On the definition of currents and the action principle in field theories of one spatial dimension

On

the Definition of Currents in Field Theories of One

and the Action Principle Spatial Dimension*

The properties of a model relativistic field theory in one apaw tiirnension arc investigated. The model consists of a massless FPL.III~OII field whose current is c.~~~lptrd to itself and also to a vector I3oson field with IIIBSS. 13~ Incans of I lw act ion principle tzhe effects of the coupling are sirnulntctl hy func*tionul tlcriva tivcs with respect, t.o external fields. These can tw r.arried out iI1 closed forn~ and yield an exact solution. It is found at an earl>- stage that tllc usual dcfinition of the Fermion current, as the simple justxpc,sition of the Fernlion field and its c*rllljug:rte is ambiguous and does not. lead to a relativistic~ two-vector. A mow precise definition of the currentf which remedies this situation is ol)tainttt t)y taking the average of the limits of a nonlocal product of the Frrrnion fields as the coordinates of those fields approach one anot h<,r fro111 :I spacelike :+II~ the art hogonat timelike direction. I~splirit examination of the (ireen’s functions shows that those conlaining 1)11ly OII~ Uoson field or Fermion current are consistellt, wit II the tirld cquat iolls. cancmic*nl commutation relations, and the definiti~ln of the rhurwnt . but that this is not true for those containing two or mc,re of thew operators. This situali~~u is clarified with the introduction of a noncovariant singlr-tinle current nhereil~ the limit of the coordinates of the FernCl)n fields is taken at, a given tilne. .i comparison of the matrix elements of this current, wit II t how of the VOvariant one shows that one of these must he considered :ts depending explicitly. in its definition, on the external fields. When the c.ov:tri:tllt current is so cow sittcrcd, t hc extra terms in its matrix elements coming frcbllr fun~tion:tt tlif’fcrentntion of t.he esptirit field dependence are foun~i to relnove the inconsistence. The original use of the action principle in solving ttle tllodet is le.es:~rnincti in tlw light of these developments and a corrected vnsio~~ 01’ the I,:cgr:tnge fun+ tion which cont,nins operators at, me tire only is tteriwti. Berausc of the massless nature of the Fermion field. t hew :tre rspwted to tw both :I current and a pseudocurrent which are (‘oIIHerved~-~‘vCII in the presenw of t hc c,xternal fields. It is found that, neit,her of the currrnts \ur c.orrcsponding 1)sc~~(lc~c’lll’rents) thus far introduced arc cc~nscrvccl. Tllc actual cotlservetl -~~__ ~ * This work was supported in part at Harvard by tlw (‘orning (~tnss Works and t,lrc Air Force Office of Scientific Research and at Yale I)!- the -\t,llnic. F:rrergy Comrllisci<,ll md the National Science Foundation. t Present address: Sloane Physics I~aborctt,orv,~-:llf~ l‘nirersity, Srw FI:lven,(‘~llllt,c.ti(,llr.

2

SOMMERFIELD

current and pseudocurrent are inferred from an examination of the matrix elements. It is then shown to what extent’ the action principle may be used to derive the conservation laws for them. It is also shown that three of the four components of the energy-momentum tensor may be derived directly from the Lagrange function without the knowledge of the effect of changing the direction of approach of the coordinates of the operators therein. These components are found to satisfy commutat,ion relations which guarantee the relativistic invariance of the theory. Since the corrected Lagrange function was written down only after complete knowledge of the solution was at hand, the question of whether it is possible to know the Lagrange function without such knowledge is discussed. It is found that in the present case the answer is no, unless the current appearing in the field equations is defined in a manner different from the one used. A choice of the conserved current for this purpose is found to simplify some of the properties of the solution. This discussion is extended briefly to more general models. I. INTRODUCTION

The one-dimensional relativistic field theory introduced by Thirring (1) has proved to be a fruitful one in admitting itself of solution by an ever-increasing number of different techniques (2). In this paper still another formalism is brought to bear on this problem. The excuse is that this formalism yields answers to questions which are not usually asked with the other methods and that the model itself yields answers to questions which are not usually asked about the formalism. We shall consider not only the Thirring model, but also its generalization to include coupling with a vector Boson fie1d.l The Thirring model consists of a charged, masslessFermion coupled to itself by means of a four-field, vectorvector (current-current) coupling. The generalization has the Fermion current coupled to the vector Boson field as well. There are two mass parameters in the extended model-the coupling constant and the Boson mass.These provide a scale of energy which is lacking in the simple Thirring model. The procedure followed is closest in spirit to that of Johnson (4), in that the Green’s functions rather than the X-matrix elements are the objects of our search. In theories not plagued by the presenceof masslessparticles the transition from the Green’s-function to the S-matrix description is the very straightforward one of identifying the latter from the asymptotic form of the former. But with masslessparticles the Green’s functions do not always possessan appropriate asymptotic form and the connection requires more precise specification. We shall, however, not discussthis aspect of the problem in any detail and shall feel content upon producing answers for all of the Green’s functions. The solution is constructed using an external-field method introduced by 1 This generalization

has also been discussed by Bialynicki-Birula

(3).

Schwittger (;;‘I. Tltc model is first extended oncr mow to ittrludr coupling of t INS l~rrmiott cturrttt, to an cxtetnal vect,or field and a rouplittg of the 13osott firl~l to a11 est,rrttal currenL Then assuming the \ralidity of thr (Ittatttum-mrcllattir;~l actiott pritiriplr (6 1, we simulate the effects of chatigrs in rithcl thr l“rrnticbtlI’rrmiott rottpling constant (r or the Bosott-lietmiott coupling rottstattt (1 OII at)? ( ;twtt’s ftuictioti hy appropriate chatigcs in thr rstrtxal fields. .\ clct.i\.a.t ivr \vith wsprrt to g or c is found to bc ecluivalrttt to futtct~iottal tlrti\-ati\-rs wit11 twprct to t,ltr rsternal firld s. These first-otdrr diffewntial rclttat~iotts iti g :IINI v may tw sol\ved, with the result that thr rsact ~~twtt’~ frttlatiott is rspt~~wrcl it1 trims of ftittctiotial derivatives of the correspotidittg (~twtt’s fitttrt~ioti for 1 Itfb tltrorg in whirli y = u = 0. The reason that this in&l is fxwtlp soht~lf~ is tllat thr (;twtt’s functions, in the presence of cstrrttal fields ottly, may 1~ for111c1as rsplicit, frtttrtiottals of these fields and that the ittdicatrd ftutrtiottal (lrri\xt i1.w ratt actually 1~ carried out,. For more grneral theories a powt,-srrirs rspattsiott is necessary and then this mrthod just, rrproduccs ordittary prrt,ttt+ation t Iwo t’y. III t,lw ptwettt case, it tnay be said that we arr summittg tlhe prrtutbatiott srrirs;. ‘l’ltr d&ails of t,his external-field technique are ptwrtttrd itt Swtiott II. AI- I ). I tt cottsidrting the theory for q = g = 0 we fittd that altholtgh it is :I sttxiglttforward matter to find the Green’s functions, thr calrttlatiott of thr clcprtttlct~w of thr \‘acIttti1i-to-Vactlunl transition amplitude on t,ltr rst.crttal firlds t)t.ittgs t () light, thr ttrrrssity of defining the F’rrmiott cutwttt tnot*c ptwisrly tltatt simply :IS &“I). l’hr singular value of thr I’etmiott attticontmttt,atiott wlatliotts rctrdrtw SIIC~I ati rxptwsioti nieat~itiglcss. In calculating ativ matrix rlcmrtit~ of tltr c~itwtlt \vfs xc fotwd to look at a matrix element of 4 (.i’ jr”+ (.c ) itt tlw limit .t’ ----* L’. ‘I’llis ih sittqtlar, hut the symmetrized comhittatiott t ?[I$ (x’ F+‘$ (.t. I - $‘I) (.I.’ 14 (.I’ i j Kivrs a fittitr result which depends, howevrr, on thr dir&ion of approac’h of ttrr two poittts attd which cannot, thereforr, wprrsrttt tlir tnat,ris rlrtwttt 01’ ;I I,oiwttz two-vector. One way of obtaining a I,ot~t~tttz~c~o~:tt~iattt rrstilt is to a\-rragr ovrt’ two directions-namely, a givrn oitr and its prt~pc~tidiriil:tt. ‘1’1ti.q clrfittitiott of thr current is t,he same as that, adopt4 t)y .Johttsott. A\lt~ltouglt t’he singularity of the currrnt has been rlitnitiatrd, n-f’ fiid that tllcrf~ is a residual cf’frct in that a matrix element of ,j” is tto lottqrr ittvatiattt uttdrr ;L coorclinatr-drprndent gauge transformation of the I~rtmiotl tirkl, as wou1d II:L\Y~ heen rspected for the formal juxtaposition $r”#. 111addit,iott, t,hc ttrrd to aflmit a timelikr as lvrll as a spacelike approach itt the limiting prowdurr mratts that \VP can no longer confidently compute the rqual-time commtttatiott wlatiotw Jjrtwrrtt .j” and # or $ from the canonical commutation rulrs. ‘I’ttrsr ronsidetnt iota,+ are discussed in Section II. I?. The properties of the solution are described in S&ion III. We discuss, it1 otdw, those tnatris elements containing no Bosott field and thosr containing otw,

4

SOMMERFIELD

two and more than two Boson-type fields. (By a Boson-type field, we mean either the Boson field itself or the Fermion current.) The simplest matrix element is just the vacuum-to-vacuum transition amplitude as a function of the external fields. By means of suitable functional differentiation, matrix elements of the Boson-type field may be derived from this amplitude. In the absence of external fields it could be set equal to 1 without any of the Green’s functions being affected thereby. This represents a redefinition of the energy of the exact vacuum to be 0. However, our calculation does not give an answer of the form efi* which could be interpreted in terms of this energy shift, but includes also a factor of the form e?, as illustrated in Section III. A. The extra factor is subsequently traced to a mistake made in our use of the external-field technique, the origin of which is discussed below. Also in the category of no-Boson matrix elements are the purely Fermionic Green’s functions. The one-Fermion Green’s function in the absence of the external fields is found to be a multiple of the free one, as indicated in Section III. B. The multiplier is an exponential of several singular functions and is, strictly speaking, zero-an indication of an ultraviolet divergence. Furthermore, since this Green’s function does not approach a constant multiple of the free one as the coordinate become large, we cannot define one-Fermi011 states. Equivalently, the Fourier transform of the Green’s function has a branch cut rather than a pole at k.’ = 0 and any attempt to renormalize the solution to a pole leads to infrared divergences. To avoid these difficulties, we shall make no attempt to define asymptotic states and shall continue to work only with unrenormalized fields. In order to make meaningful statements about the matrix elements, however, we are forced to introduce ultraviolet cutoffs. As shown in Section III. C this can be done without destroying the soluble nature of the model. From that point on all statements are understood to be true in the limit as the effective cutoff momentum becomes infinite. When the external fields are turned on, the one-Fermion Green’s function acquires an exponential dependence on these fields, similar to that which is found for the free one-Fermion Green’s function in the presence of these fields. The many-Fermion Green’s functions are basically determinants of single-particle functions, except that the exponentials multiplying each term of the determinant differ somewhat from term to term. These expressions are written out in Section III. D. The matrix elements involving just one Boson-type field and any number of Fermion fields are discussed in Section III. E. By explicit examination of these matrix elements we find that the field equations are indeed satisfied and that our definition of j” in terms of Fermion fields is equivalent to a functional derivative with respect to the external field. At the same time, we learn that the product

