Abstract
Within the framework of nonrelativistic QED, we prove that, for small values of the coupling constant, the energy function, \({E_{\vec{P}}}\), of a dressed electron is twice differentiable in the momentum \({\vec{P}}\) in a neighborhood of \({\vec{P}=0}\). Furthermore, \({\frac{\partial^2E_{\vec{P}}}{(\partial |\vec{P}|)^2}}\) is bounded from below by a constant larger than zero. Our results are proven with the help of iterative analytic perturbation theory.
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References
Bach V., Froehlich J., Pizzo A.: Infrared-finite algorithms in QED II. The expansion of the groundstate of an atom interacting with the quantized radiation field. Adv. Math. 220(4), 1023–1074 (2009)
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)
Chen T.: Infrared renormalization in nonrelativistic QED and scaling criticality. J. Funct. Anal. 254(10), 2555–2647 (2008)
Chen, T., Fröhlich, J.: Coherent Infrared Representations in Nonrelativistic QED. Proc. Symp. Pure Math. 76 (B. Simon 60-th Birthday Volume), Providence, RI: Amer. Math. Soc., 2007, pp. 25–46
Chen, T., Fröhlich, J., Pizzo, A.: Infraparticle scattering states in non-relativistic QED: I. the bloch-nordsieck paradigm. http://arxiv.org/abs/07092493v2[math-ph], 2009
Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic QED: II. mass hell properties. J. Math. Phys. 50, 012103 (2009)
Fierz M., Pauli W.: Nuovo cimento 15, 167 (1938)
Hiroshima, F., Spohn, H.: Mass renormalization in non-relativistic quantum electrodynamics. J. Math. Phys. 46(4) (2005)
Hainzl C., Seiringer R.: Mass renormalization and energy level shift in non-relativistic QED. Adv. Theor. Math. Phys. 6(5), 847–871 (2003)
Lieb, E.T., Loss, M.: Self Energy of Electrons in Non-perturbative QED. Conference Moshe Flato 1999, Vol. I, Dijon, Math. Phys. Stud. 21, Dordrecht: Kluwer Acad. Publ., 2000, pp. 327–344
Lieb E.T., Loss M.: A bound on binding energies and mass renormalization in model of quantum electrodynamics. J. Stat. Phys. 108, 1057–1069 (2002)
Pizzo A.: One-particle (improper) states in Nelson’s massless model. Ann. H. Poincaré 4(3), 439–486 (2003)
Pizzo A.: Scattering of an Infraparticle: The One-particle (improper) Sector in Nelson’s massless model. Ann. H. Poincaré 4(3), 439–486 (2003)
Schweber S.S.: An introduction to Relativistic Quantum Field Theory. Harper and Row, New York (1961)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I-II-III-IV, New York: Academic Press, 1972, 1975, 1977, 1978
Acknowledgements
The authors thank Thomas Chen for very useful discussions. A.P. was supported by NSF grant DMS-0905988.
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Communicated by H. Spohn
Also at IHES, Bures-sur-Yvette.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Fröhlich, J., Pizzo, A. Renormalized Electron Mass in Nonrelativistic QED. Commun. Math. Phys. 294, 439–470 (2010). https://doi.org/10.1007/s00220-009-0960-8
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DOI: https://doi.org/10.1007/s00220-009-0960-8