Quantum Dynamics, Minkowski-Hilbert space, and A Quantum Stochastic Duhamel Principle

TL;DR

This paper explores a Minkowski-Hilbert space framework for quantum dynamics, unifying deterministic and stochastic processes, and introduces a quantum stochastic Duhamel principle with applications to measurement and geometry in quantum mechanics.

Contribution

It presents a novel Minkowski-Hilbert space formalism for quantum dynamics, connecting it with a quantum stochastic Duhamel principle and geometric insights into measurement.

Findings

Reformulation of Schrödinger and Lindblad equations in Minkowski space

Introduction of a quantum stochastic Duhamel principle

Discussion of Lorentz transformations and Riemannian geometry in quantum measurement

Abstract

In this paper we shall re-visit the well-known Schr\"odinger and Lindblad dynamics of quantum mechanics. However, these equations may be realized as the consequence of a more general, underlying dynamical process. In both cases we shall see that the evolution of a quantum state $P_\psi=\varrho(0)$ has the not so well-known pseudo-quadratic form $\partial_t\varrho(t)=\mathbf{V}^\star\varrho(t)\mathbf{V}$ where $\mathbf{V}$ is a vector operator in a complex Minkowski space and the pseudo-adjoint $\mathbf{V}^\star$ is induced by the Minkowski metric $\boldsymbol{\eta}$. The interesting thing about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the \emph{Belavkin Formalism}; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be `unraveled' in a second-quantized Minkowski space. Working in such a space provided the author with the means to construct a QS (quantum stochastic) Duhamel principle and known applications to a Schr\"odinger dynamics perturbed by a continual measurement process are considered. What is not known, but presented here, is the role of the Lorentz transform in quantum measurement, and the appearance of Riemannian geometry in quantum measurement is also discussed.