A new perspective on functional integration
Abstract
The core of this article is a general theorem with a large number of specializations. Given a manifold N and a finite number of oneparameter groups of point transformations on N with generators Y,X_{(1)},...,X_{(d)}, we obtain, via functional integration over spaces of pointed paths on N (paths with one fixed point), a oneparameter group of functional operators acting on tensor or spinor fields on N. The generator of this group is a quadratic form in the Lie derivatives L_{X(α)} in the X_{(α)}direction plus a term linear in L_{Y}. The basic functional integral is over L^{2,1} paths x:T→N (continuous paths with square integrable first derivative). Although the integrator is invariant under time translation, the integral is powerful enough to be used for systems which are not time translation invariant. Seven nontrivial applications of the basic formula are given, and the semiclassical expansion is computed. The methods of proof are rigorous and combine AlbeverioHo/eghKrohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches, Schrödinger type equations are solved directly, rather than solving first diffusion equations and then using analytic continuation.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 May 1995
 DOI:
 10.1063/1.531039
 Bibcode:
 1995JMP....36.2237C