A general theorem is proved on the density matrix of quantum statistics. Let the de sity matrix be ρ {M} =exp [-βΣMl=1H
l], in which the operators Hl's are not always commutable. ρ {M} can be expanded in series of the form
ρ {M} =ΣMΣmi=0 n≥0 Σ {ml}
M--- {m
n} Mρ* {mM
1} ---ρ* {m
n} ρ {M-m
1----m
n}
where. the sets {m
1}, --- {m
n} are groups of operators chosen from the set {M} and operators belonging to different groups are commutable. The summation is to be taken over all the possible choice of the sets {m
1} --- {m
n}.
ρ {k} is defined by
ρ {k} = [Π {k} exp (-βH
j)] s where suffix s means the symmetrization by changing the order of the products. And ρ* {m} is defined by
ρ* {m} =Σ
{k}l (-)
m-k ρ {k} ρ {m-k}
which is proved to be 0 (β
m+1). the series expansion here proved is very useful in quantum statistics.
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