Variational perturbation theory for path integrals

Variational Perturbation Theory (VPT) of path integrals arose from a 1986 collaboration with Feynman, combining ordinary divergent perturbation series with a variational procedure. It produces exponentially-fast convergent sequences of approximations, uniformly up to infinite coupling strengths. In this way, divergent weak-coupling expansions can be converted into convergent strong-coupling expansions, whose convergence radius can be deduced from details how the variational procedure converges. A method of interpolating between weak-and strong-coupling expansions follows naturally, with interesting consequences for polarons. VPT can be continued to negative coupling constants to yield imaginary parts of amplitudes in the sliding regime. The precise imaginary parts obtained in this way can be inserted into a dispersion relation to find the large-order behavior of perturbation coefficients which turns out to be accurate down to the lowest order. A method of interpolating between weak-coupling and large-order information arises from this. A combination of the variational approach with instanton calculus leads to a Variational Tunneling Theory. This can be applied to non-Borel-summable expansions making these convergent as well.