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. (AIP)


I. INTRODUCTION
A remarkable correspondence has been noted between boson and fermion theories in one space and one time dimension. Coleman' noted that the Green's functions of the massive Thirring model and those governed by the sine-Gordon equation are related. Kogut and Susskind' provided a dictionary whereby one could translate a fermion theory into an equivalent boson theory. Finally, Mandelstam' gave the correspondence for the Fermi field operator itself. For the massless Thirring model, much of this correspondence was already noted by Dell'Antonio, Frishman, and Zwanziger, ' who constructed a solution to this model in terms of currents. For one-dimensional electron gas problems this correspondence was known to many-body physicists. ' In this article we investigate this correspondence systematically using the path-integral formalism. ' The advantages of this approach are that the delicacies of normal-ordering prominent in the other approaches become somewhat automatic. In fact we are able to present a machinery, without excessive subtleties, for translating fermion theories into boson theories, including the correct correspondence for the field operators themselves. We always work in a cutoff field theory where all mathematical expressions and manipulations are meaningful, and the limit of infinite cutoffs, both momentum and spatial, is taken at the end.
Section II is devoted to a review of the Green's functions for massless free fermion and boson theories. For the results on fermion theories, free and interacting, we rely heavily on the work of Klaiber. ' In Sec. III we set up the correspondence between free massless fermion Green's functions and path integrals over free Bose fields. These results are extended in Sec. IV to a correspondence between composite bilinear fermion operators and functionals of the Bose fields. Armed with the results of these two sections we treat in some detail in Sec. V three interacting theories. These are the massive and massless Thirring models, quantum electrodynamics, and interactions with a massive vector meson. For the massless fermion cases results on the Green's functions are obtainable in closed form; the massive fermion case cannot be solved either in the fermion or corresponding boson language.
One advantage of the present treatment that will be seen in the section on interacting fields is that whereas the definitions of currents still require care, other details of these modifications are much more automatic than in other approaches.
The magnitude of these modifications is determined by Lorentz invariance.
Several points we wish to emphasize before concluding this introduction. It is imperative to use a Hamiltonian rather than a Lagrangian formalism for the bosons. This is because we deal with functionals of derivatives of the boson fields, including time derivatives, and a naive application of Lagrangian formalism with path integrals would yield incorrect propagators lacking contact terms. ' A spatial and momentum cutoff must be provided in order to make any sense of this procedure. Dependence on the spatial cutoff disappears by itself when one looks at nonvanishing Green's functions and the momentum cutoff is absorbed in the normalization of various operators in a standard way. The last point concerns the ordering of Fermi fields. The sign of fermion Green's functions depends on their ordering. The correspondence we shall present will be valid for one definite ordering and the sign changes that may result by varying this ordering do not occur automatically in the boson language, but must be put in by hand. This difficulty becomes important in the construction of composite operators.
II. FREE-FIELD THEORIES We shall summarize the notation and basic formulas needed in subsequent sections. All deriva-13 tions are straightforward and will not be presented in detail. The properties of free massless Fermi and Bose fields will be discussed.
A. Massless Fermi fields &g,(,) "g,( ")g, *(y,)" g, *(y")& 1 &k 27ri . Owing to infrared singularities the free massless boson field does not exit. However, differences and derivatives do. We shall evaluate path integrals for functionals of these derivatives. As time derivatives will occur care has to be exercised and instead of dealjrig with the usual path integrals over fields alone we shall go to the more fundamental functional integral over fields and their canonical momenta. ' In all subsequent discussions we shall use the notations ( 2 6) (2.7) In order to make some of our subsequent discussions meaningful we softened the propagator by introducing a cutoff A. The limit of infinite cutoff will be taken at the end of all calculations. A generating functional summarizing the above is In the above Io.»I=IP»l=l Now the limit R-~m ay be taken, and a nonvanishing contribution is obtained only inthe case Z~;=Zp; =0 In order to recover (2.5) we may identify The phases are arbitrary and may be adjusted by a combination of gauge and chiral transformations. %eshallchoose~, =&, =0 and thus obtain We note that (3.7) agrees with (2.5) in the case the operators in (2.5) are in the correct ordering.
The sign changes, which would be obtained by shifting the anticommuting Fermi fields, cannot be reproduced by expressions such as (3.7). The substitution implied by (3.9) is valid only»vhen the string of (, 's and p, 's is separately timeoxdexed. This ambiguity will be important in Sec. IV when we construct composite operators.

IV. COMPOSITE OPERATORS
In order to build up interacting theories we need the correspondence between composite fields made up of fermion operators and functionals of the cnumber boson fields. This discussion will be limited to bilinear operators, namely gi'g. These operators are sums of products of the form (t(*"$8. At this state the anticommutativity of the fermions begins to plague us. The correspondence between :(t(*"(x)(t(8(x):and the boson fields will be established by substituting (3.9) for g(x) and Ps(x} and the over-all sign will be determined by ensuring the correct expression for A. Mass operators In terms of the chiral components we find 4»4'+ &2-4» (4.l) As there are no mixed contractions, normal ordering introduces no infinities and we may, up to a sign, apply (3.9) without special care. Determining the sign by the argument discussed above we obtain X xp 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 a similar expression will be used to fix the normalization of these operators. If the sign and magnitude of the Green's function for a composite operator and two fermion operators, as above, are correct then the Green's functions involving more fermion fields will likewise be correct. The one involving two Fermi operators is the only one that is connected; the more involved are made up of the simpler ones. (4.8}

B. Current operators
The situation is more complicated for operators of the form gy"g and gy"y, g. These are sums of $, *]1), and g, *g"singular products needing care in their definition. We shall choose the prescription 1 : A.r,4: =, v 7r (4.9) 1 :]j)rgr04: = v 7r If we substitute B, y for 7r we find the results quoted in Refs. 1-3, (4.5) with a, c, d constants and A the cutoff introduced earlier. c is chosen to ensure that (:g"*g":) =0; a and d will be connected to the normalization of the composite operator. We assume A is large and expand in I/A retaining the first nonvanishing term. Substituting (3.9) into (4.5) we obtain v :yr"C:=~w 'y,  (4.7) d a may be determined by ensuring proper nor-I. =T))(igm)g --j j".
( 5 1) It is tempting to let j"=7])y"g; owing to the asymmetry between space and time introduced in the definition of the currents, the interactions modify the normalization of g, relative to go. we find that the current is where c, and c2 are constants to be determined by some normalization condition.
In the case m =0 the Green's functions may be explicitly evaluated; requiring a finite answer in the limit Ayields Again, the current f" is not given simply by T))r&P.
It is defined as a gauge-invariant limit of T()(x+e)r&g (x)