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The formal path integral and quantum mechanics

### Abstract

Given an arbitrary Lagrangian function on 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.

© 2010 American Institute of Physics

Received 12 May 2010
Accepted 11 September 2010
Published online 14 December 2010

Article outline:

I. INTRODUCTION
A. Detailed outline of the paper
B. Acknowledgments
II. THE COORDINATE-FULL DEFINITION OF THE FORMAL PATH INTEGRAL
A. Finite-dimensional oscillating integrals
B. The classical action and its derivatives
C. The Green's function
D. The Morse index of a classical path
E. The diagrammatic definition of *U* _{Γ}
III. THE FORMAL PATH INTEGRAL DEPENDS ONLY ON THE VOLUME FORM
A. Oriented-volume-preserving maps are homotopic to the identity
B. Proof of Theorem 3.1
IV. A FUBINI THEOREM FOR FORMAL PATH INTEGRALS
A. The classical composition law
B. Some derivatives of the formal path integral
C. The Feynman-diagrammatic part of Eq. (4.1)
V. HANDLING ULTRAVIOLET DIVERGENCES
A. A divergent example: Geodesic motion on R in the wrong coordinates
B. Nonrelativistic quantum mechanics is divergence-free

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2010-12-14

2016-10-21

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