28 research outputs found
The formal path integral and quantum mechanics
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.Comment: 33 pages, many TikZ diagrams, submitted to _Journal of Mathematical
Physics
On the coordinate (in)dependence of the formal path integral
When path integrals are discussed in quantum field theory, it is almost
always assumed that the fields take values in a vector bundle. When the fields
are instead valued in a possibly-curved fiber bundle, the independence of the
formal path integral on the coordinates becomes much less obvious. In this
short note, aimed primarily at mathematicians, we first briefly recall the
notions of Lagrangian classical and quantum field theory and the standard
coordinate-full definition of the "formal" or "Feynman-diagrammatic" path
integral construction. We then outline a proof of the following claim: the
formal path integral does not depend on the choice of coordinates, but only on
a choice of fiberwise volume form. Our outline is an honest proof when the
formal path integral is defined without ultraviolet divergences.Comment: 9 pages, uses Tik
How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism
The Batalin-Vilkovisky formalism in quantum field theory was originally
invented to address the difficult problem of finding diagrammatic descriptions
of oscillating integrals with degenerate critical points. But since then, BV
algebras have become interesting objects of study in their own right, and
mathematicians sometimes have good understanding of the homological aspects of
the story without any access to the diagrammatics. In this note we reverse the
usual direction of argument: we begin by asking for an explicit calculation of
the homology of a BV algebra, and from it derive Wick's Theorem and the other
Feynman rules for finite-dimensional integrals.Comment: 11 pages. Final versio