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Letters in Mathematical Physics

, Volume 107, Issue 2, pp 375–408 | Cite as

On Time

  • Alberto S. CattaneoEmail author
  • Michele Schiavina
Article

Abstract

This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely, the classically equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method.

Keywords

Parametrization invariant Lagrangian Jacobi action One-dimensional gravity BV BFV AKSZ Spinning particle Supersymmetry 

Mathematics Subject Classification

Primary 81T70 Secondary 70805 83C47 81T45 

Notes

Acknowledgements

We thank the referee for a number of very valuable comments.

References

  1. 1.
    Alexandrov, M., Kontsevich, M., Schwarz, A., Zaboronsky, O.: “The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A12, 1405–1430 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barbour, J.: The nature of time. arXiv:0903.3489
  3. 3.
    Batalin, I.A., Fradkin, E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. B 122, 157–164 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102, 27–31 (1981)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Batalin, I.A., Vilkovisky, G.A.: Relativistic S-matrix of dynamical systems with boson and fermion costraints. Phys. Lett. B 69, 309–312 (1977)ADSCrossRefGoogle Scholar
  6. 6.
    Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127–162 (1975)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonechi, F., Cattaneo, A.S., Mnëv, P.: The Poisson sigma model on closed surfaces. JHEP 2012(99), 1–27 (2012)zbMATHGoogle Scholar
  8. 8.
    Brink, L., Deser, S., Zumino, B., Di Vecchia, P., Howe, P.: Local supersymmetry for spinning particles. Phys. Lett. B 64, 435–438 (1976)ADSCrossRefGoogle Scholar
  9. 9.
    Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundaries. Commun. Math. Phys. 332, 535–603 (2014)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    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. arXiv:1207.0239
  11. 11.
    Cattaneo, A.S.: Mnëv, P., Reshetikhin, N.: Perturbative quantum gauge theories on manifolds with boundary. Commun. Math. Phys. arXiv:1507.01221
  12. 12.
    Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity: Einstein–Hilbert action. J. Math. Phys. 57, 023515, 17 (2016)Google Scholar
  13. 13.
    Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity II: Palatini–Holst action (in preparation) Google Scholar
  14. 14.
    Geztler, E.: The Batalin-Vilkovisky formalism of the spinning particle. arXiv:1511.02135
  15. 15.
    Geztler, E.: The spinning particle with curved target. arXiv:1605.04762
  16. 16.
    Hartle, J.B., Hawking, S.W.: Wave function of the Universe. Phys. Rev. D 28, 2960 (1983)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Jacobi, C.G.J.: Das Princip der kleinsten Wirkung. Ch. 6. In: Clebsch, A. (ed.) Vorlesungen über Dynamik. Available at Gallica-Math. Reimer, Berlin (1866)Google Scholar
  18. 18.
    Kijowski, J., Tulczyjew, M.: A Symplectic Framework for Field Teories, Lecture Notes in Physics, vol. 107. Springer, New york (1979)Google Scholar
  19. 19.
    Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys. 176, 49–113 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Roytenberg, D.: AKSZ-BV formalism and courant algebroid-induced topological field theories. Lett. Math. Phys. 79, 143–159 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schiavina, M.: BV-BFV Approach to general relativity, PhD thesis, Zurich (2015)Google Scholar
  22. 22.
    Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism. Lebedev Institute preprint No. 39 (1975). arXiv:0812.0580

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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