Wigner-Kirkwood expansions

This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function $F$. For an $N$-body quantum system with Hamiltonian $H$ and inverse temperature $\beta$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\overrightarrow{\mathrm{x}}\left|e^{-\beta H}\right|{\overrightarrow{\mathrm{x}}}^{\prime }〉}{〈\overrightarrow{\mathrm{x}}\left|e^{-\beta H_0}\right|{\overrightarrow{\mathrm{x}}}^{\prime }〉}$. Expansion of $\ln F$ in Planck's constant $h$ leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\beta$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.