Dark side of quantum mechanics: categorification, reinterpretation, and all that in the sheaf framework
SPIE 2013, Paper 8832-36
Antonina Fedorova and Michael Zeitlin

We present an attempt of reinterpretation of the standard folklore of the elementary Quantum Mechanics, considered as some ground zero approximation for Local Quantum Field Theory. We get rid of the structureless geometrical point as a proper model for the real quantum/physical particle like electron or photon. To do that we need to substitute standard (wave) functions describing states in orthodox formulations by more complicated objects to have a chance to encode information about the possible internal structure of quantum objects in our formal mathematical description. Here we continue our program of reconsideration of Quantum Mechanics via some categorification in the powerful sheaf framework. It allows us to introduce hidden structures, like hidden degrees of freedom and hidden symmetries, into the game and, as a result, to have a chance for a more careful description and interpretation of the well-known set of standard phenomena, including, e.g., self-interference/interaction, entanglement, decoherence. We start with the description of our base categories also proper for more analytical/computation tools described in companion paper. After that, we introduce various families of sheaves which allow us to describe our version of the Quantum World, where the category of coherent sheaves play the crucial role in the qudit case while quasi-coherent sheaves are proper for the continuous variable case. We consider bridges to the more tradition quantization procedures like geometrical and deformation ones. In companion paper, we apply that machinery to an analysis of the hidden multiscale structure of quantum Wigner-Weyl-Moyal dynamics beyond of the gaussian approximation.