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Asymptotic Completeness for Compton Scattering


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’’.

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  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)

    Article  MATH  Google Scholar 

  2. Arai, A.: A note on scattering theory in nonrelativistic quantum electrodynamics. J. Phys. A 16(1), 49–69 (1983)

    MATH  Google Scholar 

  3. Bach, V.: Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998)

    Google Scholar 

  4. Bach, V., Klopp, F., Zenk, H.: Mathematical analysis of the photoelectric effect. Adv. Theor. Math. Phys. 5(6), 969–999 (2001)

    MATH  Google Scholar 

  5. Bloch, F., Nordsieck, A.: Note on the radiation field of the electron. Phys. Rev. 52, 54–59 (1937)

    Article  MATH  Google Scholar 

  6. Chen, T.: Operator-theoretic infrared renormalization and construction of dressed 1–particle states. Preprint,, 2001

  7. Davies, E.B.: The functional calculus. J. London Math. Soc. (2), 52(1), 166–176 (1995)

  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. Dereziński, J., Gérard, C.: Spectral and scattering theory of spatially cut-off P(φ)2 Hamiltonians. Commun. Math. Phys. 213(1), 39–125 (2000)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré, 3, 107–170 (2002)

    Google Scholar 

  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)

  13. Fröhlich, J.: Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys. 22, 159–198 (1974)

    Google Scholar 

  14. Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys. 14, 1165–1280 (2002)

    Article  MathSciNet  Google Scholar 

  15. Hunziker, W., Sigal, I.M.: The quantum N–body problem. J. Math. Phys. 41(6), 3448–3510 (2000)

    Article  MATH  Google Scholar 

  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, 1965

  17. Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)

    Google Scholar 

  18. Pauli, W., Fierz, M.: Zur Theorie der Emission langwelliger Lichtquanten. Nuovo Cimento 15, 167–188 (1938)

    MATH  Google Scholar 

  19. Pizzo, A.: One particle (improper) states and scattering states in Nelson’s massless model. http://arxiv:org/abs/math-ph/0010043, 2000

  20. Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38(5), 2281–2296 (1997)

    Article  MATH  Google Scholar 

  21. Spohn, H.: The polaron model at large momentum. J. Phys. A: Math. Gen. 21, 1199–1211 (1988)

    Article  MathSciNet  Google Scholar 

  22. Sigal, I.M., Soffer, A.: Local decay and propagation estimates for time–dependent and time–independent Hamiltonians. Princeton University preprint, 1988

  23. Reed, M., Simon, B.: Methods of modern mathematical physics: Scattering Theory. Vol. 3. New York: Academic Press, 1979

  24. Dereziński, J., Gérard, C.: Scattering theory of classical and quantum N-particle systems. Berlin-Heidelberg-New York: Springer, 1997

  25. Kato, T.: Perturbation theory for linear operators. New York: Springer-Verlag, 1966

  26. Yennie, D., Frautschi, S., Suura, H.: The infrared divergence phenomena and high-energy processes. Ann. Phys. 13, 379–452 (1961)

    Article  Google Scholar 

  27. Spencer, T., Zirilli, F.: Scattering states and bound states in Commun. Math. Phys. 49(1), 1–16 (1976)

    Google Scholar 

  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 

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Correspondence to J. Fröhlich.

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Communicated by H. Spohn

Dedicated to Freeman Dyson on the occasion of his 80th birthday

Work partially supported by U.S. National Science Foundation grant DMS 01-00160.

Acknowledgement. We thank V. Bach for his hospitality at the University of Mainz, where part of this work was done, and we are indebted to Gian Michele Graf for pointing out a serious gap in an earlier version of this paper. We also thank one of the referees for pointing out many typos and some small errors.

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Fröhlich, J., Griesemer, M. & Schlein, B. Asymptotic Completeness for Compton Scattering. Commun. Math. Phys. 252, 415–476 (2004).

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