Integral–geometrical consideration of density matrices

The ensemble N ‐representability problem for the k ‐th order reduced density matrix ( k ‐ RDM ) as well as the problem of reconstruction of the N ‐particle system density matrices ( N ‐ DM ) from a given k ‐ RDM are studied. The spatial parts of the k ‐ RDM expansion in terms of spin tensorial operators Θ   λ kare represented using particular values (at specially chosen ) of the Radon transform of the N ‐ DM spatial parts (or their sums) Nλ ( x ′ | x ″) (here, is a d ‐plane in the n ‐space ℝ n of x = ( x ′, x ″)), with n = 6 N , d = 3 ( N − k ), x ′ ≡ (r′ 1 , ⃛, r′ N ), x′ ≡ ( r 1 ″, ⃛, r ″ N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM , it is proved that are bounded functions. The properties of implied by the N ‐ DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N ‐ DM corresponding to a given k ‐ RDM is proposed. © 1995 John Wiley & Sons, Inc.