Dirac multimode ket-bra operators’ $\mathfrak{Q}$-ordered and $\mathfrak{P}$-ordered integration theory and general squeezing operator

Abstract

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$-ordering or $\mathfrak{P}$-ordering to multimode case, where $\mathfrak{Q}$-ordering means all Qs are to the left of all Ps and $\mathfrak{P}$-ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$-ordered and $\mathfrak{P}$-ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k }$. Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle$ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.