Weyl-ordered polynomials in fractional-dimensional quantum mechanics

Abstract

We develop algebraic properties of Weyl-ordered polynomials in the momentum and position operators P, Q which satisfy the R-deformed Heisenberg algebra, representations of which describe quantum mechanics in fractional dimensions. By viewing Weyl-ordered polynomials as tensor operators with respect to the Lie algebra sl₂(C) we derive a specific form for these polynomials, including an expression in terms of hypergeometric functions, and determine various algebraic properties such as recurrence relations, symmetries, and also a general product formula from which all commutators and anti-commutators may be calculated. We briefly discuss several applications to quantum mechanics in fractional dimensions.