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 operatorinto phase-space quantum mechanics we invent the technique ofintegration within $s$-ordered product of operators (which considersnormally ordered, antinormally ordered and Weyl ordered product ofoperators as its special cases). The $s$-ordered operator expansion(denoted by $circledS cdots circledS)$ formula of densityoperators is derived, which is$$rho=frac{2}{1-s}intfrac{d^2beta}{pi}left langle -beta right vert rho leftvert beta right rangle circledS exp Big{ frac{2}{s-1}left(s|beta|^{2}-beta^{ast}a+beta a^{dagger}-a^{dagger}aright) Big}circledS.$$The $s$-parameterized quantization scheme is thus completelyestablished.