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

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 λHI by a suitable ’potential’ λV (φ) which satisfies the following two requirements: (i) the ’Hartree Hamiltonian’ Ho generated by V (φ) is exactly solvable i.e, the eigen states |n > and the eigenvalues En are known and (ii) the ’quantum averages’ of the two are equal, i.e. < n|HI |n > = < n|V (φ)|n > for arbitrary n . The leading-order results for |n > and En , which are already accurate, can be systematically improved further by the development of a ’Hartree-improved perturbation theory’(HIPT) with Ho as the unperturbed part and the modified interaction:λH ′ ≡ λ(HI − 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 λHI chosen as the perturbation, which diverges even for infinitesimal λ ! 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, anharmonicsextic and octicoscillators leads to very accurate results for the energy levels. In case of λφ4 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.