and may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in Fourier space, moments can be calculated from these functions in the following way: Here denotes symmetric ordering. These representations are all interrelated through convolution by Gaussian functions, Weierstrass transforms, or, using the property that convolution is associative, It follows that an often divergent integral, indicating P is often a distribution. Q is always broader than P for the same density matrix. For example, for a thermal state, one has Time evolution and operator correspondences Since each of the above transformations from to the distribution functions is linear, the equation of motion for each distribution can be obtained by performing the same transformations to . Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions. For instance, consider the annihilation operator acting on . For the characteristic function of the P distribution we have Taking the Fourier transform with respect to to find the action corresponding action on the Glauber P function, we find By following this procedure for each of the above distributions, the following operator correspondences can be identified: Here or 1 for P, Wigner, and Q distributions, respectively. In this way, master equations can be expressed as an equations of motion of quasiprobability functions. Examples Coherent state By construction, P for a coherent state is simply a delta function: The Wigner and Q representations follows immediately from the Gaussian convolution formulas above, The Husimi representation can also be found using the formula above for the inner product of two coherent states, Fock state The P representation of a Fock state is Since for n>0 this is more singular than a delta function, a Fock state has no classical analog. The non-classicality is less transparent as one proceeds with the Gaussian convolutions. If Ln is the nth Laguerre polynomial, W is which can go negative but is bounded. Q, by contrast, always remains positive and bounded, Damped quantum harmonic oscillator Consider the damped quantum harmonic oscillator with the following master equation, This results in the Fokker–Planck equation, where κ = 0,