Advertisement

Communications in Mathematical Physics

, Volume 364, Issue 1, pp 203–232 | Cite as

N-Particle Scattering in Relativistic Wedge-Local Quantum Field Theory

  • Maximilian Duell
Article

Abstract

Multi-particle scattering states are constructed for massive Wigner particles in the general operator-algebraic setting of wedge-local quantum field theory. The apparent geometrical restriction of the conventional wedge-local Haag–Ruelle argument to two-particle scattering states is overcome with a swapping symmetry argument based on wedge duality.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

I am deeply indebted to Wojciech Dybalski for many valuable suggestions and his continuous support. Further I would like thank Detlev Buchholz for comments and communicating Lemma 3, Daniela Cadamuro for helpful discussions, and Yoh Tanimoto for comments on swapping and the foundations of Tomita-Takesaki theory. I also gratefully acknowledge funding by the DFG within Grant DY107/2-1.

References

  1. A.
    Araki, H.: Mathematical Theory of Quantum Fields, No. 101 in International Series of Monographs on Physics. Oxford University Press, Oxford (1999)Google Scholar
  2. AD17.
    Alazzawi S., Dybalski W.: Compton scattering in the Buchholz–Roberts framework of relativistic QED. Lett. Math. Phys. 107, 81–106 (2017).  https://doi.org/10.1007/s11005-016-0889-8 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. AG01.
    Albeverio S., Gottschalk H.: Scattering theory for quantum fields with indefinite metric. Commun. Math. Phys. 216, 491–513 (2001).  https://doi.org/10.1007/s002200000332 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Arv82.
    Arveson, W.: The harmonic analysis of automorphism groups. In: Operator Algebras and Applications, Part I (Kingston, Ont., 1980), no. 38 in Proceedings of Symposia in Pure Mathematics, pp. 199–269. AMS. (1982)  https://doi.org/10.1090/pspum/038.1 zbMATHGoogle Scholar
  5. BDN15.
    Bachmann S., Dybalski W., Naaijkens P.: Lieb–Robinson bounds, Arveson spectrum and Haag–Ruelle scattering theory for gapped quantum spin systems. Ann. Henri Poincaré 17, 1737–1791 (2015).  https://doi.org/10.1007/s00023-015-0440-y ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. BaW84.
    Baumgärtel H., Wollenberg M.: A class of nontrivial weakly local massive Wightman fields with interpolating properties. Commun. Math. Phys. 94, 331–352 (1984).  https://doi.org/10.1007/BF01224829 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. BiW75.
    Bisognano J.J., Wichmann E.H.: On the duality condition for a Hermitian scalar field. J. Math. Phys. 16, 985–1007 (1975).  https://doi.org/10.1063/1.522605 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Bor95.
    Borchers H.-J.: When does Lorentz invariance imply wedge duality?. Lett. Math. Phys. 35, 39–60 (1995).  https://doi.org/10.1007/BF00739154 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. BBS01.
    Borchers H.-J., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001).  https://doi.org/10.1007/s002200100411 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. BR1.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1. Springer, Berlin (1987)  https://doi.org/10.1007/978-3-662-02520-8 CrossRefGoogle Scholar
  11. Bu77.
    Buchholz D.: Collision theory for massless bosons. Commun. Math. Phys. 52, 147–173 (1977).  https://doi.org/10.1007/BF01625781 ADSMathSciNetCrossRefGoogle Scholar
  12. Bu17.
    Buchholz, D.: private communications, (2017)Google Scholar
  13. BLS11.
    Buchholz D., Lechner G., Summers S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011).  https://doi.org/10.1007/s00220-010-1137-1 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. BS08.
    Buchholz, D., Summers, S.J.: Warped convolutions: a novel tool in the construction of quantum field theories. In Seiler, E., Sibold K. (ed.) Quantum Field Theory and Beyond. Essays in Honor of Wolfhart Zimmermann. World Scientific, Singapore, pp.107–121 (2008).  https://doi.org/10.1142/9789812833556_0007
  15. DFR95.
