Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 094, 23 pages      arXiv:0706.2831      http://dx.doi.org/10.3842/SIGMA.2007.094
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Vacuum Energy as Spectral Geometry

Stephen A. Fulling
Department of Mathematics, Texas A&M University, College Station, Texas, 77843-3368, USA

Received June 21, 2007, in final form September 14, 2007; Published online September 26, 2007

Abstract
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximation and other methods.

Key words: Casimir; periodic orbit; energy density; cylinder kernel.

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