FIELI, THEORIES OF ONE SP.\TI.\L I)IMENslOS

.,

of a Hoson-type field and a I;ermion field at t’ltc same point is adequately reprf~sented by simple symmekization and that there is 110 latetrt tlc~pendrttce OII tlrcb directiott of approach. J. I”~tt~tltct~tuot~t~, the pseudocwt-ent j’ (defined itt the sanw way as j” except tlt:tt 7’ is rcpl:wd by ypy5j is not cottswved, cl-en it1 the atwttw of ext8etwalfields. \\-P filld, by explic’it’ cOniparisoii of matrix eb?tiietit~s,tllat) a,,?”is Wtllally a tiulltiplf’ (Jk t11t tm1s0r paL’t,, H,, ) of the 13osotifield (which, iii otif diniet~sioti, is jilS;t 3 pf~sttflo scalar). J~Jow~\~~r, t)y adding an appropriate ntult~iple of the J3osottfield to ,iL VY~ can find a pseudocurrent which is cottaer\red, attd t)g adding approprint~c~multiples of the csterttal fields to j” attd this new psettdoc’ttt.t.Pttt.,we c1at1find :I ctttwttt :tt1(1 n psf~fidoc,ttt.t,~~ttt, which arf’ hotli coilserved ~\TII itr tlifx prfwii~e f~f f~stt~r~rt:~l fields. ‘I’ltis is slto~~n in Swtiott III. (;. At this poiiit, \vfl Ir:t\x~ two c~iitwt~ts: ,I’, tlrfitwtl itt tenus of tltt I’etmiott fields and a cottset~\wl otte wltic~lt \ve call 0”; a11tlt,wo ~,“rlldf~crlrrellts: zP, defined it1 terms of the ~~‘et~tniott fields, ntttl :I WJllwrvcfl ottf1 which we call f?‘. 1111 four of thw fltlmtitif~s I)f~lia\~r~ l&f> tu7J-\~fvtrJt~. tttidcr pi’opt~r lmvtitx tlatisfolmat~ioris. When \vf’ conic to csatniiie the properties of niattk cletiwiits fwitaitrittg t\vo Bosott-t’\;pe fields, we would fittd tltat it is tto longer true tltat we get the s:ttw answer by repnwntitg j” as a futhottal derivative with respect to the estetual field as wf’ do by defining it, iii Mnis of the l~ertiiioti field. l~‘rtrtlietmot~e, tllf. field wlttutions and the cottser\-atiott cquatiotts for ()” and I!?” would ttot, tw \.:lli(l as cottst~twctcd from eith definition. Ilat,lwr tltatt tlcmottstt~atc these cotttt~;l dihiotts explicitly we proceed immediately to the I~WOltl~~ifJll of ttte difficwlty 11) going tuwk to the one-Bosott type matrix elcnirttts. Sitiw ,j” dots not, pfJS;sfw dcsitath rottscwatiotl or cotnmutat.ion propet,ticbs,we define (in Section 11I. I I j

G

SOMMERFIELD

the simpler, single-time current S” in which the Fermion fields are kept at the same time as they approach one another. As we remarked earlier, this current does not behave like a two-vector under Lorentz transformations. By explicit examination of the matrix elements of S’ we find that it is linearly related to jV, the Boson field A”, and the external fields. Thus both j” and S” cannot be considered to be explicitly independent of the external fields. The commutation properties of S’ with II, and 4 are those we would expect of the formal juxtaposition &“#. If we then take S” to be independent of the external fields and define j” in terms of it, we see that j” will have an explicit dependence on the external fields. This must be taken into account “by hand” whenever we attempt to find a matrix element containing two j%, by functionally differentiating the element with one j’ with respect to the external field. Because of the fact that Bo is not an independent field for a vector Boson, we find that it, too, depends explicitly on the external fields. Taking these explicit dependences into account, we present in Section III. I, the more interesting properties of the two-Boson-type matrix elements. In particular, we determine the equal-time commutation relations among these Boson-type fields. For the various vector fields, these turn out to involve derivatives of &functions. Included among the matrix elements is the Green’s function for the vector Boson field, which is found to have a part with a physical mass differing from the bare mass, and a part with zero mass, which can be thought of in terms of a bound state of two Fermions. All of the difficulties associated with the two ways of defining j” are resolved when we take into account its explicit dependence on the external fields. The various operator equations we have discussed-the field equations, the current conservation equations, and the relation between X” and ,j”-are found to hold in all matrix elements and apart from this no new information is learned from the examination of matrix elements containing more than two Boson-type fields. We observe, in fact, that our theory is really just a generalized-free-field theory. Having satisfied ourselves with the consistency of the solution we return to the method of solution. In Section III. K, we reconstruct the Lagrange function from operators of a single time only so that all of the action-principle properties we have used are satisfied. It is found to differ from the formal Lagrange function in that it does not look relativistically covariant and in that the coupling constants appear in an unusual way. We then find that the first step in the solution, whereby derivatives with respect to the coupling constants were written as functional derivatives with respect to external fields, is incorrect, for we did not account for the explicit dependence of the coupling terms on the external fields. The effect of the correction, as computed in Section III, L, is to provide factors which just cancel bhe real part of the exponential in the vacuum-to-vacuum transition amplitude.

Tllr I>agrarqr function has been constructed so as to supply the proper (111x11tizcd field eclnations and dependences on the external ficl~ls. If, in addition. it possesses syinnwtry properties, we expect to he able to wise the action principle to derirt cc&n operatoi “conservation” ecluatioti. L: n-it,hout havillj?: to 100k ai ttir

sol~it,ioi~

of

the

problem.

At

the

same

time,

wol~ld I)(>Id

w

to

ttw

Iillit~ilr~.

generating the svmnetry tra~lsforrnutions. l’or esamplc, \\Y’ lliL\.(’ ohsc~r\-rtl that our solution contains coilservatifm laws for a f*rirrfliit aiifl a psfv~flf~cwrent. In Section IV. Ai we address ourscl\-es to t,lw derivation of t hew Ia\\ s wit~trout wfcrencr to the solution. ITsing the corrected l,agrange function, w (XII almost do this except for the fact that the appropriate ~IMIL~;Pin ,q” r~~dt~r:I ~oordiiiatr-depeii~leiit gauge transformation is iiotj kirowi. That, is, as \\-itI1 j”. \vf’ cspect ,S” to be not invariant under sac11 a trallsfo~ln:ttioll even ttloliglr it is so formally, hecanse none of its mat,ris elements are ilr\-ariant. We are foiwtl to cheat sonwwhat and to take a look at the change ill a nut rir clenm~ t of A’” i\vhich is fortunately not dependent, on the coupling mlstant and helwc is tlw same as for free fields) t#o determine t,he appropriat,e variut,iotl ill the opcwt 01 ,s”. 111 S&ion I\‘. 13 we attempt to derive the cvwgy, nlotnentwn, anti t\vtIflimctisioiial angular rnornent~urn opelxators, which xrf tltc generators of infitiitf~si1na1,illllomo~ellet,us Lorentz transformations. \Ye arc again faced with an itlahilit? io pro\~ittc ail a-priori value for the Trariatiotl of S”. this tim rindcr 3 sp:i~‘fl fif~pciidrrit woidinate t~anHfo~matioi1. The difficvlky is to :wcolmt for t,hr cah:tlge iit tliiwtioii of approach. Ilowe\-er, we are able to write, don-ii tllfx drpc~uflctrf~f~ of (3K 011 almost all of the parameters of the t~ailsfol.inatiotl atld thris t,fJ tlf$f~rirliiw operat,ors

r”“‘, Tnl, and 7’” of the f~lle~gy-mot~lf~tltll~~l tt1c cfJlllpollcllts 1 rllfl I GUI ttw11 1~1 dcri\.ed indirectly if KC intrymte

tcstlsfjr.

tlw

IIw

associatccl

compollotlt

c*otwr\-at

iota

kL\W. ‘1’1w explicit matrix elrnieiit s of the theory st10w tStint it is rf~lati\~istif~all~~ itl\,ai*iatlt in the absence of external fields, although t,he tioncovariatlt fortn Of’ rtlcs I,agrange function might lead us to suspect ot~hrrwise. .\nother indication of ttw relat,ivistic in~ariancc is given in Section I Y. (‘, wlif~rf~ n-c ask if thr two-tnonl~~1~t)unl we ha\.c derived hehaves like a tu-o-\wt,fJr 1111der I,owntz trarlsfol,nlntiotls. \Ve awi\r at nrcessary equal-time ~ornrnatat~iot~ wlat.ioiw IJf$wel~ 7”“’ ;i11(1 itwlf al~l hrt~wcetl 7”‘” and T”‘, which are found to he satisfied. The aho\T uses of the action prinriplc are still tiot, umiplett~ly lcgit~iinate siri(*e UT had to know the solution of the prohlem hefore w co11ldTvritc tlo\\-~l tlw I,agrange function. That is, m-e made use of’ t,he elation )Jetween j” a11t1 S”, n-hich was inferred from an examination of the matrix rlrmr~nts. Tlms, itI ,qcfatif)t> I\-. I) we twn to the problem of the a-priori construction of the I,agraiige f\lllrt,ioll. -\ssurning that) it’ has essent~ially the same form as t,he previous OIIF’, \~f’ itit roducc

asking Iw>nt

scveti

uiikr~owii

corficierits.

tmllatmthe field cqrlations of relativisti?

invariance

We

he satisfied. as espwssed

find

fix-f>

A sist’h iii t,cniis

quatiotis

amfjiig

ttww

is tlet~cm~ined

hy the rcclllire

of t Iif> f>f Iii:&time

f~ommllt:tt,~

1)y )I‘

8

SOMMERFIELD

of T” with itself. This last step involves only the cannonical commutation relations and [S, (JJ), X, (x’)]. Since this commutator is constructed from canonical operators, all at the same time, we might expect that it should not depend on the dynamics of the theory. But to find its value, we are forced to look at matrix elements. Doing so, we discover that the commutator is a c-number and does not depend on the coupling constant, so we can say that use of the free-field value is sufficient. The seventh equation among the coefficients is to be determined from the definition of j’ in terms of the Fermion fields. Since the definition we have been using involves a timelike approach, we are unable to use this seventh condition except in matrix elements of the solution and hence we are, in fact, unable to derive the Lagrange function for this theory. On the other hand, it is possible to choose a di$erent definition for j” so that the Lagrange function is determined a priori. A particularly simple way is to choose the time component, j”, to be equal to So. This has the effect of reducing the number of “currents” by one since j” for this new theory is then found to be the same as Q” for this theorynamely, the conserved current. A comparison of the two theories is provided and it is found that the new one is considerably simpler than the old one. Finally, in Section V we discuss the possibilities of extending this work to more general cases-one-dimensional theories with a J’ermion mass, and threedimensional theories. II. SOLVING

THE

MODEL

The characteristics of the model under consideration are summarized in the Lagrange function

Several points of notation must be established. We take ii = c = 1. We shall distinguish contravariant from covariant indices with the diagonal metric tensor gliv, for which g’l = - go0= 1. A repeated index implies summation from 0 to 1. We will occasionally use the symbol x to represent the space coordinate CI?.The Boson field, with “bare” mass parameter p, is described by the vector operator A” and the antisymmetric tensor operator B’“. Since B” has only one independent component we introduce B = Bo, = ji~,,~B”” so that B,, = -BE,,“, where cpvis the completely antisymmetric tensor of rank two (co1= 1) . With only one space dimension the Fermion field operator # can be taken to have two components and hence for the Dirac matrices y”, we can select twodimensional Pauli spin matrices, as follows: yo = c2 zz

0 (i

-i 0 >;

y’ = iLTl =

FIELD

The product

THEORIES

OF

of any two y-matrices

ONE

introduces

!)

