The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.
The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner.
The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit). Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.
The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as Kontsevich's deformation-quantization (see Kontsevich quantization formula) and noncommutative geometry.
Phase-space distribution
The phase-space distribution of a quantum state is a quasiprobability distribution. In the phase-space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.
There are several different ways to represent the distribution, all interrelated. The most noteworthy is the Wigner representation, , discovered first. Other representations (in approximately descending order of prevalence in the literature) include the Glauber–Sudarshan P, Husimi Q, Kirkwood–Rihaczek, Mehta, Rivier, and Born–Jordan representations. These alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified.
The phase-space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta and over the position interval:
If is an operator representing an observable, it may be mapped to phase space as through the Wigner transform. Conversely, this operator may be recovered by the Weyl transform.
The expectation value of the observable with respect to the phase-space distribution is
A point of caution, however: despite the similarity in appearance, is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory. Moreover, it can, in general, take negative values even for pure states, with the unique exception of (optionally squeezed) coherent states, in violation of the first axiom.
Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than , and thus renders such "negative probabilities" less paradoxical. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of quantum optics this equation is known as the optical equivalence theorem. (For details on the properties and interpretation of the Wigner function, see its main article.)