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.
The Casimir effect is a term used for quantum effects resulting from the precise form and the finite extension of systems. The continuing miniaturization of technical devices makes this effect increasingly more important. In microelectromechanical systems it is known to be responsible for up to 10% of the forces encountered and an optimal design needs to take the effect into account. Given this ubiquitous character, a complete understanding of the effect is warranted. However, presently the origin of the sign of the Casimir energy is not well understood, and the change of the Casimir energy is largely unknown when the shape of objects and their material properties are altered. The goal of this proposal, which is a collaborative proposal with Klaus Kirsten of Baylor University, was two fold: (1) Add bits of understanding to the knowledge of Casimir forces based on specific calculations for specific configurations. Several configurations were considered within this grant including singular configurations such as inverse square law potentials. (2) Analyze the change of the Casimir energy under a deformation of the space. Analytical surgery, a geometric analysis method designed to analyze changes in spectral quantities, was newly employed in the field of the Casimir effect as part of this grant. A better understanding of fundamental aspects of the Casimir effect is likely to help find an optimal design of microelectromechanical devices. Contributions arising out of this grant are expected to add information that will help to achieve this goal. The main publication during this grant period was the paper, joint with Klaus Kirsten, Analytic surgery of the zeta function, in the journal Communications in Mathematical Physics. The paper can be described as a preliminary result concerning analytical surgery as it was presented for the special case of smooth manifolds; this work in particular did not contain Casimir's original work. After this paper was published, I was able to solve the analytical surgery formula for the zeta function in full generality for manifolds with corners, which in particular contains Casimir's original work as a corollary, as well as scores of many other works as special cases. I currently have three Ph.D. students and a younger student who has asked me to be his Ph.D. advisor. During the past couple years, all these students have been taking independent study classes with me to learn the basics of pseudodifferential operators and heat kernels, some of the main tools needed to understand the analytical surgery of zeta functions. The interest of these students is the first step to using this grant project as a springboard to attract new and current graduate students to study analysis here at Binghamton University. Perhaps the most significant educational activity this grant has helped initiate is the start of a new graduate ``Analysis Program'' and ``Analysis Seminar'' here at Binghamton University. In order to recruit and retain students interested in analysis we need to provide courses in analysis. During this grant period, for the first time in a couple decades, we offered a two semester (Fall-Spring) course in ``Graduate Analysis.'' The objective was to teach a one year unified course in integration, functional analysis and complex analysis. To bring about this new curriculum, we had to write a report explaining why the current system was not working and why the new system is superior. Also, for the first time (to the best of my knowledge) in the history of Binghamton University we have an official ``Analysis Seminar''. I was the chairman of the seminar during its inaugural period. One purpose of the seminar is to present introductory talks on various research topics with the goal of recruiting graduate students. I used this opportunity to discuss areas related to the zeta function and the Casimir effect. I am currently developing new educational materials for advanced undergraduate/graduate students through four books, Lebesgue's remarkable theory of measure and integration with probability, Amazing and aesthetic aspects of analysis, both written by myself, and two other books, A pedestrian guide to path integrals and An introduction to complex analysis and zeta-regularized determinants, written together with Klaus Kirsten. The first two books have been accepted for publication by Springer-Verlag. I was also named one of the directors of the MAT (Masters of Arts in Teaching) program here at Binghamton University. This is a program for college students desiring to teach elementary, middle and high school mathematics. As one of the directors I get to teach topics classes specifically geared toward these students. One particular topics course I get to teach is an analysis topics course, which will provide a venue to discuss aspects of the Casimir effect. Finally, I am brainstorming an after-school mathematics program for elementary to middle school-aged children. I have exchanged emails on the topic with other professors and the after-school program will hopefully come to fruition soon.
