over to the form required for the Lagrangian derivat#ive in (5). However, the Euler equations following from t,he application of Hamilton’s principle are precisely Bl,,
= 0,
I‘ = 1, 2, ...
, N.
(8‘)
Hence, when the functions cpssatisfy these Euler equations of motion, the identity ( 5) becomesthe conservation equation
0 = a,w”. Thus W’ given by Eq. ( 6) is a conserved current associated with the symmetry.
SY~I~IETHIES
ANI) CONSEHVEI)
CURRENTS
451
An elementary example of the c*onstru&ion is found for the case of a fixed space-t,ime t,ranslation in t’he free spin-0 1,agrangian 2 = ,!~(c?,,(FX’~- p’qq). Here the mapping is usually written in the form 2‘ + E’ = 2’ + 2’0, 4-r) + +,-(.?1 = p(s), with t,he definition of an invariance of the Lagrangian in mind. WC WII writ,e this in t,he form ( 1) preferred in this paper as P(.P) + cj3(x) = p( .P - ot ) = lp(x) - a,&?? + O( 8). Corresponding to ( 3), we obtain
either by direct substitution and computation or expanding the left-hand side 01 the identity C( +( c?), J,@(2) ) = C( p( s ), a,~( Y)) about I = .r. Thus we identify the divergencaet,erm of our analysis as D’ = JZE%,and the conserved current, of Eq. ( 0) is familiarly W” = dl;pdpq8 - 22’. This is identical with the usual result from use of Soether’s theorem. A secaondcx:m~ple for a symmetry of the system whicahis not an inv:ukm(.c of the Lagranginn is t’hah noted above for c = ,!~~,,qY’~. Here we cm (*ompute directly that, Z(d,+(s))
-
-C(ffl,p(.r))
= d,(pd”X(x)19).
We thus find the divergence term
and the conserved caurrent of Eq. (6) is w” = a$X(x)
- gPX(x).
I\‘ot,e bhat in this derivation of the conserved current, we use the requirement, of the invarinnc*e of the Euler equations from the original and t,ransformed Lagrangians, but we never make explicit use of t,he fact that the mapping pr( x) + &(r) is a symmet,ry of the system. In Se&on 3, we show that, every mapping which exactly preserves the Euler equations of mot,ion, or equivalent,ly, for which ( 3 ) holds, is indeed a symmetry Of the syat,em. We summarize t,he construction of a conserved current from a continuous symmetry of the system as follows. Zj’ a rrqpz’ng p,.(:r ) + &(s) = P,.(X) +&,c~(.r,) (?/ the forw ( 1) gives ISq. (3) for some jixerl junctions D’ and arbitrary qv , then the mapping is a symmetry t~ansfomation of’ the stlstsw oJ’ the t!ype giaing served curwnf W’, which cwrenf is given by Eq. (6). 3. DEMVATION CONSERVEI)
We
now
OF -4 COXTINUOUS CURRENT
SYI\I;\IETRY TRANSFORMATION
vise to a
con-
FROM A I,o~AI,
turn t,o the converse of the previous sec+tion.WC will show that from
4:i2
BOYEH
some local conserved currents, one can deduce continuous symmetry transformations which in turn give rise to the original local current by the construction indicated in t,he previous section. Considering the same system of fields pl- and Lagrangian ax,
4%(z), 4fPr(2),
. . . ),
we now assume the existence of a local current W”( 5, C&.X), dppT(x), . . ) c011served by virtue of the Euler equations of motion satisfied by the fields p,. . Then W is immediately associated with a symmet,ry of the system of the form indicated in the previous section, if and only if for some non-vanishing functions SAT PAJJt:), 4&z), . .) -[~l,J?
= d,W”
(9)
holds as an identity for all fields + . However, in general we are given only the equation 0 = a,W ‘, valid when the equations of motion hold, and we can not conclude that we can write an identity of the form (9). The obvious suggestion by comparison of Eq. (9) with (5) is to try to define the symmet,ry transformation as the mapping cpr(x) + (p7(X) = cpV(t) + &,( .r j with 6~~ = Ofr + O(IF?“) where the f are the functions appearing in t,he identity (9). The procedure of proof consists in reversing the steps in the construction of W ‘. The Lagrangian density &‘(X, &(x), a,,&-(x), * . . ) can again be expanded to first) order in e in the manner of Eq. (4). Then using Eq. ( 9) and the identities (71, we have
which, with the appropriate designation of the functions D', is precisely the Eq. (3 ) required for the derivation of the conserved current W’. We note that therestill remains someambiguity. If W” contains any terms which vanish identically when we take t’he divergence d,W’, then t,hesewill not be reflected in the functions jr . Hence we arrive back from the mapping cp,(I) + C++(X) + @4x, CPA JJ:), . . . ) + 0( 0’) only if we depart from the choice of the fundamental functions D ’ t#oinclude these further terms in 191’. We thus have proved that if a conserved current W” can appear in the identity (9)) wit,11not. all the functions f, vanishing, then there is a mapping (PJXf + &CC) which gives t,he conserved current in the manner of the previous section. It remains to be shown that this mapping doesindeed preserve identically t)he form of tQe Euler equations from the initial and transformed Lagrangians, and also that it is a symmetry transformation by our definition.
