  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.
