Quantization of (2+1)-spinning particles and bifermionic constraint problem

TL;DR

This paper develops a consistent quantization method for the 2+1 dimensional Dirac particle, addressing unique challenges like bifermionic constraints, and successfully reproduces the one-particle quantum theory without negative-energy issues.

Contribution

It introduces a novel quantization approach for 2+1 Dirac particles, handling bifermionic constraints and differences from 3+1 dimensions, aligning with the one-particle quantum theory.

Findings

Constructed a classical effective phase space with constraints and gauges.

Realized an unambiguous operator algebra and Hilbert space.

Reproduced the one-particle sector of 2+1 quantum spinor theory.

Abstract

This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to a classical model of a spinless relativistic particle as well as to the Berezin-Marinov model of 3+1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present article we apply a similar approach to the problem of quantizing the massive 2+1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3+1 dimensions. The point is that in 2+1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2+1 dimensions is also interesting from the physical viewpoint (e.g. anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2+1 quantum theory of a spinor field.