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Current Definition and Mass Renormalization in a Model Field Theory

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Current Definition and Mass Renormalization in a Model Field Theory IC/67/24 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CURRENT DEFINITION AND MASS RENORMALIZATION IN A MODEL FIELD THEORY C. R. HAGEN 1967 PIAZZA OBERDAN TRIESTE IC/67/24 INTERNATIONAL ATOMIC ENERGY AdENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CURRENT DEFINITION AND MASS HEJTORMALIZATION BT A MODEL FIELD THEORY * C.R. Hagen ** TRIESTE April 1967 *To be Bubmitted to "Nuovo Cimento" **On leave of absenoe from the Univereiiy of Rochester,NY, USA ABSTRACT It is demonstrated that a field theory of vector mesons inter- acting with massless fermions in a world of one spatial dimension has an infinite number of solutions in direot analogy to the case of the Thirring model. Each of these solutions corresponds to a different value of a certain parameter £ whioh enters into the definition of the ourrent operator of the theory. The physical mass of the veotor meson is shown to depend linearly upon this parameter attaining its bare value u, for the exceptional oase in which the pseudo-vector current is exactly conserved. The limits jA-o —> °o and JLLQ -» 0 are examined, the former yielding results previously obtained for the Thirring model while the latter implies a unique value for ^ and defines the radiation gauge formulation of Schwinger's two-dimensional model of electrodynamics. -1- CURRENT DEFINITION AND MASS RENORMALIZATION IN A MODEL FIELD THEORY 1. VECTOR MSSOITS IN ONE SPATIAL DIMENSION Although the soluble relativistic models known at the present time are remarkably few in number, they have long been cherished by field theorists for their value as potential testing areas for new techniques in particle physics. The study of such models has however, been handioapped by the faot that the only theories which are both non-trivial and soluble in a four-dimensional space-time refer to non-renormalizable interactions, a circumstance which thus far has made it extremely difficult to attach much significance to some of the formal results which have been obtained from these models. On the other hand the soluble field theories in a world of one spatial dimension are generally much less singular than their three-dimensional counterparts and are therefore considerably more amenable to calcula- tion. In particular the four fermion interaction (THIRRING model ') 2 1 and electrodynamics with a massless fermion (SCHWINGER model ') are outstanding examples of soluble renormalizable theories in one dimension and as such have received a considerable amount of attention in the literature. Somewhat less familiar perhaps is the case of a vector meson interacting with a massless fermion field in one dimension acoording to the Lagrangian 1/4 o>% fW ^ where B'(x) and ^(x) are hermitian operators which desoribe vector and spinor fields respectively. For the present, the ourrent is assumed to be defined by the formal operator expression where the Birao matrices oCmay be taken to be the two-dimensional set -2- and q is the anti-symmetrical matrix which acts in the two-dimensional charge spaoe of the field Yi[x). For later convenience we have inoluded in (l.l) the coupling of o^fx) and B"(x) to the sources A^(x) and J^(x) respeotively. By using this external field device it is possible to generate all the vacuum expectation values of the theoryf thereby effecting a oojnplete solution. The action principle oan "be formally applied to the Lagrangian (l.l) to derive the equations ^B (1.2.) and the equal-time commutation relations -3- It is useful to note that of the above field eqs.,(l.2a), (l.2b) and r , I are equations of motion for V(x) B^x) and Q0\x) respectively, while the constraint equation * *B =ei' serves merely to define B(x)in terras of the true dynamioal variables of the theory. Before proceeding to the task of computing the general struoture of the Greens functions it is well to remark briefly upon the work of BROWN ^ and SOMMERFIELIr' in connection with this model. In particular it has been observed ' that the definition of the current operator used by Sommerfield is not unobjectionable inasmuch as it requires the application of a time-like limiting procedure. This, of course^ re- presents a significant departure from the usual concept of a canonical theory acoording to whioh all operators are required to be defined solely in terms of the fields on a given spaoe-like surface. Brown's definition of jA*(3c) as , -k with the limit x —•*• x'to be taken after setting x° = x clearly avoids this oriticism and leads to a consistent solution of the model. However, it is interesting to note that