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