Refined formulation of quantum analysis,q-derivative and exponential splitting

Abstract

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential df(A) satisfying the Leibniz rule. This formulation with use of the inner derivation ?A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q-derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q-derivative is also given.