Collaborative Research: Quantum Vaccum Energy

量子场论中的真空(Casimir)能量引起了物理界的兴趣，它与谱理论中的一些数学问题以及微分算符的渐近性的联系最近变得更加清晰。因此，物理学家和数学家之间的这种合作是取得进一步进展的有效途径。研究人员打算在他们之前合作的结果的基础上，在两个领域取得进展：(1)真空能量的引力意义：之前的工作已经证明，真空能量与任何其他形式的能量一样具有引力。对于平面几何，发散使物质的质量重新规格化，从而限制了场。这一分析将扩展到一般几何。这需要(Schwartz)分布的渐近性的数学理论以及声音物理。(2)改进的计算方法：在真空能量和本征值分布的经典路径分析中，由于存在曲面，高阶修正很难计算。当有锋利的边或角(不是直角)时，半经典方法会严重破坏(绕射)。该项目将开发和应用更基本和更准确的分析，称为多次反射或多次散射。以前通过多次散射处理介质体之间的力的成功将得到扩展，并将应用于实际配置，如非接触齿轮和多层表面。此外，多重反射展开将用于精确计算曲面和边缘和角落附近的真空能量密度和压力。该项目的更广泛影响部分源于其跨学科性质。这门学科不仅结合了物理和数学，而且还结合了物理学和数学中成熟的富有成效的互动的主题。真空能源既与新的纳米技术设备有关，也与宇宙学问题(暗能量)有关。从数学上讲，周期轨道理论和真空能量之间的联系直到最近才被开发出来，而真空能量对光谱理论的影响几乎没有被探索过。将招聘来自物理和数学的本科生和研究生研究助理，并加强他们的教育，正在培养多样化的学生群体。该项目还将促进未来与其他领域(引力含义、纳米技术、量子图、光谱几何)的研究互动。

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).