Canonical Quantization of Non-Abelian Gauge Fields

Abstract

The procedure of the canonical quantization of non·abelian gauge fields presented in a previous paper is applied to the case of the axial gauge condition. In this case the problem with respect to the ordering of operators does not occur and the quantized Hamiltonian takes the same form as that of the classical one. The equivalence of our consequences for the Coulomb and the axial gauge conditions to those given by the Feynman path integration procedure is investigated. Although the results given by the both methods coincide with each other in case of the axial gauge condition, it seems that there is some inequivalence between them in case of the Coulomb gauge condition. Our results suggest that the naive change of the integral variables in the path integration method cannot remove the disagree­ ment between the both procedures. § 1. Introduction In a previous paper 1l we presented a reliable prescription for the canonical quantization of a non-abelian gauge field on the Coulomb gauge condition. We began with the familiar classical Lagrangian and introduced to it an additional term with a Lagrangian multiplier field r/Ja in order to avoid the difficulty of the vanishing canonical conjugate momenta. The constraints were imposed on the phys­ ical state vectors to eliminate this superfluous field ¢a· By means of a non-singular operator, these constraints were transformed into those which allo-vved a simple interpretation that the longitudinal components of field variables should not be included in the physical states. In other words, the Coulomb gauge condition was introduced by the above transformation, which was shown to be unitary in the physical Hilbert space. In the present article we take another choice of the transformation, by means of which the constraints upon the physical states are changed into new ones which show that the physical states do not depend on the third component of the field variables. It is seen that this transformation is also unitary in the physical Hilbert space. Another aim of the present paper is to investigate directly the equivalence of our results to those given by Feynman's path-integration formulation_ It seems that these formulations are equivalent to each other on the axial gauge condition, while on the Coulomb gauge condition there is some disagreement between the method of the canonical quantization and that of Feynman. The discrepancies may occur in the Feynman diagrams which contain two or more closed loops.