A tower of hidden scales: non-Gaussian patterns in quantum mechanics
SPIE 2013, Paper 8832-42
Antonina Fedorova and Michael Zeitlin

We present an application of our sheaf-based categorification framework considered in the companion paper to analysis of quantum dynamics inside the Wigner-Weyl-Moyal set-up. The key ingredient of our approach is related to a construction of functorial correspondence between two categories: the first one is described by families of quantum states realized as sections of some category of sheaves, allowing to cover the standard set of quantum phenomena while the second one is a proper arena for the investigation of (non-trivial, non-gaussian) solutions of Wigner-like equations in (at least) the minimal possible pseudodifferential set-up according to the ideology of microlocal analysis. To do that we need to consider together with the usual Hilbert space some underlying topological Hilbertian space with some filtration into subspaces, generated by the action of internal hidden symmetry. As a result, we obtain the analytical description of the whole tower of underlying scales together with the possibility to describe the full quantum evolution, taking into account the abundant hidden quantum structure. It seems that the existence of such rich internal structure is an essential part of any quantum phenomenon in contrast with the classical one. So, we decompose quantum evolution into a full set of multiscales according to orbits constructed by the action of internal hidden symmetry on the properly filtrated underlying Hilbertian space of states. Numerical modeling demonstrates the appearance of a big zoo of nontrivial patterns which enlarge an orthodox set of standard states like coherent ones, gaussians, etc. and allow us to have an additional chance for a possible reinterpretation and description of standard things like entanglement, decoherence, (von Neumann) measurements together with an attempt to understand the main features, e.g., hidden content, of base Quantum Mechanics.