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Noncommutative deformations of quantum field theories, locality, and causality

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Abstract

We investigate noncommutative deformations of quantum field theories for different star products, particularly emphasizing the locality properties and the regularity of the deformed fields. Using functional analysis methods, we describe the basic structural features of the vacuum expectation values of star-modified products of fields and field commutators. As an alternative to microcausality, we introduce a notion of θ-locality, where θ is the noncommutativity parameter. We also analyze the conditions for the convergence and continuity of star products and define the function algebra that is most suitable for the Moyal and Wick-Voros products. This algebra corresponds to the concept of strict deformation quantization and is a useful tool for constructing quantum field theories on a noncommutative space-time.

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References

  1. S. Doplicher, K. Fredenhagen, and J. E. Roberts, Comm. Math. Phys., 172, 187–220 (1995).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. R. J. Szabo, Phys. Rep., 378, 207–299 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. S. Galluccio, F. Lizzi, and P. Vitale, Phys. Rev. D, 78, 085007 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  4. A. P. Balachandran and M. Martone, Modern Phys. Lett. A, 24, 1721–1730 (2009).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. M. Chaichian, P. P. Kulish, K. Nshijima, and A. Tureanu, Phys. Lett. B, 604, 98–102 (2004).

    Article  MathSciNet  ADS  Google Scholar 

  6. M. Riccardi and R. J. Szabo, JHEP, 0801, 016 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  7. M. A. Rieffel, Comm. Math. Phys., 122, 531–562 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. M. A. Soloviev, Phys. Rev. D, 77, 125013 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  9. H. Grosse and G. Lechner, JHEP, 0809, 131 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  10. Yi Liao and K. Sibold, Phys. Lett. B, 549, 352–361 (2002).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. L. Álvarez-Gaumé and M. A. Vázquez-Mozo, Nucl. Phys. B, 668, 293–321 (2003).

    Article  MATH  ADS  Google Scholar 

  12. C.-S. Chu, K. Furuta, and T. Inami, Internat. J. Mod. Phys. A, 21, 67–82 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. M. A. Soloviev, Theor. Math. Phys., 153, 1351–1363 (2007).

    Article  MATH  Google Scholar 

  14. M. A. Soloviev, J. Phys. A, 40, 14593–14604 (2007).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. F. A. Berezin and M. A. Shubin, Schrödinger Equation [in Russian], Moscow State Univ. Press, Moscow (1983); English transl., Kluwer, Dordrecht (1991).

    MATH  Google Scholar 

  16. L. Hörmander, The Analysis of Linear Partial Differential Operators (Grundlehren Math. Wiss., Vol. 256), Vol. 1, Distribution Theory and Fourier Analysis, Springer, Berlin (1983).

    Google Scholar 

  17. R. F. Streater and A. S. Wightman, PCT, Spin, and Statistics, and All That, Princeton Univ. Press, Princeton, N. J. (2000).

    MATH  Google Scholar 

  18. N. N. Bogolyubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov, General Principles of Quantum Field Theory [in Russian], Nauka, Moscow (1987); English transl. (Math. Phys. Appl. Math., Vol. 10), Kluwer, Dordrecht (1990).

    Google Scholar 

  19. A. P. Balachandran, T. R. Govindarajan, G. Mangano, A. Pinzul, B. A. Qureshi, and S. Vaidya, Phys. Rev. D, 75, 045009 (2007).

    Article  ADS  Google Scholar 

  20. G. Fiore and J. Wess, Phys. Rev. D, 75, 105022 (2007).

    Article  MathSciNet  ADS  Google Scholar 

  21. H. Grosse and G. Lechner, JHEP, 0711, 012 (2007).

    Article  MathSciNet  ADS  Google Scholar 

  22. P. Aschieri, F. Lizzi, and P. Vitale, Phys. Rev. D, 77, 025037 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  23. J. Mund, B. Schroer, and J. Yngvason, Comm. Math. Phys., 268, 621–672 (2006).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. O. W. Greenberg, Phys. Rev. D, 73, 045014 (2006).

    Article  MathSciNet  ADS  Google Scholar 

  25. B. S. Mitjagin, Tr. Moskov. Mat. Obshch., 9, 317–328 (1960).

    MathSciNet  Google Scholar 

  26. M. Chaichian, M. N. Mnatsakanova, A. Tureanu, and Yu. A. Vernov, JHEP, 0809, 125 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  27. A. G. Smirnov and M. A. Soloviev, “On kernel theorems for Fréchet and DF spaces,” arXiv:math.FA/0501187v1 (2005).

  28. A. G. Smirnov, Integral Transforms Spec. Funct., 20, 309–318 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  29. M. A. Soloviev, Theor. Math. Phys., 121, 1377–1396 (1999).

    Article  MATH  Google Scholar 

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  1. Lebedev Physical Institute, RAS, Moscow, Russia

    M. A. Soloviev

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  1. M. A. Soloviev
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Correspondence to M. A. Soloviev.

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Soloviev, M.A. Noncommutative deformations of quantum field theories, locality, and causality. Theor Math Phys 163, 741–752 (2010). https://doi.org/10.1007/s11232-010-0058-7

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