Weak Coherent State Path Integrals

TL;DR

This paper develops a rigorous path integral formulation for weak coherent states, which lack a resolution of unity, focusing on affine quantum gravity and linear Hamiltonians, and addressing challenges with more complex Hamiltonians.

Contribution

It introduces a continuous-time regularization approach to construct well-defined path integrals for weak coherent states, extending the methodology beyond standard coherent states.

Findings

Path integral with Wiener measure is successfully constructed for weak coherent states.

The approach is rigorously validated for linear Hamiltonians.

Challenges with non-linear Hamiltonians are discussed and partially addressed.

Abstract

Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.