Quantum mechanical particle's wavefunction are always described by position or momentum representation. Then, I found the so-called quantization on phase space. So, what does it really mean? Does it mean that we make momentum and position as basis in Hilbert space? Since position and momentum are non-commuting operator, how could we do that? Thank you
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1Please see this wiki article on Wigner quasi-distributions: http://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution – ZeroTheHero Mar 20 '17 at 08:10
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Possible duplicates: http://physics.stackexchange.com/q/31894/2451 and links therein. – Qmechanic Mar 20 '17 at 11:27
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Term quantization on phase space usually appears when one studies semiclassical behaviour of the quantum system. The most widely used methods for this kind of analysis are quasiprobability functions: Wigner distribution and Husimi function. In this case we still define our system on Hilbert space (so we have non-commuting momentum and position) and using some approximations (usually calculating expectation values) we get functions of position and momentum, which behave similarly to quantum probability distributions.
Quantization on phase space may also correspond to some integral quantization methods which use coherent states. Usually those methods map functions on phase space to operators on Hilbert space in the following way
$f(q,p) \rightarrow \hat{A_f}=\int d\mu(q,p) f(q,p) | \psi \rangle \langle \psi |$
Nice introduction to coherent state integral quantization is in paper by Klauder
