Operator method for calculating $Q$ symbols and their relation to Weyl–Wigner symbols and symplectic tomogram symbols

We propose a new method for calculating Husimi symbols of operators. In contrast to the standard method, it does not require using the anti-normal-ordering procedure. According to this method, the coordinate and momentum operators $\hat q$ and $\hat p$ are assigned other operators $\widehat X$ and $\widehat P$ satisfying the same commutation relations. We then find the result of acting with the $\widehat X$ and $\widehat P$ operators and also polynomials in these operators on the Husimi function. After the obtained expression is integrated over the phase space coordinates, the integrand becomes a Husimi function times the symbol of the operator chosen to act on that function. We explicitly evaluate the Husimi symbols for operators that are powers of $\widehat X$ or $\widehat P$.