Generalized Geometrical Phase in the Case of Continuous Spectra

TL;DR

This paper derives an explicit formula for a generalized geometrical phase in quantum systems with continuous spectra, illustrating its application to relativistic particles and scattering phases.

Contribution

It introduces a new explicit formula for the generalized geometrical phase in systems with continuous spectra, expanding the understanding of geometric phases in quantum mechanics.

Findings

Derived an explicit formula for the generalized geometrical phase.

Calculated the phase for relativistic spinning particles in electromagnetic fields.

Connected the generalized phase to S-matrix and scattering phase shifts.

Abstract

A quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported round a circuit C by varing parameters in its Hamiltonian, will acquire a generalized geometrical phase factor. An explicit formula for a generalized geometrical phase is derived in terms of the eigenstates of the Hamiltonian. As an illustration the generalized geometrical phase is calculated for relativistic spinning particles in slowly-changing electromagnetic fields. It is shown that the the S-matrix and the usual scattering (with negligible reflexion) phase shift can be interpreted as a generalized geometrical phase.