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Journal of Physics A Mathematical and General

Gauge-invariant reference section and geometric phase

A -K Pati

Published Apr 7, 1995• 52 citations
Publisher

Abstract

We use a gauge-invariant 'reference section' and define the geometric phase for all quantum evolutions in a closed form. This geometric phase is obtained by integrating the inner product of the 'reference section' and its path derivative along the evolution curve which is valid for non-cyclic, non-unitary and non-Schrodinger evolutions of quantum systems. Two non-trivial examples are studied to realize our new expression.

DOI
10.1088/0305-4470/28/7/027

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We use a gauge-invariant 'reference section' and define the geometric phase for all quantum evolutions in a closed form. This geometric phase is obtained by integrating the inner product of the 'reference section' and its path derivative along the evolution curve which is valid for non-cyclic, non-unitary and non-Schrodinger evolutions of quantum systems. Two non-trivial examples are studied to realize our new expression.
The geometric phase is defined for any arbitrary quantum evolution using a ``reference section'' of the bundle covering the curve in the projective Hilbert space. A canonical one-form is defined whose line integral gives the desired geometric phase. It is manifestly gauge, phase, and reparametrization invariant for all quantum evolutions. A simple proof of the vanishing nature of the geometric phase along the geodesic is given. Also, an elementary proof of the nonadditive nature of the geometric phase is given. In the limit of cyclic evolution of a pure quantum state, this phase reduces to the Aharonov and Anandan phase, precisely. It is observed that in addition to the geometric phase, other geometric structures exist, such as the ``length'' and ``distance'' during any arbitrary quantum evolution. The relations among all of these geometric quantities are pointed out. Finally, two simple examples are studied to illustrate the ideas introduced in this paper.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.
The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.
This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both nondegenerate as well as degenerate states. Starting with the nondegenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an expression for the geometric phase that is gauge invariant. The parallelity conditions are shown to be easily derivable from this expression. We also extend our formalism to states that exhibit degeneracies. Here with the holonomy taking on a non-Abelian character, we provide an expression for the geometric phase that is manifestly gauge invariant. As in the case of the nondegenerate case, the form also displays the parallelity conditions clearly. Finally, we furnish explicit examples of the geometric phases for both the nondegenerate as well as degenerate mixed states.