# Path Integrals of the Vector Field --- Covariance of a Path Integral ---

## TL;DR

This paper develops a new method for deriving path integrals of a massive vector field with scalar field components, addressing covariance issues in the context of indefinite metric Hilbert spaces.

## Contribution

It introduces a novel approach based on canonical quantization on a lattice to obtain eigenvectors and formulate path integrals for vector fields with indefinite metrics.

## Key findings

- Derived time-sliced path integral formulas for vector and scalar fields.
- Showed the effective action reproduces the original action despite indefinite metrics.
- Explored the covariance properties of the path integral in this framework.

## 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.