Method for Solving Quantum Field Theory in the Heisenberg Picture
Abstract
This paper is a review of the method for solving quantum field theory in the Heisenberg picture, developed by Abe and Nakanishi since 1991. Starting from field equations and canonical (anti)commutation relations, one sets up a (qnumber) Cauchy problem for the totality of ddimensional (anti)commutators between the fundamental fields, where d is the number of spacetime dimensions. Solving this Cauchy problem, one obtains the operator solution of the theory. Then one calculates all multiple commutators. A representation of the operator solution is obtained by constructing the set of all Wightman functions for the fundamental fields; the truncated Wightman functions are constructed so as to be consistent with all vacuum expectation values of the multiple commutators mentioned above and with the energypositivity condition. By applying the method described above, exact solutions to various 2dimensional gaugetheory and quantumgravity models are found explicitly. The validity of these solutions is confirmed by comparing them with the conventional perturbationtheoretical results. However, a new anomalous feature, called the ``fieldequation anomaly'', is often found to appear, and its perturbationtheoretical counterpart, unnoticed previously, is discussed. The conventional notion of an anomaly with respect to symmetry is reconsidered on the basis of the fieldequation anomaly, and the derivation of the critical dimension in the BRSformulated bosonic string theory is criticized. The method outlined above is applied to more realistic theories by expanding everything in powers of the relevant parameter, but this expansion is not equivalent to the conventional perturbative expansion. The new expansion is BRSinvariant at each order, in contrast to that in the conventional perturbation theory. Higherorder calculations are generally extremely laborious to perform explicitly.
 Publication:

Progress of Theoretical Physics
 Pub Date:
 March 2004
 DOI:
 10.1143/PTP.111.301
 Bibcode:
 2004PThPh.111..301N