Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.
Corresponding to optical Fresnel diffraction, we show that the exponential quadratic operator exp is actually a generalized single-mode Fresnel operator （GFO） in compact form, where [Q,P]=ih. We also demonstrate that exp{iα[（Q1＋Q2）2＋（p1-P2）2]＋iβ[（Q1-Q2）2＋（p1＋p2）2]＋iy（Q1P2＋Q2P1）} is a two-mode GFO. Their disentangling formula and normal ordering form are derived with the use of technique of integration within an ordered product （IWOP） of operators and the coherent state representation. The squeezed states generated by these two GFOs are obtained.
We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product （IWOP） of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some twobody dynamic problems. Moreover, the pure-state density matrix ｜ξ1,ξ2｜ C,D C,D（ξ1, ξ2｜ is a Radon transform of Wigner operator.
A new kind of four-mode continuous variable coherent-entangled state is proposed in the Fock space by using the technique of integration within an ordered product, which exhibits both the properties of a coherent state and an entangled state, and spans a complete and orthonormal representation. The conjugate state of the four-mode continuous variable coherent-entangled state is derived by using the Fourier transformation. Moreover, a simple experimental protocol of generating a four-mode continuous variable coherent-entangled state is proposed by using beam splitters. As applications of this four-mode continuous variable coherent-entangled state, a four-mode entangled state and a four-mode squeezing-Fresnel operator are constructed.
We investigate how an optical squeezed chaotic field（SCF） evolves in an amplitude dissipation channel. We have used the integration within ordered product of operators technique to derive its evolution law. We also show that the density operator of SCF can be viewed as a generating field of the squeezed number state.
A new entangled state ｜η 0） is proposed by the technique of integral within an ordered product. A generalized Hadamaxd transformation is derived by virtue of η; θ）, which plays a role of Hadamard transformation for （a1 sinθ - a2 cosθ） and （a1 cosθ ＋ a2 sin θ）.
Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the mnltimode coordinate-momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the rnultimode phase shifting operator.
The coherent-entangled state |α, x; λ> with real parameters λ is proposed in the two-mode Fock space, which exhibits the properties of both the coherent and entangled states. The completeness relation of |α, x; λ> is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operator is derived, with its own squeezing properties. Furthermore, generalized P-representation in the coherent-entangled state is constructed. Finally, it is revealed that superposition of the coherent-entangled states may produce the EPR entangled state.
