Choices of Quantum Representations of Classical Dynamical Systems

Gerard McCaul, Dmitry V. Zhdanov, Denys I. Bondar

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.

Jan Govaerts A

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

David S. Park

David S. Park

Alessandro Sergi, Francesco Petruccione

The principle of energy conservation leads to a generalized choice of transition probability in a piecewise adiabatic representation of quantum(-classical) dynamics. Significant improvement (almost an order of magnitude, depending on the parameters of the calculation) over previous schemes is achieved. Perspectives for theoretical calculations in coherent many-body systems are opened.