To a density matrix p of a Fermi or Bose system one can associate a one-particle density matrix Pl " = β’m, m' Ira') (ml Tr pa*am, and a distribution p(q, k): = T r pa~q, k)a(q, k) in the one-particle phase space. Here Im) E L 2 ( R ) form an orthonormal basis and Iq, k) %g(x -q)e ikx for some g E L2(R a) are a complete set,
The a/= f d3xa(x)f*(x) and a~ are the destruction and creation operators corresponding t o f
Then the entropy for the whole system is bounded by the following one particle expressions: THEOREM S(p): = -T r p l n p ~<-tr[pllnpl -+ (1 T-p, )ln(1 T-pl)] =: S l ( P l ) ~< _ [ daqdak (21r)3 [p(q, k)lnp(q, k) +-(1 ~ p(q, kj)ln(1 T-p(q, k))]. J Here tr is the trace in L2(R 3) and the two signs refer to Fermi (resp. Bose) systems. Proof First part: Use the basis Im) which diagonalizes 01: Trpa*am, = nrn6mm ' 9 The density matrix with Tra*am ~ = n m and maximal entropy is = exp(-Z 7ma*am )/Tr exp(-Z 7ma*am) m m with n m = [e ;'m _+ l ] -1. It has the entropy S(p) = -~ (nmlnn m +(1 T-nm)ln(1 Ynm) ) m = --tr(pl lnpl + (1 -7-Pl )ln(t u Pl )) =: Sl (Pl). * Work supported in part by Fonds zur Forderung der wissenschaftlichen Forschung in Osterreich, Project No. 3569.