The wave operator representation of quantum and classical dynamics

Abstract

The choice of mathematical representation when describing physical systems is of great consequence, and this choice is usually determined by the properties of the problem at hand. Here we examine the little-known wave operator representation of quantum dynamics, and explore its connection to standard methods of quantum dynamics, such as the Wigner phase space function. This method takes as its central object the square root of the density matrix, and consequently enjoys several unusual advantages over standard representations. By combining this with purification techniques imported from quantum information, we are able to obtain a number of results. Not only is this formalism able to provide a natural bridge between phase and Hilbert space representations of both quantum and classical dynamics, we also find the wave operator representation leads to novel semiclassical approximations of both real and imaginary time dynamics, as well as a transparent correspondence to the classical limit. This is demonstrated via the example of quadratic and quartic Hamiltonians, while the potential extensions of the wave operator and its application to quantum-classical hybrids is discussed. We argue that the wave operator provides a new perspective that links previously unrelated representations, and is a natural candidate model for scenarios (such as hybrids) in which positivity cannot be otherwise guaranteed.

Projective representation of the Galilei group for classical and quantum–classical systems*

“…Not only is the KvN Theory interesting in itself in a purely classical context, but it is also connected to the quantum-classical correspondence. The KvN theory is related to geometric quantization [13] and the classical limit of the Wigner function [17,18].…”

Evading Quantum Mechanics á la Sudarshan: Quantum-Mechanics-Free Subsystem as a Realization of Koopman-Von Neumann Mechanics

“…The renewed interest in KvN mechanics was caused by the need to build a suitable framework for hybrid classical-quantum systems (see, for example, [12,13] and references therein). I hope that the realization of the fact that quantum-mechanics-free subsystems are described by KvN mechanics will boost this interest in KvN mechanics.…”