Expansion Theorems on the Density Matrix in Quantum Statistics.

C. Di Castro, Roberto Raimondi

Ryogo Kubo

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.

Ibrahim Adawi

A method for expanding the single-electron density matrix is sketched by using an expansion theorem due to Schafroth for the matrix element of a general function of an operator.

Kuen-Der Lain

Joseph E. Willett

Mathias Wagner

We present a Green's-function formalism generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix. On the basis of a simultaneous expansion of the usual real-time development operator and the density matrix, we define a Green's function ordered along an extended contour. In this way we are able to treat real-time many-particle Green's functions and initial correlations on the same footing. Both can be decomposed by Wick's theorem. Thus we may utilize familiar diagrammatic analysis for evaluating the Green's function, and finally arrive at a generalized Dyson's equation. The Green's function can be represented by a (3\ifmmode\times\else\texttimes\fi{}3) matrix, which includes the Green's functions used in the Feynman, Matsubara, and Keldysh theories, if the corresponding statistical average is taken. From a matrix representation of Dyson's equation, a basic set of five coupled equations is derived, which explicitly shows the corrections to the Keldysh theory due to an arbitrary initial many-particle density matrix.