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 cut-off 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 place strong constraints on the non-commutative parameter and coordinate algebra to avoid conflict with observation and may provide a stringent observational test for this scenario of non-commutativity.
Highlights
The structure of space-time at short length scales and the emergence of space-time as we perceive it at long length scales are probably the most challenging problems facing modern physics [1]
One expects that the modified dynamics implied by the noncommutativity can cause precession of elliptic orbitals, and we investigate this possibility here
2ħλv0, the dynamics must be described by the plus branch, which does not reduce to Newtonian dynamics in the commutative limit
Summary
The structure of space-time at short length scales and the emergence of space-time as we perceive it at long length scales are probably the most challenging problems facing modern physics [1] These issues are at the core of the struggle to combine gravity and quantum mechanics into a unified theory and probably links closely with the observational challenges of dark matter and energy. One other possible manifestation of noncommutativity on the macroscopic level may be in the modification of classical dynamics and gravity This has seemingly not yet been explored systematically, which is the motivation for the current paper that aims to fill this gap, at least in the case of a fuzzy-space scenario.
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