Symmetry groups and geometrical phases in quantum mechanics

Maurice A. de Gosson

Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states.

Joy Christian

D. J. Simms

Abstract This is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

Bipin R. Desai

Florian A. Scheck

fredrick michael

We develop a unified framework in which quantum mechanics is assumed to emerge from combinatorial configuration spaces under evolution, curvature, and symmetry constraints. Starting from general discrete configurations such as permutations, tilings, and causal sets, we show that entropy and statistical ensembles arise naturally from counting arguments. Dynamical evolution leads to diffusion and irreversible behavior, while the presence of defects and holonomy introduces geometric phase structure. We demonstrate that invariance under compact symmetry groups, enforced via Haar measure, necessitates a transition from probabilities to amplitudes, yielding unitary and time-reversal symmetric dynamics. This provides a possible derivation of quantum mechanics from combinatorial and geometric principles, with applications to transport theory, nonequilibrium systems, and emergent spacetime.