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.) An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of the Segal–Bargmann transform. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the Segal–Bargmann transform is a holomorphic function of . There is a quasiprobability density associated to the phase-space wave function; it is the Husimi Q representation of the position wave function. Star product The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the phase-space distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner–Weyl representation. For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and g, the left and right derivatives are defined as The differential definition