Macroscopic Quantum Electrodynamics

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Abstract
The quantities which relate the propagation of the electromagnetic field variables to experimentally measurable quantities are the expectation values of products of the field operators, 〈a(1)〉 , 〈a(1)a(2)〉 , etc., where a(1) denotes the vector potential at space–time point 1. These expectation values can be evaluated from a suitably defined generating functional. The set of equations which determines this functional in the presence of matter is particularized to the case in which matter exhibits a linear response to the transverse components of the radiation field. An explicit form is found for the generating functional, and in this linear realm, the close similarity of the propagation of the quantum fields to that determined by the classical phenomenological equations is pointed out. In the case in which matter exhibits a nonlinear response to the radiation field, the equations which determine the generating functional are reduced to an equation of motion for the expectation value of the vector potential. The result is a generalization of the classical nonlinear phenomenological Maxwell equation to include quantal effects. An alternative approach to the inclusion of higher‐order effects is by the introduction of a formally exact form for the generating functional. From this exact form, a systematic perturbation expansion which uses the generating functional found in the linear case as the zeroth‐order solution is given. The perturbation procedure gives a method for the direct calculation of higher‐order corrections to the expectation values of products of the field operators. The technique is illustrated by deriving the first nonlinear corrections to 〈a(1)〉 and 〈a(1)a(2)〉 . These corrections correctly include the quantal aspects of the radiation field and represent generalizations of the corresponding classical results. The expression found for 〈a(1)a(2)〉 gives a theoretical description of Raman and Rayleigh scattering of radiation by matter in bulk, including nonlocal time and space effects. In order to deal with problems associated with the breakdown of the above approximations in the case of strong fields, a procedure is given by which either classical fields or operator fields can be introduced so as to mimic the effects of the strong fields in the zeroth approximation. In this latter situation an effective susceptibility which determines the propagation of the radiation field is found in the case of linear response to incident radiation.