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arXiv:hep-th/9504116 · Journal of Mathematical Physics

The canonical connection in quantum mechanics

Péter Lévay, David McMullan, Izumi Tsutsui

Published Feb 1, 1996• 15 citations
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Abstract

In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G invariance is central to the emergence of the H connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-Abelian generalization of Berry’s phase, can also be found in systems that have slow variables taking values in such a coset space.

DOI
10.1063/1.531432

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In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G invariance is central to the emergence of the H connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-Abelian generalization of Berry’s phase, can also be found in systems that have slow variables taking values in such a coset space.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.
A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.
Arguments are presented that make the choice of the connection, which gives rise to the Berry phase not only natural, but unique, both in the Abelian and the non-Abelian cases. Invariance is invoked under unitary transformations of the probability amplitudes in quantum systems to force the connection to be invariant under the unitary group. Because the action is not free, the horizontal subspace chosen by the connection has to be invariant under the little group, and that makes it orthogonal to the fiber’s direction, yielding the conventional Berry connection. The argument works just as well for the non-Abelian case, where the fibers are orthonormal frames (Stiefel manifolds), and the state space a Grassmanian, with a transitive but not free unitary action.
Gauge transformations of the potentials for a prescribed electromagnetic field acting on a non-relativistic charged particle are shown to induce canonical transformations in the extended phase space, in which the time and its conjugate momentum are treated as dynamical variables. The extended Hamiltonian is invariant and the extended canonical equations are covariant under a gauge transformation. The usual time-dependent wave equations in different gauges are obtained from the formalism by using Schrödinger representations.