Star-Product Formulation of Quantum Mechanics

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Star-product, \((f*g)(q,p)\) of two functions f(q, p) and g(q, p) on the phase space is a non-commutative product corresponding to the Hilbert space product \(\hat{f}\hat{g}\) of operators \(\hat{f}\) and \(\hat{g}\). The quantum theory can be developed in analogy with the classical mechanics on the phase space, not with a Poisson bracket, but with a Moyal bracket \((f*g-g*f)/i\hbar \). This formulation is also called the deformation theory of quantization.