We sketch a group-theoretical framework, based on the Heisenberg–Weyl group, encompassing both quantum and classical statistical descriptions of unconstrained, non-relativistic mechanical systems. We redefine in group-theoretical terms a kinematical arena and a space of statistical states of a system, achieving a unified quantum-classical language and an elegant version of the quantum-to-classical transition. We briefly discuss the structure of observables and dynamics within our framework.
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Korbicz, J.K., Lewenstein, M. Remark on a Group-Theoretical Formalism for Quantum Mechanics and the Quantum-to-Classical Transition. Found Phys 37, 879–896 (2007). https://doi.org/10.1007/s10701-007-9130-z
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DOI: https://doi.org/10.1007/s10701-007-9130-z