I)lMESSIOS

turns out to be diagonal:

y”y’ = -y’Y This statement

SP.ITlAL

+ yy5 .

the diagonal psrudosralar

(“.I

I

matrix

The Hermitian conjugate of + is fii and the adjoint is $ = #‘-y”. As usual. ij, means ai’&:“. In cases where the derivative might, apply to one of several \wiahles, we make the convent’ion that, unless otherwise specified, ap acts OII t hc first coordinate variable that follows. The bar over a \wtor cluantit,y, as it\ (35” means FC klitre are t\vL coupling constants : y (with dimensions of a mass i and f7 (tlimtbt1sionless). The proper definition of t,he I’ermion current ,j” is one of the (I~wstiorw at hand. We ask, for the moment, that, it he gi\-rlr t>y j” (.I,) = lim $ (.r iy”ll, (L’ 1 I-l’ wliere the limit is to he taken in a sense yet t,o tx: determined and whtw t Ii0 rrlat,ivc ordering of I$ and I& may have to lx specified furt,her. The ordering of tlw other operators in 1: will also he indicated later. 13.

I”IELI)

kjI-.4TIONh

.\NI)

~hMMlY’.~TIOX

I:EI,.\TIOSS

I,agrange function has lwelk colistructcti so that Ivlrcli classical manner the following field ecluations are ohtail&: Tlic

- i-y"a,fi

d,.l, - &A,

= -y'

it is varied iti thf:

i!qAlg + u,jy I*

= IZ,,

d,B’” = cjj” - ~‘~1’.

i 2.2 1 (2.:; 12.1

Thf~ canonical commutation relations arc taken to t)e the standard ones for apitror

a1~1vector fields. The only llonvanishing equal-time commutators conimut~ators for kvo 12crmion fields ) inwl1.ing #, $‘, -1, . and R are

Since /lo is dekmined

in terms of R and j,, ty the e(lIlation

we arc not at lihcrt,y to assign commubation relutioiis for it a priori.

(or atIt i

10

SOMMERFIELD

C. THE ACTION

PRINCIPLE

Our immediate aim is to compute the vacuum expectation values of timeordered products of field operators, that is, the Green’s functions. Let such a time-ordered product be represented for the moment by (R), . The effect of a change in the dynamics on any given matrix element of (R) + (and, in particular, on the vacuum expectation value) can be calculated from the corresponding change in the Lagrange function by means of the quantum-mechanical action principle as follows : qo I (R), I 0) = i

(0 1j ~%fw4)+

10) + (0 / 6(R)+ IO).

The second term on the right involves any explicit appearance of the quantity being varied in the operator (R) + . The vacuum states (01 and IO) are the lowest eigenstates of the energy operator at infinitely remote times in the future and past, respectively. The procedure we will use is to simulate the effects of the two coupling terms in the Lagrange function by introducing external fields +,, and jr. These are included in the Lagrange function in the form

The field equations (2.2) and (2.4) are then modified to read -ir”a,J/

= Y” (gA, + uj, + 4,)9

(2.7)

&BP” = gjp + J” - p2Ap.

(2.8)

(Equation (2.3) is unaffected.) The usefulness of the external fields becomes apparent when we apply the action principle to variations of J’ and 4”:

[6/6c&t(x)l(O 1(R), IO) = i(O I (jp@)W+ IO) + (0 j P(R)+/%t(x:)l [s/aJ’(x>](O

1(R), ( 0) = if0 I (A,(x)R)+

IO) + (0 j P(R)+/6J”(~)l

IO>

(2.9)

IO). (2.10)

The derivatives of a matrix element with respect to the coupling constants g and U, given by

(a/ad (0 / CR)+ I Qerg = i (0 / j d2.dRf’(4A,(4

)+ j O)eco

(a/au>@ I (RI+ I Werg = can then be expressed in terms of functional derivatives of the matrix element

FIELD

THEORIES

OF

ONE

SP.ITI.\L

I1

DIMENSIOY

with respect to J” and 4’ as

(a/ag)(O /(Ii‘)+ 1O),,,, =-is

d”.~[6~6~,(.~)1(6,~6./“(.~~)l(O

i if?).+ /Oh,

(d/&7)(0 i CR)+1O),,, = -!$i J ri~.r16!6~,(.r~)][6./6~“(.r~)](O CR,, O,,.,,, The subscripts e, (T,and g indicate t’hat the external fields and the coupling CWIstants are to he considered as “on.” We have assumed that neither .j,, IIOI’ .I, csplicitly involves C, g, & , or .I, in its definition. These first-order differential equations may he itltegratcd with the result

x (0 ~(R I+ 0 ), where the lack of subscripts ‘T, g on the right indicates that this matrix elcmcn~ is to he computed in the absence of coupling. Thus if wr are ahlc to cwrstnwt a matrix element for the uncoupled problem as a funrtiollal of the estcrnal fields, hy then taking the indicated functional drrivat,ivcs we will hare simulated tlw efl’rcts of the coupling. In the model under consideration all of this can tw clone csplicitly whereas in a more general problem o~te has to resort to po\ver-series expansions in the external fields. Then this method would reproduce the cwtlillqperturbation series. It is sufficient for our purposes to ronsider vacuum espcctst8iolt values of t inw orderrd products of I’ermion fields only. The other (;reen’s flmctions cast 1)~ obtained from these by means of appropriate funct~ional derivatives with respect to the external fields according to Q.0) and (2.10 J. Tlw rl,-I~ermion C;twt~ ‘Y f 11 twtSionis defined hy

where the time-ordering symhol ( I + includes :L factor of - I if tllc tiiw ordered sequence (operators of later time t,o t,lle left,) of the coordinates is ;1r1 odd permutation of the one shown. This Grwn’s fru&oll satisfies “causal” tx)unclary conditions (indicated hy the subscript c j it1 that as a function of it:: latest (or correspondingly earliest) time it has only positix.e (or corresponditql~ negat,ivc ) frequencies. Application of l?q. CL.111 yields ,,u) (16“(,,’ =

12

SOMMERFIELD

D. FIELD-DEPENDENT

SOLUTION

WITHOUT

COUPLING

In the absenceof coupling G,‘,“’ is just the determinant of one-Fermion Green’s functions : Gcn) ec =

Gec(a, XI’> . . . Gec(x1,~‘1 :

(2.13)

G,i xn , x1’) . . . G,,(x, , 2,‘) From the field equation for $ and the canonical commutation relations we obtain an equation for G,, (x, x’): -iy’a,G,,

(x, x’) - -f& (x)G,, (x, x’) = 6 (x - 2’).

(2.14)

The solution of this requires the free Fermion Green’s function with no external field, G, , which, in its turn, satisfies -i+O,G,

(x - x’) = 6(x - e’).

The solution for G’, , with causal boundary conditions, is G,(s, x’) = (2~)~’ / d”p( -y“p,,)(pz = (27iyyP(x:

- x’>,[(x

= ir”a,(-(i/47r)

- iv)-’ exp [ip’(x

x’>,l

-

(2.15)

- x’)? + i&l

log [(z - 2’)2 + iv]\ = iy”d,D,(z

- z’),

which equation also serves as a definition of D, (CC- 5’). The positive infinitesimal constant 7 indicates how the various singularities are to be treated. If we multiply (2.14) by y” from both left and right we obtain a diagonal matrix equation for G,,(x, z’)-y’ which is of a standard first-order form. The causal solution is = exp[iF(x)]G,(x

GetC-c,ho

-

x’)y’

exp[-iiF(

or G,, (x, x’)

= exp[iF(x)

-

iF(x’)]G,(z

-

x’)

(2.16)

where F (3) is the diagonal matrix function F(z)

= -i

s

&Gc(x

- E)r’bo(E) + -wh(f)l (2.17) = yac

s d2EDc(x - Eh”U).

Inserting this form for the one-Fermion Green’s function into the determinant

FIELD

THEORIES

OF

ONE

SPATIAL

1::

DIMEKSION

of (2.13) we ohtain

111the taspansioll of the determinant we hare introduced tot. f’. ‘I’hc s\mi is over all permutations of tlw J,‘. 1~:. I?~CPIX~TION

0F

THE

tlw pr>rmutation

operas-

CL~RRENT

To dct’crminr the dependence of (0 1O)( on tlw cst,t~rllal fklds fuiwtioiial tlrri~~atJive with respect to 9, (:r ) : [6,‘6& (x)] (0 / 0),

=

we csamillcs it.s

ii0 ( .j” (.r I! O),

Tentati\-ely expressing j” (II.‘) as $(.c’)r”$ (.I::) in the limit of some mcaiis of approach of the points x and R.‘, we may compute it’s l.acuuni csprctation va111c~ from the one-12ermionGreen’s function. If we restrict t]hc direction of approacll to he not along the relatixre light COIW of s and .Y’, wr lia\~, referring t,o (2.1.-I1 (‘2.17)

mliert?:rp - .I.‘@= CL<‘,a > 0, 1[’ I = 1 and whew “tr” indicates a Draw IJ~Y~I’ the spinor indices. This expression is, of course, singular as a,-+ 0. A more satisfactory definition of the current which removes the singularity and which prclserves the essential symmetry of 1,5and tit (charge s.vmmetJry) involves considwation of the operators multiplied in the reverse order as well. That, is, w look at i (0 ql(.c’ )y”l+b(.c‘) - ~“+(.r’,lp(s, 2 (0 1Oh

/ O),,

We may expand the difference of esponentials in powers of a. This difference is zero in the limit cc- 0 hut has a term proportional to (1n-hich must he ret,aincvl

14

SOMMERFIELD

The result is then

once we make use of the explicit form of F (x) and the relation (2.1). We note that this expression depends on the direction of approach and hence is not a satisfactory definition for a Lorentz two-vector. A covariant object is obtained if we choose not a single direction of approach but if we instead average the results along two directions: namely, along 1” and r. Then [Xb”/t2 is replaced by

Kr”r”/r’) + wml

= x%7”“.

This last relation holds because 1” and r, for a given p, form a complete set of two-vectors. The precise definition of the current is then j” (xl = w& -

iii

r4a (x - %G-)i&

tis(x -

M-)4cY(~

Using this new definition

(x + 4/i@-) + qa (x - ~&z~)~, (Lx + >$zj-)

+ 3&i-)

-

$0 (J -

$$&4a(z

+ emus-)].

