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Modern Physics Letters A

BERRY'S PHASE, FRACTIONAL STATISTICS AND THE LAUGHLIN WAVE FUNCTION

Stefano Forte

Published Nov 10, 1991• 9 citations
Publisher

Abstract

We determine the path-integral for particles with fractional spin and statistics in the adiabatic limit, and we discuss the identification of the spin-changing term with a Berry phase. We show that the standard proof that the Laughlin wave function describes excitations with fractional statistics holds only on the basis of a tacit assumption of questionable validity.

DOI
10.1142/s021773239100364x

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We determine the path-integral for particles with fractional spin and statistics in the adiabatic limit, and we discuss the identification of the spin-changing term with a Berry phase. We show that the standard proof that the Laughlin wave function describes excitations with fractional statistics holds only on the basis of a tacit assumption of questionable validity.
We develop the relativistic quantum mechanics of particles with fractional spin and statistics in 2 + 1 dimensions in the path-integral approach. We endow the elementary excitations of the theory with fractional spin through the coupling of the particle number current with a topological term. We work out the dynamics of the spin degrees of freedom, and display the relation between the spin action and the knot invariants of the paths contributing to the path integral. We show that the explicit spin-changing interaction can be traded for multivaluedness of the wave function, and we relate this to the representation theory of the Lorentz and Poincaré groups in 2 + 1 dimensions. We discuss the multiparticle dynamics and derive the spin–statistics theorem.
A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.
These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a path-integral viewpoint. The group-theoretical underpinnings of relativistic fractional spin are discussed, then the path-integral quantization of spin and of massive fermions without using spinors is reviewed. The path integral for a system of n relativistic particles with fractional spin is constructed, the spin-statistics relation and the Lorentz and Poincare' representation content of physical states are discussed. Some problems in formulating a field theory with fractional statistics are presented, and their resolution in the operator cocycle approach is reviewed.
The phase change of the wave function of a superfluid film as a vortex moves around a closed path counts the number of superfluid particles enclosed by that path. This result is used to investigate whether such vortices obey ``fractional statistics.'' We conclude that this is not the case in compressible superfluids such as 4He^{4}\mathrm{He}4He films.
I revisit a longstanding question in quantum optics; when is the rotating wave approximation justified? In terms of the Jaynes–Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes–Cummings model. The source of this surprising result is traced back to different topologies in the two models.