Generalized Hartree Method: A Novel Non-perturbative Scheme for Interacting Quantum systems

Abstract

A self-consistent, non-perturbative scheme of approximation is proposed for arbitrary interacting quantum systems by generalization of the Hartree method.The scheme consists in approximating the original interaction term $\lambda H_I$ by a suitable 'potential' $\lambda V(\phi)$ which satisfies the following two requirements: (i) the 'Hartree Hamiltonian' $H_o$ generated by $V(\phi)$ is exactly solvable i.e, the eigen states $|n>$ and the eigenvalues $E_n$ are known and (ii) the 'quantum averages' of the two are equal, i.e. $< n|H_I|n>$ = $<n|V(\phi)|n>$ for arbitrary $'n '$. The leading-order results for $|n>$ and $E_n$, which are already accurate, can be systematically improved further by the development of a 'Hartree-improved perturbation theory' (HIPT) with $H_o$ as the unperturbed part and the modified interaction:$\lambda H^{\prime} \equiv \lambda (H_I-V)$ as the perturbation. The HIPT is assured of rapid convergence because of the 'Hartree condtion' : $<n| H'| n> = 0$. This is in contrast to the naive perturbation theory developed with the original interaction term $\lambda H_I$ chosen as the perturbation, which diverges even for infinitesimal $\lambda$ ! The structure of the Hartree vacuum is shown to be highly non-trivial. Application of the method to the anharmonic-and double-well quartic-oscillators, anharmonic- sextic and octic- oscillators leads to very accurate results for the energy levels. In case of $\lambda \phi^{4}$ quantum field theory, the method reproduces, in the leading order, the results of Gaussian approximation, which can be improved further by the HIPT. We study the vacuum structure, renormalisation and stability of the theory in GHA.