An algebraic formulation of quantum mechanics in fractional dimensions

Abstract

I formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P, Q satisfy the R-deformed Heisenberg relations, and find representations of P, Q in which the angular momentum l and the dimension d, which can by any real positive number, appear as parameters. These representations lead to corresponding representations of paraboson operators which can be used, for example, to solve the time-dependent harmonic oscillator for any d>0 using the method of Lewis and Riesenfeld. I develop algebraic properties of Weyl-ordered polynomials in P, Q by viewing them as tensor operators with respect to the Lie algebra sl2, and also discuss the q-analogue deformation of these properties.