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Path integrals on curved manifolds

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Zeitschrift für Physik C Particles and Fields

Abstract

A general framework for treating path integrals on curved manifolds is presented. We also show how to perform general coordinate and space-time transformations in path integrals. The main result is that one has to subtract a quantum correctionΔV∼h 2 from the classical Lagrangian ℒ, i.e. the correct effective Lagrangian to be used in the path integral is ℒeff = ℒ−ΔV. A general prescription for calculating the quantum correction ΔV is given. It is based on a canonical approach using Weyl-ordering and the Hamiltonian path integral defined by the midpoint prescription. The general framework is illustrated by several examples: Thed-dimensional rotator, i.e. the motion on the sphereS d−1, the path integral ind-dimensional polar coordinates, the exact treatment of the hydrogen atom inR 2 andR 3 by performing a Kustaanheimo-Stiefel transformation, the Langer transformation and the path integral for the Morse potential.

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Authors and Affiliations

  1. II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-2000, Hamburg 50, Federal Republic of Germany

    C. Grosche & F. Steiner

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  1. C. Grosche
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  2. F. Steiner
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Supported by Graduiertenstipendium Universität Hamburg

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Grosche, C., Steiner, F. Path integrals on curved manifolds. Z. Phys. C - Particles and Fields 36, 699–714 (1987). https://doi.org/10.1007/BF01630607

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  • DOI: https://doi.org/10.1007/BF01630607

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