Commutators, anti-commutators and Eulerian calculus

Abstract

Consider operators obeying the commutation rule Cy = 1 + qyC generalizing the rule Dx = 1 + xD, D denoting d/dx. The case q -1 corresponds to boson creation-annihilation operators, q --1 to fermion operators. We derive Leibniz' rule for general q. We find that the canonical representation of C such that CI = 0 is given by the Eulerian derivative. The basics of Eulerian calculus are discussed and an indication of a discrete Hamiltonian theory analogous to the Heisenberg representation in quantum mechanics is given.