Abstract
In the previous chapter we presented the general theory of quantum deformations of classical Poisson algebras. In the following chapter we develop a deformation procedure applied to classical statistical Hamiltonian mechanics (described in Sect. 3.3) in order to construct its quantum analogue on the phase space. First, we define quantum states as appropriate deformations of classical states and their time development through the respective deformation of the classical Liouville equation. Then we introduce quantum Hamiltonian equations of motion being a deformation of classical Hamiltonian equations and time development of quantum observables. With particular care we present the theory of quantum flow and quantum trajectories on a phase space together with a wide range of examples which illustrate the presented formalism. Such constructed quantum theories (each related with an appropriate quantum algebra) reduce to a common classical counterpart as deformation parameter \(\hslash \) tends to zero: \(\hslash \rightarrow 0.\)
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Błaszak, M. (2019). Quantum Hamiltonian Mechanics on Symplectic Manifolds. In: Quantum versus Classical Mechanics and Integrability Problems. Springer, Cham. https://doi.org/10.1007/978-3-030-18379-0_7
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