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Wave Relations

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Abstract

The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly, there is still something to say about this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian evolution are not problematic from the viewpoint of relations. In the two-dimensional Minkowski case, these relations are shown to be Lagrangian. This result is then extended to a wide class of metrics and is conjectured to be true also in higher dimensions for nice enough metrics. A counterexample where the relation is not Lagrangian is provided by the Misner space.

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References

  1. Barbero, J.F., Prieto, J., Villaseñor, E.J.S.: Hamiltonian treatment of linear field theories in the presence of boundaries: a geometric approach. Class. Quant. Grav. 31, 045021 (2014). arXiv:1306.5854 (math-ph)

  2. Cappell, S., DeTurck, D., Gluck, H., Miller, E.: Cohomology of harmonic forms on Riemannian manifolds with boundary. Forum Math. 18(6), 923–931 (2006). arXiv:math/0508372 (math.DG)

  3. Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundary. arXiv:1201.0290 (math-ph)

  4. Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. In: Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity, PoS(CORFU2011)044 (2011)

  5. Deligne, P., Freed, D.S.: Classical field theory. In: Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, Part 1. AMS, Providence, RI (1999)

  6. Gotay M., Nester J., Hinds G.: Presymplectic manifolds and the Dirac–Bergmann theory of constraints. J. Math. Phys. 19, 2388 (1978)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. May J.P.: The Geometry of Iterated Loop Spaces. Springer-Verlag, New York (1972)

    MATH  Google Scholar 

  8. Misner, C.W.: Taub-NUT space as a counterexample to almost anything. In: Ehlers, J. (ed.) Relativity Theory and Astrophysics I: Relativity and Cosmology. Lectures in Applied Mathematics, vol. 8. American Mathematical Society, Providence, p. 160 (1967)

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Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Alberto S. Cattaneo & Pavel Mnev

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  1. Alberto S. Cattaneo
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  2. Pavel Mnev
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Correspondence to Alberto S. Cattaneo.

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Communicated by N. Reshetikhin

\({^\star}\) A. S. C. acknowledges partial support of SNF Grant No. 200020-131813/1. P. M. acknowledges partial support of RFBR Grant No. 13-01-12405-ofi-m-2013 and of SNF Grant No. 200021-137595.

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Cattaneo, A.S., Mnev, P. Wave Relations. Commun. Math. Phys. 332, 1083–1111 (2014). https://doi.org/10.1007/s00220-014-2130-x

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