The fact, that t’he mapping leaves invariantS the Euler equations follows from the theorem of the caalculus of variations quot,ed above. The proof follows immediately from t,he property i-hat when regarded as functions of ~.(.r ), a,(~,( .r), . . t,he two Lagrangians differ only by :r divergence term. This can be cSonverted to a surface integral in t)he action, and so makes no (~~JIit~rihutiOtl in t,hc application (Jf Hamilton’s principle which requires that the variation of qFrand all suihably high derivatives should vanish on the boundary surface. Finally, we must, prove t,hat, tho mapping is indeed a symmetry traIlsfornl:ttioll of the syst,em in the sense of the clcfinit~ion given above; i.e., whenever qr sat isties the Euler equations
[C(s, ys(.rj, ‘. . ,I,, = 0, then zF satisfies
where
This is the statement that +,( .r) is a solution of the original Euler equations whenever ppr(.r) is a solution of these same equations. We form the action int8egral (Jf 13~. ( ?I) with the second term tfransposed to the right-hand side over the full region B of physical interest, and apply Hamiltjon’s principle for a variation ++( .r) ---f qra( .r:) = p?(s) + NT,(X) where q( .r) and a suitable number of derivatives vanish on the boundary for 2, but is arbit’rary inside S. When qF1(.r ) satisfies the Euler equations of motion, t,he right-hand side of the expression has a stationary value and hence so does t,he left side. We have thus when qc,( x) satisfies the equat~ions of motion
where +5”(s) 3 qFra(.r) + a~,.~( P). Carrying out the derivative in CLand then integrating by part,s in .r where we can discard the surface terms since qr( .r‘) and it,s (first several) derivatives vanish on the boundary, Eq. ( 10) becomes
We now consider Eq. ( 11 ) as a power series in the paran&er 0. By separating the contributions to the terms which are zero and first order in 0 and using the assumption that the Euler equations of motion are satisfied for cpSwit,h respect to
454
BOTER
a,.c, (-PI(X),. . . ), wc
see that we must have
0 = s, f14.dC(X, d-L.), . . ~)l&h,%(X)
(12)
at, least to first order in 0. Analysis for higher orders of 0 gives the result for all orders. Since Q(X) was assumed arbitrary in the interior of Z, we deduce that the quantity multiplying Q(Z) must vanish. Thus we have, as required, that, Mz, dd, . . .)I&, = 0 holds whenever (p7satisfies the equations of motion. The invertibilit,y of the transformation follows from the assumed expansion to first order in 0 and the continuation of the transformation by iteration. Note that we have made the tacit assumption that the functions 6~~ = 0f7 + 0( 8) are welldefined when t#he fields satisfy the equations of motion. We conclude that any transformation (p7(zr) 3 &(z) = qr(z) + 6~, satisfying (3), and in particular t.he transformation derived from the conserved current 157” in (9)) is a local oneparameter continuous symmet’ry transformation of the system. We noted that it is not generally true that every current W’ conserved when t,he equat,ions of motion are valid is of this form. Since W” is conserved when t,he fields satisfy the equations of motion, it might seemnatural to allow changes in the functiona for~z of W’ to an “equivalent current,” 151’”which still maintains the samevalue asW” when the fields obey the Euler equations. (This is an entirely different question from that involving the ambiguity of the functions D’. In general, a change of W” by a function dpTBP(z, p,(z), . . . ) would change the zlalueof t’he current..) One example would be to change t,he fun&)nal form of W” over to 191’”= w” + [E],,hr + a,[C],,h,P + ‘. . . In spin 3; electBrodynamicswith
R = $8( $‘a”#
- m#j - J&F”( d&A, - &A,)
+ xF”“F,y
+ e&$‘$Ap
the current W’ = -a,( 8‘A” - d”A”) is indeed conserved but a,,W” = -8# (#A” - #A”) = 0 holds as an identity and we can not find any nonzero functions jr for Eq. (8). However, we can writ,e 151” = wp - 2&[C],“’ + [Z],, = e$-f”l), and so obtain
-[S]+(
-ie#)
-
(if$)[~]~
= a,w’“.
