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Cambridge University Press eBooks

Summary and preamble to the quantum theory

Herbert E. Spohn

Published Jul 13, 2023• 0 citations
Publisher

Abstract

and preamble to the quantum theoryWithin the framework of specific models for the coupling between charges and the electromagnetic field we have presented a fair amount of rather detailed arguments and computations.Thus before embarking on the quantized theory, it might be useful to summarize our main findings.
DOI
10.1017/9781009402286.013

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The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit, already obtained by Noja and Posilicano [Ann. I.H.P. Phys. Theor. 71, 425 (1999)], is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behavior of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behavior is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibits in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.