# Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics - Mathematical Physics

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Abstract: We describe the -Feynman diagram- approach to nonrelativistic quantummechanics on R^n, with magnetic and potential terms. In particular, for eachclassical path \gamma connecting points q 0 and q 1 in time t, we define aformal power series V \gammat,q 0,q 1 in \hbar, given combinatorially by asum of diagrams that each represent finite-dimensional convergent integrals. Weprove that expV \gamma satisfies Schr\-odinger-s equation, and explain inwhat sense the t\to 0 limit approaches the \delta distribution. As such, ourconstruction gives explicitly the full \hbar\to 0 asymptotics of thefundamental solution to Schr\-odinger-s equation in terms of solutions to thecorresponding classical system. These results justify the heuristic expansionof Feynman-s path integral in diagrams.

Author: ** Theo Johnson-Freyd**

Source: https://arxiv.org/