Boundary Hilbert spaces and trace operators

TL;DR

This paper develops a framework for defining boundary Hilbert spaces in complex physical systems using trace operators, enabling quantum boundary dynamics without traditional tensor product decompositions.

Contribution

It introduces a novel approach to boundary Hilbert spaces utilizing trace operators, expanding the mathematical tools for quantum systems with non-factorizable state spaces.

Findings

Boundary Hilbert spaces can be constructed without tensor product factorization.

Trace operators facilitate the analysis of boundary quantum dynamics.

Fock space functoriality aids in defining boundary quantum states.

Abstract

We discuss the introduction of boundary Hilbert spaces for a class of physical systems for which it is not possible to factor their state spaces as tensor products of Hilbert spaces naturally associated to their boundaries and bulks respectively. In order to do this we make use of the so called trace operators that play a relevant role in the analysis of PDE's in bounded regions. By taking advantage of these operators and some functorial aspects of the construction of Fock spaces, we will show how to obtain quantum dynamics at the boundaries defined in appropriate Hilbert spaces associated with them.