Towards Nonperturbative Solution of Quantum Dynamics : A Hamiltonian Mean Field Approximation Scheme with Perturbation Theory for Arbitray Strength of Interaction

TL;DR

This work introduces the Non-perturbative General Approximation Scheme (NGAS), a universal Hamiltonian framework that maps interacting quantum systems to exactly solvable approximations via a self-consistent, nonperturbative leading order. NGAS defines an approximating Hamiltonian $H_0$ with adjustable parameters by enforcing equal quantum averages to the true Hamiltonian, yielding a leading spectrum $E_0(g)$ that captures nonlinear coupling effects and analytic structure in $g$. It then develops Mean Field Perturbation Theory (MFPT), an improved perturbation framework around $H_0$ that remains well-behaved for arbitrary coupling, utilizing tools like the Hyper-Virial and Hellmann–Feynman theorems and Borel-sum techniques to compute higher-order corrections. The method is demonstrated on one-dimensional anharmonic oscillators (QAHO, QDWO, SAHO, OAHO) and extended to a gφ^4 quantum field theory in the massive symmetric phase, reproducing Gaussian effective potential results at leading order and revealing a nontrivial, condensate vacuum structure; MFPT is shown to offer accurate, convergent perturbative sums with greater generality than standard perturbation theory. The work thus provides a practical, systematically improvable nonperturbative approach with broad applicability to quantum mechanics and quantum field theory, including potential extensions to finite temperature, fermions, and gauge theories.

Abstract

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.