In this manner, we associate the current with the gauge symmetry t)(x) + exp [-ieel+(
&cc> * G(x) exp [+iee].
SYMMETRIES
AND CONSERVED
43.5
CURRENTS
We remark also that if we are able to write d,,W’ as -[J3],,gr - a,[cl,,gr”
-
. . . = drW’(x, R(X), . . j
(13)
for some nonvanishing functions gr( .c, ‘py(z), . . ’ ), gTP(Ic,pFr(CC),. . . ), . . ., theu we could rewrite this in the form -[s],,fr
= d,(W”
+ [C],,g,p - . . . ) = drw”qx,
$&(.I-), . . )
(14)
where jV = gT - d,,gPP+ . . . Here again W NPis a new function which has the samevalue as W’ when the fields satisfy t,he equations of m&ion. The symmetry corresponding t,o ( 14) would give a (*onserved current in the fun&ma1 form W”‘.
In the last, section of this paper (Subsection II.4), we will prove that for the casesof interest in quantum field theory, it is always possible to find an equivalent, current W” and an associated symmetry such t,hat W” is generated by the symmetry in t,he manner of Subsection 1.2. 4. INVARIANCE NOETHER's
OF THE ACTION THEOREM
INTEGRAL
AND (:ONSERVEI)
CURRENTS
FROAL
At this point it would seem helpful to review briefly the content’ of Noether’s theorem (9) in order to clarify its relevance to t,he problem of relating symmet,ries and conserved rurrents. We arc concerned here with transformations dependent, upon a single real parameter 0. Hence the part of Noether’s paper which concerns us is the proof that the action integral is invariant under a oneparameter continuous transformation if and orlly if an identity of t’he form (5) holds for suitable functions W’. Specifically, the hypotheses assume a function e(.r, G(X), 4dfd~), GLd.r~, . . ) and a continuous mapping .r + 2 = h( x, (FT(x), d,p,( x), . . . ) = x + 6x,
pJx) ----t&.s(fO)= RSx, P.(T), d,cpx(x),...I = dx)
+ ~*PAx),
( 15)
where 6~ and 8*pprcontain terms that are first, order in e, and O(e”), such that
for an arbitrary region !J and image region fi for all values of the continuous real parameter e. Sote in part,icular that .Z may be a function of the fields cFr(zj, 4i’Dr(~!, . . . This is not the casefor what we have defined asan invariance of the Lagrangian. The analysis proceeds as follows. We combine the integrals over the single arbitrary region Q to first order in 0 by approximating the integral over the differ-
1.5Ci
BOPEII.
01’ tht two regions a - I! by s a,1do,,&r”~ where C?Qis the boundary surface of !?, and then using Gauss’ theorem to convert, t,o :L volume integral over fi. (These elementary manipulations are carried out in det,ail in the paper mentioned in refcrencae ( R ).) We obtain to first order in 0
enw
0=
t/“r[GLr, s 11
$&(.c), dp&(“r),
. . . 1 - se(.r, (FJ.I.1, dpppr(.r), . ” ) + a,(cwd:(.r,
q%(r), d,qdJT),
(17) . . . I,).
i\‘ow the bracket may be simplified by the same procedure given above in Subsection 1.2, t,hrough the expansion (4) and ident,ities (7), to give 0 = s, ri4.r(,b.,.6c,
-
d, (g-“,
a r
+ ‘. . -
again to first, order in 0. Since Eq. ( 1s) holds for arbitrary must vanish and we derive the ident,it,y -[[C]&LYr
C&r”
)I
)
(1s)
regions a, the integrand
= d,W”8
(19)
where ‘W’O = (d6/ddpcp,)6q3, + . . . -
62s.
(20)
This result is precisely of the form of the identity (5) when we identify 62’~ wit’h D'. As of yet, we have made no reskictions on t,he character of the functions P~( x). If, however, we now assume that the fields (p7(s) satisfy the Euler equaCons from the applicat8ion of Hamilt80n’s principle relative to s 1 t14”rC(“r, @4x), dr(Fr(.r.),
. . ),
where S is the entire physical region, then we have [,c],, = 0, 1‘ = 1, 2, . , N so t,hat t,he left-hand side of ( 19) now vanishes, and we derive that the local current ‘157’ is conserved, d,W’ = 0. The queet,ion of the converse problem for _\oet,her’s theorem (somesto t
(22)
With these definitions, it is easy to reverse the procedures of Noether’s derivation out,lined above to obtain the equality of the action integrals in Eq. ( 16).
