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arXiv:hep-th/0605218 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum-classical crossover in electrodynamics

János Polonyi

Published Sep 18, 2006• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.74.065014

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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.
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.
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.
This paper examines how Maxwell electrodynamics can be recovered as the field-theoretic structure required to consistently couple quantum matter to electromagnetism, while distinguishing what follows from quantum mechanics from what must be supplied independently. Local phase invariance introduces electromagnetic potentials through minimal coupling and yields the homogeneous Maxwell equations as geometric identities, while quantum probability conservation provides a conserved electric current and Ehrenfest dynamics recover the Lorentz-force form. The inhomogeneous Maxwell equations require an additional Lorentz- and gauge-invariant electromagnetic field action, whose vacuum constants satisfy μ0ε0=1/c2\mu_0\varepsilon_0=1/c^2, linking the construction to the SQC/MSC electric–magnetic prefactor relation. Hydrogen is used as an explicit microscopic source: the 1s1s state generates a quantum charge distribution, 2p2p states provide orbital currents and magnetic moments, and 1s1s–2p2p superpositions produce radiative dipole transitions. The treatment is then extended from individual atoms to a hydrogen gas, where polarization and magnetization generate the macroscopic fields DD and HH. The resulting framework shows how quantum sources and material response fit consistently within Maxwell electrodynamics without claiming that quantum mechanics alone uniquely derives the complete electromagnetic field dynamics.
The self-force problem of classical electrodynamics has two closely linked facets: The ill defined dynamics of a point charge due to the divergent self field at the position of the charge, and the divergence of formally conserved quantities, such as the energy, associated with symmetries of the corresponding Lagrangian. Fixing the self-force problem amounts to the construction of a \emph{new} theory, which is free of the above pathologies and yet "sufficiently close" to the immensely successful original. In a recent paper by the present author such a proposal, dubbed extended charge dynamics (ECD), was presented. The essential ingredients of classical electrodynamics preserved by ECD (and, among the plethora of solutions to the problem, only by ECD) are: - Ontology. The electromagnetic field is the same unquantized classical field, while charges are sufficiently localized conserved currents, accounting for the manifest corpuscular nature of elementary charges. - Symmetries. ECD enjoys the full symmetry group of classical electrodynamics, most importantly the hidden symmetry of scale covariance. - Conservation laws. All ECD conservation laws formally coincide with their classical counterparts, and yet lead to finite conserved quantities. Despite this seemingly classical setting, and the reduction of ECD to classical electrodynamics in the latter's domain of validity, it is shown in the present paper that ensembles of ECD solutions could, in principle, reproduce the statistical predictions of quantum mechanics. Exclusively quantum mechanical concepts, such as interference, violations of Bell's inequalities, spin and even photons (despite the use of a classical EM field), all emerge as mere statistical manifestations of the self interaction of ECD charges.
We develop a coherent-field formulation of quantum electrodynamics (QED) based on coherent-state boundary conditions, in which laser fields are represented by asymptotic coherent states rather than by prescribed classical background fields. Starting from coherent-state boundary conditions and the displacement-operator formalism, we construct operator and path-integral descriptions of transitions between distinct electromagnetic coherent states. This formulation provides a coherent-field extension of conventional background-field QED and incorporates the quantum dynamics of the coherent field itself. The corresponding path-integral representation contains a functional integral over coherent-field histories in addition to the usual functional integral over quantum fluctuations, thereby extending the conventional background-field description. The resulting path-integral representation naturally leads to an effective action for the stationary coherent-field configuration. The corresponding saddle-point condition yields an effective Maxwell equation containing contributions from vacuum polarization, photon fluctuations, and coherent-field fluctuations. In the limit where the latter two contributions vanish, the formalism reduces to the conventional Heisenberg-Euler description.
The macroscopic quantum theory of the electromagnetic field in a dielectric medium interacting with a dense collection of embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-field condition. Specifically, macroscopic quantum electrodynamics predicts that the Lorentz redshift of the resonance frequency of the atoms will be enhanced by a factor of the refractive index nnn of the host medium. However, an enhancement factor of (n2+2)∕3({n}^{2}+2)∕3(n2+2)∕3 is derived using the Bloembergen procedure in which the classical Lorentz local-field condition is applied to the optical Bloch equations. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. Microscopic quantum electrodynamics confirms the classical local-field-based results. Then the application of macroscopic quantum electrodynamic theory to embedded atoms is proved false by a specific example in which both the correspondence principle and microscopic theory of quantum electrodynamics are violated.