The Quantum Microlocal Geometry of Distributional Singularities

Revista, Zen, MATH, 10

Published Dec 6, 2025• 0 citations

Abstract

This paper explores the intricate relationship between quantum mechanics and microlocal analysis, specifically focusing on the geometrical description of distributional singularities within a quantum framework. We delve into how microlocal techniques, traditionally used to localize singularities in phase space for classical partial differential equations, can be extended and reinterpreted in the context of quantum systems. The core objective is to develop a robust theoretical framework that allows for a precise characterization and propagation analysis of wavefront sets for quantum states and operators. We investigate the quantum analogues of classical wavefront sets, leveraging the theory of pseudodifferential operators, symbol calculus, and deformation quantization. This approach provides a deeper understanding of how the singular structure of quantum distributions—representing wavefunctions, density matrices, or correlation functions—evolves under quantum dynamics. Furthermore, we examine the implications for semiclassical analysis, where the transition from quantum to classical microlocal geometry is studied. The insights gained from this quantum microlocal geometry offer new perspectives on phenomena such as quantum chaos, the propagation of quantum coherence, and the interpretation of quantum measurements, providing a powerful mathematical lens for probing the fine structure of quantum phenomena.

DOI: 10.5281/zenodo.17838876

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