Sc)tc
that’
Noether’s
c*ormerved This only
is the
the
identity
REI,ATIONS
the
arc
BE’I’\VEEN
:md invariant
c~hangcts
and
terms
Eq. real
out
Euler
equntious
other
for
one
terms because
differeut
symtnetrp for
The
itlterest
q 4
lies
selves,
the
from
the ;tntl
are
futictiotis
when 9J.c)
the
from
arise
forth a&ion
the
divergence theory,
derivation,
it
terms
have
tcrnl
it1 t,he
c~onserved of
divergence
in no
no
c+urretltJ the
identit,y term
I)“.
( 5) always
sirlcae
+,( .c)
one
We
that
this
would the
of
make
linear
the
iderttic~ally.
vanishes
identically.
Any
contribution
tto
( The
of such I,agrangim
All change,
question to
t,he is a
current. that
in
generators, cahartgcs
Lagrangian
currerlt.
c*ontribution
vanishes
rise
of
mapping
of coordinate.
mlserved
secondary
cshauges
operator
), gives
cahattge
alteratiotts
differential
a coordirtat,e this
crucial
OIltO t,hem-
a caouserved on
ex-
example the
Euler t,he
from
as an aside
paper
to my
earlier
this
a c~ontinuous of
whkh
an
il’o
from
of time-independent of
change
is that
of the
appear
ttotc
for
Lagrangiatts.
whet,hcr
divergetlc*e
of motion
separate
CJf
case
recaonsider
form
cmstructiotl
associated
C itself W”,
+
of coordinates
equations the
Forrnulatiott
II.3
on t’he
Lsgrattgi:u~
the
referet1c.e clear
effect
it1
test
applic*atiotl
Subsection
hec*otnes
&( .r )
use of coordinate Hence
author
transformed
changes
t,ertns
immediate
(V)II-
+
appropriate
SOIlle
will
present’
preserves and
Xoethcr
the
the
allow
work
integral. by
We
the
of the
it1 the or
in t.he
from
make
and
transformed
P, pFr( x)
The
hetwecn
absorbed
t ~d,qY‘q. by
initial
relevant
field
illustrated
whicah
=
mordiuatc
ever
which,
x
mapping
the
to
and
.r +
Lagrangim.
of the
a/mz,?/.s
of solutions
tr:ttisforni:ttioti, S.rP,
quantum
side
preferred
mapping
qFr are
tcrnms
(as
X in
involving
synintetry here
(o +
it1 the
c:onsiderutions
if and
D,“. invarinttc~e
view
of
for
:t
IA~v~4m4n-c~~s
pr( x)
origittal
transfornmtiot~
h:tc*k
they
point’
cspecinlly
equations
ANI)
:tpproac~hes
6x, even if it’ is necessary to make n change on the vnlues of the field. q qF7( .I’), as is indeed the
below.
holds
symmetries
ttm1sfortn:~tion
qr from
:I Noet)her
couvcrt
is it1 the
appear
itivolvitig
( 16)
C:UIZ~~ENTS,
two
for
integral
cm
divergeme
ample
whet1
interest.
of caoordinates depends
as to
( 19). about
of Soether’s
w’c’ presented
current,
the
ittterest
rterle~
question
identity
theorem:
CONSERVEI)
the
that
Soether’s
c~ruckl
of the statements
scope
SYMAIE~~UES,
t,he a(+ion
to apprcciat,e
I~urther
the
sec*ticms,
iuvariant
leaving
the
on
form
INTEGIZAL
previous
I,ugr:utginns,
hint
of Soet.her’s
holds.
beyond
of :t c*otlserved
lcavitlg
no in the
c~otltent
( 19)
:~(‘TtON
structiott
gives
be writktl
entire
czurrettts
OF THE Iti
K~II
then if
conserved 5.
theorem
caurrent
terms
.r
with
by
divcrgettc*e
pure
to the
&(
Also,
j
this
divergence fundutnetttal on
derivat.ive
the
left of :t
)?
L Olte furt,her advantage in omitting ally reference to :t coordinate change 6.z might be that it worJd reorder less likely remarks whose talltological nature is hidden under the variety of symbols. The note by Romnll (5) 011 ihe effect of divergence terms in 2 seems entirelv misdirected.
