The correspondence between boson and fermion field theories in one space and one time dimension is examined in the context of a path-integral formulation of these theories. The advantage of this formulation is that the translation, both for the Lagrangians and the field operators, is fairly automatic. Normalization of products of fields, which in more conventional formulations required careful manipulation of singular quantities, in this approach is a straightforward consequence of Lorentz invariance. © 1976 The American Physical Society

Bander, M

Myron Bander

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Boson-fermion correspondence in two-dimensional field theories

The correspondence between boson and fermion field theories in one space and one time dimension is examined in the context of a path-integral formulation of these theories. The advantage of this formulation is that the translation, both for the Lagrangians and the field operators, is fairly automatic. Normalization of products of fields, which in more conventional formulations required careful manipulation of singular quantities, in this approach is a straightforward consequence of Lorentz invariance

Bander, Myron

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UC IrvineUC Irvine Previously Published WorksTitleBoson-fermion correspondence in two-dimensional field theoriesPermalinkhttps://escholarship.org/uc/item/64d8h01gJournalPhysical Review D, 13(6)ISSN2470-0010AuthorBander, MyronPublication Date1976-03-15DOI10.1103/physrevd.13.1566Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPHYSICA L REVIE W D VOL UME 13, NUMBER 6 15 MAR CH 197 6Boson-fermion correspondence in two-dimensional field theoriesMyron Bander*Department of Physics, University of California, Irvine, California 92664(Received 17 November 1975)The correspondence between boson and fermion field theories in one space and one time dimension isexamined in the context of a path-integral formulation of these theories. The advantage of this formulation isthat the translation, both for the Lagrangians and the field operators, is fairly automatic. Normalization ofproducts of fields, which in more conventional formulations required careful manipulation of singularquantities, in this approach is a straightforward consequence of Lorentz invariance.I. INTRODUCTIONA remarkable correspondence has been notedbetween boson and fermion theories in one spaceand one time dimension. Coleman' noted that theGreen's functions of the massive Thirring modeland those governed by the sine-Gordon equationare related. Kogut and Susskind' provided a dic-tionary whereby one could translate a fermiontheory into an equivalent boson theory. Finally,Mandelstam' gave the correspondence for theFermi field operator itself. For the masslessThirring model, much of this correspondence wasalready noted by Dell'Antonio, Frishman, andZwanziger, ' who constructed a solution to thismodel in terms of currents. For one-dimensionalelectron gas problems this correspondence wasknown to many-body physicists. 'In this article we investigate this correspon-dence systematically using the path-integral for-malism. ' The advantages of this approach arethat the delicacies of normal-ordering prominentin the other approaches become somewhat auto-matic. In fact we are able to present a machin-ery, without excessive subtleties, for translatingfermion theories into boson theories, includingthe correct correspondence for the field operatorsthemselves. We always work in a cutoff fieldtheory where all mathematical expressions andmanipulations are meaningful, and the limit ofinfinite cutoffs, both momentum and spatial, istaken at the end.Section II is devoted to a review of the Green'sfunctions for massless free fermion and bosontheories. For the results on fermion theories,free and interacting, we rely heavily on the workof Klaiber. ' In Sec. III we set up the correspon-dence between free massless