although it has been shown ' that in the case of the Thirring model the results of JOHNSON ' and Sommerfield can be derived from a temporally local definition of j^(x) the conclusions of Brown and Sommerfield differ with regard to the physioal mass ^u of the veotor meson. In particular,Brown has found whereas, Sommerfield's mass renormalization is only half as great. -4- •f-- ' This leads one to anticipate that the ambiguity in the definition of j^(x) whioh has previously been noted by the author ' in the oase of the Thirring model might well be responsible for the conflicting results obtained by BROWN and SOMMERFIEID' '. It is in fact the intent of this paper to disouss in some detail the consequences of using the ourrent definition of Ref. 5 &&&• to demonstrate the existence of a more general olass of solutions of the theory described by (l.l) than has previously been known. 2. BOSON MATRIX ELEMENTS Since the vacuum expectation values of all operators contain- ing j^(x) and B^(x) in any arbitrary combination can be obtained from the vacuum-to-vacuum transition amplitude \ 00*J 00^ (or more simply <C 0 I oX -j-J merely by taking the appropriate functional derivatives with respeot to A^(x) and J^(x), it is the calculation of this quantity to which we direot our immediate attention. To this end we note that the aotion prinoiple implies from whioh one can immediately deduce the result where The effect of the interaction can thus be extracted entirely into an exponential factor containing derivatives with respect to the source terms. One consequently has only to solve the external field problem and evaluate the functional derivatives in order to complete the calculation of <C 0 | 0 •> _ e There is one further -5- Bimplifioation whioh follpwe from the removal of the interaction term, namely the factorization of <^0 1 0 ^> as indioated by The calculation of <^0 | 0/_. can be carried out by straight- forward application of the field equation In partioular one finds that reduces with the aid of (2.1) to J (2.2) where G^(x),the free field Green's funotion of the meson field,is given by and satisfies the differential equation subject to the usual causal boundary conditions. It is now trivial to integrate eq. (2.2) and obtain 0 1 0 \ \ J^U) G£*t*r*:) J, <X' -6- By contrast the,task of determining the structure of <f 0 j 0 ^> is oonsiderably more dlfficult inasmuoh as it requires a detailed consideration of the structure of the operator j/*(x). In particular, one must be oareful not to infer from the formal definition (2.3) and the apparent invariance of the theory under the infinitesimal gauge transformations (2.4b) 1 (Ye being the pseudo-soalar matrix (X ) that j^x) and oj^x) = £**" jy(x) are oonserved. Since, however, the problem of calculating <^ 0 I 0 > from the oondition and the field equation for i V has been disoussed in some detail in I, it is sufficient merely to quote the result. Thus we define the matrix element of j^x) to be - <«If (2.5) v 0 where A^ s €^* Ay and the lim x • x'is ^fco be taken after setting x° - x '. The parameters £ and M (which are required to Batisfy the oonstraint in order to preserve Lorentz invarianoe ) apppear as a direct consequence of the fact that there is no unique way in which to lend meaning to the ill-defined operator expression (2«3). With the prescription (2.5) one obtains ' which immediately implies the result where we have set and D(-x) is the causal Green's function defined by One has now to evaluate the expression \ This can be accomplished by relatively straightforward techniques ^* and we shall consequently not reproduce here these somewhat tedious calculations. -The result one obtains can be- written in the form -8- where C L tit and ^(x) is the caueal Green's function with the renormalized meson mass yu being given by the expression (2.6) It is, of course, necessary to require that both (l + VJT^) and >) •'be positive in order to ensure that the metric'of the Hilbert space be positive definite. The dependence of jx on the parameter ^ as expressed by eq. (2.6) forcefully illustrates the possibility of having physioally observable consequences associated with the freedom one has available in defining the current. Although the solutions of the Thirring model were also found to depend upon ^the appearance of this parameter in the mass renormalization of the vector meson is perhaps a somewhat more oonvinoing effeot. It is of interest to note that in the case <f = 0 one has a -9- no-renormalization theorem for the meson mass • By inspection of the structure of the current-current correlation function in the source- free limit (2.7) one can easily verify that £ = 0 corresponds to the case in whioh both j^x) and jA*) are conserved.
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