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 (P₁ + P₂) ↔ (Q₁ + Q₂) ordered operators.