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arXiv:1501.00231

Fundamental groupoids in quantum mechanics: a new approach to quantization in multiply-connected spaces

Klil H. Neori, Philip Goyal

Published Jan 1, 2015• 1 citation

Abstract

Quantization of multiply-connected spaces requires tools which take these spaces' global properties into account. Applying these tools exposes additional degrees of freedom. This was first realized in the Aharonov-Bohm effect, where this additional degree of freedom was a magnetic flux confined to a solenoid, which an electron cannot enter. Previous work using Feynman path integrals has either only dealt with specific cases, or was limited to spaces with finite fundamental groups, and therefore, was in fact inapplicable to the Aharonov-Bohm effect, as well as to interesting systems such as anyons. In this paper we start from the fundamental groupoid. This less familiar algebraic-topological object is oriented towards general paths. This makes it a more natural choice for the path integral approach than the more commonly known fundamental group, which is restricted to loops. Using this object, we provide a method that works for spaces with infinite fundamental groups. We adapt a tool used in the previous, restricted result, in order to clearly delineate physically significant degrees of freedom from gauge freedom. This allows us to explicitly build phases which take account of symmetries of the space, directly from topological considerations; we end by providing a pertinent example relating to anyons.

DOI: 10.48550/arxiv.1501.00231

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Quantization of multiply-connected spaces requires tools which take these spaces' global properties into account. Applying these tools exposes additional degrees of freedom. This was first realized in the Aharonov-Bohm effect, where this additional degree of freedom was a magnetic flux confined to a solenoid, which an electron cannot enter. Previous work using Feynman path integrals has either only dealt with specific cases, or was limited to spaces with finite fundamental groups, and therefore, was in fact inapplicable to the Aharonov-Bohm effect, as well as to interesting systems such as anyons. In this paper we start from the fundamental groupoid. This less familiar algebraic-topological object is oriented towards general paths. This makes it a more natural choice for the path integral approach than the more commonly known fundamental group, which is restricted to loops. Using this object, we provide a method that works for spaces with infinite fundamental groups. We adapt a tool used in the previous, restricted result, in order to clearly delineate physically significant degrees of freedom from gauge freedom. This allows us to explicitly build phases which take account of symmetries of the space, directly from topological considerations; we end by providing a pertinent example relating to anyons.

The motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's path-integral approach to quantum mechanics. In particular, the authors study the simple case when the multiply-connected region is obtained by excluding a disc from the plane. If a nonzero magnetic flux is confined inside the disc, this is shown to include a change in the one-dimensional unitary representation of the fundamental group of the space which enters a proper definition of the path-integral. In this way, a simple explanation of the Aharonov-Bohm effect is shown to arise.

The Aharonov–Bohm effect in general gauge theories, for particles in gauge-curvature-free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from ℋ, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non-Abelian, according to whether ℋ*, the image of this homomorphism, is Abelian or non-Abelian. For a non-Abelian Aharonov–Bohm effect, it is necessary that both ℋ and G be non-Abelian. Simple examples illustrate the theory.

On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group. 1 Introduction Within a physical theory there are often effects or phenomena studied in idealized form in order to gain insight into the peculiarities of the theory. In Quantum Mechanics one such effect is the appearance of topological quantum phases in the wave functions of particles moving freely in multiply connected space-times the prototype of this effect being the Aharonov-Bohm (AB) effect [1], the appearance of a phase factor in the wave function of an electron which moves around a magnetic flux line. A similar effect is the Aharonov-Casher (AC) effect [2] which is obtained from the AB effect by replacing the flux line and the electron by a charged line and a neutral particle with magnetic moment, respectively. There have al...

The Aharonov-Bohm effect is investigated in the Feynman path-integral formulation of quantum mechanics. We consider an idealized situation with an electron moving in a magnetic-field-free region outside a solenoid whose radius to length ratio is very small. The nonvanishing vector potential term in the Lagrangian is written as an angular-velocity-dependent potential. In order to account for a singularity due to the presence of the solenoid itself, a periodic constraint is imposed on the path integral. The propagator can then be evaluated using the polar-coordinate methods of Peak and Inomata. It is found that the propagator has the general form $K({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})=\ensuremath{\Sigma}{n}^{}{\ensuremath{\chi}}_{n}{K}_{n}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}};\ensuremath{\tau})$ where the sum is taken over all classes of homotopic paths and the ${\ensuremath{\chi}}_{n}$ are a one-dimensional representation of the homotopy group. This is the form of the propagator as conjectured by Schulman.

Transitions out of non-Abelian topological order are difficult to describe in microscopic quantum models with numerical methods that remain tractable at large scales. We develop a sign-free path-integral framework for Kitaev quantum doubles

\mathcal D(G)

by organizing single-link perturbations in terms of non-invertible electric and magnetic 1-form symmetries that proliferate distinct anyon species. For any finite group

G

, an exact

\textit{matterization}

isometry introduces vertex degrees of freedom and maps the link-only model onto a

G

gauge--Higgs theory. Furthermore, the 1-form symmetries of the fixed point allow us to define generalized Fredenhagen--Marcu order parameters that become finite when the corresponding anyons condense. For

G = S_3

, quantum Monte Carlo simulations show that proliferating a non-Abelian electric anyon drives a first-order transition in which all nontrivial electric anyons condense. In the purely magnetic limit, the model reduces to a

(2+1)

D pure

G

gauge theory; for

G=S_3

, it exhibits a first-order confinement transition, diagnosed by the onset of a Wilson-loop area law and the restoration of an emergent magnetic 1-form symmetry. These results provide a unified numerical framework for non-Abelian anyon condensation, confinement, and generalized symmetry breaking.