# A Convergent Continuum Strong Coupling Expansion For Quantum Mechanics & Quantum Field Theory

## TL;DR

This paper introduces a convergent strong coupling expansion method for quantum mechanics and quantum field theory, enabling accurate eigenenergy calculations at all positive couplings and improving electron g-factor estimates.

## Contribution

It develops a novel rescaling and subtraction technique to achieve a convergent expansion applicable at strong coupling in quantum theories.

## Key findings

- Accurate eigenenergies for anharmonic oscillators at all positive couplings.
- Effective calculation of the electron g-factor at one loop using the new formalism.
- Convergence demonstrated for double well potential at strong coupling.

## Abstract

We generalize the notion of an asymptotic weak coupling expansion about an exactly solvable model in quantum mechanics and quantum field theory to an all positive value coupling convergent expansion. This is done by rescaling the variables available in the theory by free parameters, then adding and subtracting the exactly solvable model. The rest (initial rescaled theory by free parameters + the subtracted exactly solvable model) is expanded about the added exactly solvable model. Evaluating finite orders of this expansion at its extremum points with respect to the free parameter(s) gives a sequence that converges to the result of the previous asymptotic expansion, with a good convergence rate, at relative strong coupling. We solve for the eigenenergies of the anharmonic, pure anharmonic and double well potential problems using this method by expanding about the symmetrical point of these potentials. Accurate results for the eigenenergies can be obtained for all positive values of the coupling for the anharmonic and pure anharmonic oscillators and at strong coupling for the double well potential. To provide confirmation for the convergent formalism developed for $\phi^4$ theory and QED we improve the electron g-factor calculation at the one loop level using the convergent formalism. Applications of this method are not limited to quantum mechanics or quantum field theory, for example it can also have applications in the context of differential equations.