(2.21)

we find

where u”‘(‘$ Integrating,

g)

= - (2*)-l

(gPUgYT-

tpa~Yr)d~dvDc(~ -

l’).

we obtain (0 I Oh = (0 10)~ em

[

Mi 1 & &‘d~>~““(t

- i-‘MI’)]

= (0 I 0)~ exp F4&&$1. In the second form we have introduced a matrix notation wherein coordinates, integral signs, and indices are suppressed. The subscript J indicates that & is off while J, remains on during the computation of the matrix element. Careful attention is required when the product of fi and 4 at the same point is as singular as we have found it to be. We note that the casual assumption that a purely spacelike approach will yield a current with the desired Lorentz-transformation properties is not correct. If <” is a spacelike vector then y is timelike and the current must then involve a timelike direction of approach. This, as we

FIELD

THEORIES

OF

ONE

SP.~TI.~L

13

DIMEiVSION

shall see, gives rise to commutation relations of the current’ with other operators which are not simple consequences of the canonical commutation rules. I;urthv more, the singularity of the current is strong enough to force one to disallow some For example, utidcr an infinitesimal formal properties of operator products. iiunwrical gauge transformation wherein # (.t. I ----) $ i.r / + i6d (J I# (.l, 1 :111(1 q (1. i + IJ (x ) - i&1 (.rj$ (n ), the current, as drtint4 hy (2.~21 1 will he cha~~gc~l and hcncc it is not gauge invariant. ITor, in computing ally matrix f~lf~nnf~iltoi j” (.rj, wt’ find that 6h (~1 contributes additively t,o F (.c.I ill (2.20 1. There rcmaiiis the calculation of (0 / O)., as a frulctioiial of .I” (.r I. I’rowfvlitig ill a manner similar to the above, we considrr [~,~W(.r)](O I-sing the field rcluations

1O)., = i(0 I -I,(x)

(2.8) and (2.8), n-e find

+ Fcl’)(0 / d,(FI

(-a”

SO that upon intcgrat’ion

Oil.

o),

= [yGy -

(1 ‘ML?)dpdy].~Y(d (0 [ o),,

we obtain

WC have introduced %I”( t - E’ ) = IY,” =

(l//.b,,&]A,(~;

(2SY’

I

tl”/i[fJ+,

+

$ - .$‘) (l/&l”)/i~k,][/~

+

/J,’ -

( t’.‘,”--

iv1-l

‘)

x esp lil;?E - &I which is the Green’s function for a free vector Boson of mass M in two dimellsions. The quantity AC (CL; { - <‘) is the Green’s function for a scalar Boson of m:w p in t,wo dimensions. III.

PROPERTIES

OF

THE

SOLt-TIOS

Now that we have computed all of t,he ingredients that go into (2.12) w must proceed to take exponentials of quadratic functional derivat,ives of CSponentials of quadratic functionals. This is a straightforward task and is awonplished in .\ppendix A. We shall reproduce here only the results.

-1. \'.~cT:I.M-TO-VACUUM

TRANSITION

l:or the exact vacuum-to-vacuum fields we find the simple form (0 I O),,,

AMPLITUDE matrix

= fit0 esp [!$$I’+

element as a furlction +

i$C7.J +

1,i./Ir7./j

of t]he external

16 in which

SOMMERFIELD

three new symmetric

functions

have been introduced:

(3.1) X(/.&h” +

(1

+

Cal

-

a,&)

+

a

+

X)

A&‘;

F -

5’)

-

+S(l

a

- F’)]

_ Et)1

+ (1 + ~(cL’~s~~ - ad,) 1+(Y+x The coupling

constants

now appear in the rationalized

12

(3.3) IO]

.

forms

a = u/2a

(3.4)

x = g2/27rpz

(3.5)

and there has been a change in the mass parameter P

Ac(pLI; t _

(3.2)

= /J2(1 + cy + X)/(1

of the Boson function + (.y).

(3.6)

The constant a0 = (0 1O),, which represents the amplitude that the vacuum remain the vacuum when the external fields are off is calculated as O,=exp{-~~d?i[*“‘(O)

log[(l-a--A)(l+cr)] - ?4 s,:’

It is to be noted that this is not of the form ?* or e+im to be expected if the exact vacuum state has a different energy than the bare vacuum state, but that it is actually e-im-m which is zero. This apparent inconsistency presents no difficulty as far as the Green’s functions are concerned since Q0 appears in both numerator and denominator of (2.12). Furthermore, in Section III.L, we shall see that there is an additional multiplicative contribution to (0 1 O),, which precisely cancels the part involving 6’“’ (0). The remaining factor will be discussed there as well.

FIELD IS. O~-E-I~ERMI~N

The one-b’ermion

THEORIES

GREEN'S

OF

OiVE

SPATI,\L

17

I)IhlENSIO~

lccscTron-

Green’s function

with

the external field “off”

is foutld to tw

Now both II, (0) and A, (P’; 0) are infinit.e, an indicat,ioll that the theory :is it stands is ultraviolet, divergent. A4lso since fj, (.r - I’ I = - (; ~4~ 1 log [ (.r - A.‘)’ + ,ivJ, there is no approach of t)he coupled (~reen’s fwct,iou to ;I multiple of the free one for large (s - .r 1’ and hence in the I<‘ourier transform of (:*!,( (2., X’ I there is no pole at k” = 0, hut only a branch w-~-a sympt,om ot infrared difficulties. We shall not concern ourscl\~es here nit,11 the infrared proI )lem and the t,ask of defining asymptotic I’crmion st’ates. ()n t,he other hand, itI order to make sense out of t,he limiting procws which defines the c~wrent~ \v(’ shall hc forwd to supply an ultraviolet clltofl’.

We can do t,lris in au exact’ way by asking t,hat the coltpling of ,j” t,o it,sclf atl(1 to -.l, ill the I,agrange function be nonlocal in the spare coordinate.‘i Tlrur; w malw the following replacement in the space integral of the coupling terms:

+

! 9jP(

.)’ iu(

x -

x’

,,j,(

.r’ 11,

( J”

The result of spreading out the interaction in this way is expressed most in terms of t,hc I’ourier transforms: zc g(k) = dx g(x) esp( -ikxi s- -xa u(k) = (lx u(x) rspi -ikxi s-z X(k)

= [g(k)]‘,

a(k) = u(k);%.

%rP”

=

,,.“’ 1,

c:tsil>

18

SOMMERFIELD

Then if without the cutoff we had a term of the form F(a, X)h(x - x’) = (2~)-‘F(a,

X) / d2kh(k) exp [ilc’(x: - x’),]

the transcription to the case with cutoff is accomplished by bringing f (a, X) under the integral sign and including the dependences of CYand X on k: F(cY, X)h(x - x’) + (27r)-2 /

d%h(Ic)F[a(k),

X(k)] exp[ik’(z

- x’),].

(3.9)

Thus all we need do is to interpret CYand X as they occur in the local theory as coordinate integral operators in the senseof (3.9) and we will have achieved the results of the cutoff theory without complicating the notation. It must be remembered, in this connection, that since P’ depends on the coupling constants, it, too, must be interpreted as a coordinate integral operator. Since D,(O) and AC(p’; 0) are logarithmically divergent integrals we ask that. the cutoffs be such that m dk ol(k)/k < Q), O”dkX(k)/k < co. s s For simplicity we may use functions satisfying the relations 0 5 cz(k),‘a 5 1 0 5 X(k)/X

5 1.

Then if the cutoff functions are to introduce no further singularities into the various integrals in which they occur, we must have (referring to (3.8)) l>

la+XI

1+(Y>o. D. ~LIANY-FERMION

GREEN'S

FUNCTION

The dependence of the one-Fermion function on the external fields occurs in exponential factors:

where

+ (1 + ?;;(I

[A&L’; .$ - x) - A&‘; + CX)

E- x’)l

FIELD

THEORIES

-

1 +‘k”;

OF

ONE

SPATIIL

l!l

DIMENSION

x bA/.LLI; E - .s) - .I,.(/*‘; E - .r“)]

The second spinor index in the diagonal matrix y5 is summed index associated with the coordinate .I’ in GC(x, m’j. In general, t’he n-Fermion Green’s function is given 1)~

1 .

n-it,h thf> spinor

r;l.$c ,CI . .I‘,A) .L.‘l . . . .I.‘,, 1

&(/.L’;.(‘,, .s:>, ; .SI, .$j) = &(/A’;s, - .(‘,)- 1,.(/l’;.I’,- x;, I - L&IL’;.s:,,- .(.,))+ 1,.(/d’; .r;, - .I$,I. superscript on yhi’ Indicates . that this matris \vit,h the spinor index of .r, . The

is associated in multiplic~atiotl

‘I’hc task of finding one-Boson matrix rlemcnts (wc include j” among the> Hoson operators) is accomplished 1,~ taking the appropriate flulctiorial drri\-a ti\w.

?‘ltlw

n-c find

i~(.l,(,t)~(.I.)~i.~,‘)

)+)co<,= [(A4p(f))rru + Jfp!(

.r,.s’ljt;,, !,,‘(.\.,.S’l

(jP(,o)pv!, = I’ fr”~‘wwc,t - E’h#J,:‘) + I Tp”(: - E’ I./“( (’ I I i((.j,i,t‘i~(.s)1ci(.s“,

i :1.1:: 1

)+),oy = I(j,C t: )lpr, + .lVH( ,t 1 s, .s' )I(;,.,, 1.r, .s' )

\vhrre w have introduced the notation ((R)+),,,

= (0 1CR)i : O),,!, ‘(0 I (I),,,

I:sing these relations it is possible to check the l’f~rmiol~ field ecluation (2.7 1 and canonical commutation relations. That is, it is nmv possible to det~rnlil~~~

20 explicitly

SOMMERFIELD

whether

-ir”a,[i((J/(~)4(~‘))+),,,1 = isr”((A,(2)~(x)~(~‘))+),,, + iur”((j,(x’)~(II:)~(x’))+),,o + ir”~~(x)((~(x)~(x’))+),,,

+ 6(x - x’),

a matrix equation obtained from the field equations and commutation relations. The s-function arises when -i-/8,, is applied to Gc(x - x’). All of the factors multiplying this d-function conspire to be 1 when x = xl, a fact made quite rigorous if the local theory is thought of as the limit of the non-local, cutoff theory as described in Section 1II.C. The other terms on both sides are then identical provided we make the identification

d,D, (0) = &,Ac (p’; 0) = 0. This is guaranteed if we understand the products of jp and A” with $Jin the field equations, to be the following symmetrized expressions:

~,(JJ)J,(J) 4 ?4 z-z’ lim h(x’M(x> lim M,(x’M(x> A, @M(x) --f 92’ x+I

+ +(~‘h(~>l + ~(x’)A,(x)l.