fermion Green'sfunctions and path integrals over free Bose fields.These results are extended in Sec. IV to a cor-respondence between composite bilinear fermionoperators and functionals of the Bose fields.Armed with the results of these two sections wetreat in some detail in Sec. V three interactingtheories. These are the massive and masslessThirring models, quantum electrodynamics, andinteractions with a massive vector meson. Forthe massless fermion cases results on the Green'sfunctions are obtainable in closed form; the mas-sive fermion case cannot be solved either in thefermion or corresponding boson language.One advantage of the present treatment that willbe seen in the section on interacting fields is thatwhereas the definitions of currents still requirecare, other details of these modifications aremuch more automatic than in other approaches.The magnitude of these modifications is deter-mined by Lorentz invariance.Several points we wish to emphasize before con-cluding this introduction. It is imperative to usea Hamiltonian rather than a Lagrangian formalismfor the bosons. This is because we deal with func-tionals of derivatives of the boson fields, includingtime derivatives, and a naive application of La-grangian formalism with path integrals wouldyield incorrect propagators lacking contact terms. 'A spatial and momentum cutoff must be providedin order to make any sense of this procedure.Dependence on the spatial cutoff disappears byitself when one looks at nonvanishing Green'sfunctions and the momentum cutoff is absorbedin the normalization of various operators in astandard way. The last point concerns the order-ing of Fermi fields. The sign of fermion Green'sfunctions depends on their ordering. The corre-spondence we shall present will be valid for onedefinite ordering and the sign changes that mayresult by varying this ordering do not occur auto-matically in the boson language, but must be putin by hand. This difficulty becomes important inthe construction of composite operators.II. FREE-FIELD THEORIESWe shall summarize the notation and basic for-mulas needed in subsequent sections. All deriva-1313 FFRMION CORRESPONDENCE IN TWO DIMENSIONAI. . . .tions are straightforward and will not be presentedin detail. The properties of free massless Fermiand Bose fields will be discussed.A. Massless Fermi fields&g,(,) "g,( „)g,*(y,)" g,*(y„)&1 &k27ri . g (x,+ -y„')j,-k(2.5)&o=O'x ~ Zo ~ ~~ gthe chiral componentsThe massless Dirac equation isif'P= 0.With the convention(2.1)(2 2)&[[),(x,)" 4.(x„)P,(y,)" 4.*(y„)&2&k2mi g (x[ )-y[ ~)with x' =x +x', and no mixed eontraetions be-tween diff erent chiral components.(2.3)satisfyt( Bo+Bg)(g= IBd. $) = 0 )i(B, B,)y,-= iB y, =0. (2 4)A compact form of the Green's function of thistheory was obtained by Klaiber':B. Path integrals for massless bosonsOwing to infrared singularities the free mass-less boson field does not exit. However, differ-ences and derivatives do. We shall evaluate pathintegrals for functionals of these derivatives. Astime derivatives will occur care has to be exer-cised and instead of dealjrig with the usual pathintegrals over fields alone we shall go to the morefundamental functional integral over fields andtheir canonical momenta. ' In all subsequent dis-cussions we shall use the notationsf [dip dw]Iexp[i f d'x(ms, q ——,' [w'+ (B,y)'])]F [w, y]]f[dydee]{exp[if d'x(7) B,p — [)T'+ (B,y)')]].The basic propagators of the theory areiA', 2k, (k,+ k )e '~'i" '[&I ( )'B,y( )][ (y)'B,y(y)]&= &(,)( -y)- —(2 ). dk(k. . )(k, A. . ),&[~(x)~ B,q(x)] [~(y)+ B,q (y)]& = 0.(2 6)(2.7)In order to make some of our subsequent discussions meaningful we softened the propagator by introducinga cutoff A. The limit of infinite cutoff will be taken at the end of all calculations.A generating functional summarizing the above isexp ~' d'x ex px +B,yx + x px —B,yx=exp —2d'xd'y n x &~,~x-y n y + x &~ ~x-y p y . 