Path integrals and boundary conditions

TL;DR

This paper examines the limitations of the path integral approach in quantum mechanics and field theory, especially regarding the description of boundary conditions, highlighting its inability to handle highly non-local boundaries.

Contribution

It analyzes the origin of boundary condition limitations in the path integral method and discusses implications for quantum mechanics and field theory.

Findings

Path integrals cannot describe highly non-local boundary conditions.

The limitations stem from the formalism's structure and affect field theory applications.

The paper clarifies the scope of path integral quantization in bounded domains.

Abstract

The path integral approach to quantum mechanics provides a method of quantization of dynamical systems directly from the Lagrange formalism. In field theory the method presents some advantages over Hamiltonian quantization. The Lagrange formalism preserves relativistic covariance which makes the Feynman method very convenient to achieve the renormalization of field theories both in perturbative and non-perturbative approaches. However, when the systems are confined in bounded domains we shall show that the path integral approach does not describe the most general type of boundary conditions. Highly non-local boundary conditions cannot be described by Feynman's approach. We analyse in this note the origin of this problem in quantum mechanics and its implications for field theory.