Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state

TL;DR

This paper develops a functional integral framework for quantum models with non-L2 ground states, illustrating the approach with a particle in a periodic potential and exploring thermodynamic limits and superselection sectors.

Contribution

It introduces a spectral stochastic process representation for quantum models with non-L2 ground states, extending the functional integral approach to these systems.

Findings

Existence of a unique ground state as a state on the Weyl algebra

Construction of a spectral stochastic process on trajectories in the spectrum

Analysis of thermodynamic limit and superselection sectors

Abstract

A functional integral representation is given for a large class of quantum mechanical models with a non--L2 ground state. As a prototype the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite volume measures.