Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators (which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s-ordered operator expansion (denoted by \circledS \cdots \circledS) formula of density operators is derived, which is \rho=\frac21-s\int \frac\rm d^2\beta\pi\left \langle -\beta \right \vert \rho \left \vert \beta \right \rangle \circledS \exp \Big\ \frac2s-1\left( s'\beta'^2-\beta^\asta+\beta a^\dagger-a^\daggera\right) \Big\ \circledS. The s-parameterized quantization scheme is thus completely established.