Vacuum Energy as Spectral Geometry

Abstract

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.

Vacuum energy and closed orbits in quantum graphs

“…One very convenient regularization is to insert a factor e −tω k in the sum (taking t → 0 at the end). This method is discussed in more detail in [21]. An asymptotic expansion of the sum at small t (which has a close connection to both the spectral distribution of the operator in the field equation -− ∂ 2 ∂x 2 in the example (1) -and the geometry of the configuration) displays divergent terms that are independent of the configuration and hence disappear when the energies are subtracted.…”

“…Physical background. Although vacuum energy can be posed as a purely mathematical problem of intrinsic mathematical interest [21], readers with some knowledge of quantum theory will want to know something of its physical origin and significance. Unfortunately, an adequate treatment would require inserting a graduate course in quantum field theory.…”

“…See [21].) In general, each Robin quantity is equal to the corresponding Neumann quantity plus a correction dependent on α.…”

“…can be localized in space, the study of vacuum energy promotes attention to localized spectral information, a somewhat neglected topic. The present article is complementary to [21]; both are directed primarily to a mathematical audience.Much of the article is based on the undergraduate research [43] of J. Wilson under the direction of S. Fulling and G. Berkolaiko. Some results of that research have been published in [9,22,23], to which we refer for details.…”

Boundary shape and Casimir energy

“…In this sense the so-called pointwise summation methods too can be considered as proximity force approximation [7]. The second approach is the one called multiple scattering expansion [8] which relies on the connection between closed classical paths and Casimir energy [9]. It formulates the vacuum energy for the electromagnetic field in terms of the successive scatterings from the conducting boundaries [10].…”