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Physical Review A

Geometric phases in the asymptotic theory of coupled wave equations

Robert G. Littlejohn, William G. Flynn

Published Oct 1, 1991• 199 citations
Publisher

Abstract

Traditional approaches to the asymptotic behavior of coupled wave equations have difficulties in the formulation of a consistent version of the Bohr-Sommerfeld quantization conditions. These difficulties can be circumvented by using the Weyl calculus to diagonalize the matrix of wave operators. In analyzing the diagonalized wave equations, geometric phases enter in an important way, especially in the development of Bohr-Sommerfeld quantization rules. It turns out that a version of Berry's phase is incorporated into the symplectic structure in the ray phase space, influencing the classical Hamiltonian orbits, the construction of solutions to the Hamiltonian-Jacobi equation, and the computation of action integrals. Noncanonical coordinates in the ray phase space are useful in carrying out these calculations and in making the construction of eigenvalues and wave functions manifestly gauge invariant.

DOI
10.1103/physreva.44.5239

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Traditional approaches to the asymptotic behavior of coupled wave equations have difficulties in the formulation of a consistent version of the Bohr-Sommerfeld quantization conditions. These difficulties can be circumvented by using the Weyl calculus to diagonalize the matrix of wave operators. In analyzing the diagonalized wave equations, geometric phases enter in an important way, especially in the development of Bohr-Sommerfeld quantization rules. It turns out that a version of Berry's phase is incorporated into the symplectic structure in the ray phase space, influencing the classical Hamiltonian orbits, the construction of solutions to the Hamiltonian-Jacobi equation, and the computation of action integrals. Noncanonical coordinates in the ray phase space are useful in carrying out these calculations and in making the construction of eigenvalues and wave functions manifestly gauge invariant.
A linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters is considered. Its asymptotic solutions are constructed to an arbitrary accuracy in the adiabaticity parameter ε. By extending the phase space of the system not only with the space of parameters, like in the theory of Berry’s phase, but also with the space of their derivatives, it proves possible to represent the phase of the solution as a sum of the dynamic phase and a generalized geometric phase (determined by a contour integral in the space of parameters and derivatives). Hence, it is possible to obtain the asymptotic results to any degree of accuracy in ε, while earlier they were obtainable only in the first-order adiabatic approximation. Namely, for linear oscillatory adiabatic systems, the quantizability, parametric stability, and the Hamiltonian nature of the system are equivalent properties. As a consequence, one can obtain the well-known result regarding conservation with accuracy to any power of ε of adiabatic invariants in a Hamiltonian system on a torus. An important point is that the generalized geometric phase can appear, in contrast to Berry’s phase, with only one real varying parameter.
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.
Newly developed methods for finding the semiclassical or WKB eigenvalues and eigenfunctions for vector wave fields are applied to spinning particles moving in central potentials subject to spin-orbit forces. The new methods are like the familiar Bohr-Sommerfeld or Einstein-Brillouin-Keller quantization methods for scalar wave fields, but involve additional issues relating to Berry's phase, gauge structures, and monopolelike singularities. All these extra issues occur in the asymptotics of spin-orbit coupling. The role of angular momentum and conservation laws is particularly interesting, due to the role played by gauge fields in the classical-quantum correspondence. The work presented is a contribution to ongoing semiclassical studies of shell structure in nuclei and other fermion systems such as metal clusters, as well as an example of the general methods of semiclassical vector wave quantization.
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 level crossing problem and associated geometric terms are neatly formulated by the second-quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second-quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path-integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval TTT. The integrability of Schr\"odinger equation and the appearance of the seemingly nonintegrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and TTT is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.
In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group.