The approach of the points :r and x’ may be taken along any direction except the light cone. It is to be noted that these products present none of the difficulties associated with the product of 9(x) and G(x). It is now just a matter of complicated detail, which will not be reproduced here, to show that (2.7) and the commutation relation (2.5) are valid as operator equations in all matrix elements otherwise containing only Fermion fields. It can also be shown that the computation of a matrix element of j” (containing no other Boson-type operators) as the appropriate limit, previously described, of a matrix element of &$‘# yields the same result as that obtained when we take a functional derivative with respect to & . F. BOSON-FERMION COMMUTATION RELATIONS The matrix elements (3.12) and (3.13) may be used to calculate the equaltime commutators of A, and j, with 9. For this purpose, we examine the discontinuity in matrix elements containing A,, and + as the time coordinate of A, becomesequal to that of +. We then find, for example, that

x0 = ~‘~:[A,(z),#(x’)]

= #(x’) disc,,z,M,(x

/ x’, x”) = 5 1 -‘{

_ x

x J/(x)6(x - x’> (3.14) [jp(x),~(x’)l

= #(x’) disc,,,,N,(z

1x’,L”)

=

1 -gfC

1

x

rzy6 &x)6(x - x’) +---1+ff

FIELD

THEORIES

OF

OiYE

SP4TI31,

‘2 1

1JIMENSIOS

It is to he noted that the relation for 111is precisely t#hecanonical olw, uamc*ly, a vanishing commutator with +. l;urt,hermore, that for ,,I,, is oh\-iously not sonu’ tiling that8 could have been written down in advance. I~i~lally, alld most inport,aut,, it is seen that the relation for j, is not what one \vodfl qwt for tllc, simple jklstaposit,ion $-y”# 011 the basis of the caanoiiicnl ITf~nniolt cvxim~r~t:lt iwl relations, luamcly, the right-hand term of (3.1-k) wit11 CY= A = 0. This k IIO~ sIlrprising ill vic\f of the necessity of including a t,imelilw wparatioll of 1IN’ I~‘r~~~~~io~~-fif~ltl cwordinates in the definit,ion of j”, :I procedwc whicah itl\xlitl:l tw thcl usfx of the equal-hime commutation relatIions. Tllat is, it is tlw tlyiiamic~ ot t81wprohlenl which is responsible for the changed c~ommlltatictlr whtiotw. I,‘or t,h

Cad?

1

=

0,

+hS

m3ldt

agw?s

Wit,11

that

Of

.~o~lllSOli.

‘I’he matrix elcment~sof B will he found from tllose of -lP t)y 111c;ms of tile fifAItl ccluatiorl (2.3 ). We have already seen that --II comnkutcs \vitll fi and 4 at (V!II;I] tinw,

so

t11at

ii cK(,Ej~(.r,J/(.l.‘)j+)~,.,, = I (H( $,l<,,,- lg./i 1 + a:)Iy&L( /J’; < - ,1.)- A,.(CL’;< - .c’ I] ) G,.,,,,1,I.,.!.’1. These expressions are consistent with the eclual-timr commutatoi~ 10 .I,” =x: [B(.rl, yqr’j] = 0 and t,lic field equation (2.1‘). ( ;.

(:~~~~I~~:N~~-C)ON~~RV-~TIOS

Ij.4ws

The fact, t.hat the Lagrange function seemsto he invariant transformation

4 ---)-p [i (q + %~~)I~,

$-

$ esp [-i

under t,hc gauge

(y - y6/’ I]

leads one, at least formally, to expect conservation of hot11 a current and a pseudocurrent. To inrest,igatc the possible conser\:ation of j” and ,T”we conside a,ii( (,;“(~)~(.r.)~(r’)j+)eb,J

= i( (~,j”(,t)~(.r)$Cr”~ -

Cl - 01- x‘,-‘[sc,t - .(‘i - I?(:” - AJG,.,,,. (x,.1.‘)

~~,[i((,i’(5)~(-(.)tc/(1.‘~~+),,,1 = i( (a,~“(~)~(2-)~(s“)) -

l+)t.Oy

+)cc,,

(1 + a)-% C,t - .(’) - 6 C,: - .r’ )]y&,,,,

(J, I’).

22

SOMMERFIELD

Comparison of the explicit expressions of the left-hand entries on the right-hand sides allows us to infer

4.m)

= J d*t’h VPV(E- t’Mf’) =

(1/27r)(l

sides with

the second

+ a, UPY(5, t’K(Ol

- QI- c%&#m)

+ (slPpv’(~)l;

a/47”(5) = - I &‘[%A V”‘G- t’Mt’> + atU”“(E- ‘$‘)J&‘)I(3.15) = (g/2v2)(1

+ &‘%t

1 d2.6&;

- (g/29)(1 + &#wN = -(gl2T)(l

+ aYB([)

E - d)~*[J”(f’)

+ (Wa)(l

+ a)-‘&$w

-

+ &‘a,$YO.

(l/29)(1

We notice that even when 4 = J = 0, the pseudocurrent is not conserved. This raises the question of whether the apparent symmetry of the Lagrange function under the 75 gauge transformation of the J’ermion fields is deceptive. That is, does there exist any conserved pseudovector quantity? There does, and it can be obtained immediately once we notice that the field equation (2.3) can be written a,A” = B. The linear combination (1 + (~)j” + (g/Ba)A’ then has the property that a,[(1 + a)jP +

(g/27&4”]

= - (1/2n)aJ

and is a conserved pseudovector when $ = 0. We may then define a pseudocurrent l?, conserved under all circumstances, by I? = (1 + cy)j” + (g/27+4’ where K” is the vector equation, namely

appearing

+ (1/27r)@

on the right-hand

= j” + (l/274? side of the Fermion

field

K, = gA, + uj, + +p A suitable vector which is conserved even in the presence of external fields is found as follows. Knowing that B,, is antisymmetric in LLand v, we may conclude from the field equation (2.8) that

,&,A’

= a,(gj’

+ J’)

~)-‘[(g/2~P2)Q$’

+

which implies that a,,A” = (1 a result which

requires

OL-

(I/CL’) (1 -

the use of (3.15). A comparison

a)a,J”]

(3.16)

of (3.15) and (3.16)

FIELD

THEORIES

OF

ONE

SPATIAL

DIMENSIOS

(l/‘?r)qS’

= j” -

shows that, Q” = (1 - cyy -

(g/&r)A

-

(I 27r;)K”

is conserved. T1.

SIXGLE-TIME

CURRENT

‘l’he facf that j” neither is conserved nor has simple commutation rnlt~s sug.. gcsts that it is not as desirable an object to work with as might haw iwen alht,icaipatcd. Its sole simple property, achieved at the expense of introducing a t.inxdikc~ limiting process into its definition, is that it behaves like a two-\-&or r11111t*r 1,or~1it.z tl~anaforlnatioils. IJet, us return, then, to the consideration of the 4mplcst limiting procrduw for &“$, namely, when the P’ermion fields have t.h(a same time coordil1at.e. \\‘f* clrfinc

The matrix elements of 8“ containing no other Hoson fields may he calculnt~c~tl directly from the Vermion Green’s frmct,ions. KP find? Iwing t&he oue-I~crtniotl frictions

and using the two-Fermion

function

21

SOMMERFIELD

The derivation of (3.17) depends on the identity Gc(x - E)r,Gc (t - 2’) = [Gc(x - El + Gc(E - x’>lr,Gc (x - ~‘1. By inspection of the several matrix elements we conclude that the following are valid operator equations, at least for the restricted class of matrix elements we have been considering: So = (1 - aljo 5’1 = (1 + ah

(1/2~)4, = Qo

(3.18)

+ (1/2a)4, = RI.

(3.19)

(g/2*)Ao -

+ (g/2a)Al

These show explicitly that S,, does not behave like a two-vector. They show further that not both So and& nor both S1 and j, can be considered to be explicitly independent of the external field @P. Because of the fact that it is defined by operators taken at the same time, we shall take the current S,, as basic and consider j, as defined in terms of it. By the same token, the field equation (2.6) may be interpreted as a definition of A0 in terms of j, and &B. Solving this and (3.18) simultaneously, we obtain Ao = (lh’) $7 = (1 -

(1 - a - A)-l[gSo - (1 - ol)dlB + (1 - a)Jo + (g/2a)401 a! -

A)-‘[&

-

(g/2n$)W

+

(g/2W2)Jo +

(1/2s)&J

(3.20)

Correspondingly, we rewrite (3.19) as jl =

(1 + a)-‘[&

-

(g/2n)A1 -

(1/2~)&1.

(3.21)

From the matrix elements of (3.17) we can conclude that x0=x

,O

:

[So(x),sea = +(x)6(x - x’)

(3.22)

[Sl(X), YG'N = -Y5v+(x)~(x - x'),

(3.23)

which are the “normal” commutation relations for a current. The interpretation of (3.20) and (3.21) as definitions of j, and j, shows the origin of the strange commutation relations of j, with +. Our remarks concerning the one-Boson matrix elements are now complete. The realization that j. , A0 , and j, must be considered as explicit functions of the external fields has it,s immediate and most important effects, however, on those matrix elements containing two BOSOII fields. I. TWO-B•

SON ~IATRIX

ELEMENTS

If we want to calculate ;( (j, (.$)j, ($))+)eag it is not sufficient to consider -W&J’ (~>lW~‘(,t’)l(O I Oh alone, but we must correct for the fact that

