Fractional Particle and Sigma Model

TL;DR

This paper develops a classical fractional particle model incorporating the fractional Laplacian, explores its properties, solutions, and symmetries, and constructs a related sigma model that is quantized to analyze vacuum energy.

Contribution

It introduces a new fractional particle model, derives its equations of motion, and constructs a quantized sigma model based on the extension problem, extending classical mechanics with fractional calculus.

Findings

Derived equations of motion for fractional particles

Solved the model using Green's functions

Quantized the sigma model and computed vacuum energy

Abstract

We introduce a classical fractional particle model in $\mathbb{R}^{n}$, extending the Newtonian particle concept with the incorporation of the fractional Laplacian. A comprehensive discussion on kinetic properties, including linear momentum and kinetic energy, is provided. We further derive the equations of motion and discuss the symmetries of the system. The Green's function method is employed to solve the equations of motion in a general case. We illustrate the theory with three important examples: the free fractional particle, the fractional harmonic oscillator, and the charged fractional particle that interacts locally with the electromagnetic field. We use the results of the extension problem by Caffarelli and Silvestre, to construct the associated classical local sigma model for the fractional particle. The sigma model is then quantized using the canonical quantization method, and we compute the vacuum energy at the boundary.