Vacuum (Casimir) energy in quantum field theory is of interest in the physics community and its connections with some mathematical topics in spectral theory and asymptotics of differential operators have recently become clearer. Thus this collaboration between physicists and mathematicians is an effective approach to further progress. The investigators intend to build on the results of their previous collaboration to make progress in two areas: (1) Gravitational significance of vacuum energy: The previous work has demonstrated that vacuum energy gravitates just as does any other form of energy. For plane geometries the divergences renormalize the masses of the material bodies confining the field. This analysis will be extended to general geometries. This requires the mathematical theory of the asymptotics of (Schwartz) distributions as well as sound physics. (2) Improved calculational methods: In the presence of curved surfaces, higher-order corrections are hard to calculate in classical-path analyses of vacuum energy and eigenvalue distribution. In the presence of sharp edges or corners (other than right angles) the semiclassical methods break down quite seriously (diffraction). The project will develop and apply more fundamental and accurate analyses, known as multiple reflection or multiple scattering. Previous success in treating the forces between dielectric bodies by multiple scattering will be extended and applications to practical configurations such as noncontact gears and multilayer surfaces are being pursued. Also, the multiple-reflection expansion will be used for accurate calculation of vacuum energy density and pressure near curved surfaces and edges and corners.
The broader impact of the project stems partly from its interdisciplinary nature. The subject not only combines physics and mathematics, but also combines topics within physics and within mathematics that are ripe for productive interaction. Vacuum energy is relevant both to new nanotechnological devices and to cosmological issues (dark energy). Mathematically, the connection between periodic-orbit theory and vacuum energy has only recently been exploited, and the implications of vacuum energy for spectral theory have barely been explored. Undergraduate and graduate student research assistants from both physics and mathematics will be recruited and their educations will be enhanced, and a diverse student population is being trained. The project will also foster future research interactions with other fields (gravitational implications, nanotechnology, quantum graphs, spectral geometry).
Milton's research group has been studying the effect of the quantum vacuum. That is, the vacuum is not empty space, but is active, with particles such as electrons and photons, popping into and out of existence for short periods of time. This results in physical effects, such as the van der Waals force between neutral atoms and molecules. Mesoscopically, these quantum fluctuations give rise to forces between rather large bodies, such as metallic plates. In 1948 Casimir discovered the so-called Casimir force of attraction between parallel neutral conducting plates. In the last 15 years there has been much activity in studying such forces, with a large number of important experiments and improved theoretical calculations. We are studying quantum fluctuation forces between various shaped bodies which may have various electric and magnetic properties. It is possible that in some cases the forces may be repulsive rather than attractive. This can happen between ordinary conductors if they are sufficiently anisotropic. For example, an elongated conducting cylinder will be repelled from a circular hole in a conducting plate. Similarly, a very anisotropically polarizable atom can be repelled by an aperture. We are also looking a torques that tend to rotate objects, such as the disk above a plane shown in the figure. Even more interesting is when the disk is replaced by a rectangular plate. Then cusps appear in the torque when the corners of the rectangle pass over the edge of the half-plane. These are examples of calculations we can carry out, with the aim of eventually creating nano-scale devices that will rely on the quantum vacuum to transmit energy and perform useful functions.
