Skip to main content
SpringerLink
Log in

On Adiabatic Pair Creation

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

Abstract

We give here a rigorous proof of the well known prediction of pair creation as it arises from the Dirac equation with an external time dependent potential. Pair creation happens with probability one if the potential changes adiabatically in time and becomes overcritical, which means that an eigenvalue curve (as a function of time) bridges the gap between the negative and positive spectral continuum. The potential can be thought of as being zero at large negative and large positive times. The rigorous treatment of this effect has been lacking since the pioneering work of Beck, Steinwedel and Süßmann [1] in 1963 and Gershtein and Zeldovich [8] in 1970.

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. Beck F., Steinwedel H., Süssmann G.: Bemerkungen zum Klein’schen Paradoxon. Z. Phys 171, 189–198 (1963)

    Article  ADS  MATH  Google Scholar 

  2. Bhabba H.J.: The Creation of Electron Pairs by Fast Charged Particles. Proc. R. Soc. London Ser. A 152, 559–586 (1935)

    Article  ADS  Google Scholar 

  3. Brezin E., Itzykson C.: Pair Production in Vacuum by an Alternating Field. Phys. Rev. D. 2, 1191–1199 (1970)

    Article  ADS  Google Scholar 

  4. Cowan T. et al.: Observation of correlated narrow-peak structures in positron and electron spectra from superheavy collision systems. Phys. Rev. Lett. 56, 444–447 (1986)

    Article  ADS  Google Scholar 

  5. Dirac P.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1930)

    MATH  Google Scholar 

  6. Dolbeault, J., Esteban, M.J., Loss, M.: Relativistic hydrogenic atoms in strong magnetic fields. http://arxiv.org/list/math/0607027v1, 2006

  7. Dürr D., Pickl P.: Flux-across-surfaces Theorem for a Dirac-particle. J. Math. Phys. 44, 423–465 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Gershtein S., Zeldovich Y.: Positron Production During the Mutual Approach of Heavy Nuclei and the Polarization of the Vacuum. Sov. Phys. JETP 30, 358–361 (1970)

    ADS  Google Scholar 

  9. Greiner W., Müller B., Rafelski J.: Quantum Electrodynamics of Strong Fields. Springer Verlag, Berlin (1985)

    Google Scholar 

  10. Hainzl C., Lewin M., Solovej J.P.: Mean-field approximation in Quantum Electrodynamics. The no-photon case. Comm. Pure Appl. Math. 60, 546–596 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heisenberg W., Euler H.: Consequences of Dirac’s Theory of the Positron. Z. Phys. 98, 714 (1936)

    Article  ADS  Google Scholar 

  12. Ikebe T.: Eigenfunction expansions associated with the Schrödinger operators and their application to scattering theory. Arch. Rat. Mech. Anal. 5, 1–34 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jensen A., Kato T.: Spectral Properties of Schrödinger operators and time-decay of the wavefunctions. Duke Math. J. 46(3), 583–611 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klaus M.: On coupling constant thresholds and related eigenvalue properties of Dirac operators. J. Reine Angew.Math. 362, 197–212 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Klein O.: Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157 (1929)

    Article  ADS  Google Scholar 

  16. Müller B.: Positron creation in superheavy quasimolecules. Ann. Rev. Nucl. Science 26, 351–383 (1976)

    Article  ADS  Google Scholar 

  17. Müller B., Peitz H., Rafelski J., Greiner W.: Solutions of the Dirac Equation for Strong External Fields. Phys. Rev. Lett. 28, 1235–1238 (1972)

    Article  ADS  Google Scholar 

  18. Nenciu G.: On the adiabatic limit for Dirac particles in external fields. Commun. Math. Phys. 76, 117–128 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  19. Nenciu G.: Existence of spontaneous pair creation in the external field approximation of Q.E.D. Commun. Math. Phys. 109, 303–312 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. O’Connell R.F.: Effect of the Anomalous Magnetic Moment of the Electron on Spontaneous Pair Production in a Strong Magnetic Field. Phys. Rev. Lett. 21, 397–398 (1968)

