Physics – Mathematical Physics
Scientific paper
2010-04-24
Physics
Mathematical Physics
33 pages, many TikZ diagrams, submitted to _Journal of Mathematical Physics_
Scientific paper
Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.
No associations
LandOfFree
The formal path integral and quantum mechanics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The formal path integral and quantum mechanics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The formal path integral and quantum mechanics will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-387404