A new perspective on functional integration

@article{Cartier1995ANP,
  title={A new perspective on functional integration},
  author={Pierre Cartier and Cecile DeWitt-Morette},
  journal={Journal of Mathematical Physics},
  year={1995},
  volume={36},
  pages={2237-2312}
}
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 LX(α) in the X(α)‐direction… Expand
Characterizing Volume Forms
Old and new results for characterizing volume forms in functional integration. 1 The Wiener measure Defining volume forms on infinite dimensional spaces is a key problem in the theory of functionalExpand
A Rigorous Mathematical Foundation of Functional Integration
TLDR
This work will present a new approach that is in part a synthesis of what has been accomplished over the past decades and in part an extension of functional integration to a larger class of ftinctionals. Expand
Liberating the Dimension for Function Approximation and Integration
We discuss recent results on the complexity and tractability of problems dealing with ∞-variate functions. Such problems, especially path integrals, arise in many areas including mathematicalExpand
Functional Integration: Action and Symmetries
Acknowledgements List symbols, conventions, and formulary Part I. The Physical and Mathematical Environment: 1. The physical and mathematical environment Part II. Quantum Mechanics: 2. First lesson:Expand
Functional Integration: Action and Symmetries
Acknowledgements List symbols, conventions, and formulary Part I. The Physical and Mathematical Environment: 1. The physical and mathematical environment Part II. Quantum Mechanics: 2. First lesson:Expand
Representations of the Renormalization Group as Matrix Lie Algebra
Renormalization is cast in the form of a Lie algebra of infinite triangular matrices. By exponentiation, these matrices generate counterterms for Feynman diagrams with subdivergences. AsExpand
A note on the limiting procedures for path integrals
Limiting procedures for the Feynman type path integral are considered. The evolution operator is approximated with operators corresponding to the exponential of the Hamiltonian's symbol. The proof ofExpand
The Dyson–Feynman conjectures
In this article, we survey recent work on a suggestion by Freeman J. Dyson in his famous “Missed Opportunities” Gibbs Lecture at the annual AMS meeting in 1972. There he offered the open problem ofExpand
Path Integral Solution of the Dirichlet Problem
Abstract A scheme for functional integration developed by Cartier/DeWitt-Morette is first reviewed and then employed to construct the path integral representation for the solution of the DirichletExpand
PHYSICS ON AND NEAR CAUSTICS
Physics on caustics is obtained by studying physics near caustics. Physics on caustics is at the cross-roads of calculus of variation and functional integration. More precisely consider a space P M dExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 24 REFERENCES
Transformations of Wiener integrals under a general class of linear transformations
Introduction. Let C be the space of all real-valued functions x(t) continuous in 0 < t ?1, and vanishing at t = 0. Wiener has defined a measure over the space C and in terms of this measure he hasExpand
Realization of compact Lie algebras in Kähler manifolds
The Berezin quantization on a simply connected homogeneous Kahler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in aExpand
The semiclassical expansion
Abstract Techniques based on a new definition of path integrals are given to compute the semiclassical expansion of the probability amplitude K(B; A) of a transition A → B K(B; A) = K o (B; A) ∑ n=oExpand
A stochastic scheme for constructing solutions of the Schrödinger equations
2014 Stochastic differential equations on fibre bundles are used to suggest path integral solutions for certain Schrodinger equations. Three examples are discussed in detail: motion in curved spaces,Expand
Berezin quantization and unitary representations of Lie groups
In 1974, Berezin proposed a quantum theory for dynamical systems having a K\"{a}hler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. ForExpand
Mathematical theory of Feynman path integrals
Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, includingExpand
Stochastic Differential Equations on Manifolds
A. The title is designed to indicate those particular aspects of stochastic differential equations which will be considered here: these are almost equally valid when the manifold in question is ℝ nExpand
Feynman's path integral
AbstractFeynman's integral is defined with respect to a pseudomeasure on the space of paths: for instance, letC be the space of pathsq:T⊂ℝ → configuration space of the system, letC be the topologicalExpand
Path integrals in polar co-ordinates
  • S. Edwards, Y. Gulyaev
  • Mathematics
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1964
Functional integrals of the usual diffusion type in x(t), y(t), z(t) are discussed when transformed into polar co-ordinates r(t), φ(t), ϑ(t). It is found that the functional integration can beExpand
Quantum Mechanics in Curved Spacetimes Stochastic Processes on Frame Bundles
It is customary, nowadays, to pay homage to path integration at the beginning of a course in Quantum Physics—but later on to use only its most obvious properties. So much so that it is often saidExpand
...
1
2
3
...