E.g., one may define a noncommutative torus as a deformation quantization through a ★-product to implicitly address all convergence subtleties (usually not addressed in formal deformation quantization).
There results a complete phase space formulation of quantum mechanics, completely equivalent to the Hilbert-space operator representation, with star-multiplications paralleling operator multiplications isomorphically.
[1] Expectation values in phase-space quantization are obtained isomorphically to tracing operator observables Φ with the density matrix in Hilbert space: they are obtained by phase-space integrals of observables such as the above f with the Wigner quasi-probability distribution effectively serving as a measure.
correspondence principle) of classical mechanics, with deformation parameter ħ/S.
Conversely, group contraction leads to the vanishing-parameter undeformed theories—classical limits.)
Classical expressions, observables, and operations (such as Poisson brackets) are modified by ħ-dependent quantum corrections, as the conventional commutative multiplication applying in classical mechanics is generalized to the noncommutative star-multiplication characterizing quantum mechanics and underlying its uncertainty principle.