Consistent Linearization for Quasi-probabilities

It is shown that naive linearization leads to different answers when applied to different quasi-probability distributions such as P W or Q. A consistent linearization scheme is developed and applied to the problem of sub-second-harmonic generation. In particular, it is shown that, in the Wigner representation, linearized equations contain no third-order derivatives. In the nonlinear regime, however, third-order derivatives can only be neglected if the nonlinearity is small.