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Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 415–476 | Cite as

Asymptotic Completeness for Compton Scattering

  • J. FröhlichEmail author
  • M. Griesemer
  • B. Schlein
Article

Abstract

Scattering in a model of a massive quantum-mechanical particle, an ‘‘electron’’, interacting with massless, relativistic bosons, ‘‘photons’’, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of ‘‘photons’’ by the ‘‘electron’’; but ‘‘electron-positron’’ pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed ‘‘electron’’ is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the ‘‘electron’’ and the ‘‘photons’’.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Electron State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ammari, Z.: Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3(3), 217–285, (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arai, A.: A note on scattering theory in nonrelativistic quantum electrodynamics. J. Phys. A 16(1), 49–69 (1983)zbMATHGoogle Scholar
  3. 3.
    Bach, V.: Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998)Google Scholar
  4. 4.
    Bach, V., Klopp, F., Zenk, H.: Mathematical analysis of the photoelectric effect. Adv. Theor. Math. Phys. 5(6), 969–999 (2001)zbMATHGoogle Scholar
  5. 5.
    Bloch, F., Nordsieck, A.: Note on the radiation field of the electron. Phys. Rev. 52, 54–59 (1937)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, T.: Operator-theoretic infrared renormalization and construction of dressed 1–particle states. Preprint, http://www.ma.utexas.edu/mp-arc/01-310, 2001
  7. 7.
    Davies, E.B.: The functional calculus. J. London Math. Soc. (2), 52(1), 166–176 (1995)Google Scholar
  8. 8.
    Dereziński, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(4), 383–450 (1999)Google Scholar
  9. 9.
    Dereziński, J., Gérard, C.: Spectral and scattering theory of spatially cut-off P(φ)2 Hamiltonians. Commun. Math. Phys. 213(1), 39–125 (2000)CrossRefGoogle Scholar
  10. 10.
    Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164(2), 349–398 (2001)CrossRefGoogle Scholar
  11. 11.
    Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré, 3, 107–170 (2002)Google Scholar
  12. 12.
    Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré, Sect. A XIX(1), 1–103 (1973)Google Scholar
  13. 13.
    Fröhlich, J.: Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. 22, 159–198 (1974)Google Scholar
  14. 14.
    Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys. 14, 1165–1280 (2002)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hunziker, W., Sigal, I.M.: The quantum N–body problem. J. Math. Phys. 41(6), 3448–3510 (2000)CrossRefzbMATHGoogle Scholar
  16. 16.
    Jost, R.: The general theory of quantized fields. In: M. Kac, (ed.), (Proceedings of the Summer Seminar, Boulder, Colorado, 1960), Volume IV, Lectures in Applied Mathematics, 1965Google Scholar
  17. 17.
    Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)Google Scholar
  18. 18.
    Pauli, W., Fierz, M.: Zur Theorie der Emission langwelliger Lichtquanten. Nuovo Cimento 15, 167–188 (1938)zbMATHGoogle Scholar
  19. 19.
    Pizzo, A.: One particle (improper) states and scattering states in Nelson’s massless model. http://arxiv:org/abs/math-ph/0010043, 2000
  20. 20.
    Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38(5), 2281–2296 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Spohn, H.: The polaron model at large momentum. J. Phys. A: Math. Gen. 21, 1199–1211 (1988)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Sigal, I.M., Soffer, A.: Local decay and propagation estimates for time–dependent and time–independent Hamiltonians. Princeton University preprint, 1988Google Scholar
  23. 23.
    Reed, M., Simon, B.: Methods of modern mathematical physics: Scattering Theory. Vol. 3. New York: Academic Press, 1979Google Scholar
  24. 24.
    Dereziński, J., Gérard, C.: Scattering theory of classical and quantum N-particle systems. Berlin-Heidelberg-New York: Springer, 1997Google Scholar
  25. 25.
    Kato, T.: Perturbation theory for linear operators. New York: Springer-Verlag, 1966Google Scholar
  26. 26.
    Yennie, D., Frautschi, S., Suura, H.: The infrared divergence phenomena and high-energy processes. Ann. Phys. 13, 379–452 (1961)CrossRefGoogle Scholar
  27. 27.
    Spencer, T., Zirilli, F.: Scattering states and bound states in Open image in new window Commun. Math. Phys. 49(1), 1–16 (1976)Google Scholar
  28. 28.
    Spencer, T.: The decay of the Bethe-Salpeter kernel in P(φ)2 quantum field models. Commun. Math. Phys. 44(2), 143–164 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Theoretical Physics, ETH–HönggerbergZürichSwitzerland
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  3. 3.Department of MathematicsStanford UniversityStanfordCAUSA

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