45s
UOYEH
It, is significant that although we have considered when cpr(s) and &(x) satisfy the same Lagrangian equations of motion, nothing has been said about the equations satisfied by the functions &( 3). Indeed &(z) considered as a function of L is in general not a solution of the same equation of mot(ion as (C,(X). Nor do we obtain any easily ident8ifinble equation of mot,ion if we apply t,he full Noether mapping (15) to the different8ial equation of motion for &z). If we consider the Lagrangian 6: = J,$,,&‘(p for one of our examples, then t,he symmetry of the syst,em P(X) + 1;1( x:) = q(e) + 0x(r) + 0( 0’) where a%( X) = 0 is associated with the very cumbersome Soether mapping
which is an invariance of the action integral and gives rise to the sameconserved current W” = d”(oX - (~8% obtained earlier. However, ~(3) considered as a function of z is not a solution of the original equation of motion; here a,a”(p(%) # 0. Also, the full Noether mapping above applied to $J”p(s) = 0 gives a complicated equation of motion for +( 2). We may summarize the difference in att#itudes by viewing t#hemas diverging from the identit,y (5) or [ 19) which is the pnint, common to both. Invariance of 1,agrangian Present, Analysis Invariance of Euler Equations
IdenMy
Noct,her’s Theorem Invariance of (5) or (19) s Action Integral Yields satisfy Euler Equations
I Symmetry of System
I Conserved Current’
To the knowledge of the present writer, virtually all authors, [seefor example Hill (4) or Roman (S)], have tacitly adopted an attitude intermediate between the two above-mentioned positions; they allow coordinate changes 6x(.x) independent of the fields cpr(x) but also include divergence terms separately into the new Lagrangian. Our feeling is that in a general analysis, any reference t,o a coordinate change is superfluous in the actual derivation of t’he local conserved current.
SYMMETRIES II. RESULTS 1.
SOME
AN11 CONSERVED
AN11 EXTENSIONS QUANTIJM
LIMITATIONS
RELEVANT FOR FIELD THEORY
OF THE STRWTURE
459
CURREBTS LOCAL
OF QUANTIZED
LAGRANGIAN
FIELD
THEORIES
ln the previous sect,ions, we have tried to present with some generality the problem of connecting symmetries and conserved current’s. Here we wish to review and extend t,hese results for the c*aseof local Lagrangian quantum field theories as formulated by n:iive cxronical quantiza.tion and by the Schwinger action principle. WC restrict, ourselves to the following t’wo syst,ems of classiral or quantum 13,&c)). fields I&X‘), i’ = 1, 2, . . . , N with Lagrangian density 2(x, opt, Case ( i) (Naive Canonical Quantixation). The Euler equations of motions are second order partial differential equat,ions in which the second order t,ime derivatives of the fields are regarded as dependent, and all fields, space derivatives and first time derivat’ives are independent at a point. Because of the form of t,he Euler equations from C( .z, qr( 21, a,,~.( x)), if &&P, appears in [ZIPr , then &&~l- appears in [J-?],~; consistency requires t,hat the identities
can be solved for the second time derivatives in the form dodo~r- ‘I’,( x, pFs,d,~ , d,d,d
= Ark .I’, pr , a,(~s)[c],, ,
I’ = 1, 2, . . . , N
(21)
for all functions p’r . An immediate example is the free spin-0 I,agrangian c = !,*~(a,,&~ - p’~pF). The identity (24) is here
Case (ii) (Schloinge, Action Principle). All t,he Euler equations are first order partial differential equations from the Lagrangian d: = f i(‘pr.@:,&cp,~ d,,+ .@$,c6) - 6( cp&) . If any field pr must be regarded as dependent, or if any first derivative of an independent field, (this includes all first time derivatives of independent fields), is dependent, when t,he equations of motion hold, then we can write as an identit)y holding for all fields qr (dependent quantJity) -3 = a,[~],,
+ &‘i3k[&J,,
(25)
where 3, a, , and atk are functions of only those quantities which can be regarded as independent at a point z, even when t,he fields satisfy the equations of motion. Quantixation of t,he field theories, of course, involves many further restrictions. [See Schwinger (IO), p. 919 (1951); p. 716 ( 1953).] However, the limitations not,ed here are those which will play a role in our analysis, especially in deriving symmetries from conserved currents.
4tio
HOYEH
The previous analysis of Subsection I.? for t#heconst~ru~tion of corlservcd NI'rents may be taken over in its entirety; limitat~ions WI the strurt,ure of t,hetheory here add 110 new information. Simply, given my continuous mapping pr(.r) -
&(K)
= pr(.r) + &p,(s) = SJS) + B.&-r, p?.(.xC,a,p,(s‘,) + fh02),
we construct C( .L’, +,( .r), a,,;,(x) ) but, now regard it, as a function of @Js 1, ap(F?4 d) d,&d .r) , . . ; i.e., everywhere +,.(.r ) appears, we replace it by qr( s ) + 0fr( s, pr( s), a,~,.( .I:)) to first’ order in 0. Kow subtract C( s, pr( .I’ ), 8,1pr(x) ). If we can find some funct’ions D' such t’hat, to first, order in 0, C(r, &(x1, d&(-c))
- C(r, pArI,
ap4d.r) 1 = a&“(s,
p,(s), d,p,(r))
t “6)
holds as an identity for all fields (F~(.I’J, then the mapping is a cont8inuoussymmet,ry transformat,ion of the class assocktt~edwith conserved currents, and the conserved L current W’ is W”% = ( a~:jaa,Lc,)jyo - 1)“.