FIELD

j, clepends explicitly

THEOHIES

OF

OSE

sPATLIL

2-A -

- ~~~~~- - I316 (x -

2.)

DIMESSlOS

on $, . That is,

[j,, (.r), ,jI (.r' )] =

[=lo(.r),jI(2:‘)]

'

___-

27r(l + cr)(l - (Y - A)

= [Ijo(

S,(.r’)]

X’l

= icy, L)~p’)(l - LY- X)--‘816(x - x’)

[:1,,(.1:), S”(3.‘)] = i(1Q.P) (1 - a) (1 - u - A) ‘&6(x - x’i. I’siug field equation (2.3) and these commutation relatiow we can compute the matrix elements i( (B (,t)S, ($)) +)esyand i( (B (c)j, ({“j I +jFC,,. \Vr rrproduct~ t)rlow only the associated commutation relations : .J.‘)= .I’Ill: [B(x), .4&q = [B(J), j,(x’iJ = 0 [B(x),

.41(.r’)] = -d(x

[B(s), j,(n’)]

- x’)

(3 .-“7 1

= i(g;thr) (1 + cu)-‘6(x - ~‘1.

lPiilaIly, one is able to compute

1.1’zz J’IO:

[B(n),B(n’:)]

i Tlw general lack of conunutativit,y I!)).

= 0.

(329 I

between j,, and j, has been pointed out, hy Sichwinper

26

SOMMERFIELD

Equations (3.27) and (3.29) are consistent with the expectation that the canonical commutation relations are not affected by the coupling. Calculations of two-Boson matrix elements containing the operators S, may be made from the one-Boson elements by the appropriate treatment of ST,+. One then finds that the relations (3.18, 19) between S,, and the other operators remain valid. The commutation relations for S,, can then be calculated from those we have just derived. We find, in particular, x0 = z’O:

[So(I(:),So( = [XI(X), S,(.~‘>l = L&(z), AI = [So(x), Al( = [S,(x), B(d)] = 0 [So(z),

Xl(d)]

= (i/7r)&6 (x

-

= [So(z),B(x’)] (3.30)

x’)

(3.3 1)

Equations (3.22), (3.23), and (3.30) are just what are to be expected from the interpretation of S, as the formal juxtaposition $y,# and the use of the canonical commutation rules. In this sense, S,, is a canonical operator along with B and A1 . The relation (3.31) is independent of the coupling constants and holds for the free fields as well. It cannot be computed directly from the canonical commutation rules but must be inferred by evaluating its matrix elements. These statements reinforce our belief that S, is to be considered independent of the external fields and j,, dependent on them. Another indication of this comes from observing that the matrix element ( (j, ([)jY (t(‘) ) +)erQ , as we have computed it, agrees with that obtained from lim,,t ( (4 (x)Y,+ (E)jy (l’) ) +)eog where the limit is taken by symmetrizing spacelike and timelike approaches. Had we not made the correction for the dependence of j, (0 on $., (.$), there would have been a contradiction. Furthermore, the conservation of the currents Q, and &, as operator equat.ions would not have been valid in matrix elements containing another Boson field. We conclude this section by referring to Eq. (3.3) for IV,, , the Green’s function of the Boson field. We have already shown that this is simply related (although not exactly equal) to i( (A, (x)A” (x’))+),, . Thus the spectrum of the operator A, is seen to include a zero-mass term representing a bound state (in some sense) of two Fermions, as well as a term for a vector Boson of mass P’. Since it has only one independent component, the vector Boson is best thought of in terms of the pseudoscalar field B/p', whose Green’s function is given by (3.28). An operator having only the zero-mass term in its spectrum is R, . We have, for example d((R&)R&‘))+)w

- (Rl(E))e~,(R1(4’))eoQl 1 & =- ll+cr+X 2?r 1 - a! - x (27r)Z s

kl’

k2q

exp W(.t

- 5’Ll.

l’liosc matrix elements containing Permian fields as we11 as t\vo Hos01r titbl(ls ptwetit, 110 ttew difficulties and they may lw easily csprt~ssrtl itt tSct*nis of tltoic~ matrix elements already computed. -1Iatris cleniet~ts twtttaitiing 1n0w tlratt t\v~r Hosoti fields are gixw directly iti terms of those of pairs of these fieltls. ‘I’ltis simplificat~iott is most easily expressed by tlte st,utemettt tlut the gt~ticlal v0ii1 tntitator (tiot titcessarily at, equal times 1 of t,wo ISosoii fields (or the :ttttic*ottt mtitator of t,wo I~crmion fields i is a c-tiunihei~ and that l~c~twwtt n Hosott licxltl aii(1 a I~crmion field is a c-number times the ICertnioii field. The mo(lrl is tlrt~s sw11 to 1)~ a generalized-free-field theory. ;Zn immetliatr wtisc(~uetict~ is t,lrnt all of those operator relat,ions t,hat we have deduecd 1)~ lookittg at :t twtrktc~d V~;IW of matrix vlcmettts remain valid for all matrix eletwttts.

\\‘c now ohser~c that the original Lagrange futwtiou and the fashion itt nltielt t,lw field e(Iuat~ions were derived from it come under suspicion. (hc of the assump t,iotts rwd in these derivations was that S&j“ = C&Jly”$. But, n-e ha\-c seen t Itat ,j” cwtttot, he viewed simply as the juxtaposition of $r”#. Itt fart it, is S” \vtticalr must tw so considered. We shall remedy this situation 1))~ rewriting the l,:tgrattge friitctiott iti terms of S, rather than j, so that the same field ccluatiorts as tdow are obtained when we use (3.X) and (3.21 ) t,o defittr j, . We shall not f~limitiatc~ .I,, since the equat)ion giving it in t,erma of the other opcrat.ors can hc derivecl as otte of the field equations. Furt~hrrmore, to gusrantJec that, the Lagrange frmct iott coittaitia operators of a single time only, we shall interpret all produck of field operators itt it as being symmetrized limits of an approach at a givttt titntl. I”inally, we will retain the property that, t’tie \-ariations of the Lagrange ftttivtiott with respcvt, t,o the est,ernal fields are givett t,y

&2(,t) = GJ,(
C:i.Xi

1

Thesr rquatious are to he treated on the same footing as field equations. IIcttw t,lie field equation givin, (7 .4, in terms of JO and j,, is tiot to tw taken into acrouttt in making these variations. If the new Lagrange function is nssttmrd to Ix> of t trc> fom

where I’ is a homogeneous quadratic function of the fields B, .dy, A’,, , ,I, and $J, (not necessarily of a covariant structure), t,lwii it, is uniquely determined t q-

28

SOMMERFIELD

the conditions just, described. We find

To put this into a form which may be directly compared with the old Lagrange function, we use Eqs. (3.18) and (3.19) for S, and find

-

~/.?A,A”

+

gA,j”

+ %u&

+ id’

(3.34)

+ J’“A, + (1/47r) K; + (1/47r)K,,? The differences occur only insofar as the kinematical terms for the Fermion field are no longer manifestly covariant, since an equal-times approach of the operators is implied and insofar as this is compensated by the addition of the last two noncovariant terms. That this Lagrange function is appropriate for a relativistic theory has already been shown, since all of the matrix elements derived from it have the proper transformation properties. Another indication of the relativistic invariance will be presented in Section IV where we derive the energy-momentum tensor. L. CORRECTED

VACUUM-TO-VACUUM

TRANSITION

AMPLITUDE

We must now reexamine the steps leading to the expression of a matrix element with coupling in terms of one without. Using the new Lagrange function we find that it is still true that 6,d: = (8u)$$jlj’p

Now, however, the matrix elements of two Boson operators are not given directly by two variational derivatives with respect to the external fields, but the explicit dependence on these fields must be accounted for as well. Thus for an operator R depending explicitly on neither external fields nor coupling constants

FIELD

THEORIES

This may IF integrated

IMfbentiating

with

OF

OXE

SPATIAL

DIMESSIOS

“!I

to yield

respect to g we find

0n comparing this with our previous expression (2.11’) we notice that there is a~I (dra factor which just cancels the 6’” (0) part, of !l,, as g.iren in (3.7’).

30

SOMMERFIELD

The remaining factor in Q0, which is certainly not unity, may be understood as follows: We consider a free vector Boson field of mass P and notice that the action principle supplies a dependence of the vacuum-to-vacuum transition amplitude on the mass: (3.36) Understanding the equal-times product as the limit of a time-ordered product, we may compute the latter from wPy({ - 6’) (see (2.22)) by accounting for the explicit dependence that Ao would have on an external current Jo in the same way as we found Eqs. (3.24)- (3.26) from V,, by taking into account the explicit dependence of j, on 9, . We have i(0 ) (A1 (z)A1 (z’) )+ I 0) = (0 1O)(l/p’)[(-8:

i(O I (A0 (XMO (d>+ + $)A&‘;

I 0)

z - 2’) + d:A,(p’; =

J: - z’)J

(0 1O)A, (I’;

z -

x’).

Inserting this expression into Eq. (3.36) and integrating, we obtain (0 10)~ = (0 I O), exp [ -45 /” c?x:J”

d?A,(r;

o)]

,

Hence, for a TreeBoson field the energy of the vacuum state changes as the mass is changed. We call attention to the fact that the part of 00 which has survived is just the exponential factor of (3.37). This is not surprising in view of the fact that when the coupling is turned on the vector Boson cquires a new mass pLI. Thus the energy of the vacuum state automatically becomes adjusted to this. IV.

THE

A. GAUGE

ACTION

PRINCIPLE

AND

INFINITESIMAL

TRANSFORMATIONS

TRANSFORMATIONS

The new Lagrange function is still formally invariant under the infinitesimal gauge transformation

where q and r‘ are infinitesimal numerical constants. Do there exist conservation conditions on a current and a pseudocurrent that follow from this invariance? To answer this question we let q and T be coordinate-dependent and ask for the change in the Lagrange function. Since A1 and B are canonical variables, independent of +, they will not change under this transformation. Furthermore, since the time derivatives of A, do not appear in the Lagrange f:mtion and since

FIELD

THEORIES

OF

05-E

SP.~TI;\L

I)IhlES5LOS

:; 1

tmhe Lagrange function is already stationary with respect to chauges ill .I,, , terms which alp pure spatial divergences would arise from \-ariat’ious of ;lo nud t,lws;c would not contribute to the calculatiou. In considering A’,, ) howe~-c~~, WC’ must, rerncn~her t,hat it does depeud OH $. Becauw of the siugrllar Itat.ure of’ tlw matrix elements of S, , we cannot assume that there will Ix a c~aucellatiolt of tlits effect~s of the rhauges due to $ and 4. By rsplicit ~alculntiotl we fi~ltl

where we ]~a\-c applied Gauss’ theorem arid taken all \rariatious to \-anish ~116 cirntly rapidly at large distances. If the gauge trausfomation is t.o 1~ equivalex~t to a lmitarp t,rausformation on the st,ates generated by an operator defiued at the same time as the states, then terms dependent on the interior times in the matrix element cannot survive. That is, we must have d,Q” = d,R’ = 0 which is tfw desired result. The generator of the rulitary t.rallsformat,ioll :tt :I pi\-en time is then

32

SOMMERFIELD

The change in any operator X equivalent to this infinitesimal unitary transformation is given in terms of the generator by 6,.,X (x) = - i[X (x), G,,r (x0)].

(4.1)

