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Frontiers in Optics

Entanglement from longitudinal and scalar photons

J. D. Franson

Published Jan 1, 2010• 0 citations
Publisher

Abstract

The quantization of the electromagnetic field in the Lorentz gauge produces longitudinal and scalar photons in addition to the usual transverse photons. It is shown that these additional photons can produce entanglement between distant atoms.
DOI
10.1364/fio.2010.ftug6

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The covariant quantization of the electromagnetic field in the Lorentz gauge gives rise to longitudinal and scalar photons in addition to the usual transverse photons. It is shown here that the exchange of longitudinal and scalar photons can produce entanglement between two distant atoms or harmonic oscillators. The form of the entangled states produced in this way is very different from that obtained in the Coulomb gauge, where the longitudinal and scalar photons do not exist. A generalized gauge transformation is used to show that all physically observable effects are the same in the two gauges, despite the differences in the form of the entangled states. An approach of this kind may be useful for a covariant description of the dynamics of quantum information processing.
The minimal-coupling Hamiltonian for a nonrelativistic hydrogenic atom interacting with a photon field in the \ensuremath{\alpha}-Lorentz gauge is canonically gauge transformed to one with four arbitrary degrees of gauge freedom. The theory, nonrelativistic in the charge dynamics, is wholly covariant in its treatment of the scalar, transverse, and longitudinal electromagnetic fields. One degree of gauge freedom of the transformed Hamiltonian comes from the arbitrary scalar parameter contained within the \ensuremath{\alpha}-Lorentz gauge, while the others come from arbitrary parameters governing the gauge transformation. A Jaynes-Cummings-type model is constructed by suitable restrictions to the gauge freedom, and found to display an admixture of transverse and scalar interactions but no longitudinal component. The model gives the Lamb shift, correct to the second order of the electronic charge consistent with relativistic quantum electrodynamics.
The problem of the relation between the Ahronov-Bohm effect and traditional postulates of electrodynamics, which claim that only electric and magnetic fields are observable, is resolved by denial of the statement about validity of the Maxwell equations for microscopic fields. We proceed from the idea that the Maxwell equations, as the generalization of experimental data, are valid only for averaged values. We show that microscopic electrodynamics should be based on postulation of the d'Alembert equations for four-vector of the electromagnetic field potential. The Lorentz condition is valid only for the averages and provides the implementation of the Maxwell equations for averages. This concept eliminates the problem of electromagnetic field quantization and provides the correctness of all known results of quantum electrodynamics. Therefore, the "virtuality" of the longitudinal and scalar photons has a formal mathematical character, conditioned by the Maxwell equations for averaged fields. The longitudinal and scalar photons provide not only the Coulomb interaction of charged particles, but also allow the electrical Aharonov-Bohm effect.