On extended umbral calculus, oscillator-like algebras and generalized Clifford algebra

Abstract

Some quantum algebras built from deformed oscillator algebras a la Jordan-Schwinger may be described in terms of a particular case of (umbral) ψ-calculus. We give here an example of a specific relation between certain such quantum algebras and generalized Clifford algebras, also in the context of Levy-Leblond’s azimuthal quantization of angular momentum, which was afterwards interpreted as a finite dimensional quantum mechanics by Santhanam et all. ψ-calculus, used here as a framework, is a example of the classical operator calculus of Rota. By its nature ψ-umbral calculus supplies a simple mathematical underpinning for ψ-deformed quantum-like oscillator algebras and, at least for theψn (q) = [nq!]−1 case [1–3], it provides the natural underpinning for quantum group investigation. Moreover, the other way around, one may formulateq-extended finite operator calculus with the help of the “quantumq-plane”q-commuting variables. The ψ-calculus is expected to be useful in theC* algebraic [4] description of “ψ-quantum processes” with various parastatistics [5].