Title:

Nonperturbative Energy Corrections in Quantum Mechanics Due to Instantons

For the nonrelativistic quantum mechanics of a particle in one dimension, the exact nonperturbative energy spectrum for a particle moving in a potential with degenerate minima, whose locations are specified by a small parameter g, has been obtained in certain examples using the so-called exact-WKB method. However, the exact-WKB method does not extend to higher dimensional cases. So, I have calculated the energy correction from the one instanton background using path integral methods, which are able to be applied to higher dimensional cases. I elected to use a symmetric double-well potential as it has been studied rigorously in the past and has degenerate minima. The instanton contributions lift the degeneracy on the the minima which allows us to find the difference in energy of the new distinct minima. To calculate this value, I first calculated the Green function for the one-instanton background. This function was used as the Feynman generator for the two loop disjoint Feynman diagrams, which I evaluated. After subtracting the contribution of the free background, I was left with the final energy correction of -71g/12. This result matches with the literature using the exact-WKB method, verifying my result is accurate.

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