New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula (Hong-YiFan, emph{ J. Phys.} A {bf 25} (1992) 3443) this paper finds akind of two-fold integration transformation about the Wigneroperator $varDelta left( q',p'right) $($mathrm{q}$-number transform) in phase space quantum mechanics,$iint_{-infty}^{infty}frac{{rm d}p'{rm d}q'}{pi}varDelta left( q',p'right) e^{-2ileft(p-p'right) left( q-q'right) }=delta left(p-Pright) delta left( q-Qright),$and its inverse%$iint_{-infty}^{infty}{rm d}q{rm d}pdelta left( p-Pright)delta left( q-Qright) e^{2ileft( p-p'right) left(q-q'right) }=varDelta left(q',p'right),$ where $Q,$ $P$ are the coordinateand momentum operators, respectively. We apply it to study mutualconverting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weylordering of operators. In this way, the contents of phase spacequantum mechanics can be enriched. The formula of the Weylordering of operators expansion and the technique of integration within the Weylordered product of operators are used in this discussion.