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Progress of Theoretical Physics

Vanishing Theorems for Hannay Angle and Berry Phase

Minoru Hirayama, Toshihiko TAJIMA

Published Apr 1, 1988• 0 citations
Publisher

Abstract

The classical Hannay angle and the quantal Berry phase are discussed for systems involving neither dimensionless adiabatic parameters nor vector potentials. Both of them are shown to vanish generally in a flat space. They are also shown to vanish in any n -dimensional curved space which can be embedded in an ( n + 1)-dimensional Euclidean space.

DOI
10.1143/ptp.79.746

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The classical Hannay angle and the quantal Berry phase are discussed for systems involving neither dimensionless adiabatic parameters nor vector potentials. Both of them are shown to vanish generally in a flat space. They are also shown to vanish in any n -dimensional curved space which can be embedded in an ( n + 1)-dimensional Euclidean space.
A new relationship between Berry’s phase and Hannay’s angle (which is its classical counterpart) is established. This relationship is exact and does not depend on semiclassical arguments, though in this note the aforementioned relationship is proved only for a restricted class of Hamiltonians. It is shown that if a Hamiltonian, for which Berry’s phase and Hannay’s angle are nonvanishing, can, in classical theory, be canonically transformed into one for which both are zero, the generating function of this canonical transformation yields, through appropriate averages, both the quantal geometrical phase and its classical analog.
The Berry phase, which was discovered more than two decades ago, provides very deep insight into the geometric structure of quantum mechanics. Its classical counterpart, Hannay's angle, is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By the term quantum-classical hybrid system, we mean a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem couplings on the Berry phase and Hannay's angle are explored. The results show that the Berry phase has been changed sharply by the couplings, whereas the couplings have a small effect on the Hannay's angle.