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Journal of Physics A Mathematical and General

Single-valuedness of wavefunctions from global gauge invariance in two-dimensional quantum mechanics

Michel Bawin, André Burnel

Published Aug 1, 1985• 9 citations
Publisher

Abstract

The authors consider in two-dimensional non-relativistic quantum mechanics an arbitrary static magnetic field confined within an inaccessible region D of arbitrary geometry. It is shown that global gauge invariance implies that the wavefunction psi (x, y) for a charged particle outside D is single-valued in a single-valued gauge.
DOI
10.1088/0305-4470/18/11/033

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Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values ⟨ψ|A|ψ⟩ are invariant under a global unitary transformation acting on the wavefunction |ψ⟩→U|ψ⟩ and operators A→UAU† , and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions |ψJ⟩ , one for each patch of space J . The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches I and J are related to each other by dynamical unitary transformations UIJ . The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions |ψJ⟩ and connections UIJ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.
The requirement that the quantal wave function be single-valued is examined in the light of two recent developments: The effect of a magnetic vector potential on a particle moving in a multiply connected field-free region (Aharonov-Bohm effect), and the flux quantization inside a superconducting ring. Both of these effects are new applications of conventional elementary quantum mechanics. The single-valuedness problem is considered, with its historical background, for particles with no spin, particles with spin 12, and rigid bodies. It is shown that the single-valuedness condition, if properly adapted to the particular physical model, is deeply rooted in the foundations of quantum mechanics. However, wave functions which superficially appear to be double-valued are useful in an helicity representation of spinors and in the construction of nuclear wave functions.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
If a distinction is made between kinematical and dynamical symmetries of a quantum mechanical localizable system it is possible to extend the principle of Galilei-invariance to elementary particles, subject to arbitrary external forces. This principle implies a severe restriction on the possible types of external forces which can act on a particle. In fact we show in this paper that for a scalar particle the only forces which are compatible with this principle are those which derive from a scalar and vectorpotential. The Hamiltonian takes on the usual gauge-invariant form of a charged particle in interaction with arbitrary electromagnetic forces.
For a quantum mechanical particle in a classical electromagnetic field, the active and passive views of gauge invariance are compared. In the active point of view, both the wavefunction and the potentials are transformed so that the Schrödinger equation is form invariant. In the passive point of view, a change in the representation of the canonical momentum is made which just cancels the change in the gauge of the potentials. The condition that a gauge-invariant operator has the same expectation value in all gauges gives a functional equation, the solution of which shows that gauge-invariant operators depend only on the kinetic momentum or coordinates.
Abstract The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).