Some Theorems on Uniqueness and Reconstruction of Higher-Order Density Matrices

Abstract

The 2-matrix corresponding to an N-particle wave function Ψ is defined by the relation

$$ ^{2}{\Gamma _{{ij;kl}}} \equiv \int {\varphi _{i}^{*}\left( {X_{1}^{'}} \right)\varphi _{j}^{*}\left( {X_{1}^{'}} \right)} {\Gamma _{2}}\left( {X_{1}^{'},X_{2}^{'};{X_{1}},{X_{2}}} \right){\varphi _{k}}\left( {{X_{1}}} \right){\varphi _{1}}\left( {{X_{2}}} \right)dx_{1}^{'}dx_{2}^{'}d{x_{1}}d{x_{2}} = \left\langle {\Psi \left| {\hat{a}_{i}^{ + }\hat{a}_{j}^{ + }{{\hat{a}}_{l}}{{\hat{a}}_{k}}} \right|\Psi } \right\rangle . $$

((1.1))