This Collaborative Research is proposed by Klaus Kirsten, BaylorUniversity, and Paul Loya, Binghamton University. Casimir effect is a term used for quantum effects resulting from the finite extension of systems. The continuing miniaturization of technical devices makes this effect increasingly more important; e.g. in microelectromechanical systems it is responsible for up to 10% of the forces encountered. Also on cosmological scales this effect is relevant to the dark energy and to the stabilization of extra dimensions of the universe. However, presently not even the origin of the sign of the Casimir energy is well understood. In addition, the change of the Casimir energy is largely unknown when the shape of small objects and their material properties are altered. The goal of the present proposal is to considerably improve this situation by employing two completely different strategies. 1.) For highly symmetric situations, e.g. spheres, cubes and tori, the Casimir energy is well understood. Using contour integral methods this pool will be significantly increased to include configurations related to any separable coordinate system. 2.) At present, no practicable technique is known to find the change of the Casimir energy when the geometry of objects or their material properties are changed. This proposal entails a completely new approach. Analytical surgery, a geometric analysis method designed to analyze changes in spectral quantities, will be newly employed in the field of the Casimir effect. Broader Impacts. The deeper understanding of the Casimir effect is necessary for the optimal design of microelectromechanical devices. Furthermore this project will involve the collaboration of undergraduate and graduate students from various backgrounds and different departments. The techniques that the PIs will use to study the Casimir effect are accessible to advanced undergraduate students with a complex analysis background. An undergraduate text book and class on complex analysis, path integrals, and zeta regularized determinants will be developed by the PIs within the next two years. The newly established mathematical physics seminar at Baylor University started by Kirsten will serve to communicate results obtained under this grant to attract Baylor graduate students. Loya as the adviser of the Undergraduate Math Club at SUNY Binghamton will inform those students. This project will serve as a springboard to attract graduate students to this field of research to both involved universities. Finally, coming from the areas of mathematical physics (Kirsten) and analysis (Loya), the collaboration of the PIs in this project is multidisciplinary and will be a seed for new ideas combining physics and analysis.
Casimir effect is a quantum effect first predicted by the Dutch physicist Hendrik Casimir in 1948. It is known to have significant impact on physical processes at very small scales. The continuing miniaturization of technical devices makes this effect increasingly more relevant, for instance in micromachines it is responsible for up to 10 percent of the forces encountered and an optimal design needs to take the effect into account. Not only on small, but also on cosmological scales its impact might be extremely relevant. It relates to the dark energy and to the stabilization of extra dimensions of the universe. Given this ubiquitous character, a complete understanding of the Casimir effect is warranted. In particular, the question we need to answer is how the resulting Casimir force depends on the precise shape and properties of the systems involved. However, presently not even the origin of the negative or positive sign of the Casimir energy, which decides between the attractive or repulsive nature of Casimir forces, is well understood. In addition, the change of the Casimir energy is largely unknown when the shape of small objects and their material properties are altered. The goal of our project has been to improve this situation by employing two completely different strategies. The first strategy is to compute the Casimir energy for as many as possible configurations such that a larger picture emerges. The second strategy uses techniques common in global analysis, a mathematical discipline, to understand how Casimir energy changes under deformations of the geometry of objects, or systems. The most important outcome of the funded project is that we laid the mathematical foundation to understand how the Casimir energy changes under a specific deformation of systems. Furthermore, we enlarged considerably the category of treatable configurations. This was possible because we captured many of the important mathematical subtleties which are now understood at a deeper level. One of these configurations can be used to rule out inappropriate cosmological models based on current Casimir experiments making it a viable tool for physicists working on theories of the universe. This NSF grant allowed us to engage three PhD students and one postdoc at Baylor University in the research related to mathematical aspects of the Casimir effect. One dissertation has been successfully finished in 2012, the other two are expected to be completed in 2014. The PI has participated in eleven workshops and conferences, he has been co-editor of two books as well as co-organizer of two special sessions at conferences of the American Mathematical Society in a subject area related to the projects of our grant. This significantly helped to increase the much needed discussion and interaction about the Casimir effect among mathematicians and physicists. We gratefully acknowledge that the above research and related activities have fully or partially been made possible by our NSF grant.