2.8tion.This difficulty is overcome by modifying theintegral in (3.1) through the introduction of aspatial cutoffIII. FERMION-BOSON CORRESPONDENCE-FREE FIELDSFollowing Mandelstam' it would be tempting tomake the identificationJx' x~d((x*e,p)- J d(e'r (x+e, rp).x~p, ,(x)=CexpI-(dx J d [()dxx, (dxe, e)]Ip. p,(3.1)However, it is straightforward to show that ex-pectation values of such expressions are singularowing to the infinite range of the spatial integra-At the very end of the calculations we will setIt should be noted that this procedure wouldhave been necessary even if the fields were mas-sive.1568 MYRON BANDE RAll our expressions will be built up fromX~», (~(=j d(~""(~(*',((+8 p(*'((I , (& N&4.(x)4 (y)&=o,withThe Green's functions are generated by the basictwo-point function [evaluated with the help of (2.7)] lny = e "1nydy =0.577 ~ ~~ . (3.6)(3.4)%e are thus led to consideriN Nexp -i p +4+ x, +,.4 y&—i=1 j-1(N+N )/2(-ir~) '~"» "(-ir&) '~"'.. [Ix»"-x,'"I "»"'.[],Iy» '-y,' 'I'»' .i&j i&g(3.7)In the above Io.»I=IP»l=lNow the limit R-~ may be taken, and a non-vanishing contribution is obtained only inthe caseZ~;=Zp; =0In order to recover (2.5) we may identifyi/20, ( )= '" exp[- i W»» 4, (x)],2ryi/a(t, (x) =e"2 exp[-iW»r@ (x)].27ty(3.8)The phases are arbitrary and may be adjusted bya combination of gauge and chiral transforma-tions. %eshallchoose ~, =&, =0 and thus obtain(,(x) = exp[- i v»»@, (x)],27ry(C(, (x)= - exp[-tWir4' (x)].27ry(3.9)We note that (3.7) agrees with (2.5) in the casethe operators in (2.5) are in the correct ordering.The sign changes, which would be obtained byshifting the anticommuting Fermi fields, cannotbe reproduced by expressions such as (3.7). Thesubstitution implied by (3.9) is valid only»vhenthe string of (,'s and p, 's is separately time-oxdexed. This ambiguity will be important in Sec.IV when we construct composite operators.IV. COMPOSITE OPERATORSIn order to build up interacting theories we needthe correspondence between composite fields madeup of fermion operators and functionals of the c-number boson fields. This discussion will belimited to bilinear operators, namely gi'g. Theseoperators are sums of products of the form (t(*„$8.At this state the anticommutativity of the fermionsbegins to plague us. The correspondence between: (t(*„(x)(t(8(x):and the boson fields will be estab-lished by substituting (3.9) for g(x) and Ps(x} andthe over-all sign will be determined by ensuringthe correct expression forA. Mass operatorsIn terms of the chiral components we find4»4'+ &2—4»(4.l)As there are no mixed contractions, normalordering introduces no infinities and we may, upto a sign, apply (3.9) without special care. Deter-mining the sign by the argument discussed abovewe obtainXxp 2i~1r d~e~t 9 q{x ~)(4.2}Xg: (t,*(x)(t,(x):= exp -2ivw dg ' "etys( xg),or equivalently+ g,*(y)P,(~) + P, (y)4.*(e)1&,with p'&z'. In the case of singular products asimilar expression will be used to fix the normal-ization of these operators. If the sign and magni-tude of the Green's function for a composite opera-tor and two fermion operators, as above, arecorrect then the Green's functions involving morefermion fields will likewise be correct. The oneinvolving two Fermi operators is the only one thatis connected; the more involved are made up ofthe simpler ones.13 BOSON- FERMION CORRESPONDENCE IN TWO- DIMENSIONAL. . . 1569A:T()(x)p(x): = cos 2v w dg e'~"a,y(x0, g)7ry OO(4 3)malization by comparing&:4.*(x)4.(x):[4,(y)4,*(e)+P.