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Physical Review A

Bargmann invariants, null phase curves, and a theory of the geometric phase

Narasimhaiengar Mukunda, Arvind, Elisa Ercolessi, Giuseppe Marmo, G. Morandi, R. Simon

Published Apr 29, 2003• 28 citations
Publisher

Abstract

We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves defined entirely in terms of the (third order) Bargmann invariant, and it is shown that these are the curves natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.

DOI
10.1103/physreva.67.042114

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We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
We present a study of the properties of Bargmann Invariants (BIs) and Null Phase Curves (NPCs) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator-based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters whose algebraic properties as functions of Hilbert space dimension are analyzed using elegant group theoretic methods. The BI-geometric phase connection, extended by the use of NPCs, is explored in detail, and interesting new experiments in this subject are pointed out.
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The geometric phase in quantum mechanics was originally elucidated in the context of the adiabatic theorem. This and other limitations were later removed. We show how this phase can be presented as the simplest functional of smooth open curves of unit vectors in Hilbert space, invariant under the two groups of local phase changes and monotonic reparametrisations. This viewpoint clarifies the role of the early Pancharatnam criterion for two Hilbert space vectors to be `in phase', in the later geometric phase context. Noncyclic evolution is handled in a very direct manner. Connections to the Bargmann invariants of quantum mechanics are immediate and can be exploited. A class of geometric phases arises in connection with unitary representations of Lie groups. In this case a surprisingly complete analysis of the structure of the geometric phase is possible, motivated by the Wigner-Eckart theorem. The interplay between geometric phase ideas and Lie group unitary representations is extraordinarily rich; the differential geometric structures of Lie groups and coset spaces play very important roles and allow classification of all possible geometric phases associated with a given Lie group. Examples involving SU(2), SU(3) and the metaplectic group Mp(2) are presented.