Newton-Leibniz integration for ket-bra operators (II)-application in deriving density operator and generalized partition function formula

Via the route of applying Newton-Leibniz integration rule to Dirac's symbolic operators, we show that the density operator e−βH, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring convenience in deriving partition functions.