(S)4,*(e))&,:g(x)y, g(x}:= sin 2)(w7rydg e'~"a, p(x0, g) .both by using (2.5) by substituting (4.V) and (3.9).The results are:T[)y,g: = sin2&wy .iA7ry(4 4)However, it must be emphasized that this limitmay not always be taken before the end of all cal-culations.If we naively let R- at this stage we mould obtainthe results of Refs. 1-3, namely,Acos2vw p,7ry1:0*]1):= 2-~- (w+ a V),:0, 4. :=-2~ (w-a, V}The current operators have the form1:4r,4:=, a, y,v7r1:T])y,g:= ~ w,(4.8}B. Current operatorsThe situation is more complicated for operatorsof the form gy„g and gy„y, g. These are sums of$,*]1), and g,*g„singular products needing care intheir definition. We shall choose the prescription1:A.r,4:=,v 7r(4.9)1:]j)rgr04: =v 7rIf we substitute B,y for 7r we find the results quotedin Refs. 1-3,(4.5)with a, c, d constants and A the cutoff introducedearlier. c is chosen to ensure that (:g„*g„:) =0;a and d will be connected to the normalization ofthe composite operator. We assume A is largeand expand in I/A retaining the first nonvanishingterm. Substituting (3.9) into (4.5) we obtainv:yr„C:=~w'y,1:Or](r04: = rv7r(4.10)V. FERMION-BOSON CORRESPONDENCE-INTERACTINGTHEORIESA. Massive Thirring model@ (w+s, p) —)I,(4.6)x'+0/Aexp ivw d&(w —a, y) —1~ .2'd, A&exp i& 1T2wy (For large A this may be reduced toid~a~ ("+ai()0) ~2yv 7r2yv 7r(4.7)d a may be determined by ensuring proper nor-I. =T))(ig —m)g ——j j". (5 1)It is tempting to let j„=7])y„g; owing to the asym-metry between space and time introduced in thedefinition of the currents, the interactions modifythe normalization of g, relative to go. Thus in(5.1) we shall let j,=]ty0(; however, j, =&T[)y,g andlet Lorentz invariance choose ~ for us. Imple-menting the correspondence discussed in the lasttwo sections the Green's functions of this theoryare obtained, up to an over-all normalization,fromThe Lagrangian for the massive Thirring model'is(s'[(„(,I)=mf [dydee](exp ) d'x ws rp—0 (a,y)' mJ].2 7ryxlcos 2 7rx E — exp(- i v w C', ), exp(-i)tw C' )27ry 27ry -] (5.2)1570 MYRON BANDERWe shall now rescale n' by (1 —&g/v) ~2 and y by (1+g/v) '~2. The terms quadratic in v and (],qr have theusual coefficients; however, ve, y is multiplied by (1+g/]r) '~2(1 —g&/w) ~2. Lorentz invariance is re-covered only if this product is unity. Imposing this condition we find~ = i/(i +g/s) .Intr oduc ingp=2v~v(1+g/]]) '~2we find that the current is(5.3)(5.4)which agrees with known results.Returning to (5.2) we obtain]]2 (a, y)2(P[P„P,]) re J[drp —de] [expi rl'x ee rp ———2 2(5.5)xy expexp -s a'g e"" (5.8)From this we note that the Green's functions ofthe massive Thirring model are obtainable asthose of the boson theory with the LagrangianmA x1=d-', ( )e're+ ee,e d e' "e,r)r(x', ()) (5.))ny ~ ()0with Mandelstam's identification12v p(I)~ =c~ exp —2 dl e — v+ —s~pImay solve fore81(5.11}e2f pff = 244 — 2 pro& 2 (I)ro(f) ~1(5.12)The correspondence with the boson fields yieldsan actionand find the effective Lagrangian for the fermions2]]' pP, =e, exp -i die'r e-pe, p)(5.8)A =@0&0y—0 0(s y)2 e2 11(5.18}where c, and c2 are constants to be determinedby some normalization condition.In the case m =0 the Green's functions may beexplicitly evaluated; requiring a finite answer inthe limit A- yieldsA(ii /8 +8 A, eit)Cat CL (5.9)B. Quantum electrodynamicsand the Green's functions agree with those pre-sented in Ref. 7 for currents defined in the mannerof Schwinger. '](2 (s ~)2 N 2A =7TB p—2 2 2(5.14)(5.15)The substitution for the fermions remains as in(3.9). The Green's functions may be evaluated asin the free-field case except that the propagatorsare modified:which may be simplified to that of a massive bosonfield with massI. = i Pg(I) —,'E„,E2' —eT(]r„(I)A" . — (5.10)It is convenient to work in the gauge A., =O. WeWe shall first consider a massless fermion in-teracting with the electromagnetic field 10,11 Intwo dimensions there are no photons and thus weshall look only at the fermion Green's functions.The Lagrangian of this theory is(c' d(x}@.(y))o,o = (@.(x)c', (y))„.,+E, (x -y)+c in]uR+d,(4),(x)C (y)) =E'(x-y)+clnpR,where c, d are some constants and(5.16)BOSON- F E RMION CORRE S POND E NC E IN TV(0- DIME NSIONA L. . . 15Viip' „,& (e""—1)(k,' —e')(2w)' (k' —»i'+ ie)(k, '+e')' '(5.17)(t.+u, )2(e-»' 1)(t,2(2w)' (k' —ie)(k' —»»'+»e)(k '+e')' 'Thus the Green's functions for the interactingtheory are equal to those for free fields multipliedby an exponential of a sum of the I's. Terms inlnp, A cancel as long as the number of conjugatedand unconjugated fields is the same. Instead ofwriting down the most general expression we shallgive as an example a four-point function:&41 (xi I'i*(&l )»1,(x.))I),*(s.)