General Formulation of Quantum Analysis

Abstract

A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the G\^ateaux derivative or commutators. This yields a unified formulation of quantum analysis, namely the invariance of quantum derivatives, which are expressed by multiple integrals of ordinary higher derivatives with hyperoperator variables. Multivariate quantum analysis is also formulated in the present unified scheme by introducing a partial inner derivation and a rearrangement formula. Operator Taylor expansion formulas are also given by introducing the two hyperoperators $ \delta_{A \to B} \equiv -\delta_A^{-1} \delta_B$ and $d_{A \to B} \equiv \delta_{(-\delta_A^{-1}B) ; A}$ with the inner derivation $\delta_A : Q \mapsto [A,Q] \equiv AQ-QA$. Physically the present noncommutative derivatives express quantum fluctuations and responses.