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Journal of Mathematical Physics

Flat connections on quantum bundles and fractional statistics in geometric quantization

Yue Yu, Zhongyuan Zhu, H. C. Lee

Published Mar 1, 1993• 2 citations
Publisher

Abstract

It is shown that in geometric quantization formulation, fractional statistics in a quantized system of N indistinguishable particles in two spatial dimensions arise from the nontrivial cohomology of the flat connection on the quantum line bundles as well as from the nontrivial homology of the configuration space. The propagator of a nonrelativistic interacting system with fractional statistics is derived.

DOI
10.1063/1.530205

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It is shown that in geometric quantization formulation, fractional statistics in a quantized system of N indistinguishable particles in two spatial dimensions arise from the nontrivial cohomology of the flat connection on the quantum line bundles as well as from the nontrivial homology of the configuration space. The propagator of a nonrelativistic interacting system with fractional statistics is derived.
We study the relation of the spin-statistics theorem to the geometric structures on phase space, which are introduced in quantization procedures (namely a U(1) bundle and connection). The relation can be proved in both the relativistic and the nonrelativistic domain (in fact for any symmetry group including internal symmetries) by requiring that the exchange can be implemented smoothly by a class of symmetry transformations that project in the phase space of the joint system system. We discuss the interpretation of this requirement, stressing the fact that any distinction of identical particles comes solely from the choice of coordinates — the exchange then arises from suitable change of coordinate system. We then examine our construction in the geometric and the coherent-state-path-integral quantization schemes. In the appendix we apply our results to exotic systems exhibiting continuous "spin" and "fractional statistics." This gives novel and unusual forms of the spin-statistics relation.
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Abstract We develop a geometric formulation of nonrelativistic quantum mechanics in Randers–type backgrounds based on the effective quantum connection induced by the Randers one-form. Starting from the classical Randers action, we derive the nonrelativistic Hamiltonian through the Legendre transformation and obtain the corresponding Schrödinger equation by canonical quantization. The resulting formalism naturally yields the kinetic momentum, probability current, effective force, and the curvature associated with the induced quantum connection. We show that the nontrivial quantum effects associated with the spatial effective connection are characterized by its local curvature and global holonomy, leading to three complementary geometric regimes. Globally trivial flat connections leave the local dynamics unchanged apart from uniform energy shifts, flat connections with nontrivial holonomy generate geometric phases, twisted boundary conditions, and persistent probability currents, whereas nonvanishing effective curvature gives rise to noncommuting kinetic momenta, Landau-type quantization, a characteristic geometric length, and curvature-controlled degeneracy. The formalism is further extended to nonuniform backgrounds through the first Born approximation, where the vector contribution to elastic scattering is shown to probe the transverse component of the effective connection and, consequently, the Fourier components of its effective curvature. These results provide a unified geometric framework for nonrelativistic quantum mechanics in Randers–type backgrounds and identify holonomy and effective curvature as the fundamental global and local mechanisms through which the induced quantum connection modifies quantum behavior.
Abstract The quantum Hall effect is an important physical phenomenon which has been analyzed both theoretically and experimentally for a few decades by now. In this chapter, we discuss this effect in relation to geometric quantization. We will also see how fractional statistics for quasiparticles is related to nontrivial H1(M,R)\mathcal {H}^1(M, \mathbb {R})H1(M,R) of the phase space.