Hartree-Fock one-body dynamics and U(n) co-adjoint orbits

Abstract

Hartree-Fock theory is basic to our microscopic understanding of many-fermion systems [1,2]. Its fundamental approximation restr icts the states of A-particle systems to be Slater determinants, i .e. simple vectors @ = @l ^ @2 . . . . . @A in the exterior product of A-copies of the single-part ic le space. The ground state is approximated by the Slater determinant which minimizes the expectation of the energy. Although the atomic many-electron ground state may be approximated for most purposes quite adequately by a Slater determinant, the same cannot be said for the atomic nucleus. Thus, the restr ict ion to determinantal wavefunctions must be el iminated i f a satisfactory microscopic theory of nuclei is to be achieved. However, i f the simplifying restr ict ion to determinants is dropped, then we are faced with an Intractable many-body problem. Fortunately, thls pessimistic conclusion is unwarranted. To understand how the restr ict ion to determinants can be eliminated ent i re ly, and yet a tractable generalization of HF achieved, I t is necessary to appreciate the c r i t i ca l role played by the unitary group U(n) In HF. The group U(n) of unitary transformations in the n-dimensional s ingle-part ic le space Is known to be important for the many-body problem in general and HF in par t i cular [3,4]. For example, Matsen and NelIn have observed that a Slater determinant is a highest weight vector of U(n) [5,6]'. Here, however, we need to exploit two features of U(n). First note that the group U(n) acts t ransi t ive ly on the Slater determinants, i .e. i f @ is any fixed normalized determinant, then every other normalized determinant is of the form g@ = g@l ^ 9@2 . . . . . g@A Hence, the group U(n) characterizes the states available in the HF approximation, viz. the HF states constitute a single orbi t of U(n) in the exterior product space. Secondly, recall that the Lie algebra u(n) consists of the hermitian one-body