    Article  ADS  Google Scholar 

  21. Pickl, P.: Existence of Spontaneous Pair Creation, Dissertation, 2005

  22. Pickl P.: Generalized Eigenfunctions for Dirac Operators Near Criticality. J. Math. Phys. 48, 1 (2007)

    Article  MathSciNet  Google Scholar 

  23. Pickl P., Dürr D.: Adiabatic Pair Creation in Heavy Ion and Laser Fields. Eur. Phys. Lett. 81, 40001 (2008)

    Article  ADS  Google Scholar 

  24. Popov V.S.: Positron Production in a Coulomb Field with Z > 137. Zh. Eksp. Teor. Fiz. 59, 965–84 (1970)

    Google Scholar 

  25. Prodan E.: Spontaneous transitions in quantum mechanics. J. Phys. A: Math. Gen. 32, 4877–4881 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Rafelski J., Fulcher L.P., Greiner W.: Superheavy Elements and an Upper Limit to the Electric Field Strength. Phys. Rev. Lett. 27, 958–961 (1971)

    Article  ADS  Google Scholar 

  27. Reed M., Simon B.: Functional Analysis. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  28. Rein D.: Über den Grundzustand überschwerer Atome. Z. Phys 221, 423–430 (1969)

    Article  ADS  Google Scholar 

  29. Reinhardt J., Müller U., Müller B., Greiner W: The decay of the vacuum in the field of superheavy nuclear systems. Z. f. Physik A 303, 173–188 (1981)

    Article  ADS  Google Scholar 

  30. Riesz F., von Sz.-Nagy B.: Functional Analysis. F. Ungar. Publ. Co., New York (1955)

    Google Scholar 

  31. Roberts C.D., Schmidt S.M., Vinnik D.V.: Quantum Effects with an X-Ray Free-Electron Laser. Phys. Rev. Lett. 89, 153901 (2002)

    Article  ADS  Google Scholar 

  32. Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time dependent potentials. Invent. math. 155, 451–513 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Sauter F.: Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs. Z. Phys. 69, 742 (1931)

    Article  ADS  MATH  Google Scholar 

  34. Scharf G., Seipp H.P.: Charged Vacuum, Spontaneous Positron Production and all that. Phys. Lett. 108B, 196–198 (1982)

    ADS  Google Scholar 

  35. Schweppe J. et al.: Observation of a Peak Structure in Positron Spectra from U+Cm Collisions. Phys. Rev. Lett. 51, 2261–2264 (1983)

    Article  ADS  Google Scholar 

  36. Schwinger J.: On Gauge Invariance and Vacuum Polarization. Phys. Rev. 82, 664–679 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Smith K., Peitz H., Müller B., Greiner W.: Induced Decay of the Neutral Vaccum in Overcritical Fields Occurring in Heavy-Ion Collisions. Phys. Rev. Lett. 32, 554–556 (1974)

    Article  ADS  Google Scholar 

  38. Teufel S.: Adiabatic Perturbation Theory in Quantum Dynamics. Springer Verlag, Berlin (2003)

    MATH  Google Scholar 

  39. Teufel, S.: The flux-across-surfaces theorem and its implications for scattering theory. Dissertiation an der Ludwig-Maximilians-Universität, München, 1999

  40. Thaller B.: The Dirac equation. Springer Verlag, Berlin (1992)

    Google Scholar 

  41. Yamada O.: Eigenfunction expansions and scattering theory for Dirac operators. Publ. RIMS. Kyoto Univ. 11, 651–689 (1976)

    Google Scholar 

  42. Zeldovich Ya.B., Popov V.S.: Electronic Structure of Superheavy Atoms. Sov. Phys. Usp. 14(6), 673–694 (1972)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090, Vienna, Austria

    Peter Pickl

  2. Mathematisches Institut der Universität München, Theresienstraße 39, 80333, München, Germany

    Peter Pickl & Detlef Dürr

Authors
  1. Peter Pickl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Detlef Dürr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Pickl.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pickl, P., Dürr, D. On Adiabatic Pair Creation. Commun. Math. Phys. 282, 161–198 (2008). https://doi.org/10.1007/s00220-008-0530-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0530-5

Keywords

Search

Navigation