    Doplicher S., Fredenhagen K., Roberts J.E.: The quantum structure of space-time at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187–220 (1995).  https://doi.org/10.1007/BF02104515 ADSCrossRefzbMATHGoogle Scholar
  16. Du17.
    Duell M.: Strengthened Reeh–Schlieder property and scattering in quantum field theories without mass gaps. Commun. Math. Phys. 352, 935–966 (2017).  https://doi.org/10.1007/s00220-017-2841-x ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. Dy05.
    Dybalski W.: Haag–Ruelle scattering theory in presence of massless particles. Lett. Math. Phys. 72, 27–38 (2005).  https://doi.org/10.1007/s11005-005-2294-6 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. DG14.
    Dybalski W., Gérard C.: A criterion for asymptotic completeness in local relativistic QFT. Commun. Math. Phys. 332, 1167–1202 (2014).  https://doi.org/10.1007/s00220-014-2069-y ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. DT11.
    Dybalski W., Tanimoto Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Commun. Math. Phys. 305, 427–440 (2011).  https://doi.org/10.1007/s00220-010-1173-x ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. FGR96.
    Fredenhagen K., Gaberdiel M.R., Rüger S.M.: Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory. Commun. Math. Phys. 175, 319–335 (1996).  https://doi.org/10.1007/BF02102411 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. GL07.
    Grosse H., Lechner G.: Wedge-local quantum fields and noncommutative Minkowski space. JHEP 11, 012 (2007).  https://doi.org/10.1088/1126-6708/2007/11/012 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. Ha58.
    Haag R.: Quantum field theories with composite particles and asymptotic conditions. Phys. Rev. 112, 669–673 (1958).  https://doi.org/10.1103/PhysRev.112.669 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. Hep65.
    Hepp K.: On the connection between the LSZ and Wightman quantum field theory. Commun. Math. Phys. 1, 95–111 (1965).  https://doi.org/10.1007/BF01646494 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. Her13.
    Herdegen A.: Infraparticle problem, asymptotic fields and Haag–Ruelle theory. Ann. Henri Poincaré 15, 345–367 (2013).  https://doi.org/10.1007/s00023-013-0242-z ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. KR2.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II, Advanced Theory, vol. 16 of Graduate Studies in Mathematics. American Mathematical Society, Providence (1997)Google Scholar
  26. Le03.
    Lechner G.: Polarization free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003).  https://doi.org/10.1023/A:1025772304804 MathSciNetCrossRefzbMATHGoogle Scholar
  27. Le06.
    Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D. thesis, Universität Göttingen, (2006). arXiv:math-ph/0611050
  28. Le15.
    Lechner, G.: Algebraic constructive quantum field theory: integrable models and deformation techniques. In: Brunetti R., Dappiaggi C., Fredenhagen K., Yngvason J. (eds.) Advances in Algebraic Quantum Field Theory, pp. 397–448. Springer, Berlin (2015)  https://doi.org/10.1007/978-3-319-21353-8_10 CrossRefGoogle Scholar
  29. LTU17.
    Longo, R., Tanimoto, Y., Ueda, Y.: Free products in AQFT. arXiv:1706.06070
  30. Mo18.
    Morinelli V.: The Bisognano–Wichmann property on nets of standard subspaces, some sufficient conditions. Ann. Henri Poincaré. 19, 937–958 (2018).  https://doi.org/10.1007/s00023-017-0636-4 arXiv:1703.06831 MathSciNetCrossRefzbMATHGoogle Scholar
  31. RS2.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 2, Fourier Analysis and Self-Adjointness. Academic Press, San Diego (1975)Google Scholar
  32. RS3.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 3, Scattering Theory. Academic Press, San Diego (1979)Google Scholar
  33. Ru62.
    Ruelle D.: On the asymptotic condition in quantum field theory. Helv. Phys. Acta. 35, 147–163 (1962).  https://doi.org/10.5169/seals-113272 MathSciNetCrossRefzbMATHGoogle Scholar
  34. Smi.
    Smirnov, F. A.: Form Factors in Completely Integrable Models of Quantum Field Theory, vol. 14 of Advanced Series in Mathematical Physics. World Scientific, Singapore (1992)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany

Personalised recommendations