Skip to main content
SpringerLink
Log in

Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider a class of translation invariant models of non-relativistic QED with net charge. Under certain natural assumptions we prove that ground states do not exist in the Fock space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amour, L., Grébert, B., Guillot, J.: The dressed mobile atoms and ions. J. Math. Pures Appl. (9) 86(3), 177–200 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Arai, A., Hirokawa, M., Hiroshima, F.: On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168(2), 470–497 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999)

    Article  MATH  ADS  Google Scholar 

  5. Chen, T.: Operator-theoretic infrared renormalization and construction of dressed one-particle states in non-relativistic QED. preprint mp_arc 01-310

  6. Chen, T.: Infrared renormalization in non-relativistic QED and scaling criticality. http://arxiv.org/list/math-ph/0601010, 2006

  7. Chen, T., Fröhlich, J.: Coherent infrared representations in non-relativistic QED. In: Proceedings of “Spectral Theory and Mathematical Physics”, a Festscrift in honor of Barey Simon’s 60th birthday, Vol. 1, Proc. Symp. Pure Math. 76, Providence, RI: Amer. Math. Soc., 2007, pp. 25–46

  8. Dereziński, J., Gérard, C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 5(3), 523–577 (2004)

    Article  Google Scholar 

  9. Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré Sect. A (N.S.) 19, 1–103 (1973)

    Google Scholar 

  10. Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Forts. Der Phy. 22, 159–198 (1974)

    Article  Google Scholar 

  11. Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Hiroshima, F.: Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of the coupling constants. Ann. Henri Poincaré 3, 171–201 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hirokawa, M.: Infrared catastrophe for Nelson’s model—non-existence of ground state and soft-boson divergence. Publ. Res. Inst. Math. Sci. 42(4), 897–922 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)

    Google Scholar 

  15. Kulish, P., Fadeev, L.D.: Theor. and Math. Phys. 4, 247 (1970)

    Google Scholar 

  16. Loss, M., Miyao, T., Spohn, H.: Lowest energy states in nonrelativistic QED: atoms and ions in motion. J. Funct. Anal. 243(2), 353–393 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lörinczi, J., Minlos, R.A., Spohn, H.: The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3(2), 269–295 (2002)

    Article  MATH  Google Scholar 

  18. Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4(3), 439–486 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pizzo, A.: Scattering of an infraparticle: The one particle sector in Nelson’s massless model. Ann. Henri Poncaré 6(3), 553–606 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schweber, S.: An introduction to relativistic quantum field theory. Evanston, IL/Elmsford, NY: Row, Peterson and Company, 1961

  21. Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  22. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Press, Princeton, NJ (1971)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904-4137, USA

    D. Hasler & I. Herbst

Authors
  1. D. Hasler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Herbst
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Hasler.

Additional information

Communicated by I.M. Sigal

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hasler, D., Herbst, I. Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED. Commun. Math. Phys. 279, 769–787 (2008). https://doi.org/10.1007/s00220-008-0444-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0444-2

Keywords

Search

Navigation