A New Kind of Integration Transformation in Phase Space Related to Two Mutually Conjugate Entangled-State Representations and Its Uses in Weyl Ordering of Operators

Based on the two mutually conjugate entangled state representations |ξ> and |η>, we propose an integration transformation in ξ-η phase space , and its inverse transformation, which possesses some well-behaved transformation properties, such as being invertible and the Parseval theorem. This integral transformation is a convolution, where one of the factors is fixed as a special normalized exponential function. We generalize this transformation to a quantum mechanical case and apply it to studying the Weyl ordering of bipartite operators, regarding to (Q₁-Q₂)↔(P₁-P₂) ordered and simultaneously (P_1⁺P_2^)↔(Q₁+Q_2⁾ ordered operators.