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).
Eugene Wigner famously observed that mathematics is "unreasonably effective" in physics and other natural sciences. Both detailed predictions and conceptual understanding are obtained from simple mathematical models that are always approximations to reality. However, overreliance on a favorite mathematical formulation may lead to sterility or just empirical inadequacy. We must constantly check whether we have fulfilled Einstein's instructions to make things "as simple as possible but not simpler." The principal physical phenomenon addressed by this project is the Casimir interaction, an attraction between bodies that is attributed to the influence of their positions on the energy of the electromagnetic field in its vacuum state, where no real photons are present. In the simplest model, where the bodies are perfectly conducting and the electric field is replaced by a scalar field, the energy density and pressure in the field are calculable from the "cylinder kernel", a Green function associated with the equation of motion of the field; this function forms a natural fourth member of the family of Green functions containing the more familiar fundamental solutions of the heat, wave, and Schrodinger equations related to the particular field equation and geometry under study. Total energy is rather easily calculated but is plagued by "divergences" -- apparent infinities that must be argued away. The physical reasons for these unwelcome terms, and the judgment on various strategies for getting rid of them, should be sought in (more difficult) calculations of energy density at every point. A distribution of energy that integrates to infinity may locally be perfectly plausible on physical grounds, and a modification of the model (making the boundaries less than perfectly conducting in some way) can provide a physically acceptable improvement in the theory. This project has been largely concerned with responding to the observation that rendering the divergences finite by ad hoc damping of the contributions of high frequencies yields formulas for energy density and pressure that are inconsistent with a fundamental aspect of conservation of energy called the "principle of virtual work". This problem, which was totally invisible in the global calculations of total energy, means that leaving this "ultraviolet cutoff" finite gives results that are not acceptable as a serious model. The project has made progress toward a remedy in two directions: replacing the sharp boundary by a "soft" or spread-out one, and introducing the cutoffs in a manner that treats different dimensions in space-time in similar ways (instead of singling out time as special). On the other hand, when the cutoff is regarded purely as a mathematical device that should be taken to infinity at the end of a calculation, we have shown (in the simplest situations) that the results are consistent and plausible, using techniques of distribution theory. An improved understanding of the pressure anomaly having been achieved, it was natural to look for a similar situation in the torque related to angular motion of the walls of a wedge. Surprisingly, it appears that there is a torque anomaly that has nothing to do with the cutoff used to tackle divergences, but appears in the finite values of energy density and pressure for the wedge geometry. (These formulas have been known for decades, but the discrepancy was not noticed until now.) The resolution of this puzzle presumably has some relation to the behavior of the field quantities near the vertex of the wedge, where a certain divergence remains that is not completely handled by conventional cutoff procedures. Global energy calculations in a sector with the vertex region removed do not show any torque anomaly. Full understanding will require local calculations of energy density and pressure, which are difficult and still under way. The mathematical techniques involved in this work are applicable in many fields besides vacuum energy. Participation in the research has provided valuable experience to a number of undergraduate and graduate students.
