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MOMENTO

QUANTUM MECHANICS AND TOPOLOGY

J. Gamboa

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Physics defined on multiply connected manifolds is an old topic in theoretical physics. In the context of the path integral formalism it was studded by the first time by Schulman in 1968 and rigorously formulated by Laidlaw and the Witt and Dowker in 1971. The central point is that in a multiply connected manifold the paths have different weights in the sum over histories and the problema - which does not exist in the "standard" quantum mechanicsis how to define a quantum theory taking this fact into account.

Related papers

Quantum mechanics on multiply connected manifolds with applications to anyons in one and two dimensions

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arXiv:hep-th/9602158•AIP conference proceedings•Jan 1, 1996•0 citations

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Homotopy and Path Integrals

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Reformulation of Quantum Theory

Seyed Ebrahim Akrami

We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.

arXiv:2201.01187•Jan 3, 2022•0 citations
Physics defined on multiply connected manifolds is an old topic in theoretical physics. In the context of the path integral formalism it was studded by the first time by Schulman in 1968 and rigorously formulated by Laidlaw and the Witt and Dowker in 1971. The central point is that in a multiply connected manifold the paths have different weights in the sum over histories and the problema - which does not exist in the "standard" quantum mechanicsis how to define a quantum theory taking this fact into account.
We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on R3N\mathbb{R}^{3N}R3N to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.
This is an introductory review of the connection between homotopy theory and path integrals, mainly focus on works done by Schulman [23] that he compared path integral on SO(3) and its universal covering space SU(2), DeWitt and Laidlaw [15] that they proved the theorem to the case of path integrals on the multiply-connected topological spaces. Also, we discuss the application of the theorem in Aharonov-Bohm effect given by [20,24]. An informal introduction to homotopy theory is provided for readers who are not familiar with the theory.
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.