( “7 )
We note from the construction of W”, using t,he fundamental functions 11” of Subsection 1.3, t,hat#because both t,he functions c and jr involve no derivatives of order higher t,han first, the same must be true of W”. 3. INTEGHALH SYMMETRIES
OF CONSERVED
C~RIZENTS
AS QUANTUM
GENERATORS
FOR
If the current, ‘W’ is conserved when the fields satisfy the equations of motion, then for fields which can be regarded as vanishing at spatial infinity,
is a time-independent quantity. One proof makes use of a four-dimensional Gauss’ theorem
taking t,wo of the bounding surfaces as rn = cl , xc = c2 for cl aud c2 conskmts, and allowing the remaining surfaces to receed to spatial infinity. In the case of quantum fields, the condition on the behavior of the fields at spatial infinity should be viewed as a restriction on the allowed domain of states. [See Schwinger (IO), footnote p. 917 ( 1951).] It is usual in quantum field theory to regard the operators W arising from symmetries as the generators of the original symmet’ries. Hence it is of interest here to ascertain the circumstances under which the int,egrals of the local conserved currents from these local cont,inuous sym-
SY3lMETIZIES
AND
CONHEHVEU
l(il
CUHKENTS
metries do indeed serve consistently as t#he operator generators of these same The analysis presented is more general than that symmetry transformat,ions. available for invariances of the Lagrangian---which invariances require that 6~ is independent) of the fields. k~ analysis of quantum operators for symmetry transformation both local and nori-local strictly from t,he point of view of canonically quantized field theories has been given elsewhere” by the present author. In the case of quant,um field theories resulting from naive canonical quarttization, the answer to this question of consistenc~y is that for symmetries of the form
involving no higher than first derivatives of the fields, the integrals of the currents always generate the original symmetry. We saw in Subsection I.2 that the functional form of d,W’ associat~ed with the symmetry was given by the ident,ity (5) which here takes the form
- _cas - aP$1 ( aa
fX. c,pa(x),a, cps(;c))= a, w’(.r, $%(:I:),a, q,<(x)1.
(29’1
Since, as we noted above, W’ is a function only of t.he fields and first derivatives we can determine the functional form of dW’/d&cp, by t’aking the deriva.tive with respect to d&~,,. of both sides of (29). We t’hus find
again as an identity theory, we have
However,
holding for all funct,ions pr . In the quantized
from the original canonical
3 T. H. Boyer, Nzcovo Ci’mento 44A,
quantization,
613 (1966).
we require
version of the
42
BOTEH
bl-(~),
WI
=
+c-6
e(2),
(33)
Gf4~))
precisely as required. Since W is invariant under time and space translations, it also generates the appropriate transformations for the derivatives of the fields. In the case of quantum field theories from the Schwinger action principle, t,he int’egral of the current acts as the generator of the original symmetry only in the case that all of the fields are given in t’erms of the canonical variables. In simple Loreritz transformations are particular, for quantum elect’rodynamics, not reproduced by the quantum generators in the radiat’ion gauge quantization but rather further terms are int’roduced; there are also gauge symmetries giving rise to conserved currents whose integrals do not act as generators for the symmetry. For spin+ qua&urn electrodynamics specifically, the symmetry of the system 1J4.s) -3 exp [-;eN,.r)0]+(z), A,(x)
-j A,(.c)
-
$(.r) &A(x:)e,
4 $(.c) exp [+;eh(X)e], P”“(x)
+ F’“(z)
produces the conserved current W’ = e$-f#(x)X(x) - F”‘(.r)d,X(x). However, the space integral of the zero component will not generate t#he required quantum symmetry. The failure of this generator W is symptomatic of the difficulties in quarkizing the electromagnetic field. In the Schwinger radiation gauge formalism only t,he t,hree-transverse parts of the fields are canonical variables. We remark that the integrals of the quantum currents from gauge transformations in the Gupta-Bleuler formalism, where na’ive canonical quantization applies, do indeed generate the required gauge transformations. Again we take W“ from the fundamental functions D” and so conclude that if c and fr contain derivatives of at most first order, t’hen so does TV”. But now from the identity (5)) or ( 29) here, which again holds for t.he construction, we seethat the left-hand side involves only first derivatives, since all the equations of motion are first order. But then (,dw”/d&zcor)drax’Fr = 0
(34)
