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Register a new userClassical dynamics on three dimensional fuzzy space: Connecting the short and long length scales
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Authors
FG Scholtz
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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.
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We use the recently derived density of states for a particle confined to a spherical well in three dimensional fuzzy space to compute the thermodynamics of a gas of non-interacting fermions confined to such a well. Special emphasis is placed on non-commutative effects and in particular non-commutative corrections to the thermodynamics at low densities and temperatures are computed where the non-relativistic approximation used here is valid. Non-commutative effects at high densities are also identified, the most prominent being the existence of a minimal volume at which the gas becomes incompressible. The latter is closely related to a low/high density duality exhibited by these systems, which in turn is a manifestation of an infra-red/ultra violet duality in the single particle spectrum. Both non-rotating and slowly rotating gasses are studied. Approximations are benchmarked against exact numerical computations for the non-rotating case and several other properties of the gas are demonstrated with numerical computations. Finally, a non-commutative gas confined by gravity is studied and several novel features regarding the mass-radius relation, density and entropy are highlighted.
We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrodinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.
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