Abstract
A photon of momentum k can have only two polarization states, not three. Equivalently, one can say that the magnetic vector potential A must be divergence-free in the Coulomb gauge. These facts are normally taken into account in QED by introducing two polarization vectors ɛ λ (k) with λ ∈ {1,2}, which are orthogonal to the wave-vector k. These vectors must be very discontinuous functions of k and, consequently, their Fourier transforms have bad decay properties. Since these vectors have no physical significance there must be a way to eliminate them and their bad decay properties from the theory. We propose such a way here.
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Bach, V., Fröhlich, J., Sigal, I.M., Soffer, A.: Positive commutators and spectrum of Pauli-fierz Hamiltonians of atoms and molecules. Commun. Math. Phys. 207, 557–587 (1999)
Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3, 107–170 (2002)
Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics, J. Funct. Anal. 210, 321–340 (2004)
Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145, 557–595 (2001)
Lieb, E.H.: The stability of matter and quantum electrodynamics. In: Proceedings of the Heisenberg symposium, Munich, Dec. 2001, Fundamental Physics – Heisenberg and Beyond, G. Buschhorn and J. Wess, eds., pp. 53–68, Springer (2004). arXiv math-ph/0209034.
Lieb, E.H., Loss, M.: Existence of atoms and molecules in non-relativistic quantum electrodynamics. Adv. Theor. Math. Phys. 7, 667–710 (2003). arXiv math-ph/0307046
Lieb, E.H., Loss, M.: Analysis, 2nd edn. Providence, RI: Amer. Math. Soc., 2001
Lieb, E.H. Loss, M.: The thermodynamic limit for matter interacting with Coulomb forces and with the quantized electromagnetic field: I. The lower bound. arXiv math-ph/0408001
Planck, M.: Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verhandlung der Deutschen Physikalischen Gesellschaft 2, 237–245 (1900)
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Communicated by J.L. Lebowitz
Dedicated to Freeman Dyson on the occasion of his eightieth birthday
Work partially supported by U.S. National Science Foundation grant PHY 01-39984.
Work partially supported by U.S. National Science Foundation grant DMS 03-00349. 2003 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
Acknowledgement We thank Herbert Spohn and Jakob Yngvason for many useful discussions about this work. After completing this work and submitting it to CMP it was brought to our attention that the last section, 10.3, of the paper [2] by Fröhlich, Griesemer and Schlein contains the same idea in the context of Rayleigh scattering in the dipole approximation. The three-component concept enables them to extend the results in the rest of their paper from scalar fields to vector fields, but, as we see here, the concept works in much greater generality.
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Lieb, E., Loss, M. A Note on Polarization Vectors in Quantum Electrodynamics. Commun. Math. Phys. 252, 477–483 (2004). https://doi.org/10.1007/s00220-004-1185-5
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DOI: https://doi.org/10.1007/s00220-004-1185-5