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arXiv:1002.2773 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Subclassical fields and polarization in electrodynamics

Mathieu Planat, János Polonyi

Published Aug 19, 2010• 7 citations
arXivHTMLPDF

Abstract

Expectation values of the electromagnetic field and the electric current are introduced at space-time resolution which belongs to the quantum domain. These allow us to approach some key features of classical electrodynamics from the underlying QED. One is the emergence of the radiation field in the retarded solution of the Maxwell equation, derived from an action principle. Another question discussed is the systematic derivation of the polarizability of a charge system. Furthermore, the decoherence and the consistency of the photon field are established by a perturbative calculation of the reduced density matrix for the electromagnetic field within the closed time path formalism.

DOI
10.1103/physrevd.82.045021

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Expectation values of the electromagnetic field and the electric current are introduced at space-time resolution which belongs to the quantum domain. These allow us to approach some key features of classical electrodynamics from the underlying QED. One is the emergence of the radiation field in the retarded solution of the Maxwell equation, derived from an action principle. Another question discussed is the systematic derivation of the polarizability of a charge system. Furthermore, the decoherence and the consistency of the photon field are established by a perturbative calculation of the reduced density matrix for the electromagnetic field within the closed time path formalism.
A classical field theory is proposed for the electric current and the electromagnetic field interpolating between microscopic and macroscopic domains. It represents a generalization of the density functional for the dynamics of the current and the electromagnetic field in the quantum side of the crossover and reproduces standard classical electrodynamics on the other side. The effective action derived in the closed time path formalism and the equations of motion follow from the variational principle. The polarization of the Dirac-sea can be taken into account in the quadratic approximation of the action by the introduction of the deplacement field strengths as in conventional classical electrodynamics. Decoherence appears naturally as a simple one-loop effect in this formalism. It is argued that the radiation time arrow is generated from the quantum boundary conditions in time by decoherence at the quantum-classical crossover and the Abraham-Lorentz force arises from the accelerating charge or from other charges in the macroscopic or the microscopic side, respectively. The functional form of the quantum renormalization group, the generalization of the renormalization group method for the density matrix, is proposed to follow the scale dependence through the quantum-classical crossover in a systematical manner.
Analysis of the interaction of the electromagnetic field and matter is a topical area of research in modern physics. The goal of this paper is to move from mathematical postulates and formalisms of quantum mechanics to equations that include macroscopic variables. In this case, the macroscopic observables are determined by the mean values of the operator corresponding to the observable value, and the mean value is deter-mined by the density matrix. Then the equations are written only through the mean values, by substituting equivalent expressions for the density matrix. As a result, we come to a system of equations expressed in terms of such macroscopic variables as polarization and electromagnetic field. In the semi-classical approach, when the electromagnetic field is not quantized, it is necessary to add the Maxwell's equations of the electromagnetic field to the equations for the mean values. This forms a self-consistent system of equations.
In an exact quantum-mechanical framework, we show that expectation values of the second-quantized electro-magnetic fields in the Coulomb gauge, and in the presence of classical sources, automatically lead to causal and retarded electro-magnetic field strengths. The classical ℏ\hbarℏ-independent Maxwell's equations naturally emerge from this fundamental quantum-mechanical approach in terms of expectation values of quantum fields, and are therefore also consistent with the special theory of relativity. The fundamental difference between interference phenomena due to the linear nature of the classical Maxwell theory as, e.g., in classical optics, and interference effects of quantum states is clarified. The framework outlined also provides for a simple approach to, e.g., spontaneous photon emission and/or absorption processes as well as to the classical Vavilov-Cherenkov radiation. The inherent and necessary quantum fluctuations, limiting a precise space-time knowledge of expectation values of the quantum fields considered, are, finally, recalled.
Using the Schwinger-Keldysh (closed time path or CTP) and Feynman-Vernon influence functional formalisms we obtain an expression for the influence functional in terms of Bogoliubov coefficients for the case of spinor quantum electrodynamics. Then we derive a CTP effective action in semiclassical approximation and its cumulant expansion. Using it we obtain a equation for the description of the charged particle creation in electric field and of backreaction of charged quantum fields and their fluctuations on time evolution of this electric field. Also an intimate connection between CTP effective action and decoherence functional will allow us to analyze how macroscopic electromagnetic fields are ``measured'' through interaction with charges and thereby rendered classical.
Abstract 1. The procedure usually adopted in the Quantum Theory for the estimation of the intensities of a spectrum line includes two stages : (1) the electric moment, p, of the distribution responsible for the spectrum line is calculated by means of Schrodinger’s integrals or some equivalent method ; (2) the rate of loss of energy due to radiation is calculated by the classical formula due to H. Hertz, viz., 2p2/3c3, just as if the distribution were concentrated in a dipole of precisely the same moment p. The first stage involves no difficulties beyond those met with in the evaluation of the integrals or matrices with which we are concerned, the only principles needed being those of the particular quantum theory adopted. But the second stage is open to grave objections and can only be justified on the ground of necessity owing to the absence of any completely satisfactory quantum theory of the electromagnetic field. Sometimes an appeal is made to Bohr’s Correspondence Principle, but this itself involves a somewhat hazardous extrapolation from infinitely large down to small quantum numbers and cannot take the place of a rigorous calculation. A logically consistent method is possible, based on Maxwell’s theory as represented by his electromagnetic equations and their solutions in terms of the classical retarded potential integrals. It may be objected that we do not know how far Maxwell’s equations can be applied to the interiors of atoms, in particular to distributions like those of Schrodinger, but precisely the same objection applies to the customary use of the classical expression for the radiation from an oscillating dipole. Moreover, the attempts made by Heisenberg and Pauli, Dirac and others to formulate a quantum theory of the electromagnetic field have all been based on the assumption that such a theory must be consistent with Maxwell’s equations. The direct application of the methods of classical electromagnetic theory to Schrodinger’s distributions can be justified at least as a rigorous attempt to determine their electromagnetic field and the radiation from them, and the validity of the procedure will have to be judged by the agreement, or otherwise, of the results obtained with experience. In this way we avoid the assumption that a spatially extended distribution can be treated as if its electric moment were concentrated in a point; in fact, the results obtained in this paper for the more important spectrum line series show definitely that the assumption is false, because it leads to results in many cases quite inconsistent with a rigorous application of Maxwell’s theory.
A systematic description of a spin one-half system endowed with magnetic moment or any other two-level system (qubit) interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED (albeit in a greatly simplified form). The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two level system leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study two-level systems (qubits) can be extended to many-level systems (qudits). As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one SSS state and three degenerate PPP states.