Princeton University High Energy Theory Seminar

Abstract: Perturbation theory is used extensively for solving problems in quantum mechanics and quantum field theory. In most cases, the perturbative series in powers of the coupling is an asymptotic series (it ultimately diverges). This is not an issue at weak coupling where one can make precise predictions by computing a few lower orders. However, this procedure fails completely at strong coupling. In this work, we show that one can obtain an absolutely convergent perturbative series in powers of the coupling if one uses finite path integral limits. In particular, the series can be used to calculate observables at strong coupling. In some cases, one can prove that there is a duality between the perturbative series and a different series based on inverse powers of the coupling (which converges quickly at strong coupling).  We illustrate the procedure for λ ϕ⁴ theory in lower dimensions:  a basic integral involving quadratic and quartic terms and the more dynamical scenario of the quantum anharmonic oscillator. In particular, we obtain an energy series expansion at strong coupling for the anharmonic oscillator and the result agrees with the exact energy (obtained numerically) to within 0.1%, a remarkable result in light of the fact that at strong coupling the usual perturbative series diverges badly right from the start.