We investigate truncation schemes for the many-body dynamics of finite fermion systems beyond the time-dependent Hartree-Fock (TDHF) approximation. One approach starts from the quantum Bogolyubov-BornGreen-Kirkwood-Yvon (BBGKY-)hierarchy of equations of motion for density matrices. It is shown that simple truncation of higher correlations within this scheme is inconsistent because essential exchange correlations are lost. As an alternative, we study the (extended) exp(S) or coupled-cluster formalism. It provides a wider range of applicability. But it also runs into problems with non-unitary propagation after truncation.

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We consider the truncation of exact equations of motion in a finite fermion system. The truncated von Neumann hierarchy for the densities and correlations violates the probabilistic interpretation if one neglects three-particle and higher correlations. Even for a two-body system, truncation may induce a chaotic behavior on the Hartree-Fock level.In statistical mechanics the time evolution of an Nbody system is described by the Liouville equation for the X-body probability density f~, or in a completely equivalent manner, by a hierarchy of equations which couple the n body -distribution f"-f"+). In a practical calculation one closes the hierarchy of equations by trurication. The consequences of a truncation on the resulting n-body subdynamics (SUBn) have recently been investigated in the framework of an exactly soluble classical many-body system. 'In this Brief Report we want to investigate the effect of a similar truncation for quantum fermion systems. There the time evolution of an X-body system is described by the von Neumann equation for the N-body density matrix or by the equivalent hierachies of equations of motion for Green's functions, density matrices, or n-particle, nhole amplitudes. Most investigations of quantum many-body theory have been concerned with approximation schemes for stationary states, and one has developed expansion schemes in many fields of application. Much less is known about the time evolution of quantum many-body systems and the proper approximations to it, although there are many attempts to go beyond the lowest order, the time-dependent Hartree-Fock (TDHF) approach.In this Brief Report we want to comment on two problems of time-dependent approximation schemes. First, we examine whether truncation is consistent with a probabilistic interpretation of the theory, in particular, we have to look for possible complications in the quantum problem due to exchange correlations. Second, we note that the von Neumann equation and the corresponding exact hierarchies are deterministic, linear equations for the probability densities (matrices), whereas truncation will generally induce nonlinearities.We investigate in a simple two-fermion system the dynamical consequences of the truncation and find that it may lead to a chaotic time evolution on the SUB1 level whereas the full dynamics is regular.We start from the hierarchy of the equations of motion for the n-body density operators P"(Ref. 3) ip, =[t(1), P, ]+Tr(z)I [v(1,2), Pz]I, n -1 ip"= g [t(i ),p"]+ g [v(i,j), p"] +Tr("+, ) g [v(i, n +1), p"+,] which is obtained from the von Neum ann equation ipv =[H, P& ] by using the partial trace relation for the various densities. 1 P» ) (» + ), . . . , x) IPx Tr -n .(2)where the operator S ( A ) symmetrizes (antisymmetrizes) the following expressions with respect to their arguments. The factorized forms (3) are inserted into the hierarchy (1) and the hierarchy is then truncated by neglecting all c"above a given order n, , For example, in SUB1 we neglect all correlations c, & 2 which yields the TDHF equat...

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