for all a,~~ . This can hold only if dW”/&3,p, is antisymmetric in p and 0~.In particular, this implies dW”/&30pp = 0. In the Schwinger action principle, the conjugate variables are pairs of fields pr and (P+such that in d: all time derivat,ives of these fields appear as >$(Q+.d~cp,~- I~~P~.c,+‘).But then we can consider the partial derivative of the identity ( 29) with respect to &cp,t and so derive (35)
SYMMETRIES
AND
COSSERVED
CURRESTS
463
In deriving the terms on the right-hand side, we have made use of the nntisymmetry noted above, and also the fact that W” involves derivnt,ives of at most first order, to find t’he relations
The effect of the generator
W on p,. is computed as
where we have made use of the equal t’ime commutation
rules including
[cpr(x),cpd~jl* IL,I=L”r= ;63(x - $7,
(37)
of an integration by parts in three-space, and of t,he identity ( 35) now employed when the fields satisfy the equations of motjion so that [s], = 0. (The analysis can be followed easily in t’he example of space translations of the free spin-J+j field,
The time and space-independence of W means that, the generator acts appropriately on the derivatives of the canonical fields. -Vow the dependent fields arc connected by the equations of motion to the independent fields and derivatives of these fields as indicated in the identities (,2.5). Thus W also generates the required transformation for all t,hose dependent fields cormected to the canonical fields. The generator fails only for those unquantjized fields which are not, connected to the canonical fields. In considering t,he consistency of the quantum operator generators arising from symmetries through conserved currents, we have taken the current arising from the fundamental divergence terms. However, the current could be modified by terms aP”(.~, (O,(X), . . . ) whose divergence vanishes identically due to the nntisymn&ry of T”” in p and V. When considered in the integral ( ‘28) for IV
40-l
HOTER
such terms contribute J” d3x8~7’“” which may be replaced by a spatial surface integral. In the case that hhis surface integral can be regarded as vanishing, the t,ime-iildependellt quantity W is t,he same as that obtained by omitting such terms. We note also that if we are not concerned with the tensor aspects of the currents, then from a conserved current W” we can obtain other conserved currents merely by taking derivatives W:,) = d,W”, etc. However, on taking the space integral of the zero component,, the new quant’itfy becomes a spaGal surface integral or vanishes. 4.
THE
ASSOCIATION
OF
C~ONHERVEU
CURRENTS
WITH
SYMMETRIES
The second half of the problem considered in this paper, t,hat of finding symmetries associat,ed with conserved currents, allows substantial comment when restricted to the relatively simple forms required for quantum field theory. Here we can decide that any nontrivial local conserved current has associated a symmetry transformation producing a conserved current of magnitude equal t’o t,he original current when the equations of motion are satisfied. The analysis bears a basic resemblance to a theorem of Fletcher [see (51, p. 741 which the author discovered after carrying out the very natural procedures described here. Also, during the revision of the present paper to move the examples from footnotes into the body of the text, an article by Dass* appeared in print. This paper purports t’o carry out the converse construction of symmetries from conserved currents for arbitrary Lagrangian systems. The crucial st,ep in Dass’ general analysis turns out to be the first, where it is asserted that every conserved current W’ WII be written as an identity d,W” = f([61p,). In our analysis to follow, we will note that for some current’s W’, for example all those associated with what Fletcher terms “strong” conservat’ion laws and also for some trivial cases, we have d,W” = 0. The restrictive assumptions we introduced earlier were made so as to allow more than a heurist’ic argument’ in the converse construction. In the case of quantum field theory, it seems ent’irely natural to change the functional form of the current 157’ while maintaining its value. The quantized version of the theory contains meaning only when the equations of motion hold, and indeed t,he action of operators on the fields is defined only through the equation of mot,ion by the action on the canonical variables, which are again determined by the form of the equations of motion. Thus a natural procedure for finding a corresponding symmetry transformntion is as follows. First we separate out all the distinct conserved currents WI;,,. ,4 power-series expansion of W” in the fields p, , arcpr , . 1. would allow one t’o 4 T. Dass,
Phys.
Rev.
160, 1251 (1966).
