Fractional Canonical Quantization: a Parallel with Noncommutativity

TL;DR

This paper extends the Faddeev-Jackiw quantization method to fractional calculus using the Modified Riemann-Liouville approach, exploring parallels with noncommutative geometry in a coarse-grained charged particle model.

Contribution

It introduces a consistent fractional canonical quantization framework employing the MRL approach, linking fractional derivatives with noncommutative structures.

Findings

Successful extension of the Faddeev-Jackiw algorithm to fractional derivatives.

Establishment of a parallel between fractional calculus and noncommutative geometry.

Application to a charged particle system in a magnetic field.

Abstract

Adopting a particular approach to fractional calculus, this paper sets out to build up a consistent extension of the Faddeev-Jackiw (or Symplectic) algorithm to carry out the quantization procedure of coarse-grained models in the standard canonical way. In our treatment, we shall work with the Modified Riemman Liouville (MRL) approach for fractional derivatives, where the chain rule is as efficient as it is in the standard differential calculus. We still present a case where we consider the situation of charged particles moving on a plane with velocity $\dot{r}$, subject to an external and intense magnetic field in a coarse-grained scenario. We propose an interesting parallelism with the noncommutative case.