Commutation relations for the electromagnetic field in the presence of dielectrics and conductors (original) (raw)
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On Casimir forces for media with arbitrary dielectric properties
Revista Mexicana De Fisica, 2002
We derive an expression for the Casimir force between slabs with arbitrary dielectric properties characterized by their reflection coefficients. The formalism presented here is applicable to media with a local or a non-local dielectric response, an infinite or a finite width, inhomogeneous dissipative, etc. Our results reduce to the Lifshitz formula for the force between semi-infinite dielectric slabs by replacing the reflection coefficients by the Fresnel amplitudes.
Microscopic theory of the Casimir effect
2005
Based on the photon-exciton Hamiltonian a microscopic theory of the Casimir problem for dielectrics is developed. Using well-known many-body techniques we derive a perturbation expansion for the energy which is free from divergences. In the continuum limit we turn off the interaction at a distance smaller than a cut-off distance a to keep the energy finite. We will show that the macroscopic theory of the Casimir effect with hard boundary conditions is not well defined because it ignores the finite distance between the atoms, hence is including infinite self-energy contributions. Nevertheless for disconnected bodies the latter do not contribute to the force between the bodies. The Lorentz-Lorenz relation for the dielectric constant that enters the force is deduced in our microscopic theory without further assumptions. The photon Green’s function can be calculated from a Dyson type integral equation. The geometry of the problem only enters in this equation through the region of integration which is equal to the region occupied by the dielectric. The integral equation can be solved exactly for various plain and spherical geometries without using boundary conditions. This clearly shows that the Casimir force for dielectrics is due to the forces between the atoms. Convergence of the perturbation expansion and the metallic limit are discussed. We conclude that for any dielectric function the transverse electric (TE) mode does not contribute to the zerofrequency term of the Casimir force.
The Casimir force between real mirrors at non zero temperature
The Casimir force between dissipative metallic mirrors at non zero temperature has recently given rise to contradictory claims which have raised doubts about the theoretical expression of the force. In order to contribute to the resolution of this difficulty, we come back to the derivation of the force from basic principles of the quantum theory of lossy optical cavities. We obtain an expression which is valid for arbitrary mirrors, including dissipative ones, characterized by frequency dependent reflection amplitudes.
Physical Review A, 2010
A general, exact formula is derived for the expectation value of the electromagnetic energy density of an inhomogeneous absorbing and dispersive dielectric medium at zero temperature, assuming that the medium is well approximated as a continuum. From this formula we obtain the formal expression for the Casimir force density. Unlike most previous approaches to Casimir effects in which absorption is either ignored or admitted implicitly through the required analytic properties of the permittivity, we include dissipation explicitly via the coupling of each dipole oscillator of the medium to a reservoir of harmonic oscillators. We obtain the energy density and the Casimir force density as a consequence of the van der Waals interactions of the oscillators and also from Poynting's theorem.
Scattering theory approach to electrodynamic Casimir forces
Physical Review D, 2009
We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, nonzero temperatures, and spatial arrangements in which one object is enclosed in another. Our method combines each object's classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. The method is illustrated by re-deriving the Lifshitz formula for infinite half spaces, by demonstrating the Casimir-Polder to van der Waals cross-over, and by computing the Casimir interaction energy of two infinite, parallel, perfect metal cylinders either inside or outside one another. Furthermore, it is used to obtain new results, namely the Casimir energies of a sphere or a cylinder opposite a plate, all with finite permittivity and permeability, to leading order at large separation.
Quantization of the electromagnetic field in dielectrics
Physical Review A, 1992
We present a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on a microscopic model, in which the medium is represented by a collection of interacting matter fields. We calculate the exact eigenoperators for the coupled system and express the electromagnetic field operators in terms of them. The dielectric constant of the medium is explicitly derived and is shown to satisfy the Kramers-Kronig relations. We apply these results to treat the propagation of light in dielectrics and obtain simple expressions for the electromagnetic field in the medium in terms of space-dependent creation and annihilation operators. These operators satisfy a set of equal-space commutation relations and obey spatial Langevin equations of evolution. This justifies the use of such operators in phenomenological models in quantum optics. We also obtain two interesting relationships between the group and the phase velocity in dielectrics.
Surprises in Theoretical Casimir Physics: Quantum Forces in Inhomogeneous Media
Despite more than half a century of theoretical work, the Casimir effect is still not as fully understood as some suppose. In this treatise, the author uncovers new puzzles and paradoxes concerning this mysterious phenomenon. In particular, he clearly demonstrates that the most sophisticated theories fail when confronted with dielectrics in which the refractive index is not uniform but gradually changes.
Casimir force between two dielectric slabs
Physical Review A, 2001
We examine the Casimir effect between two dispersive dissipative slabs in three dimensions. The dielectric function of the slabs is assumed to be an arbitrary complex function of frequency satisfying Kramers-Kronig relations. The Maxwell stress tensor is used to evaluate the vacuum radiation pressure of the electromagnetic field on each slab in terms of the field correlation functions. These correlations are simply related to the imaginary part of the vector potential Green function by using the fluctuation dissipation theorem and Kubo's formula. The response function is obtained by employing the standard method for the present configuration. The formalism enables us to calculate the attractive Casimir force between a pair of dielectric slabs without resorting to the electromagnetic field quantization. Special attention is paid to the various limits of the general expression and the results are compared with previous work whenever possible. The Lorentz model of the dielectric function is also used to show the typical behavior of this force in terms of the slab thickness. We also consider the effect of finite temperature on the Casimir force between two dielectric slabs.
Electromagnetic Energy, Absorption, and Casimir Forces
The derivation of Casimir forces between dielectrics can be simplified by ignoring absorption, calculating energy changes due to displacements of the dielectrics, and only then admitting absorption by allowing permittivities to be complex. As a first step towards a better understanding of this situation we consider in this paper the model of a dielectric as a collection of oscillators, each of which is coupled to a reservoir giving rise to damping and Langevin forces on the oscillators and a noise polarization acting as a source of a fluctuating electromagnetic (EM) field in the dielectric. The model leads naturally to expressions for the quantized EM fields that are consistent with those obtained by different approaches, and also results in a fluctuation-dissipation relation between the noise polarization and the imaginary part of the permittivity. Our main result is the derivation of an expression for the QED energy density of a uniform dispersive, absorbing media in thermal equil...
Casimir force and the quantum theory of lossy optical cavities
Physical Review A, 2003
We present a new derivation of the Casimir force between two parallel plane mirrors at zero temperature. The two mirrors and the cavity they enclose are treated as quantum optical networks. They are in general lossy and characterized by frequency dependent reflection amplitudes. The additional fluctuations accompanying losses are deduced from expressions of the optical theorem. A general proof is given for the theorem relating the spectral density inside the cavity to the reflection amplitudes seen by the inner fields. This density determines the vacuum radiation pressure and, therefore, the Casimir force. The force is obtained as an integral over the real frequencies, including the contribution of evanescent waves besides that of ordinary waves, and, then, as an integral over imaginary frequencies. The demonstration relies only on general properties obeyed by real mirrors which also enforce general constraints for the variation of the Casimir force.