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arXiv:quant-ph/0605081 · International Journal of Modern Physics A

GEOMETRIC PHASES, GAUGE SYMMETRIES AND RAY REPRESENTATION

Kazuo Fujikawa

Published Oct 20, 2006• 7 citations
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Abstract

The conventional formulation of the nonadiabatic (Aharonov–Anandan) phase is based on the equivalence class {eiα(t)ψ(t,x)} which is not a symmetry of the Schrödinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the nonadiabatic phase. This reformulation reproduces all the successful aspects of the nonadiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and nonadiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The nonadiabatic phase may well be regarded as a special case of the adiabatic phase in this reformulation, contrary to the customary understanding of the adiabatic phase as a special case of the nonadiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this reformulation.

DOI
10.1142/s0217751x06033799

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The conventional formulation of the nonadiabatic (Aharonov–Anandan) phase is based on the equivalence class {eiα(t)ψ(t,x)} which is not a symmetry of the Schrödinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the nonadiabatic phase. This reformulation reproduces all the successful aspects of the nonadiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and nonadiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The nonadiabatic phase may well be regarded as a special case of the adiabatic phase in this reformulation, contrary to the customary understanding of the adiabatic phase as a special case of the nonadiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this reformulation.
The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry, which is associated with the arbitrariness of the phase choice of a complete orthonormal basis set, becomes explicit in this formulation (in particular, in the adiabatic approximation) and specifies physical observables. The choice of a basis set which specifies the coordinate in the functional space is arbitrary in the second quantization, and a subclass of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. We discuss the implications of this hidden local gauge symmetry in detail by analyzing geometric phases for cyclic and noncyclic evolutions. It is shown that the hidden local symmetry provides a basic concept alternative to the notion of holonomy to analyze geometric phases and that the analysis based on the hidden local gauge symmetry leads to results consistent with the general prescription of Pancharatnam. We however note an important difference between the geometric phases for cyclic and noncyclic evolutions. We also explain a basic difference between our hidden local gauge symmetry and a gauge symmetry (or equivalence class) used by Aharonov and Anandan in their definition of generalized geometric phases.
The analysis of geometric phases is briefly reviewed by emphasizing various gauge symmetries involved. The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry becomes explicit in this formulation and specifies physical observables; the choice of a basis set which specifies the coordinates in the functional space is arbitrary in the second quantization, and a sub-class of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. It is shown that the hidden local symmetry provides a basic concept which replaces the notions of parallel transport and holonomy. We also point out that our hidden local gauge symmetry is quite different from a gauge symmetry used by Aharonov and Anandan in their definition of non-adiabatic phases.
The second quantized approach to geometric phases is reviewed. The second quantization generally induces a hidden local (time-dependent) gauge symmetry. This gauge symmetry defines the parallel transport and holonomy, and thus it controls all the known geometric phases, either adiabatic or non-adiabatic, in a unified manner. The transitional region from the adiabatic to non-adiabatic phases is thus analyzed in a quantitative way. It is then shown that the topology of the adiabatic Berry's phase is trivial in a precise sense and also the adiabatic phase is rather fragile against the non-adiabatic deformation. In this formulation, the notion such as the projective Hilbert space does not appear.
All the geometric phases are shown to be topologically trivial by using the second quantized formulation. The exact hidden local symmetry in the Schrödinger equation, which was hitherto unrecognized, controls the holonomy associated with both of the adiabatic and nonadiabatic geometric phases. The second quantized formulation is located in between the first quantized formulation and the field theory, and thus it is convenient to compare the geometric phase with the chiral anomaly in field theory. It is shown that these two notions are completely different.
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.