Spectrum of the three-dimensional fuzzy well

Abstract

We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrödinger 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 ultraviolet/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.

Equilibrium properties of blackbody radiation with an ultraviolet energy cut-off

“…Also, we see similar cut-offs appearing in other candidate quantum gravity theories like noncommutative geometry, string theory, loop quantum gravity and GUP (Generalized Uncertainty Principle) etc. [2,[12][13][14][15][16][17][18][19][20][21][22][23].…”

Commutative–non-commutative dualities

“…for the space-time commutation relations is normally adopted, where θ µν are constants. Generalizations to fuzzy-and Snyder-space have been explored in [9][10][11][12] and [13] respectively. The latter offers the possibility of avoiding the breaking of rotational and Lorentz symmetries.…”

Scattering in a three-dimensional fuzzy space

“…A more direct way to overcome the breaking of rotational symmetry, at least in 3-dimensional quantum mechanics, is to adopt fuzzy sphere or su(2) commutation relations [8][9][10] of the form…”

“…where λ is the non-commutative length parameter. The Coulomb problem [8,9] and spherical well [10] have been investigated in this framework. The latter results were also used to investigate the thermodynamics of non-commutative Fermi gases [11] which was found to deviate strongly from that of the commutative case at high densities and temperatures.…”