Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics A

RELATIVISTIC PARTICLES WITH FRACTIONAL SPIN AND STATISTICS

Stefano Forte

Published Feb 20, 1992• 15 citations
Publisher

Abstract

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.

DOI
10.1142/s0217751x92000466

Related papers

Relativistic Quantum Theory with Fractional Spin and Statistics

Stefano Forte

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.

arXiv:hep-th/9307047•Jul 7, 1993•3 citations

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

Stefano Forte

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.

Modern Physics Letters A•Nov 10, 1991•9 citations

SPIN-STATISTICS THEOREM AND GEOMETRIC QUANTIZATION

Charis Anastopoulos

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.

arXiv:quant-ph/0110169•International Journal of Modern Physics A•Feb 20, 2004•5 citations

Fractional quantum mechanics

Nick Laskin

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.

arXiv:0811.1769•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 2000•883 citations

Quantization of Many-Particle Systems and Quantum Statistics in Lower Dimensions

Lucjan Jacak, Piotr Sitko, Konrad Wieczorek +1 more

This chapter describes how, in two-dimensional systems, the scalar fractional statistics may occur besides Bose–Einstein and Fermi–Dirac statistics. One-dimensional irreducible unitary representations of the full braid group for selected manifolds (Euclidean plane, sphere, torus, and Euclidean three-dimensional space) and its relation with symmetry of the wave function of the system are presented. The idea of anyons — particles obeying fractional statistics — is discussed in detail. The relation between spin and statistics in two- and three-dimensional systems and its topological nature is discussed. The chapter also shows how magnetic flux attached to particles in planar systems can change their statistical properties.

Oxford University Press eBooks•Sep 25, 2003•0 citations

Generalized Spin-Statistics Theorem

Lauri Suoranta

We derive the spin-statistics theorem in both relativistic and non-relativistic first-quantized form, extending considerably the earlier proofs. Our derivation is based on the representation theories of the groups SU (2) and SL(2,C), latter being the universal covering of the Lorentz group. We define the identity of particles. We include theories that have an internal symmetry group. We discuss consistency of the non-relativistic limit.

arXiv:1008.5382•Aug 31, 2010•1 citation
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.
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.
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.
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.
This chapter describes how, in two-dimensional systems, the scalar fractional statistics may occur besides Bose–Einstein and Fermi–Dirac statistics. One-dimensional irreducible unitary representations of the full braid group for selected manifolds (Euclidean plane, sphere, torus, and Euclidean three-dimensional space) and its relation with symmetry of the wave function of the system are presented. The idea of anyons — particles obeying fractional statistics — is discussed in detail. The relation between spin and statistics in two- and three-dimensional systems and its topological nature is discussed. The chapter also shows how magnetic flux attached to particles in planar systems can change their statistical properties.