Free scalar field theory on a Sobolev space over a bounded interval

TL;DR

This paper explores the functional analytic framework for a free, massless scalar field on a bounded interval, emphasizing Sobolev spaces and boundary effects in the Hamiltonian formulation.

Contribution

It introduces a Sobolev space-based approach that explicitly characterizes phase space fibers and analyzes boundary influences in scalar field theories.

Findings

Explicit representation of phase space fibers using Sobolev spaces

Analysis of boundary effects on the Hamiltonian formulation

Functional analytic tools for scalar field theories on bounded intervals

Abstract

This paper discusses several functional analytic issues relevant for field theories in the context of the Hamiltonian formulation for a free, massless, scalar field defined on a closed interval of the real line. The fields that we use belong to a Sobolev space with a scalar product. As we show this choice is useful because it leads to an explicit representation of the points in the fibers of the phase space (the cotangent bundle of the configuration space). The dynamical role of the boundary of the spatial manifold where the fields are defined is analyzed.