An Expansion Theorem of the Density Matrix

A general theorem is proved on the density matrix of quantum statistics. Let the density matrix be ρ{M}=exp[−β ∑ l=1MHl],where the operators Hl's are not always commutable. ρ{M} can be expanded in series of the form ρ{M}=(ΣM≧ ∑ i=1nmi, n≧0Σ{m1}M···{mn}M)ρ*{m1}···ρ*{mn}ρ̄{M−m1−···−mn},where {m1}M, ··· {mn}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}, ··· {mn}. ρ̄{k} is defined by ρ̄{k}=[Π{k}exp(−βHi)]s,where suffix s means the symmetrization by changing the order of the products. And ρ*{m} is defined by ρ*{m}=(Σ{k}m)(−)m−kρ{k}ρ̄{m−k},which is proved to be O(βm+1). Some possible applications are briefly discussed.