Low- and high-temperature series expansions for quantum coupled oscillators systems or quantum fields on a lattice

In this paper, a new formulation which can be used to systematically extend and study the low- and high-temperature series for the thermodynamic functions of systems of coupled quantum oscillators corresponding to the quantum $\lambda {\phi }^{2m}$ fields, $m=2,3\mathrm{}$,..., on a lattice is proposed. In this formulation, the low- and high-temperature series for the partition function of the systems are given in the forms $Z=Z_h\mathrm{}\left(T\right)[1+A_1(\lambda T^{m-1\mathrm{}})+A_2{\left(\lambda T^{m-1\mathrm{}}\right)}^2+\cdots ]$ and $Z=Z_a\mathrm{}\left(T\right)[1+B_1{\left(\lambda T^{m-1\mathrm{}}\right)}^{\frac{-1\mathrm{}}{m}}+B_2{\left(\lambda T^{m-1\mathrm{}}\right)}^{\frac{-2\mathrm{}}{m}}+\cdots ] ,$ respectively, where $A_n$ and $B_n$ are certain combinatorial numbers which can be expressed in terms of the numbers of ways of placing certain diagrams on the $d$-dimensional $d=1,2,3\mathrm{}$,... discrete lattice. Note that the forms of these series are new and they are different from those of the more extensively studied classical models. A systematic study of these series will, as in the corresponding cases of the Ising and Heisenberg models, reveal the critical behaviors of this class of quantum-field models which is not only interesting in its own right, but is also important in our overall understanding of the critical behaviors of many other related models from the point of view of the renormalization-group approach.