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American Journal of Physics

Nonequivalent representations of canonical commutation relations in quantum mechanics: The case of the Aharonov-Bohm effect

L. Bracci, Luigi E. Picasso

Published Feb 12, 2007• 1 citation
Publisher

Abstract

We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.

DOI
10.1119/1.2360994

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We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.
We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space.A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops.A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect.Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.
We consider a representation of canonical commutation relations (CCR) appear­ing in a (non-Abelian) gauge theory on a non-simply connected region in the two­dimensional Euclidean space. A necessary and sufficient condition for the representa­tion to be equivalent to the Schrodinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrodinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.
The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrddoto\\ddot{o}ddotodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrddoto\\ddot{o}ddotodinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.
The electric Aharonov-Bohm effect is usually explained by means of the scalar potential, which is present in a multiply connected region of space time where no electric field acts on the charged particle. An alternative interpretation is given in terms of the particle's polarization field penetrating the region occupied by the electric field but inaccessible to the particle itself. Similarly, the magnetic Aharonov-Bohm effect is usually explained by means of the vector potential, which is present in a multiply connected region of space where no magnetic-induction field acts on the charged particle. An alternative interpretation of this effect is given in terms of the particle's magnetization field penetrating the region occupied by the magnetic-induction field but inaccessible to the particle itself. Our interpretations of the electric and magnetic Aharonov-Bohm effects are both developed by using the multipolar form of the Lagrangian instead of the minimal-coupling form in the Feynman path integral.
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.