Vacuum Energy as Spectral Geometry

2007/06/30 by S. A. Fulling, Stephen A. Fulling
Mathematics · Physics and Astronomy · #Cosmology and Gravitation Theories #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect #math-ph #math.DG #math.MP #msc:34B27 #msc:58J50 #msc:81Q10

paper · pdf · doi:10.3842/sigma.2007.094

published as SIGMA 3 (2007), 094, 23 pages · This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

arxiv created 2007/09/26 · openalex publication_date 2007/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Semiclassical Electromagnetic Casimir Self-Energies
• The Casimir effect
• The spectral function of an elliptic operator
• Can One Hear the Shape of a Drum?
• Casimir interaction between two concentric cylinders: Exact versus semiclassical results
• The Casimir Effect: Physical Manifestations of Zero Point Energy