Wave operator representation of quantum and classical dynamics

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 semiclassical approximations of both real and imaginary time dynamics, as well as a transparent correspondence to the classical limit. It is then demonstrated that there exist a number of scenarios (such as thermalization) in which the wave operator representation possesses an equivalent unitary evolution, which corresponds to nonlinear real-time dynamics for the density matrix. 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.