Annales Henri Poincaré

, Volume 11, Issue 8, pp 1545–1589 | Cite as

Absence of Embedded Mass Shells: Cerenkov Radiation and Quantum Friction

  • Wojciech De Roeck
  • Jürg Fröhlich
  • Alessandro Pizzo
Article

Abstract

We show that, in a model where a non-relativistic particle is coupled to a quantized relativistic scalar Bose field, the embedded mass shell of the particle dissolves in the continuum when the interaction is turned on, provided the coupling constant is sufficiently small. More precisely, under the assumption that the fiber eigenvectors corresponding to the putative mass shell are differentiable as functions of the total momentum of the system, we show that a mass shell could exist only at a strictly positive distance from the unperturbed embedded mass shell near the boundary of the energy–momentum spectrum.

References

  1. 1.
    Angelescu N., Minlos R.A., Zagrebnov V.A.: Lower spectral branches of a particle coupled to a Bose field. Rev. Math. Phys. 17(9), 1–32 (2005)MathSciNetGoogle Scholar
  2. 2.
    Angelescu N., Minlos R.A., Zagrebnov V.A.: Lower spectral branches of a spin-boson model. J. Math. Phys. 49, 102105 (2008)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Bach V., Chen T., Fröhlich J., Sigal I.M.: The renormalized electron mass in non-relativistic quantum electrodynamics. J. Funct. Anal. 243(2), 426–535 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 254(10), 2555–2647 (2007)CrossRefGoogle Scholar
  5. 5.
    Chen, T., Fröhlich, J.: Coherent infrared representations in nonrelativistic QED. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proc. Symp. Pure Math. AMS (2007)Google Scholar
  6. 6.
    Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in QED: II. Mass shell properties. J. Math. Phys. 50, 012103 (2009)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Chen, T., Fröhlich, J., Pizzo, A.: Infraparticle scattering states in QED: I. The Bloch-Nordsieck paradigm. Commun. Math. Phys. doi:10.1007/s00220-009-0960-x
  8. 8.
    Erdös L.: Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys 107(5–6), 1043–1127 (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Cycon H.L., Froese R.G., Kirsch W., Simon B.: Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin (1987)Google Scholar
  10. 10.
    Fröhlich J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Inst. Henri Poincaré, Section Physique Théorique 19(1), 1–103 (1973)Google Scholar
  11. 11.
    Fröhlich J.: Existence of dressed one electron states in a class of persistent models. Fortschritte der Physik 22, 159–198 (1974)CrossRefGoogle Scholar
  12. 12.
    Fröhlich J., Pizzo A.: On the absence of excited eigenstates in QED. Commun. Math. Phys. 286(3), 803–836 (2009)MATHCrossRefADSGoogle Scholar
  13. 13.
    Fröhlich J., Pizzo A.: The renormalized electron mass in non-relativistic QED. Commun. Math. Phys. 294(3), 761–825 (2010)CrossRefADSGoogle Scholar
  14. 14.
    Fröhlich J., Griesemer M., Sigal I.M.: Spectral theory for the standard model of non-relativistic QED. Commun. Math. Phys. 283, 613–646 (2008)MATHCrossRefADSGoogle Scholar
  15. 15.
    Georgescu V., Gérard C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208(2), 275–281 (1999)MATHCrossRefADSGoogle Scholar
  16. 16.
    Hasler D., Herbst I.: Absence of ground states for a class of translation invariant models of non-relativistic QED. Commun. Math. Phys. 279(3), 769–787 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Schach-Møller J.: The translation invariant Nelson model: I. The bottom of the spectrum. Ann. H. Poincaré 6(6), 1091–1135 (2005)CrossRefGoogle Scholar
  18. 18.
    Pizzo A.: One-particle (improper) states in Nelson’s massless model. Ann. H. Poincaré 4(3), 439–486 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pizzo A.: Scattering of an infraparticle: the one particle sector in Nelson’s massless model. Ann. H. Poincaré 6, 553–606 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Spohn H.: The polaron at large total momentum. J. Phys. A 21, 1199–1212 (1988)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Wojciech De Roeck
    • 1
  • Jürg Fröhlich
    • 2
  • Alessandro Pizzo
    • 3
  1. 1.Institut für Theoretische PhysikUniversität HeidelbergHeidelbergGermany
  2. 2.Institute of Theoretical PhysicsETH ZürichZurichSwitzerland
  3. 3.Department of MathematicsUniversity of California DavisDavisUSA

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