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Path Integrals on Manifolds with Boundary

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.

References

  1. AD99

    Anderson L., Driver B.: Finite dimensional approximations to wiener measure and path integral formulas on manifolds. J. Funct. Anal 165, 430–498 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  2. BGV04

    Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin (2004)

    MATH  Google Scholar 

  3. BP08

    Bär C., Pfäffle F.: Path integrals on manifolds by finite dimensional approximation. J. Reine Angew. Math. 625, 29–57 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Cha06

    Chavel I.: Riemannian Geometry. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  5. Che86

    Chernoff P.R.: Note on product formulas for operator semigroups. J. Func. Anal 2, 238–242 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  6. CFS82

    Cornfeld, I. P., Fomin, S. V., Sinai, Y. G.: Grundlehren der Mathematischen Wissenschaften. Ergodic theory, vol. 245, p. 486. Springer-Verlag, New York (1982). doi:10.1007/978-1-4615-6927-5

  7. Die69

    Dieudonné J.: Foundations of Modern Analysis. Academic Press Inc., Orlando (1969)

    MATH  Google Scholar 

  8. FH65

    Feynman R., Hibbs A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill Companies, New York (1965)

    MATH  Google Scholar 

  9. Gil95

    Gilkey P.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  10. Gil04

    Gilke P.B.: Asymptotic Formulae in Spectral Geometry. Chapman and Hall, Boca Raton (2004)

    Google Scholar 

  11. Gre71

    Greiner P.: An asymptotic expansion for the heat equation. Arch. Rat. Mech. Anal. 41(3), 163–218 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  12. Hal77

    Halpern B.: Strange billiard table. Trans. Am. Math. Soc. 232, 297–305 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  13. SvWW07

    Smolyanov O., Weizsäcker H.V., Wittich O.: Chernoff’s theorem and discrete time approximations of brownian motion on manifolds. Potential Anal. 26(1), 1–29 (2007)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Matthias Ludewig.

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Communicated by M. Hairer

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Ludewig, M. Path Integrals on Manifolds with Boundary. Commun. Math. Phys. 354, 621–640 (2017). https://doi.org/10.1007/s00220-017-2915-9

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