Wigner functions and quasiprobability

The density operator has a two-position kernel. At fixed midpoint x, Fourier transforming its dependence on the separation y introduces momentum. Declare both the half-separation convention and the normalization: changing either without adjusting the prefactor causes factor-of-two mistakes. For a pure state, the kernel is a product of the wavefunction at one endpoint and its complex conjugate at the other. A mixed state's kernel enters the same linear transformation.

Hermiticity makes the Wigner function real: conjugation exchanges the two endpoints, and reversing the separation variable restores the original integral. Integrating over momentum collapses the separation to zero through the Fourier delta identity, recovering the position density. Integrating over position recovers the momentum density in the matching Fourier convention. Normalization over both axes follows, and its units are inverse action rather than inverse position alone.

The position and momentum marginals are nonnegative because they are Born densities. The Wigner function itself can have negative regions while both marginals remain valid. Integrating an arbitrary tiny phase-space cell is therefore not generally the probability that an ideal joint sharp detector finds x and p there. A physically implemented approximate joint measurement uses its own positive effects and resolution, often smoothing or otherwise modifying the phase-space description.

Expectation values can be computed with phase-space symbols when the operator-to-symbol convention is handled consistently. Symmetric operator ordering is built into the Weyl–Wigner correspondence. One cannot replace every operator expression by an ordinary classical polynomial while ignoring ordering; position and momentum do not commute. For example, the symmetric combination of XP and PX has the ordinary xp symbol, whereas the ordered product XP has symbol $xp+i\hbar/2$ under this convention.

A normalized unchirped Gaussian has a positive Gaussian Wigner function, with reciprocal widths in x and p. Increasing its position scale squeezes the momentum direction without changing total phase-space normalization. Positive Wigner values alone do not erase uncertainty or make all quantum experiments classically equivalent. Squeezed Gaussian states can be positive in this representation while retaining quantum properties relative to an optical reference or an allowed operation set.

The first excited oscillator state instead has a negative Wigner value at the origin. A classical mixture transforms linearly into the same weighted mixture of Wigner functions, while a coherent superposition introduces cross terms before transformation. Negativity is a useful nonclassicality witness in this chosen continuous-variable representation, but it is not a universal measure of every quantum resource. Distinguish this established representation from stronger claims about computational power, Bell nonlocality, or literal negative event frequencies.