The TDHF Approximation for Hamiltonians with $m$-particle Interaction Potentials

According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in N -particle systems whose dynamics are prescribed by a non-relativistic Schrodinger equation with 2-particle interactions, in the limit N tends to infinity while the strength of the 2-particle interaction potential is scaled by 1/N . In previous work we have considered the same mean field scaling for systems of fermions, and established that the error of the time-dependent Hartree-Fock (TDHF) approximation tends to 0 as N tends to infinity. In this article we extend our results to systems of fermions with m-particle interactions (m > 2). 1 The TDHF equation as a mean field approximation The time-dependent Hartree Fock (TDHF) equation [1] is an attempt to approximate the state of a system of interacting fermions by one time-dependent Slater determinant (thus discarding any “correlation” in the many electron system, cf. [4]). In our papers [2, 3] we have derived the TDHF dynamics as that of a single fermion in the mean field, in the spirit of Spohn’s derivation of the time-dependent Hartree equation [5] and refinements thereof [6, 7, 8] (see [9] for a good overview). Here we show how the theorem of [2] for 2-particle interactions may be generalized to cases where the N -particle Hamiltonian involves m-particle interactions with m > 2. Let H be a Hilbert space and let Hn denote the n tensor power of H, i.e., HN = n times { }} { H⊗ H⊗ · · · ⊗ H . For π in the group Sn of permutations of {1, 2, . . . , n}, define the unitary “permutation” operator Uπ by Uπ(x1 ⊗ ...⊗ xn) = xπ−1(1) ⊗ ...⊗ xπ−1(n) for all x1, . . . , xn ∈ H. Define An = ∑ π∈Sn sgn(π)Uπ (1) for all n ∈ N. Then 1 n!An is an orthogonal projector whose range is the space of antisymmetric vectors in Hn. Consider N identical fermions whose 1-particle Hilbert space is H. The appropriate N -fermion Hilbert space is the space of antisymmetric wavefunctions in HN , i.e., the range of the orthogonal projector 1 N !AN . If {ej}j∈J is an orthonormal basis of H then the set { 1 √ N ! AN (ej1 ⊗ ej2 ⊗ · · · ⊗ ejN ) : {j1, j2, . . . , jN} ⊂ J } ∗Univ Paris 7 and Laboratoire J.L. Lions (Univ. Paris 6), France (bardos@math.jussieu.fr). †CEA/DAM Ile de France, DPTA/Service de Physique Nucleaire, BP 12, F-91680 Bruyeres-le-Châtel, France (bernard.ducomet@cea.fr). ‡Laboratoire J.L. Lions (Univ. Paris 6), France (golse@math.jussieu.fr). §Wolfgang Pauli Inst., Nordbergstr. 15, A–1090 Wien, Austria (alex@alexgottlieb.com). ¶WPI c/o Fak. f. Math., Univ. Wien, Nordbergstr. 15, A–1090 Wien, Austria (mauser@courant.nyu.edu).