We present a new formulation of perturbation theory for quantum systems, designated here 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 non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability 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.
We present a new formulation of perturbation theory for quantum systems, designated here 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 non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability 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.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.
A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λHI(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.
Abstract It is well known that quantum-mechanical perturbation theory often gives rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground-states wave functions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically “non-perturbative”. A possible application to exact diagonalization is briefly discussed.
Introducing low-energy effective Hamiltonians is usual to grasp most correlations in quantum many-body problems. For instance, such effective Hamiltonians can be treated at the mean-field level to reproduce some physical properties of interest. Employing effective Hamiltonians that contain many-body correlations renders the use of perturbative many-body techniques difficult because of the overcounting of correlations. In this work, we develop a strategy to apply an extension of the many-body perturbation theory starting from an effective interaction that contains correlations beyond the mean-field level. The goal is to re-organize the many-body calculation to avoid the overcounting of correlations originating from the introduction of correlated effective Hamiltonians in the description. For this purpose, we generalize the formulation of the Rayleigh-Schrödinger perturbation theory by including free parameters adjusted to reproduce the appropriate limits. In particular, the expansion in the bare weak-coupling regime and the strong-coupling limit serves as a valuable input to fix the value of the free parameters appearing in the resulting expression. This method avoids double counting of correlations using beyond-mean-field strategies for the description of many-body systems. The ground state energy of various systems relevant for ultracold atomic, nuclear, and condensed matter physics is reproduced qualitatively beyond the domain of validity of the standard many-body perturbation theory. Finally, our method suggests interpreting the formal results obtained as an effective field theory using the proposed reorganization of the many-body calculation. The results, like ground state energies, are improved systematically by considering higher orders in the extended many-body perturbation theory while maintaining a straightforward polynomial expansion.
I provide a straightforward proof that a simple harmonic oscillator perturbed by an (almost) arbitrary positive interaction has a perturbative expansion for any finite-time Euclidian transition amplitude which obeys the following result: the difference of the sum of the first N terms of the series and the exact result is bounded in absolute value by the next term in the series. The proof makes no assumptions about either the strength of the interactions or the convergence of the perturbation series. I then argue that the result generalizes immediately to a much broader class of quantum mechanical systems, including bare perturbation theory in quantum field theory. The case of renormalized perturbation theory is more subtle and remains open, as does the generalization to energy levels and connected S-matrix elements.
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]