Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds

Andersson, Lars; Driver, Bruce K.

doi:10.1006/jfan.1999.3413

Cited by 72 publications

(108 citation statements)

References 58 publications

(46 reference statements)

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“…The Feynman-Kac formula for non-trivial V is then a rather simple consequence of the Trotter formula. In [AD99] the authors approximate path space C x (M, T) by finite dimensional spaces of geodesic polygons and obtain two approximations for Wiener measure. They differ by a scalar curvature term.…”

Section: V(γ(s)) Ds · U(γ(t)) Dw(γ)mentioning

confidence: 99%

Path integrals on manifolds by finite dimensional approximation

Baer¹,

Pfaeffle²

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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ABSTRACT. Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.

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Section: V(γ(s)) Ds · U(γ(t)) Dw(γ)mentioning

confidence: 99%

Path integrals on manifolds by finite dimensional approximation

Baer¹,

Pfaeffle²

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The interested reader is referred to [114,26] for rigorous estimates of the heat kernel on Riemann manifolds and to the recent paper [5] for a complete proof of the path integral construction.…”

Section: Path Integrals From Stochastic Differential Equationsmentioning

confidence: 99%

“…Since the motivation is the semiclassical approximation of path integrals, the probabilistic interpretation of these latter ones is recalled in the first section of the chapter. Further details together with an outline of the recent rigorous proof [5] of the covariant form of the path integral measure are given in appendix C. Chapter 3 summarises basic results of Morse index theory…”

Section: Introductionmentioning

confidence: 99%

Path integration over closed loops and Gutzwiller's trace formula

Muratore-Ginanneschi

2003

Physics Reports

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In 1967 M.C. Gutzwiller succeeded to derive the semiclassical expression of the quantum energy density of systems exhibiting a chaotic Hamiltonian dynamics in the classical limit. The result is known as the Gutzwiller trace formula.The scope of this review is to present in a self-contained way recent developments in functional determinant theory allowing to revisit the Gutzwiller trace formula in the spirit of field theory.The field theoretic setup permits to work explicitly at every step of the derivation of the trace formula with invariant quantities of classical periodic orbits. R. Forman's theory of functional determinants of linear, non singular elliptic operators yields the expression of quantum quadratic fluctuations around classical periodic orbits directly in terms of the monodromy matrix of the periodic orbits.The phase factor associated to quadratic fluctuations, the Maslov phase, is shown to be specified by the Morse index for closed extremals, also known as Conley and Zehnder index.

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“…Formula (1.4) is valid uniformly in x in the case that u 0 is continuous and holds in the L p sense in the case that u ∈ L p . For the C 0 case, such a result was proved already by Andersson and Driver [AD99] and was later generalized to the case of vector-valued Laplacians by Bär and Pfäffle [BP08].…”

Section: Introductionmentioning

confidence: 73%

Path Integrals on Manifolds with Boundary

Ludewig

2017

Commun. Math. Phys.

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Abstract:We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions.

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Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds

Cited by 72 publications

References 58 publications

Path integrals on manifolds by finite dimensional approximation

Path integrals on manifolds by finite dimensional approximation

Path integration over closed loops and Gutzwiller's trace formula

Path Integrals on Manifolds with Boundary

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