Investigations of Interacting Quantum Systems in a Novel General Approximation Scheme (NGAS)

Abstract

A "Novel General Approximation Scheme"(NGAS) is proposed, which is self-consistent, nonperturbative and potentially applicable to arbitrary interacting quantum systems described by a Hamiltonian. The essential method of this scheme consists of finding a "mapping" which maps the "interacting system" on to an "exactly solvable" model, while preserving the major effects of interaction through the self consistency requirement of equal quantum averages of observables in the two systems. We apply the method to the different cases of the one dimensional anharmonic-interactions (AHI), which includes the case of the quartic-, sextic-and octic-anharmonic oscillators and quartic-, sextic-double well oscillators within the harmonic approximation and demonstrated how this simple approach reproduces, in the leading order (LO), the results to within a few percent, of some of the earlier methods employing rather different assumptions and often with sophisticated numerical analysis. We demonstrate the flexibility of the proposed scheme by carrying out the analysis of the AHI by choosing the infinite square-well potential (ISWP) in one dimension as the input approximation. We extend the formalism to λφ 4 -quantum field theory (in the massive symmetric-phase) to show the equivalence of the present method to the "Gaussian-effective potential" approach. The structure and stability of the Effective Vacuum is also demonstrated. We also present a new formulation of perturbation theory based on NGAS, designated as "Mean Field Perturbation Theory (MFPT)", which is free from power-series expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system properties having nonanalytic dependence on the coupling, thus overcoming the primary limitations of the "standard formulation of perturbation theory" (SFPT). We demonstrate Borelsummability of MFPT for the case of the quartic-and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since "renormalon"-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case. The general nature and the simplicity of the formulation underlying MFPT leads us to conjecture that this scheme may be applicable to arbitrary interactions in quantum theory.