The formal path integral and quantum mechanics

Journal Article · Mon Nov 15 00:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3503472· OSTI ID:21501200

Johnson-Freyd, Theo ^[1]

---

1. Department of Mathematics, University of California - Berkeley, 970 Evans Hall, Berkeley, California 94720 (United States)

Given an arbitrary Lagrangian function on R{sup 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.

• OSTI ID: 21501200
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 11; Other Information: DOI: 10.1063/1.3503472; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICAL METHODS AND COMPUTING
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CHARGED PARTICLES
COORDINATES
ELECTROMAGNETIC FIELDS
FEYNMAN DIAGRAM
GENERAL RELATIVITY THEORY
INTEGRAL EQUATIONS
LAGRANGIAN FUNCTION
MATHEMATICAL EVOLUTION
MATHEMATICAL OPERATORS
MATRICES
PATH INTEGRALS
POWER SERIES
QUANTUM MECHANICS
RIEMANN SPACE
SCHROEDINGER EQUATION
ULTRAVIOLET DIVERGENCES
DIAGRAMS
DIFFERENTIAL EQUATIONS
EQUATIONS
EVOLUTION
FIELD THEORIES
FUNCTIONS
INFORMATION
INTEGRALS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
RELATIVITY THEORY
SERIES EXPANSION
SPACE
WAVE EQUATIONS