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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.

The generalized Wigner operator and Wigner distribution function in phase space corresponding to the field-intensity-dependent operators is constructed in the nonlinear coherent state representation. The generalized Wigner theorem is also derived. The whole discussion is based on the technique of integration within an ordered product of the field-intensity-dependent operators.

We introduce a generalized entangled state |η,λ1,λ2>, which spans a complete and orthonormal representation. Using the technique of integration within an ordered product of operators we prove that the projection operator |η,λ1,λ2>1,λ2| is just the Radon transformation of the entangled Wigner operator. The inverse Radon transformation is also derived and the tomography theory for two-mode correlated system is established.

We derive the entangled eigenstate ‖ξ> of complex scalar fields ϕ and ϕ† in the Fock space. The ‖ξ> state is found to embed the entanglement possessed by the Einstein–Podolsky–Rosen states in quantum mechanics. The ‖ξ> set spans a complete and orthonormal representation. The advantage of the new

By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.

We show that the operator $U=\mathrm{exp}[\ensuremath{-}{ir(Q}_{1}{P}_{2}{+Q}_{2}{P}_{3}{+Q}_{3}{P}_{1})]$ is a three-mode squeezing operator for the three-mode quadratures exhibiting the standard squeezing, the corresponding squeezed vacuum state in three-mode Fock space is derived by virtue of the technique of integration within ordered product of operators. The entanglement involved in such a state is explained. The optical network for producing the three-mode squeezed state is proposed.