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Phase functions and path integrals

Published online by Cambridge University Press:  16 October 2009

Joel Robbin
Affiliation:
Mathematics Department, University of Wisconsin Madison, WI 53706 USA
Dietmar Salamon
Affiliation:
Mathematics Institute, University of Warwick Coventry CV4 7AL Great Britain
Dietmar Salamon
Affiliation:
University of Warwick
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Summary

This note is an introduction to our forthcoming paper. There we show how to construct the metaplectic representation using Feynman path integrals. We were led to this by our attempts to understand Atiyah's explanation of topological quantum field theory in.

Like Feynman's original approach in an action integral plays the role of a phase function. Unlike Feynman, we use paths in phase space rather than configuration space and use the symplectic action integral rather than the (classical) Lagrangian integral. We eventually restrict to (inhomogeneous) quadratic Hamiltonians so that the finite dimensional approximation to the path integral is a Gaussian integral. In evaluating this Gaussian integral the signature of a quadratic form appears. This quadratic form is a discrete approximation to the second variation of the action integral.

For Lagrangians of the form kinetic energy minus potential energy, evaluated on curves in configuaration space, the index of the second variation is well-defined and, via the Morse Index Theorem, related to the Maslov Index of the corresponding linear Hamiltonian system. The second variation of the symplectic action has both infinite index and infinite coindex. However, this second variation does have a well-defined signature via the aforementioned discrete approximation. This signature can be expressed in terms of the Maslov index of the corresponding linear Hamiltonian system. This is a symplectic analog of the Morse Index Theorem.

Our treatment is motivated by the formal similarity between Feynman path integrals and the Fourier integral operators of Hörmander.

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Symplectic Geometry , pp. 203 - 226
Publisher: Cambridge University Press
Print publication year: 1994

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  • Phase functions and path integrals
    • By Joel Robbin, Mathematics Department, University of Wisconsin Madison, WI 53706 USA, Dietmar Salamon, Mathematics Institute, University of Warwick Coventry CV4 7AL Great Britain
  • Edited by Dietmar Salamon, University of Warwick
  • Book: Symplectic Geometry
  • Online publication: 16 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526343.013
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  • Phase functions and path integrals
    • By Joel Robbin, Mathematics Department, University of Wisconsin Madison, WI 53706 USA, Dietmar Salamon, Mathematics Institute, University of Warwick Coventry CV4 7AL Great Britain
  • Edited by Dietmar Salamon, University of Warwick
  • Book: Symplectic Geometry
  • Online publication: 16 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526343.013
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Phase functions and path integrals
    • By Joel Robbin, Mathematics Department, University of Wisconsin Madison, WI 53706 USA, Dietmar Salamon, Mathematics Institute, University of Warwick Coventry CV4 7AL Great Britain
  • Edited by Dietmar Salamon, University of Warwick
  • Book: Symplectic Geometry
  • Online publication: 16 October 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511526343.013
Available formats
×