ISRN High Energy Physics

Volume 2013 (2013), Article ID 398030, 15 pages

http://dx.doi.org/10.1155/2013/398030

Research Article

## On a Semiclassical Limit of Loop Space Quantum Mechanics

The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India

Received 28 February 2013; Accepted 10 April 2013

Academic Editors: A. Belhaj, A. Konechny, A. Koshelev, and L. Pando Zayas

Copyright © 2013 Partha Mukhopadhyay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- C. G. Callan and L. Thorlacius, “Sigma models and string theory,” in
*Particles, Strings and Supernovae*, A. Jevicki and C.-I. Tan, Eds., World Scientific, Hackensack, NJ, USA, 1989. View at Google Scholar - A. A. Tseytlin, “Sigma model approach to string theory,”
*International Journal of Modern Physics A*, vol. 4, no. 6, pp. 1257–1318, 1989. View at Google Scholar - J. Honerkamp, “Chiral multi-loops,”
*Nuclear Physics B*, vol. 36, no. 1, pp. 130–140, 1972. View at Google Scholar · View at Scopus - D. Friedan, “Nonlinear models in two
*ε*dimensions,”*Physical Review Letters*, vol. 45, p. 1057, 1980. View at Google Scholar - D. H. Friedan, “Nonlinear models in two +
*ε*dimensions,”*Annals of Physics*, vol. 163, no. 2, pp. 318–419, 1985. View at Google Scholar · View at MathSciNet · View at Scopus - L. Alvarez-Gaumé, D. Z. Freedman, and S. Mukhi, “The background field method and the ultraviolet structure of the supersymmetric nonlinear
*σ*-model,”*Annals of Physics*, vol. 134, no. 1, pp. 85–109, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Eisenhart,
*Riemannian Geometry*, Princeton University Press, Princeton, NJ, USA, 1965. - E. Witten, “Supersymmetry and Morse theory,”
*Journal of Differential Geometry*, vol. 17, no. 4, pp. 661–692, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Witten, “The Index of the Dirac Operator in Loop Space,” PUPT-1050.
- U. Schreiber, “On deformations of 2d SCFTs,”
*Journal of High Energy Physics*, vol. 2004, no. 6, article 058, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - U. Schreiber, “From loop space mechanics to nonabelian strings,” http://arxiv.org/abs/hep-th/0509163.
- E. Frenkel, A. Losev, and N. Nekrasov, “Notes on instantons in topological field theory and beyond,”
*Nuclear Physics B*, vol. 171, pp. 215–230, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Frenkel, A. Losev, and N. Nekrasov, “Instantons beyond topological theory II,” http://arxiv.org/abs/0803.3302.
- P. Mukhopadhyay, “On a coordinate independent description of string worldsheet theory,” http://arxiv.org/abs/0912.3987.
- P. Mukhopadhyay, “DeWitt-Virasoro construction in tensor representations,”
*Advances in High Energy Physics*, vol. 2012, Article ID 415634, 33 pages, 2012. View at Publisher · View at Google Scholar - P. G. C. Freund and R. I. Nepomechie, “Unified geometry of antisymmetric tensor gauge fields and gravity,”
*Nuclear Physics B*, vol. 199, no. 3, pp. 482–494, 1982. View at Google Scholar · View at Scopus - P. S. Florides and J. L. Synge, “Coordinate conditions in a Riemannian space for coordinates based on a subspace,”
*Proceedings of the Royal Society of London Series A*, vol. 323, pp. 1–10, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gray,
*Tubes*, Progress in Mathematics, Addison-Wesley, New York, NY, USA, 2nd edition, 1990. View at MathSciNet - P. Mukhopadhyay, “All order covariant tubular expansion,” http://arxiv.org/abs/1203.1151.
- U. Müller, C. Schubert, and A. E. M. van de Ven, “A closed formula for the Riemann normal coordinate expansion,”
*General Relativity and Gravitation*, vol. 31, no. 11, pp. 1759–1768, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. A. Marcus, “On the analytical mechanics of chemical reactions. Quantum mechanics of linear collisions,”
*Journal of Chemical Physics*, vol. 45, pp. 4493–4499, 1966. View at Google Scholar - H. Jensen and H. Koppe, “Quantum mechanics with constraints,”
*Annals of Physics*, vol. 63, pp. 586–591, 1971. View at Google Scholar - R. C. T. da Costa, “Constraints in quantum mechanics,”
*Physical Review A*, vol. 25, pp. 2893–2900, 1982. View at Google Scholar - P. Maraner, “A complete perturbative expansion for quantum mechanics with constraints,”
*Journal of Physics A*, vol. 28, no. 10, pp. 2939–2951, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. A. Mitchell, “Gauge fields and extrapotentials in constrained quantum systems,”
*Physical Review A*, vol. 63, no. 4, Article ID 042112, 20 pages, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. F. Dell'Antonio and L. Tenuta, “Semiclassical analysis of constrained quantum systems,”
*Journal of Physics A*, vol. 37, no. 21, pp. 5605–5624, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Wachsmuth and S. Teufel, “Effective hamiltonians for constrained quantum systems,” http://arxiv.org/abs/0907.0351.
- J. Wachsmuth and S. Teufel, “Constrained quantum systems as an adiabatic problem,”
*Physical Review A*, vol. 82, no. 2, Article ID 022112, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. Stacey, “The differential topology of loop spaces,” http://arxiv.org/abs/math/0510097, http://ncatlab.org/nlab/show/equivariant+tubular+neighbourhoods.
- M. Nakahara,
*Geometry, Topology and Physics*, Graduate Student Series in Physics, Taylor & Francis, Boca Raton, Fla, USA, 2nd edition, 2003. - S. Kobayashi and K. Nomizu,
*Foundations of Differential Geometry II*, Wiley, New York, NY, USA, 1st edition, 1969. - M. V. Berry, “Quantal phase factors accompanying adiabatic changes,”
*Proceedings of the Royal Society of London A*, vol. 392, no. 1802, pp. 45–57, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sasaki, “On the differential geometry of tangent bundles of Riemannian manifolds,”
*The Tohoku Mathematical Journal*, vol. 10, pp. 338–354, 1958. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. M. Wald,
*General Relativity*, University of Chicago Press, Chicago, Ill, USA, 1984. View at MathSciNet - A. Kriegl and P. W. Michor,
*The Convenient Setting of Global Analysis*, vol. 53 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 1997. View at MathSciNet - J. Polchinski,
*String Theory. Volume 1: An Introduction to the Bosonic String*, Cambridge University Press, Cambridge, UK, 1998. - S. Kobayashi, “Fixed points of isometries,”
*Nagoya Mathematical Journal*, vol. 13, pp. 63–68, 1958. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. S. DeWitt, “Point transformations in quantum mechanics,”
*Physical Review*, vol. 85, no. 4, pp. 653–661, 1952. View at Publisher · View at Google Scholar - K. Peeters, “Symbolic field theory with Cadabra,”
*Computeralgebra Rundbrief*, vol. 41, p. 16, 2007. View at Google Scholar - K. Peeters, “Introducing Cadabra: a symbolic computer algebra system for field theory problems,” http://arxiv.org/abs/hep-th/0701238.
- L. Brewin, “A brief introduction to Cadabra: a tool for tensor computations in general relativity,”
*Computer Physics Communications*, vol. 181, no. 3, pp. 489–498, 2010. View at Google Scholar