Holomorphic Functional Calculus approach to the Characteristic Function of Quantum Observables

TL;DR

This paper introduces a novel method using holomorphic functional calculus and complex analysis to compute the characteristic function of quantum observables, providing a new mathematical framework for quantum probability analysis.

Contribution

It applies Dunford's holomorphic functional calculus to quantum operators to derive their characteristic functions, expanding the mathematical tools available for quantum probability theory.

Findings

Derived explicit formulas for quantum characteristic functions

Applied the method to position and momentum operators

Demonstrated the approach's effectiveness in quantum analysis

Abstract

We show how Cauchy's Integral Formula and the ideas of Dunford's Holomorphic Functional Calculus (for unbounded operators) can be used to compute the Vacuum Characteristic Function (Quantum Fourier Transform) of quantum random variables defined as self-adjoint operators on $L^2(\mathbb{R},\mathbb{C})$. We consider in detail several quantum observables defined in terms of the position and momentum operators $X$, $P$, respectively, on $L^2(\mathbb{R},\mathbb{C})$.