It is readily seenthat Eq. (4.1) applied to fi, 4, A1 and S, yields just the changes under gauge transformations that have been assignedto these operators. Letting a(z) and r(x) revert to the status of coordinate-independent quantities, we observe that q$dxi? (x) + rJdxX’ (r) generates just the desired gauge transformations. In deriving these results, we were forced to examine the explicit forms of matrix elements in order to infer the correct variation of S, . Such a device is not supposed to be necessary in the use of variational principles because it presupposesa knowledge of the solution. Fortunately, however, we happen to possess the solution in the present case. On the other hand, since S’ involves # and 4 only at a single time it is difficult to seehow SS” could depend on the time derivatives d,,qand dor. Thus even without using the solution we would find the correct generator of the transformation. We have, of course, treated the Fermion kinematical terms formally and this is apparently sufficient. B. COORDINATE

TRANSFORMATIONS

These difliculties are encountered once more when we attempt to derive the generators of, translations and Lorentz transformations (two-dimensional rotations). These generators give the change in the transition matrix element (a, x,’ ) 6, xb”) between two states defined at given times, when we change the time of one or the other of the states, or allow one of the states to be defined on a spacelike surface whose normal is not along the time axis. The change in position 6x’, of a point x’ on the spacelike surface is a coordinate transformation which may be extended to points throughout the region between the two surfaces.5Using the action principle, we find the change in the matrix element from the change in the action integral under the transformation

x@(x)+x”(X)

= x’(x) + 8x@(x).

Because of the coordinate dependence of 6x” this apparent translation will, in general, also contain an effective rotation, measured by the antisymmetric part of &6x, ) namely, +~[13,6z, - &6x,]. The fields respond to the coordinate change as they would to a Lorentz transformation, so that only the rotational part has any effect. We take +, A, and 5 This method of deriving the energy-momentum dure used by Schwinger in ref. 6.

tensor follows very closely the proce-

B to bcha1.e like a spinor, vector and pseudoscalar

respcct,i\.ely,

SO that

The cwmput.:ttion of sA’;‘, (s) under this transformation requires special collsitlcw tion. ‘l’lwrc iu a readily given dependence on Sic/(.I. 1 a11c1S$ C.1.1. But in additiolr . \ve must t,alw into account the fact that if two points are at t,hr same tinw iti the new coordinate system, as are the points .I.1’ and x2’ in 4’ (,(.I’ )r,lC/’ (x2 1 Mow \ve tjakc the limit to form A$’ (s’ ), then they were not necessarily at t,hc S:illlC’ t.inw ill the old coordinak system. However, in the sprrial case of a t,ransfor1n:rt.ioii for \vhich &6x,, is zero, points at the same time iii the new system were :~Iso at, the same t,imc in the old system. Since the drpendtnc~~ of 6S, on d,&r~ is ;l linear one, we conclude that t,here is no additional contrih~~tjion tluc to the ci~:tnpc~ in dircct,ion of approach of the E’ermion fields which define S, , at least a.- far as the dcprlldrnces on d16al &6.ro and &6.rl are concerned. A knowledge of t tw dcpendelwe on &6ro clearIy requires a detailed inrwtigat,ion of the dy~~amiw of the system t,o account for an approach from different times. We shall ha\~~ to (10 without this knowledge in our deri\Tation. The explicit’ appearances of the coordinates ill t,he I,agrange functioll, ot twr than tjhrougll thr> fields, are accounted for as follow: 6,c3, = - (d&r, 1a”

tl.ti

s,+, = (&A&)6x” 6,J, = (a,./, )6x”.

c-I.71 (13)

The change in the region of integration l).Y 6, (d’x) Applying

the action principle,

1

of the Lagrange funct,ion is reprewtt wcl

= (d”x ) (d,&t.‘).

t-l.!) I

we have

Incorporating all of the above variations we expect, to find that the depcnde~w of the action integral on 6x, can be brought into the following form: ho rtI” d2.L.JYx) = c?.r[T”“( ar 6x, 1 + I~“‘d.r,j t-l.10 ) 6, s l!JQ s .h”

34

SOMMERFIELD

where neither Tp” or W’ depend on 6x, . Integrating we find

The generator

of translations

(for which P”6xp =

whereas

all contributions

from interior

by parts with respect to time,

6x’ is independent s

of x) is identified

as

dxTO’ikq,

times must vanish, so that

aPTfl” = W”

(4.11)

The generator of rigid rotations (Lorentz transformations) special choice 6x,, = -&,,x’, (o,, = - 8,) as

is obtained from the

~Jpyelru = -O,,v 1 dxT’“x’. From these relations, we conclude that the energy, momentum sional angular-moment,um operators are, respectively, P”(zo)

and two-dimen-

= 1 dxToo(r)

P1(zo) = 1 dxT”(x) J”‘(xo)

= x”P’(zo)

- 1 dxx Too(x) = -J”(x’).

It is clear that in order to calculate these operators integral on &6x0 is not needed. The results of the calculation are as follows: Too = ->&‘a,# -

gj”A,

To1 = ->$&“a,ll/

+ ~~ia&‘~ -

>&~jllj, + ~&&$~“J/

- Ao(&B) c$“j, -

J,A”

the dependence of the action

+ >$IB2 + $$.tAlrAP -

(1/47r)K;

+ Al&B

-

(1/47r)K1”

(4.12) (4.13)

T1’ = Tw - B” + c&j” + A’J,

(4.14)

Wv = $@#d

(4.15)

+ A”@vJJ.

With a knowledge of Too and WV, we may use (4.11), together with the field equations and derived current-conservation laws to calculate T”. The result is T1’ = To’ + j,&‘ + A,J” so that T”” is symmetric

in the absence of external fields.

FIELD

C’. REIATIVISTIC

THEORIES

OF

OSE

SPATIAL

.,?I.,

l)lMl??;~lOS

I~~ARIANCX

The relativistic invariance of the theory in t’hr absenceof external fields is tlot immediately obvious, for the Lagrange function and the etlergy-monlcIltall~ tensor are not expressed in terms of covariant operators. This invariance ma? be tested by observing whether or not t’he t\vo-molnelltllm I”’ transforms l&r> :I two-\-&or under the Lorentz transfol,mat,iorl KencraM by J”‘. That is, UY~ consider whether is the same as &p

= - 8”Yp” .

(1.17 I

15quation (a.11 ) guarantees that 1’” (.r”) is constant ill time when the estcrllal fields are off. Thus, it follows from a comparison of (4.16 ) wit’11 (1.17 ) that [P”, p”] = -&pp - q”“p]. (4.18 1 This is a condition involving the integrals and first moments of 2”“” and ?‘I” n,it,ll respect to the space coordinate x. It will necessarily 1,~ satisfied if the followitl,c conditions arc met by 7”’ and 7’“l themselves”: x1’ = .r “‘: [P’“(.r), TO”(.c’)] = i[T”‘(z)&‘G(X - X’i - T”‘(.r’lr116(X - x’)] (,-I-.l!ji [T")

(s),

7'"" (CT')]

=

i[TOO (.ddl'cs

(x

-

x' ) -

7'"

(.r' )a16 (x

-

x' I].

(1.W

I

The first of these equal-time commutation relat’ions (whose right-hand side is necessarily antisymmetric in 5 and .c’ ) and the coefficient of &‘6 (x - x’ 1 US indicated in the second, are all that are required to satisfy (4.18 ). The cocfficietlt, of &6(x - x’) in (-$20) is not determined by equation (1.18 ) but follows, rathrr, from equation (4.11) (with W” = 0), and :!le relation dO!l’“l(x) = - i[ T”’ (.r i ( I’” (.TI’J] which expressesthe role of P”(a”) as a generator of time translations. It is just a matter of algebra to demonstrate that (1.19) and il.20) arc satisfied by 7’“” and 7’“’ as gi\-en by (4.12) and (4.13). I). DERIVATIOX

OF THE LAGRANGE

FTTXCTION

We have shown that the new Lagrange function (3.31) has all of the properties necessary for its use in the action principle: it is composed of operators at, a single time and it has simple variations. There is still a difficulty, however, ill t’hat we had to solve the theory completely before we were able to write it down. We could construct this Lagrange function only because we knew the relatiollships among S, , A, and j, as given in Eqs. (:X20) and (3.21) which ec]uations were found only by examining the solution. Thus we seemto face the prospect,

36

SOMMERFIELD

that the actual Lagrange function corresponding to a given set of field equations, cannot be written down a priori. To enlarge on this point, we may ask for the Lagrange function of a theory which is the one treated above, but modified so as to give the Fermion field a mass parameter I?&,or such that there is an additional coupling involving p (.r)p (zz), where p (z) = 4 (x)+ (z). These changes make it impossible to find explicit expressions for the matrix elements and so we apparently have no way of knowing what the equations corresponding to (3.20) and (3.21) will be. In the present case, however, it is possible to construct the Lagrange function, by using only one very unrestrictive fact about the solution, namely, that the equal-time commutator, [S, (z), S, (x’)], is a c-number, independent of the coupling constants. We assume that the form of the Lagrange function will be unchanged from when it was incorrectly expressed in terms of j, rather than S,, , and write it, with seven unknown coefficients, as follows: d: = $$$y”d,$

-

>$id$y”+

-

f$d,AYB””

+ >~A,d,B” - ~@,/J~A,A”

The field equations

+

(q/4)B”‘BPy

+ .s,gA,S” + &d’S,,

.

are then - Va,# a,&

-

= Y“ (s,gA, + t,u&)

&A, = qB,w a,B”

= gs,X” - &,A”.

(no sum over M)

A comparison with the desired field equations (2.2), following relations uj, = ut,s, - g(l - s&L

(2.3), and (2.4) yields the

gj, = gs,S, + ~~(1 - 1;)&

(no sum over P)

(no sum over P)

q = 1. Since S,, and A, are taken to be linearly independent,

we learn that

s, = t, (g2/p2) (1 - SJ + a(1 - Y,) = 0. These conditions leave us with two unknown constants, say sir . We may hope to find them by asking that the requirements (4.19) and (4.20) for relativistic invariance be satisfied. The components Too and To’ of the energy-momentum tensor are calculated according to the method of the previous section. We find Too = -f~i$&~~+

+ ~ia&‘lC,

-

A,,(&B)

+ >$*r,APA,

-

To’ = -~$~“a~~ + ~,$i&$r”$ + A, (&B). Equation (4.19) cannot be tested until we know

gsJYA,

$&+S’S,,

the equal-time

+ $$$B’

commutation

YIELD

THEORIES

OF

ONE

BPATI.\L

.,.t,

I)IMEXSIOS

ivlations among all the operat,ors appearing in 1’““. (My [S, (.r 1S, (x’ ) 1 i> tlot kt~~~t-11 indrprndently of the solution. (Those comtntttators involving _I ,, catI t)cx fo~mcl from thr others hy means of (Lti 1.1 But a glance at, C%.30i attd (:;.::I 1 sho~vs t,hat for t,he prcsrnt. model [6, (.r ), S, (.c’ I] is just a c-tt~tmtwr attd Ctlrtlwt.mow d(JPS not depettd OII the coupling constant. Th~is a howkdgc of ttlr fubfs fkld \xlur f(Jl’ [s, (.r ), I\‘, (.r’)] is, in this casr, sufficient t#o cotttit~ue 0111’progrant. ‘lb cwmputatioti of [ 7”“’ (.I, 1, 7”“’ (.r’ i ] yields the twtilt [Y’“(.r~,

T”“(.r’ )] = i% (.r);),‘6 (x -

% (,,. ) =

7’“’ (1”)

x’ / -

;z (.I.’ )a,6 (x -

x’ 1

\\~IWW

it is ttcccssary

zz

and sufficient

-

1 $ipy”&ll, +‘~&$y$5

- fi’,,,,.4,,A1, + {/,SllA-llS,,

that s,, - St = ‘a&Sl .

‘l’his wduwa the number of unknown coefficients t,o one. ,~pplicatiott of ccluat.iotl (4.20 I is found not to yield any new informatiotl. ‘l’hc additional information to determine the coefficirnt~s must, he ohtairwtl from the tldinit,ion of ,j, in terms of # and 4. Hut since we have definrd it :IS an a\.eragc of a spacelike and fhe orthogonal timelike approach of two I;ermiotl fields w can po no further without tkig the explicit dynamics (Jf t~h solritiott to take? iilh accOllllt h? helike part of the apprOach. f~ellce W are Ilot ath to dwivc t,lw last equation among t,he coefficients a priori and otw definition of the currcitt~ is shill unsatisfactory in that we do tiot know ht>fow solvittg tlw theory wliat, the proper 1,agrange futtction is. (If we are willing to sneak a cluicli look at. tlic solution, in the form of 1~2~s. (3.20) attd (3.21) we n-ould find that for the theory n-c have heen considering .s,, = (1 - 01I I and s1 = (I + u I ‘. I WC see tlwn that we are left with one degree of freedom, which cotwspc~tttls to thrl dc+ttitiott of the current j” iu terma of the b’crmion field. .I reasonat+ IV:\!of chwsitt~ it., which can supply t.he missing coefficiwt hcfore tlte t.ltcory i.. sol\~d, WOIM he as the density of the generator of ordinary gauge t.ransforntat,ions. This is a dcsirahle property for a cutretk We ha\-c found in Section I \..\ ’ that the detisity of this generator is S”. ’1111~ we arc kd to cottsitlcr a theory \vt-wt,(l .r’ = S”, t,ltat, is, whew

