Generalised Geometric Phase

TL;DR

This paper introduces a generalized geometric phase for pure quantum states, extending the concept to observable averages and operator forms, with applications in energy shifts and scattering phenomena.

Contribution

It proposes a novel operator-based generalization of the geometric phase, linking it to holonomy and physical effects in quantum evolution.

Findings

Generalized geometric phase defined via observable matrix elements.

Manifestation as a global phase in time evolution when usual phase is undefined.

Contributes to energy spectrum shifts and scattering amplitudes.

Abstract

A generalised notion of geometric phase for pure states is proposed and its physical manifestations are shown. An appreciation of fact that the interference phenomenon also manifests in the average of an observable, allows us to define the argument of the matrix element of an observable as a generalised relative phase. This identification naturally paves the way for defining an operator generalisation of the geometric phase following Pancharatnam. The notion of natural connection finds an appropriate operator generalisation, and the generalised geometric phase is indeed found to be the (an)holonomy of the generalised connection. It is shown that in scenarios wherein the usual geometric phase is not defined, the generalised geometric phase manifests as a global phase acquired by a quantum state in course of time evolution. The generalised geometric phase is found to contribute to the shift in the energy spectrum due perturbation, and to the forward scattering amplitude in a scattering problem.