Q-Adapted Quantum Stochastic Integrals and Differentials in Fock Scale

TL;DR

This paper develops a rigorous framework for quantum stochastic calculus using Fock-Guichardet formalism, introducing Q-adapted dynamics for different quantum systems, with implications for quantum field theory and open systems.

Contribution

It introduces a new formalism for quantum stochastic integrals and derivatives, including Q-adapted dynamics for various quantum statistics, expanding the mathematical tools for quantum stochastic processes.

Findings

Proved continuity of the QS derivative.

Developed Q-adapted quantum dynamics for Bosonic, Fermionic, and monotone cases.

Provided a rigorous analysis of quantum stochastic integrals.

Abstract

In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures of the QS integration over a space-time. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative is proved. Finally, Q-adapted dynamics is discussed, including Bosonic Q=1, Fermionic Q=-1, and monotone Q=0 quantum dynamics. These may be of particular interest to quantum field theory, quantum open systems, and quantum theory of stochastic processes.