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 one-parameter 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 one-parameter 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. We give seven non trivial applications of the basic formula, and we compute its semiclassical expansion.The methods of proof are rigorous and combine Albeverio Høegh-Krohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches we solve Schrödinger type equations directly, Integrands and integrators.Splitting the quantity inside the integral sign into "integrator" and "integrand" belongs to the art of integration, but rules of thumb apply:-When the functional integral has its origin in physics try not to break up the action into, say, kinetic and potential contributions. On the other hand, do not hesitate to work with a potential which is a functional of a path rather than a function of its value (e.g. in equation (III.1), V is a functional of z, not a function of z(t)).-Look for a possible change of variable of integration; this may suggest a practical choice for the integrand.

Proof Nets as Formal Feynman Diagrams

“…Moving to infinite dimensional systems raises the mathematical stakes considerably and led to much controversy about whether this is actually well-defined. In the past two decades a rigourous theory, due to Cartier and DeWitt-Morette has appeared [CDM95] but not everyone accepts that this formalises what physicists actually do when they make field-theoretic calculations.…”

Path integral solution of linear second order partial differential equations I: the general construction

“…According to the general scheme ( [5], [8]), a path integral is defined on a separable Banach space X with a norm x where x ∈ X is a map x : Σ → M. Σ is a 1-dimensional manifold and M is an m-dimensional manifold. The dual Banach space X ′ ∋ x ′ is a space of linear forms such that x ′ , x ∈ C with an induced norm given by…”

“…Section 2 contains the outline of a framework for functional integration developed by Cartier/DeWitt-Morette in [5] (see also [8]). It allows one to define path integrals in a general setting.…”

“…In particular, stochastic methods have been used to solve inhomogeneous elliptic and diffusion-type parabolic PDEs (see e.g. [1], [2], [3]), and path integrals 1 in physics have been used to solve the Schrödinger equation ([4], [5]) and homogeneous Dirichlet problem ( [6]). …”

Stochastic mechanics and the Feynman integral

“…There exists, by now, a large body of literature investigating various aspects of the Feynman integral and its generalization, see [3]- [15] and references therein. Two years later, Nelson, elaborating on previous work of Fényes and others, laid the foundations of a quantization procedure for classical dynamical systems based on diffusion processes [16].…”