Advertisement

Annales Henri Poincaré

, Volume 18, Issue 10, pp 3325–3370 | Cite as

Hadamard States for Quantum Abelian Duality

  • Marco Benini
  • Matteo Capoferri
  • Claudio DappiaggiEmail author
Article

Abstract

Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a \(\hbox {C}^*\)-algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states on this algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three \(\hbox {C}^*\)-algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor and in the case of ultra-static globally hyperbolic spacetimes with compact Cauchy surfaces, we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors, we obtain a state for the full theory, ultimately implementing Abelian duality transformations as Hilbert space isomorphisms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to Nicolò Drago and Alexander Schenkel for stimulating discussions and valuable suggestions and would like to thank the anonymous referees for their encouragement towards improving the structure and clarity of the paper. M. C. and C. D. are grateful to the Institute of Mathematics of the University of Potsdam for the kind hospitality during the realization of part of this work. The work of M. B. has been supported by a research fellowship of the Alexander von Humboldt foundation. The work of M. C. has been partially supported by IUSS (Pavia). The work of C. D. has been supported by the University of Pavia.

References

  1. 1.
    Acerbi, F., Morchio, G., Strocchi, F.: Theta vacua, charge confinement and charged sectors from nonregular representations of CCR algebras. Lett. Math. Phys. 27, 1–11 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agullo, I., del Rio, A., Navarro-Salas, J.: Electromagnetic duality anomaly in curved spacetimes. arXiv:1607.08879 [gr-qc]
  3. 3.
    Bär, C.: Green-hyperbolic operators on globally hyperbolic spacetimes. Commun. Math. Phys. 333, 1585–1615 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bär, C., Becker, C.: Differential Characters. Lect. Notes Math. 2112, Springer, p. 198 (2014)Google Scholar
  5. 5.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave Equations on Lorentzian Manifolds and Quantization. American Mathematical Society, Providence (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Becker, C., Benini, M., Schenkel, A., Szabo, R.J.: Cheeger-Simons differential characters with compact support and Pontryagin duality. arXiv:1511.00324 [math.DG]
  7. 7.
    Becker, C., Benini, M., Schenkel, A., Szabo, R.J.: Abelian duality on globally hyperbolic spacetimes. Commun. Math. Phys. 349, 361–392 (2017). doi: 10.1007/s00220-016-2669-9 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Becker, C., Schenkel, A., Szabo, R.J.: Differential cohomology and locally covariant quantum field theory. arXiv:1406.1514 [hep-th]
  9. 9.
    Benini, M.: Optimal space of linear classical observables for Maxwell \(k\)-forms via spacelike and timelike compact de Rham cohomologies. J. Math. Phys. 57, 053502 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Benini, M., Dappiaggi, C., Hack, T.-P., Schenkel, A.: A \(C^\ast \)-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds. Commun. Math. Phys. 332, 477–504 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benini, M., Dappiaggi, C., Murro, S.: Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states. J. Math. Phys. 55, 082301 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Benini, M., Dappiaggi, C., Schenkel, A.: Quantized Abelian principal connections on Lorentzian manifolds. Commun. Math. Phys. 330, 123–152 (2014). arXiv:1303.2515 [math-ph]
  13. 13.
    Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.): Advances in Algebraic Quantum Field Theory. Springer International Publishing, Berlin (2015)zbMATHGoogle Scholar
  14. 14.
    Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31–68 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Capoferri, M.: Algebra of observables and states for quantum Abelian duality. M.Sc. Thesis, University of Pavia (2016). arXiv:1611.09055 [math-ph]
  16. 16.
    Cheeger, J., Simons, J.: Differential characters and geometric invariants. Lect. Notes Math. 1167, Springer (1985)Google Scholar
  17. 17.
    Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. Lett. Math. Phys. 101, 265–287 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dappiaggi, C., Siemssen, D.: Hadamard states for the vector potential on asymptotically flat spacetimes. Rev. Math. Phys. 25, 1350002 (2013). arXiv:1106.5575 [gr-qc]
  19. 19.
    Dimock, J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4, 223–233 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dixmier, J.: C\(^*\)-Algebras. North Holland Publishing Company, Amsterdam (1977)zbMATHGoogle Scholar
  21. 21.
    Fewster, C.J., Pfenning, M.J.: A quantum weak energy inequality for spin-one fields in curved space–time. J. Math. Phys. 44, 4480 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fewster, C.J., Lang, B.: Dynamical locality of the free Maxwell field. Ann. Henri Poincaré 17, 401–436 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Freed, D.S.: Dirac charge quantization and generalized differential cohomology. In Cambridge 2000, Surveys in Differential Geometry, pp. 129–194 [hep-th/0011220]Google Scholar
  24. 24.
    Freed, D.S., Moore, G.W., Segal, G.: Heisenberg groups and noncommutative fluxes. Ann. Phys. 322, 236–285 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Freed, D.S., Moore, G.W., Segal, G.: The uncertainty of fluxes. Commun. Math. Phys. 271, 247–274 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gérard, C., Wrochna, M.: Hadamard states for the linearized Yang–Mills equation on curved spacetime. Commun. Math. Phys. 337(1), 253–320 (2015). arXiv:1403.7153 [math-ph]
  27. 27.
    Guichardet, A.: Tensor product of C\(^*\)-algebras. Sov. Math. 6, 210–213 (1965), and Lect. Notes Series no. 12, Aarhus Universitet (1969)Google Scholar
  28. 28.
    Harvey, F.R., Lawson Jr., H.B., Zweck, J.: The de Rham–Federer theory of differential characters and character duality. Am. J. Math. 125, 791–847 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  30. 30.
    Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70, 329–452 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Classics in Mathematics, Springer, p. 440 (2003)Google Scholar
  32. 32.
    Moretti, V., Khavkine, I.: Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds.) Advances in Algebraic Quantum Field Theory, Springer (2015). arXiv:1412.5945 [math-ph]
  33. 33.
    Manuceau, J., Sirugue, M., Testard, D., Verbeure, A.: The smallest \(C^*\)-algebra for canonical commutations relations. Commun. Math. Phys. 32, 231–243 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space–time. Commun. Math. Phys. 179, 529–553 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705–731 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sahlmann, H., Verch, R.: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203–1246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sanders, K., Dappiaggi, C., Hack, T.P.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law. Commun. Math. Phys. 328, 625–667 (2014). arXiv:1211.6420 [math-ph]
  38. 38.
    Schubert, S.: Über die Charakterisierung von Zuständen hinsichtlich der Erwartungswerte quadratischer Operatoren. M.Sc. Thesis, Universität Hamburg (2013)Google Scholar
  39. 39.
    Simons, J., Sullivan, D.: Axiomatic characterization of ordinary differential cohomology. J. Topol. 1, 45–56 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Szabo, R.J.: Quantization of higher Abelian Gauge theory in generalized differential cohomology. PoS ICMP 2012, 009 (2012). arXiv:1209.2530 [hep-th]
  41. 41.
    Wald, R.M.: Quantum Field Theory on Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics, University of Chicago Press, p. 220 (1994)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marco Benini
    • 1
  • Matteo Capoferri
    • 2
  • Claudio Dappiaggi
    • 3
    • 4
    Email author
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Department of MathematicsUniversity College LondonLondonUK
  3. 3.Dipartimento di FisicaUniversità degli Studi di PaviaPaviaItaly
  4. 4.Istituto Nazionale di Fisica Nucleare - Sezione di PaviaPaviaItaly

Personalised recommendations