Path integrals of the vector field: Covariance of a path integral

Abstract

On the basis of the canonical quantization procedure of a system defined on a cubic lattice, we propose a new method, in which resolutions of unity expressed in terms of eigenvectors are naturally provided, to find eigenvectors of field operators. By making use of fundamental ingredients thus obtained, we derive time sliced path integral formulae for a massive vector field accompanied with a scalar field. Due to the indefinite metric of the Hilbert space upon which we define field operators, the action appears in the path integral looks quite different from the classical one. Nevertheless we will find that the effective action defined by introducing external sources results in the original action. By taking the effective action as the base of consideration, we study the proper meaning of the covariance of a path integral.