Phase-space wavefunctions
Phase-space representation of quantum state vectors is a formulation of quantum mechanics elaborating the phase-space formulation with a Hilbert space. It ""is obtained within the framework of the relative-state formulation. For this purpose, the Hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system. Relative-position state and relative-momentum state are defined in the extended Hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators, acting on these states, are obtained."" Thus, it is possible to assign a meaning to  the wave function in phase space, formula_1, as a quasiamplitude, associated to a quasiprobability distribution. The first wave-function approach of quantum mechanics in phase space was introduced by Torres-Vega and Frederick in 1990 (also see). It is based on a generalised Husimi distribution. In 2004 Oliveira et al. developed a new wave-function formalism in phase space where the wave-function is associated to the Wigner quasiprobability distribution by means of the Moyal product. An advantage might be that off-diagonal Wigner functions used in superpositions are treated in an intuitive way, formula_2, also gauge theories are treated  in a operator form. Phase space operators. Instead of thinking in terms multiplication of function using the star product, we can shift to think in terms of operators acting in functions in phase space. Where for the Torres-Vega and Frederick approach the phase space operators are with and And Oliveira's approach the phase space operators are with In the general case and with formula_11, where formula_12, formula_13, formula_14 and formula_15 are constants. These operators satisfy the uncertainty principle: Symplectic Hilbert space. To associate the Hilbert space, formula_17, with the phase space formula_18, we will consider the set of  complex functions of integrable square, formula_19 in formula_18, such that Then we can write formula_22, with where formula_24 is the dual vector of formula_25. This symplectic Hilbert space is denoted by formula_26. An association with the Schrödinger wavefunction can be made by formula_27, letting formula_28, we have formula_29. Then formula_30. Torres-Vega–Frederick representation. With the operators of position and momentum a Schrödinger picture is developed in phase space The Torres-Vega–Frederick distribution is Oliveira representation. Thus, it is now, with aid of the star product possible to construct a Schrödinger picture in phase space for formula_19 deriving both  side by formula_35, we have therefore, the above equation has the same role of Schrödinger equation in usual quantum mechanics. To show that formula_37, we take the 'Schrödinger equation' in phase space and 'star-multiply' by the right for formula_38 where formula_40 is the classical Hamiltonian of the system. And taking the complex conjugate subtracting both equations we get which is the time evolution of Wigner function, for this reason formula_43 is sometimes called quasiamplitude of probability. The formula_44-genvalue is given by the time independent equation Star-multiplying for formula_38 on the right, we obtain Therefore, the static Wigner  distribution function is a formula_44-genfunction of the formula_44-genvalue equation, a result well known in the usual phase-space formulation of quantum mechanics. In the case where formula_50, worked in the beginning of the section, the Oliveira approach and phase-space formulation are indistinguishable, at least for pure states. Equivalence of representations. As it was states before, the first wave-function formulation of quantum mechanics was developed by Torres-Vega and Frederick, its phase-space operators are given by and This operators are obtained transforming the operators formula_53 and formula_54 (developed in the same article) as and where formula_57 . This representation is some times associated with the Husimi distribution and it was shown to coincides with the totality of coherent-state representations for the Heisenberg–Weyl group. The Wigner quasiamplitude, formula_43, and Torres-Vega–Frederick wave-function, formula_59, are related by where formula_61 and formula_62.