}o„2()' (* -& )+»'-(» -)')-&'(» - »)+ )'(», -3'.)+&'(», - »)-P (», - »')])& (g, (x, )((),*(y,)g, (x,)$2*(y2)), (5.18)This expression agrees with that given by Brown. "Giving the fermions a mass changes the Lagran-gian of the boson field tow'+ (s, (»))'Ay —mb y—0 2(Aos, y -A, w) + e&A, '. (5.23)(s(I())' »i', A2 2 1ry(5.19) Changing variables in the functional integrationAs we are now dealing with a mas'sive boson theoryit is permissible to replace 1 d& e~~~s, y byq -q(")."e 8w-w+ ~ (8 A)Ww s'e 1(»7—~ 2 (&OA i —(» i A O)(5.24)C. Interaction with massive vector mesons yieldsL =L~+Ly, (5.2O)withThe last interacting theory we shall treat isthat of a fermion interacting with a massive vectormeson. ""The r. assless fermion case will betreated first, and the inclusion of the mass termleft for the end. The Lagrangian of the theory is+ e6A,'.+B~p e 1 ~y e2 4n "" 9' 2m(5.25)~2~ (g )2 e2OP 2 4P +Choosing & =-e/2w eliminates the noncovariantpart of A&, and combining with L„we find the ac-tion corresponding to (5.19)1 pp j.L~ =-~E~„I"" +2go~A(5.21) +—'p, 2A' (5.26)I ~ = if''»I) —ef„A" .Again, the current f„ is not given simply by T))r&P.It is defined as a gauge-invariant limit ofT()(x+e)r&g(x)exp(ieeA) For sp.acelike e we findthatThe change of variables proposed in (5.23) inducesa modification in the correspondences for thefermion fieldsexp -ivy @,(x)+e s, (A. -A, )- f27ry w 8'f, =fr.4,f, =Vr, l+5A, .(5.22)(5.27)exp -i)(w C, (x)+e s (A. +A)-27ry w 8Again, we shall let Lorentz invariance fix & forus. In the case of QED, treated in Sec. VB, nosuch modification was necessary in the gaugeA, =0. Let us discuss the correspondence for Lzfirst:It is straightforward to evaluate all Green's func-tions once the propagator for the vector fields isknown. Inverting the part of (5.25) involving A„and being careful about contact terms one findsMORON BANDER(A ( )A (0)) d2y—ik x 'All& ( ll Ulh )( I™) IJO &0 (5.28)withM' = p, '+e'/w.These results agree with those of Ref. 13. If the fermion has a mass m we have to add(5.29}mAI. = cos 2&m7ry ~ OOto (5.25).df 8 Bgp + . r 2 (BOA~ B~A0}V7T 8(5.30)ACKNOWLEDGMENTThis work was initiated while the author was a guest of the theory group at the Fermi National AccelatorLaboratory, and he is grateful for invaluable discussions with Dr. S. J. Chang, Dr. S. Ellis, andDr. B. Lee.*Work supported in part by the National Science Foun-dation.~S. Coleman, Phys. Rev. D 11, 2088 (1975).2J. Kogut and L. Susskind, Phys. Rev. D 11, 3594(1975).3S. Mandelstam, Phys. Rev. D 11, 3026 (1975).46. DeQ'Antonio, Y. Frishman, and D. Zwanziger, Phys.Rev. D 6, 988 (1972).~E. Lich and D. Mattis, Mathematical Physics in OneDimension (Academic, New York, 1966); A. Luther andI. Peschel, Phys. Rev. B 9, 2911 (1974); A. Lutherand V. Emery, Phys. Rev. Lett. 33, 509 (1974).A recent review of the path-integral formalism appliedto field theory is presented in E. Abers and B. Lee,Phys. Rep. 9C, 1 (1973).B, Klaiber, in Lectu~es in Theoretical Physics, editedby A. Q. Barut and W. Brittin (Gordon and Breach,New York, 1968), Vol. X-A, p. 141.For references to earlier works on this model see Ref.7.~J. Schwinger, Phys. Rev. Lett. 3, 296 (1959).J. Schwinger, Phys. Rev. 125, 397 (1962).~~L. Brown, Nuovo Cimento 24, 617 (1963).~~J. Frohlich, Phys. Rev. Lett. 34, 833 (1975); J. Lowen-stein and A. Swieca, Ann. Phys. (N. Y.) 68, 172 (1971).C. Sommerfield, Ann. Phys. (¹Y.) 26, 1 (1963);W. Thirring and J. Wess, iMd. 27, 331 (1964).4A solution to this problem in terms of boson fields,analogous to Ref. 3, was presented by M. Kaku, Phys.Rev. D 12, 2330 (1975).

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Boson-fermion correspondence in two-dimensional field theories

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