Continuing a program of examining the behavior of the vacuum expectation value of the stress tensor in a background which varies only in a single direction, we here study the electromagnetic stress tensor in a medium with permittivity depending on a single spatial coordinate, specifically, a planar dielectric half-space facing a vacuum region. There are divergences occurring that are regulated by temporal and spatial point splitting, which have a universal character for both transverse electric and transverse magnetic modes. The nature of the divergences depends on the model of dispersion adopted. And there are singularities occurring at the edge between the dielectric and vacuum regions, which also have a universal character, depending on the structure of the discontinuities in the material properties there. Remarks are offered concerning renormalization of such models, and the significance of the stress tensor. The ambiguity in separating “bulk” and “scattering” parts of the stress tensor is discussed.

We calculate the Casimir forces in two configurations, namely, three parallel dielectric slabs and a dielectric slab between two perfectly conducting plates, where the dielectric materials are dispersive and inhomogeneous in the direction perpendicular to the interfaces. A renormalization scheme is proposed consisting of subtracting the effect of one interface separating two inhomogeneous media. Some examples are worked out to illustrate this scheme. Our method always gives finite results and is consistent with the principle of virtual work; it extends the Dzyaloshinskii-Lifshitz-Pitaeveskii force to inhomogeneous media.

Quantum fluctuations can induce friction on a neutral but polarizable particle and cause it to radiate energy even if the particle is moving in free space filled with blackbody radiation and is not in contact with or close to any surface or other object. We explore the energetics of such a particle moving uniformly in vacuum, continuing our previous investigations of quantum friction. The intrinsic polarizability of the particle is considered to be purely real before it is dressed by radiation. The particle is then guaranteed to be in the nonequilibrium steady state (NESS), where it absorbs and emits energy at the same rate. We first calculate the quantum frictional power and force on the particle in the rest frame of the blackbody radiation from first principles, namely the Maxwell-Heaviside equations and the Lorentz force law. Then we provide a simpler method of obtaining the same quantities in the rest frame of the particle by using the principle of virtual work. The equivalence of the two approaches is illustrated. The formulas we derive for quantum vacuum frictional power and force are fully relativistic and applicable to finite temperature. In NESS, the quantum vacuum frictional force on the particle is shown to be a true drag, independent of the model for polarizability and the polarization state of the particle. Finally, we give an estimate of the quantum vacuum friction on a gold atom and comment on the feasibility of detecting such quantum vacuum frictional effects.

The classical electromagnetic self-force on an arbitrary time-dependent electric or magnetic dipole moving with constant velocity in vacuum, and in a medium, is considered. Of course, in vacuum there is no net force on such a particle. Rather, because of loss of mass by the particle due to radiation, the self-force precisely cancels this inertial effect, and thus the spectral distribution of the energy radiated by dipole radiation is deduced without any consideration of radiation fields or of radiation reaction, in both the nonrelativistic and relativistic regimes. If the particle is moving in a homogeneous medium faster than the speed of light in the medium, Vavilov-Čerenkov radiation results. This is derived for the different polarization states, in agreement with the earlier results of Frank. The friction experienced by a point (time-independent) dipole moving parallel to an imperfectly conducting surface is examined. Finally, the relativistic quantum/thermal Einstein-Hopf effect is rederived. We obtain a closed form for the spectral distribution of the force and demonstrate that, even if the atom and the blackbody background have independent temperatures, the force is indeed a drag when the imaginary part of the polarizability is proportional to a power of the frequency. The unifying theme of these investigations is that friction on an atom requires a dissipative mechanism, be it through radiation or resistivity in the environment.

Quantum vacuum friction experienced by an atom, where the only dissipative mechanism is through its interaction with the radiation field, has been studied in our recent paper [ Phys. Rev. D 104, 116006 (2021)]. Quantum vacuum friction on an intrinsically dissipative particle is different in that the friction arises not only from the field fluctuations but also from the dipole fluctuations intrinsic to the particle. As a result, the dissipative particle can be out of the nonequilibrium steady state (NESS), and therefore loses or gains internal energy (rest mass). Only if the temperature of the particle equals a special NESS temperature will the particle be in NESS. In this paper, general NESS conditions are derived which give the NESS temperature of the particle as a function of the temperature of the radiation and the velocity of the particle. Imposing the NESS conditions, we also obtain an expression for the quantum vacuum friction in NESS. The NESS quantum vacuum friction is shown to be negative definite (opposing the motion of the particle) and equivalent to that found in our previous paper if the dissipation mechanism is restricted to radiation reaction. The NESS temperature and quantum vacuum friction are calculated numerically for various models. In particular, we show that, for a gold nanosphere, the deviation of its NESS temperature from the temperature of the radiation can be substantial and it is also possible to detect the NESS quantum vacuum friction directly at sufficiently high temperatures. Out of NESS, even though the quantum vacuum friction no longer has a definite sign in the rest frame of the radiation, the friction in the rest frame of the particle is still negative definite. Also, the external force needed to keep the particle moving must be in the same direction as the motion of the particle, therefore excluding the possibility of a perpetual motion machine, which could convert the vacuum energy into useful mechanical work. In addition, we find that the deviation of the temperature of the particle from its NESS temperature causes the particle to lose or gain internal energy in such a way that the particle would return to NESS after deviating from it. This enables experimental measurements of the NESS temperature of the particle to serve as a feasible signature for these quantum vacuum frictional effects.

From the generalized fluctuation-dissipation theorem, it is known that a body at rest made of nonreciprocal material may experience a torque, even in vacuum, if it is not in thermal equilibrium with its environment. However, it does not experience self-propulsion in such circumstances, except in higher order. Nevertheless, such a body may experience both a normal torque and a lateral force when adjacent to an ordinary surface with transverse translational symmetry. We explore how these phenomena arise, discuss what terminal velocities might be achieved, and point out some of the limitations of applying our results to observations, including the Lorenz-Lorentz correction and the cooling due to radiation. In spite of these limitations, the effects discussed would seem to be observable.