38

SOMMERFIELD

In order to avoid confusion among the various symbols, we shall denote the current that is defined in this manner by & and reserve for j,, the original definition. Thus, in asking that ,$ rather than j, appear in the field equations with the values of g, u, and p unchanged, we require a Lagrange function different from the one we have been using. But we may simulate the new theory with the old one by changing the values of g, CT,and p in the old one so as to produce the new Lagrange function. The two theories will then be the same, provided [g/

(1

[g/(1 b2(1

+

~2 +

A)/

(1

+

a)lold

=

[g/

(1

+

2~)1,,,

;

[g/(1

+

a)]old

=

b/

(1

+

~~)]Mw

;

b/

+

a)]old

=

GV

+

2f-Y + 2X)/ (1 + 2a)lnew ; l/a

-

TABLE COMPARISON

Old

OF OLD

Field + ~74 + bJ9

(g/Sr)Afi

a,[(1

(gl27r)Ap

+

currents

computed

- (1/2?r)#*l= + (1/27rWl

0

= 0

y%c’)l

= (1 - 01 - x)-i$(s)S(x (1 + 01)-*~5ti(z)~(x

- x’) - x’)

IjIb), Cr’)l = [A,(z),

A,(d)]

i

1-a

= -

4 1 -

Ly -

h

a&x

-

x’)

(ig/zTr*)(i

-

a -

xl-la,6(x

-

(1 -

o1 -

Implicit

= (1 + ol)-‘[Si - (g/2T)Ar Ao = (l/&(1 - a - X)-l x [gs, - (1 - 4aIB + (1 -

+ (g/2*Mol

a)lold

=

knew

-

a)]old

=

b&v.

THEORIES

theory

of current $ = so 3’ computed

from

relativistic

invariance

equations = Y%$

+ g-4, + +dib

from

gauge

a,$

= 0

transformations +

(gl7r)Ap

+

(l/~)@l

= 0

[gob), $&‘)I = $&)6(x - x’) [$1(r), !+‘)I = - (1 + 2~)-‘r5~(2)6(x--x’) [A&),

A,fz’)]

i

= - &6(x - x’) 2

[gdz), gl(d)l = f+)(i

x)-ya,s(x

-

+ 2a)-vv(x-xf)

x’)

dependences

j, = (I - a - A)-ys, - (g/2Tpz)a,B + (g/2?rp2)Jo + (1/2r)@ol j,

-

x’)

[j,(x), j,(z’)l = (i/2*)10 + a1-l +

gnew

[Aob), &l,(x’)l = [A(z), 3oW)l = 0

[Ads), &@‘)I = [AI(~), joW1 =

NEW

a,[(1 + 243 commutators

Equal-time [jo(z),

=

aYBfiY = gpp + Jfi - $A#

Conserved

a,[(1 - U)y -

a)]old

I AND

-iva,ti

a,Brv = gjp + JP - pAp

+ 4.P

x)/(1

New

.io = symmetrized limit of 4-r~ from spacelike and othogonal time-like directions of approach = v(4

-

theory Definition

-iv&$

a

(1

-

on external

fields

go = so $1 = (1 + Z$‘[S,

-

(gln)Ai

(1/2~)&1

Ao = 4Jo

(l/p”)[gSo

+ Jo - WI

-

(l/~)+il

FIELI)

THEORIES

OF

These ecluat8iotis may he rewritten

ONE

SP.1TI.ZL

l)lMES8IO.U

::!I

more usefully as

To find a matrix clement, of the IWW t,heory involving the operators #, $, ,Y, 1 -1, attd B, we find the corresponding element in the old theory (for which we IMLY~ a. soIut,iott ) attd insert. the above \-aIues for yol,l , ~Ol
‘lb new theory has the advantage of reducing the multiplicitjy of currents 1)~ one, since the same currettt that appears in the field equations is now also relatcd to the generator of ordinary gauge transformations. The IWU- t~twory also with the coticomit.atit~ elimittatcs the dependence of 30 on .I,, and ~10 on +,, , effect that, .I, uow commut,es with c(ivat equal times. The equal-time cotnmutat.ot of -4, with &l, is also simplified in that it no longer depends on the coupling COIIstattt, but ottly ou the “bare” mass parameter h. I-.

c-f , )uc1LIwIoN~ _

As a sumtnary of the foregoing analysis we present a way of looking at ttw model we have been describing which is cottsist’ettt, wi-ith the cluatttum-mechattical action prittciple and which depends as little as possible on the fact that the model is soluble. I’irst we state the desired field equations (in n-hi& there appears :t current $‘, to 1~ defined in berms of the Fermion fields:) attd assign equal-time> aommut.atiott relations among the independent opet.ators-~natileIy, those wltow time dcrivat,ives are specified by the field equations. A Lagrange function which contains fields at a single time only is then cottstruct,cd so that the field eclttatiott:: are obtained from the requirement that the action integral he statiortaty rtttder c-tutmher \-ariationa of the cluantizcd fields. This sit&titnc I,agrange frutctiotr must tw expressed in terms of the sittgk-time current S”. Perhaps the tnost useful way of defning the co\.ariant current operator is ~II terms of the generator of ordinary gauge transformations. That is, we let, ;l” = $‘I. The fact that S” is the density of the generator of ordinary gauge trattsformat,iotl~ has ken derived without reference to the solution of t,he model. The other ronlpottent of $‘ may he found only after we have computed the elements 7”” (tf the etiergy-niomet~tum tensor. As we 1iaT.e seen, the requirement, of relat~ivist,ic~ invariattw as expressed in terms of tlte equal-time commrttat,ors formed from

40

SOMMERFIELD

among the TOPprovides the final bit of information necessary to find ,$. However, in working out these commutators we had to know that the value of [S”, S’J at equal times was independent of the interactions and hence equal to the freefield value. That this should be the case seemsvery reasonable. The interactions appear only in !P, which determines the time development of operators. However, an equal-time commutator involves only the properties of the theory at a given time and in a canonical formalism we are already supposedto know these operator relations. Thus we expect that for equal times [X”, X”] = X where X is some operator which can be expressed, in principle, in terms of the independent canonical field operators and which does not depend on either massesor coupling constants. Matrix elements of X will, however, depend on the coupling constants since the states do. The singular nature of the current operator and the problem of defining a single-time Lagrange function certainly persist when we come to consider more general theories. As examples there are the theory in one dimension where the Fermion has a mass or the corresponding theories in three dimensions. The concluding remarks in the last paragraph suggest that the program of that paragraph may be carried out for these casesas well. We shall discussthis possibility. If the Fermion has a massthere will be a term --nzp in the Lagrange function where p = ,“i[$, $1 (the limiting process is to take place at a given time). Then in addition to [S, s’] we will have to know [p, P] and [p, p] at equal times. These can all be expected to be equal to the corresponding operators for the case of masslessfree fields. Kow in calculating the vacuum expectation value of the above commutation relations for a free Fermion field with mass111,one finds that the answers are independent of m. Thus we may reasonably conclude that for one-dimensional problems these commutators are actually c-numbers and thus the task of finding the right Lagrange function is no more difficult than before. In three dimensions the singularity of the free Fermion Green’s function is cubic rather than linear. This gives rise to two new difficulties. The definition of S’ depends on the spatial direction of approach, and the vacuum expectation value of [So(z), Sk (.E’)] for free fields has two singular terms. One of these terms does not depend on the mass but is much more singular than in one dimension and may loosely be thought of as proportional to V2&S(x - x’). The second term depends on the mass and is proportional to n~~dk6 (x - x’). Thus this commutator is certainly not a c-number. Just what its explicit operator form is, is not easily ascertained. We see then that there are additional difficulties in applying our program in three dimensions. We shall not pursue these further in this paper. It goes without saying, of course, that even if Lagrange functions are written down for these more general theories, we will have come no closer to an exact

FIELI)

THEORIES

OF

ONE

SPATI.11~

41

DIJIEXSIOX

solution. However, we will have placed these theories ott a firmer cluatttttmmechanical footing. The solutiott of the soluhle model may he found from its Idagrange ftuwtiott t)?; the estert~al-field technique, applied in exactly the same way as WC applied it t,c) the old, formal T,agrange function.

IVr are led to consider a? =

the expression

esp [‘.$.(&‘6x)e(6

‘6xIj(rsp

i :xX, (esp ‘&3x)

itt which all of the symbols signifying integratiotl have hret1 sttppwssed. csttrttal fields + attd ,J have bee11 combined into a single colrtmtl matrix X=

4 0 .J *

C’ot,t,~~spottditlgly, the two coupling cottstattts

appear itt the matrix

and the drpcndencr of the T’acULtrn-tO-~aCI1ttt~i transitiott trtnal fields in t,he matrix c=

l’tlf~

I’ (0

amplitude

011 the es-

0 w’>

(Roth C?attd ‘i: aw symmetric matrices. j I:itially, the linear rxpotwntial drprtttlettcc of the multi-IPermion Green’s function 011 the cxterttal field, as gi\.fatt itr (2.18, 10 I is represented by :XX whew

‘hi llla~~t’ix lllUl~iphhl ill xx SylnhOlizrs illtcgrat~ioti O\V?L’the 5 coordittatcs attd sttmmatiott over p. The other indices and cootdittatcs MY~ptwettt, hut wm:titt sttppressed. U’r complrt~r t,lte squaw in the last two factotx of !2 t)y itttrodttcitig the tte\\. colunitt mntricw X’

so that

= x + xu

’

32

SOMMERFIELD

where d = exp [~~i@/~~‘)C?(6/6~‘)]

exp [>six’Wx’].

find the dependence of fi’ on x’, we consider the functional

TO

d/6x’

= (exp [>$i (S/6x’) (2 (6/6x’)])

(iWx’)(exp

[p$x’Wx’]) = [iwx’

This last form follows W@)

[fexp

&iiP

=

eW6x’)ll

[Pi@

(Ux’)(exp

@/6x’)

(exp

when integrated

[-X4

e @/Sx’)ll

X which,

-

[ @/SX’)

@/SX’)

e @lS~‘)ll WX’I

e (S/Sx’),

lexp] - %$I (S/6x’) e (6/6x’)]]

[-~2/i(6/6x’)e(s/sx’)]) = wx’

(A.1)

+ iwe (6/6x’).

as (6/6&l’

we integrate

= iw e (6/6x’)

from 0 to 1 with respect to p, becomes

[~~;(6/6x’)e(6/6x’)I)Wx’~exp

Rewriting

(A.1)

we@/&)]d.

from the fact that

W&X’)

Mexp

derivative

= i(1 + we)-lwx’fi’

and obtain 0’ =

Do{exp [f$x’(l

+

We)-lWx']]

where Q0 = 0’ (x’ = 0). We may compute !& by considering Qa(PI = Iexp MP dfio (p)/dp

=

fi20

@)&ii

WSx’) (S/6x')

=

e (Wx’>ll

Iexp [?!i~x’~x’ll

(13(6/sx’)l(exp

[%ix’(l

tr

(1 +

->@o(@)

=

I

d=0

-I- pWe)-‘Wx’l}

( x'=~

PWC?)’ WC!

(A.2)

(d/dP) tr log (1 + PWC)

-f@o(P)

Thus we find Qo = no(l)

= exp [-35

tr log (1 + We)].

When all of the pieces are put together we obtain the following: D = fro exp [>$i(x

+ XW-')

(1 + We)-‘W

= O. exp [-$$Xe(l

(x + W-k)]

+ ‘UC)-‘X]

exp [ix(l exp [jsix(l

The dependence

of the matrices

which

exp [-+@XW-‘X] + We)-‘X] + WC?)-‘Wx].

appear in (A.3)

(A.3)

on the suppressed

coordinates are functions of differences of these coordinates. Thus in a momrw turn-space representation they become ordinary four- hyfollr mlmerical matriwr. the manipulation of which is straightforward.