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Presheaves of Superselection Structures in Curved Spacetimes

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Abstract

We show that superselection structures on curved spacetimes that are expected to describe quantum charges affected by the underlying geometry are categories of sections of presheaves of symmetric tensor categories. When an embedding functor is given, the superselection structure is a Tannaka-type dual of a locally constant group bundle, which hence becomes a natural candidate for the role of the gauge group. Indeed, we show that any locally constant group bundle (with suitable structure group) acts on a net of C* algebras fulfilling normal commutation relations on an arbitrary spacetime. We also give examples of gerbes of C* algebras, defined by Wightman fields and constructed using projective representations of the fundamental group of the spacetime, which we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed.

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References

  1. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)

  2. Avis, S.J., Isham, C.J.: Quantum field theory and fibre bundles in a general spacetime. In: Cargese Levy, M., Deser, S. (eds.) Recent Developments in Gravitation, Springer, US (1978)

  3. Araki, H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (2009)

    MATH  Google Scholar 

  4. Brunetti, R., Franceschini, L., Moretti, V.: Topological features of massive bosons on two dimensional Einstein space-time. Ann. Henri Poincaré 10, 1027–1073 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum physics. Comm. Math. Phys. 237, 31–68 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Buchholz, D., Mack, G., Todorov, I.: Localized automorphisms of the U(1)-current algebra on the circle. An instructive example. In: Book: The algebraic Theory of Superselection Sectors. Introduction and Recent Results, pp. 356–378. World Scientific, Singapore (1990)

  7. Buchholz, D., Mack, G., Todorov, I.: The current algebra on the circle as a germ of local field theories. Nuclear Phys. B (Proc. Suppl.) 5B, 20–56 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Baumgaertel, H., Lledó, F.: Duality of compact groups and Hilbert C*-systems for C*-algebras with a nontrivial center. Int. J. Math. 15, 759–812 (2004)

    Article  MATH  Google Scholar 

  9. Brunetti, R., Ruzzi, G.: Quantum charges and spacetime topology: the emergence of new superselection sectors. Comm. Math. Phys. 87, 523–563 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  10. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology, Lecture Notes in Mathematics, vol. 1167, pp. 50–80. Springer, Berlin (1985)

  11. Dimock, J.: Algebras of local observables on a manifold. Comm. Math. Phys. 77, 219–228 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Dimock, J.: Dirac quantum fields on a manifold. Trans. Am. Math. Soc. 269, 133–147 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Doplicher, S.: Spin and statistics and first principles. Found. Phys. 40, 719–732 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Doplicher, S., Piacitelli, G.: Any compact group is a gauge group. Rev. Math. Phys. 14, 873–885 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Doplicher, S., Roberts, J.E.: Duals of compact Lie groups realized in the Cuntz algebras and their actions on C*-algebras. J. Funct. Anal. 74, 96–120 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Inv. Math. 98, 157–218 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–107 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Ehrenberg, W., Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. B 62, 8–21 (1949)

    Article  ADS  MATH  Google Scholar 

  19. Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, vol. 2. Addison-Wesley Publishing Company, Reading (1964)

    Google Scholar 

  20. Fewster, C.J.: Endomorphisms and automorphisms of locally covariant quantum field theories. Rev. Math. Phys. 25, 1–47 (2013)

    ADS  Google Scholar 

  21. Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: Kastler, D. (eds) (ed.) The Algebraic Theory of Superselection Sectors. Introduction and Recent Results, pp. 379–387. World Scientific Publishing, Singapore (1990)

  22. Guido, D., Longo, R., Roberts, J.E., Verch, R.: Charged sectors, spin and statistics in quantum field theory on curved spacetimes. Rev. Math. Phys. 13, 125–198 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Haag, R.: Local quantum physics. In: Springer Texts and Monographs in Physics, 2nd ed (1996)

  24. Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton University Press, Princeton (1987)

    Book  MATH  Google Scholar 

  25. Isham, C.J.: Spinor fields in four dimensional space-time. Proc. Roy. Soc. Lond. A 364, 591–599 (1978)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Isham, C.J.: Twisted Quantum Fields in a Curved Space-Time. Proc. R. Soc. London A. 362, 383–404 (1710)

  27. Lang, B.: Twisted Quantum Fields in Curved Spacetimes á la C.J. Isham from the point of view of Algebraic Quantum Field Theory. PhD Thesis, in preparation, Adv.: C.J. Fewster

  28. Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1998)

    MATH  Google Scholar 

  29. Quillen, D.: Higher algebraic K-theory. I. In: Algebraic K-theory, I: higher K-theories. In: Lecture Notes in Mathematics, vol. 341. Springer, Berlin (1973)

  30. Roberts, J.E.: Lectures on algebraic quantum field theory. In: Kastler, D. (eds) The Algebraic Theory of Uperselection Sectors. Introduction and Recent Results, pp. 1–112. World Scientific Publishing, Singapore (1990)

  31. Roberts, J.E.: More lectures on algebraic quantum field theory. In: Noncommutative Geometry: Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy. Lecture Notes in Mathematics, vol. 1831, pp. 263–339. Springer, Berlin (2000)

  32. Roberts, J.E., Ruzzi, G.: A cohomological description of connections and curvature over posets. Theory Appl. Categ. 16, 855–895 (2006)

    MATH  MathSciNet  Google Scholar 

  33. Roberts, J.E., Ruzzi, G., Vasselli, E.: A theory of bundles over posets. Adv. Math. 220, 125–153 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Roberts, J.E., Ruzzi, G., Vasselli, E.: Net bundles over posets and K-theory. Int. J. Math. 24, 1–34 (2013)

    Article  MathSciNet  Google Scholar 

  35. Ruzzi, G.: Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic spacetimes. Rev. Math. Phys. 17, 1021–1070 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ruzzi, G.: Punctured Haag duality in locally covariant quantum field theories. Comm. Math. Phys. 256, 621–634 (2005)

  37. Ruzzi, G., Vasselli, E.: A new light on nets of C*-algebras and their representations. Comm. Math. Phys. 312, 655–694 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Ruzzi, G., Vasselli, E.: Representations of nets of C*-algebras over S1. Comm. Math. Phys. 316, 127–149 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Vasselli, E.: Bundles of C*-categories, II: C*-dynamical systems and Dixmier-Douady invariants. J. Funct. Anal. 257, 357–387 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  40. Vasselli, E.: Presheaves of symmetric tensor categories and nets of C*-algebras. J. Noncommutative Geom, To appear. arXiv:1210.4302

  41. Vasselli, E.: A theory of gerbes over posets. In preparation

  42. Verch, R.: Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields in curved spacetime. Rev. Math. Phys. 9, 635–677 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Roma, Italy

    Ezio Vasselli

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  1. Ezio Vasselli
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Correspondence to Ezio Vasselli.

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Communicated by Y. Kawahigashi

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Vasselli, E. Presheaves of Superselection Structures in Curved Spacetimes. Commun. Math. Phys. 335, 895–933 (2015). https://doi.org/10.1007/s00220-014-2251-2

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