An Expansion Theorem of the Density Matrix

A general theorem is proved on the density matrix of quantum statistics. Let the density matrix be $$\mathrm{\rho \{M\}=}\exp \mathrm{[-\beta }{\displaystyle \sum _{\text{l=1}}^M}{H}_l\mathrm{],}$$ where the operators H_l's are not always commutable. ρ{M} can be expanded in series of the form $${\mathrm{\rho \{M\}=(\Sigma }}_{\mathrm{M\geqq }{\displaystyle \sum _{\text{i=1}}^n}{m}_i\text{,\hspace{0.17em}n≧0}}{\Sigma }_{{\mathrm{\{m}}_1{\}}_M{\mathrm{···\{m}}_n{\}}_M}{\mathrm{)\rho }}^{*}{\mathrm{\{m}}_1{\mathrm{\}···\rho }}^{*}{\mathrm{\{m}}_n{\mathrm{\}\rho ̄\{M-m}}_1{\mathrm{-···-m}}_n\mathrm{\},}$$ where {m₁}_M, ··· {m_n}_M are subsets of operators chosen from the set {M} and operators belonging to different subsets are commutable. The summation is over all the possible choices of the sets {m₁}, ··· {m_n}. $\mathrm{\rho ̄\{k\}}$ is defined by $${\mathrm{\rho ̄\{k\}=[\Pi }}_{\mathrm{\{k\}}}\exp {\mathrm{(-\beta H}}_i{\mathrm{)]}}_s,$$ where suffix s means the symmetrization by changing the order of the products. And ρ^*{m} is defined by $${\rho }^{*}{\mathrm{\{m\}=(\Sigma }}_{{\mathrm{\{k\}}}_m}{\mathrm{)(-)}}^{\mathrm{m-k}}\mathrm{\rho \{k\}\rho ̄\{m-k\},}$$ which is proved to be O(β^m+1). Some possible applications are briefly discussed.