Classical dynamics on three-dimensional fuzzy space

Abstract

We derive the path integral action for a particle moving in three-dimensional fuzzy space. From this we extract the classical equations of motion. These equations have rather surprising and unconventional features: they predict a cutoff in energy, a generally spatial-dependent limiting speed, orbital precession remarkably similar to the general-relativistic result, flat velocity curves below a length scale determined by the limiting velocity and included mass, displaced planar motion, and the existence of two dynamical branches of which only one reduces to Newtonian dynamics in the commutative limit. These features may provide a stringent observational test for this scenario of noncommutativity.

Path integral action in the generalized uncertainty principle framework

“…From the free particle analysis, we have seen that the action imposes an upper bound on the free particle velocity which depends on the mass of the particle. This feature is consistent with the results obtained in earlier [14,15,17]. Moreover the fact that the particle velocity must be real and finite leads us to a relation between the parameters α and β .…”

Fuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries

“…In a recent paper [7] the equations of motion for a particle moving in three dimensional fuzzy space were derived and the most striking features of these equations were the emergence of a limiting speed and energy. In form these equations are also very reminiscent to relativistic dynamics and it is natural to ask to what extent these equations are in fact underpinned by some Lorentz geometry.…”

“…A detailed exposition of the formulation of quantum mechanics on the Moyal plane can be found in [5] and the generalisation to three dimensional fuzzy space in [8]. In [7] a review of this formulation as well as a detailed derivation of the path integral action for a particle moving in three dimensional fuzzy space has been given and we refer the reader for the details to [7]. Sections III and IV respectively discuss the geometric interpretation of the classical equations of motions on three dimensional fuzzy space and the fuzzy sphere.…”

“…Here f i ( R) are arbitrary real dimensionless functions, which generalises the Laplacian, ∆, in [7]. It is obvious that all these observables are hermitian and commute with Γ.…”

“…It can also be readily verified that the angular momentum operators commute with the Hamiltonian. The Laplacian of [7] corresponds to the choice of functions…”