Boundary Conditions as Dirac Constraints

TL;DR

This paper demonstrates that boundary conditions in field theories can be formulated as Dirac constraints, revealing their nature as an infinite chain of second class constraints, and explores their implications for quantization and noncommutativity.

Contribution

It introduces a novel perspective by treating boundary conditions as Dirac constraints, providing a systematic method for quantization and analyzing noncommutative features.

Findings

Boundary conditions are equivalent to an infinite chain of second class constraints.

Mode expansion and quantization with boundary conditions are justified as proper methods.

Quantized fields with mixed boundary conditions exhibit noncommutative properties.

Abstract

In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are noncommutative.