SYMMETRIES
AND
(‘ONSERVED
C’URRENTS
405
select a monomial term and then t,o choose a minimum number of further terms necessary t’o obt,ain a conserved current, WY;, . Then we remove all the derivutives which allow W’;O to be written as the derivative of a conserved current %‘;;I. This last ?6’;!, is t,he current W’ which we now analyze for :t relationship t(J a symmetry transformation. We proceed by removing all dependent fields and derivatives by use of the equations of motion. Then on application of t’he conservation condition 0 = d,W”, the only dependent’ quantities arc the further derivatives of t,he fields. The need for somesuch consistent procedure in eliminating the dependent variables is illustrated in the example of spirl-,l,2 quaut,um electrodynamics. The current, \Q“ = -~3,(#‘~4” - i3”.4’) gives d,W” = 0. However, when reduced t,o an expression in terms of the independent fields, this is immediately seen to be the electric current associated with gauge transformat,ions of the first kind. In t,he case of field t,heories allowing na’ivc canonical yuantization. we can remove the dependent derivatives by the use of the identities ( a4j and so find
as an identity for all fields qI . We can rewrite thk as noted in Suhsect,ion I.3 as
(39) where W”’ has t.he same value as W’ when the fields satisfy tjhe equations of motion. We remark that here W”’ and 191” have also t’he same funct,ional form. Then Eq. ( 39 ) coincides with t)he identit,y ( 5) and so if the right-hand side does not vanish identically, we can find a symmet#ry gcnerat’ing a conserved current W”” which differs from W”’ only in terms of the form d,T’”where 7’“” is antisymmetric in /* and Y. We have noted t#hat such t)erms give spatial surface integrals when we try to form an operator generat,or II’. In the casethat t)hebracket on t,he right-hand side of (39) vanishes, we conclude that t,he original current] was kivial in the sense that even when expressed in terms of t,he canonical variables, the conservation of t,he current does not, depend upon the equations of motion for the dynamical variables. For example, the csurrent W” = 131~ W1 = --~?~q”,\Q2 = &q, W” = -&+c is trivial in this sense.The possibility that -w’ was the spatial derivative of a conserved current, in which case our transformation from W’” to WN’ would give a vanishing identity ( 39)) n-as spec3ically avoided by the initial procedures. An analogous result holds in the case of the systems allowing applicat,ion of the Schwinger action principle. In some casesof fields in the Schwinger formulation, the procedure described here is unnec*essarily circuitous, although the symmetry found is unaltered. For csample, for the free spin-0 field c = $q”.d,q
-
! gLp”.p
-
,I yp”U” - ~‘q(p, the current cp+ P + a&,
corresponding
to space translations
'pp + p* + apgtP,
w@ = ,!,p*.ap(FtP - f~dppP.EPp - (>$pY.dvp - ;&(F".p
- ?yp"pv - &p)tW
allows immediate present,ation in the form of the identit,y ( 5). However, the procedure prescribed in the test would remove the dependent fields (ok, &q and convert to W’” involving second spatial derivatives. Partial integrations of t,he form mentioned above are then necessary to arrive at the identity (39). If we had used the Lngrangian of naive canonical quantizntion A? = .1;(i)PPDd”P - M’~(F), then for CF+ CF+ d,,&’ we have
w" = avpaPpEp- j 2(a~c"as~v - p'?p'cF)tY, which does not involve any dependent quantities so that our rule gives immediately the identity (5). We conclude that nontrivial currents can be traced to symmetry transformations for these very restrictive field theories which allow quantum interpretations. ACKNOWLEDGMENT I would
like
to thank
L. DeRaad
for reading
the manuscript.
RECEIVED: October 27, 1966 REFERENCES I. 1). HORN, kn. Phys. (S. Y.) 32, 444 (1965). 8. T. D.\ss, Phys. &c,. 146, 1011 (1966). 8. P. ROMAK, X7covo Cimenlo 10,5% (1958); H. STEUDEL, Z. Sa/wforsch. l?A, 129(1962); 1). B. FAIRLIE, Suouo Cimenfo 37, 897 (1965); H. STEUUEL, X71ovo Cimento 39, 395 (1965); C. A. LOPEZ, N~tovo Cirnenfo 40, 19 (1965); T. D.\ss, Muovo Cimento 42A, 730 (196(i). 4. E. L. HILL, lieu. Mod. Phys. 23, 253 (1960). 5. J. C. FLETCHER, Rev. Mod. Phys. 32, 65 (1960). 6. P. ROMAX, “Theory of Elementary Particles.” North-Holland, Amst,erdam, 1961; T. H. BOYER, Att,. b. Phys. 34, 475 (1966). 'Y. H. STEUDEI,, 2. Natwforsch. 17a, 133 (1962). 8. R. COUR.YNT AND 1). HILBERT, “Methods of Mathematical Physics.” Interscience, New York, 1953. 9. E. NOETHER, 1Vachr. Akud. W&x. Gottingen, Kl. II, Math.-Phyik. 234 (1918). 10. J. SCHWINGER, Phys. Rev. 82, 914 (1951); Ibid. 91. 713 (1953).