On quantization of the field in dielectrics

Abstract

The precepts behind the macroscopic and microscopic quantizations of the electromagnetic ®eld in a dielectric are discussed. Using the correspondence principle, it is demonstrated that the macroscopic quantization procedure leads to incorrect equations of motion of embedded two-level atoms. The fundamental nature of the Lorentz viewpoint of electrodynamics is discussed. IntroductionThe foundation of the quantum theory of radiation, due principally to Dirac, is the quantization of Maxwell's equations in a vacuum wherein each mode of the radiation ®eld is associated with a quantized simple harmonic oscillator [1]. Following the initial quantization of the electromagnetic ®eld in a vacuum, it was natural to extend the quantization procedure to the electromagnetic ®eld in a dielectric. Quantization of linear dielectrics, begun in the 1940s by Ginzburg [2] and Jauch and Watson [3], has since become textbook material [4]. For over 50 years, then, the macroscopic Maxwell equations of classical continuum electrodynamics have been used as the basis for quantizing the electromagnetic ®eld in a dielectric [5]. Matter, however, is not a continuous medium, but is composed of discrete particles embedded in the vacuum. This fact was recognized by Lorentz, who gifted us an atomistic version of electrodynamics in which the electromagnetic ®eld, which is rooted in the vacuum, interacts with charges that are attached to the atomistic particles of matter that, in turn, generate reaction ®elds [6]. Nevertheless, Lorentz's fundamental results generally have been treated as a way of correcting the error introduced by the approximation of continuous media.In this article, it is proved that quantization of the Maxwell equations of continuum electrodynamics is incompatible with fundamental quantum mechanical principles. Equations of motion are derived for two-level atoms embedded in a dielectric utilizing (i) macroscopic quantization of the Maxwell equations in ponderable media and (ii) microscopic quantization of the Maxwell equations in vacuum with enumerated discrete oscillators [7,8]. There are substantial qualitative di erences in the results of the two techniques that are due to the di erence between the continuum and atomistic models of electrodynamics in matter. It is contended, therefore, that the macroscopic quantization of the ®eld in a dielectric

Nonlinear and Quantum Optics with Whispering Gallery Resonators

“…In many cases of interest polarization of each field is fixed with respect to the χ (2) tensor axes, and then the sum in ( 19) can be omitted along with the factor 1/3. Following the canonical quantization of scalar single-mode electromagnetic field [249] inside a dielectric [250,251], we write…”

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Quantum Gravity: A Heretical Vision

“…Look for relations between quantizations: If two such quantizations at different levels are carried out, one may then investigate the relation between them. Example: (Crenshaw 2002) have investigated the relation between microscopic and macroscopic quantizations of the electromagnetic field in a dielectric, when it is treated as a lattice of molecules and as a continuous medium, respectively.…”

“…Look for relations between quantizations: If two such quantizations at different levels are carried out, one may then investigate the relation between them. Example: (Crenshaw 2002) have investigated the relation between microscopic and macroscopic quantizations of the electromagnetic field in a dielectric, when it is treated as a lattice of molecules and as a continuous medium, respectively.If two such quantizations at the same level exist, one may investigate the relation between them. Example: (Ashtekar 1992) studied the relation between loop quantization and the more usual field quantization of the electromagnetic field: They showed that, if you "thicken" the loops, the two are equivalent.…”