A Quantum Electrodynamic Investigation of the Jaynes-Crisp-Stroud Approach to Spontaneous Emission
Motivated by the work of Jaynes and co-workers,  the old problem of atomic level shifts and widths has recently begun to be re-examined. While QED is a highly successful theory (indeed the only workable field theory we have), it is still beset with self-energy infinities ultimately associated with the point-like nature of the electron. These are removed from the public view by the process of renormalization, in the hope that eventually some high frequency cut-off will be found to make the renormalization constants finite: such modifications proposed usually imply a radius to the electron. In the usual approach to QED, perturbation theory is used. Jaynes, prompted by the great progress made in theory and experiments on atoms interacting with electromagnetic fields, has re-investigated the problem of spontaneous emission by solving the relevant semi-classical equations of motion for the interacting field-atom system directly. An interesting product of such an approach (other than the novel time-dependence) is the non-appearance of divergences: the finite size of the atomic charge distribution removes the point like singularity when retardation is taken into account.
KeywordsFrequency Shift Spontaneous Emission Level Shift Renormalization Constant Lamb Shift
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References and Footnotes
- 2.We have directly verified by explicit calculation that all of the usual results of QED (including the divergent behavior of the level shift) are obtained using the present method but neglecting retardation. However, retardation is the crucial factor in deriving the QED analogue of the Jaynes-Crisp level shift. +Google Scholar
- 3.The integrals occurring in the evaluation of A± are of the same form as that in equation (26) and are performed using the same approximations. Such a derivation closely follows those of reference 1.Google Scholar
- 4.E.A. Power, Introductory Quantum Electrodynamics ( American Elsevier Publishing Company, New York, 1965 ).Google Scholar
- 5.This is justified by the reduction in the degree of divergence found using a relativistic treatment, since at ℏck≳mc2, pair states remove the high energy contribution to ΔL which then becomes convergent; for the same reason δm becomes logarithmically divergent.Google Scholar
- 6.A result apparently well known in the older literature due to Waller shows that if one takes the non-relativistic formalism seriously and includes retardation and recoil energies, the free electron self-energy diverges only logarithmically. It is known that the corresponding ΔL for a real hydrogen atom converges; Lamb has pointed out that this effectively cuts the integrals off at twice Bethe’s cut-off and disagrees with experiment.Google Scholar
- 7.It is amusing, but without physical significance, that if the RWA is imposed consistently on (31) then the (incorrect) result found by using the RWA in perturbation theory, δωRWA=Δ+, is duplicated by our Heisenberg equation result.Google Scholar