Three-body Casimir-Polder interactions

As part of our program to develop the description of three-body effects in quantum vacuum phenomena, we study the three-body interaction of two anisotropically polarizable atoms with a perfect electrically conducting plate, a generalization of earlier work. Three- and four-scattering effects are important, and lead to nonmonotonic behavior.Comment: 10 pages, 5 figures, for the proceedings of the conference Mathematical Structures in Quantum Systems, Benasque, Spain, July 2012, to be published in Nuovo Ciment

GEOMETRICAL INVESTIGATIONS OF THE CASIMIR EFFECT: THICKNESS AND CORRUGATION DEPENDENCIES

In the quantum theory the vacuum is not empty space. It is considered as a stateof infinite energy arising due to zero point fluctuations of the vacuum. Calculationof any physically relevant process requires subtracting this infinite energy using aprocedure called normalization. As such the vacuum energy is treated as an infiniteconstant. However, it has been established beyond doubt that mere subtractionof this infinite constant does not remove the effect of vacuum fluctuations and itcannot be treated just as a mathematical artifact. The presence of boundaries, whichrestricts the vacuum field, causes vacuum polarization. Any non-trivial space-timetopology can cause similar effects. This is manifested as the Casimir effect, wherebythe boundaries experience a force due to a change in the energy of the vacuum. Tocalculate the vacuum energy we treat the boundaries or other restrictive conditionsas classical backgrounds, which impose boundary conditions on the solution of thevacuum field equations. Alternatively, we can incorporate the classical background inthe Lagrangian of the system as classical potentials, which automatically include theboundary conditions in the field equations. Any change in the boundary conditionschanges the vacuum energy and consequently the Casimir force is experienced by theboundaries.In this dissertation we study the geometric aspect of the Casimir effect. We consider both the scalar field and the physically relevant electromagnetic field. Aftera brief survey of the field in Chapter 1, we derive the energy expression using theSchwinger\'s quantum action principle in Chapter 2. We present the multiple scatteringformalism for calculating the vacuum energy, which allows us to calculate theinteraction energy between disjoint bodies and subtract out the divergent terms fromthe beginning. We then solve the Green\'s dyadic equation for the electromagneticfield interacting with the planar background surfaces, where we can decompose theproblem into two transverse scalar modes. In Chapter 3 we collect all the solutionsfor the scalar Green\'s functions for the planar and the cylindrical geometries, whichare relevant for this dissertation.In Chapter 4 we derive the interaction energy between two dielectric slabs of finitethickness. Taking the thickness of the slabs to infinity leads to the Lifshitz resultsfor the two infinite dielectric semi-spaces, while taking the dielectric permittivity toinfinity gives the well-known Casimir energy between two perfect conductors. Wethen present a simple model to consider the thin-plate limit (taking the thicknessof the slabs to zero) based on Drude-Sommerfeld free electron gas model, whichmodifies the plasma frequency of the material to include the finite size dependence.We get a non-vanishing result for the Lifshitz energy in the slab thickness going tozero limit. This is remarkable progress as it allows us to understand the infinitesimalthickness limit and opens a possibility of extending this model to apply it to grapheneand other two dimensional surfaces. The Casimir and Casimir-Polder results in theperfect conductor limit give us the expected results.In Chapter 5 we study the lateral Casimir torque between two concentric corrugatedcylinders described by -potentials, which interact through a scalar field. Wederive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicitresults for the Dirichlet case, and exact results for the weak coupling limit, inthe leading order. The results for the corrugated cylinders approach the correspondingexpressions for the case of corrugated parallel plates in the limit of large radii ofthe cylinders (relative to the difference in their radii) while keeping the corrugationwavelength fixed.In Chapter 6 we calculate the lateral Casimir energy between corrugated paralleldielectric slabs of finite thickness using the multiple scattering formalism in the perturbativeapproximation and obtain a general expression, which is applicable to realmaterials. Taking the thickness of the plates to infinity leads us to the lateral Lifshitzformula for the force between corrugated dielectric surfaces of infinite thickness. Takingthe dielectric constant to infinity leads us to the conductor limit which has beenevaluated earlier in the literature. Taking the dilute dielectric limit gives the van derWaals interaction energy for the corrugated slabs to the second order in corrugationamplitude. The thin plate approximation proposed in Chapter 4 is used to derive theCasimir energy between two corrugated thin plates. We note that the lateral forcebetween corrugated perfectly conducting thin plates is identical to the ones involvingperfectly conducting thick plates. We also evaluate an exact expression (in terms ofa single integral) for the lateral force between corrugated (dilute) dielectric slabs

How does Casimir energy fall? IV. Gravitational interaction of regularized quantum vacuum energy

Several years ago we demonstrated that the Casimir energy for perfectly reflecting and imperfectly reflecting parallel plates gravitated normally, that is, obeyed the equivalence principle. At that time the divergences in the theory were treated only formally, without proper regularization, and the coupling to gravity was limited to the canonical energy-momentum-stress tensor. Here we strengthen the result by removing both of those limitations. We consider, as a toy model, massless scalar fields interacting with semitransparent ($\delta$-function) potentials defining parallel plates, which become Dirichlet plates for strong coupling. We insert space and time point-split regulation parameters, and obtain well-defined contributions to the self- energy of each plate, and the interaction energy between the plates. (This self-energy does not vanish even in the conformally-coupled, strong-coupled limit.) We also compute the local energy density, which requires regularization near the plates. In general, the energy density includes a surface energy that resides precisely on the boundaries. This energy is also regulated. The gravitational interaction of this well-defined system is then investigated, and it is verified that the equivalence principle is satisfied.Comment: 14 pages, 4 figure

Electromagnetic Non-contact Gears: Prelude

We calculate the lateral Lifshitz force between corrugated dielectric slabs of finite thickness. Taking the thickness of the plates to infinity leads us to the lateral Lifshitz force between corrugated dielectric surfaces of infinite extent. Taking the dielectric constant to infinity leads us to the conductor limit which has been evaluated earlier in the literature.Comment: 7 pages, 2 figures, Contribution to Proceedings of 9th Conference on Quantum Field Theory Under the Influence of External Conditions (QFEXT09), Norman, OK, September 21-25, 200