Periodic orbits, spectral oscillations, scaling, and vacuum energy: beyond HaMiDeW

Abstract

Oscillations in eigenvalue density are associated with closed orbits of the corresponding classical (or geometrical optics) system. Although invisible in the heat-kernel expansion, these features determine the nonlocal parts of propagators, including the Casimir energy. I review some classic work, discuss the connection with vacuum energy, and show that when the coupling constant is varied with the energy, the periodic-orbit theory for generic quantum systems regains the clarity and simplicity that it always had for the wave equation in a cavity.While quantum gravity people were learning about the heat kernel, atomic physicists were progressing beyond it.The key property of the heat kernel expansion is its locality: the coefficients a n (x) are completely determined by the metric and potentials in a small neighborhood of x. This makes it easy to compute and universal in its applications to renormalization of ultraviolet divergences. Its formal inverse Laplace transform is an "averaged" spectral density [1][2][3] that is equally local and universal, insensitive to the detailed spacings of the eigenvalues (if the spectrum is discrete at all).The interesting part of renormalization is what is left behind when the counterterms are subtracted. For example, vacuum (Casimir) energy can be calculated from a two-point function (a kernel for the wave equation) by subtracting universal singular terms. The wave kernel and the vacuum energy are nonlocal ; they reflect boundary conditions, global topology, presence of periodic or closed classical orbits, and whether the dynamics is chaotic or integrable. This global geometrical information is encoded in the fine structure of the spectrum.More precisely, let T (t, x, y) be the integral kernel of e −t √H , where H is some positive selfadjoint second-order differential operator. (This "cylinder kernel" is analytically more tractable than kernels of wave (e.g., e

Systematics of the relationship between vacuum energy calculations and heat-kernel coefficients

“…The analogue of (4.2) yields only the terms in (3.1) involving time derivatives. The other terms can be obtained [18] by applying appropriate spatial partial derivatives to the integral kernel of the operator…”

“…The aim of this paper is to bring order and completeness into a welter of observations in the physics literature about various contributions to quantum vacuum energy (before renormalization) and their relations to the heat-kernel expansion. The primary tool is the systematic theory of Riesz means of spectral densities and their relation to the asymptotic expansions of various integral kernels associated with the central partial differential operator of the field theory (e.g., [20]); the Casimir energy, regularized by an exponential ultraviolet cutoff, is identifiable with certain terms in these expansions [18]. Both total energy and local energy density are considered; for the latter, we concentrate on the singular asymptotic behavior near a boundary, which was the subject of considerable calculational attention and physical controversy two decades ago (e.g., [12,31]), when the Casimir effect was of great interest as a model of quantum effects in cosmology (and in the bag model of hadrons).…”

Mathematical aspects of vacuum energy on quantum graphs

“…A widely employed method of calculation of the vacuum energy is expanding it into a sum over classical paths [9][10][11][12][13][14]. The expansion is usually done by the method of images, or 'multiple reflections', leading to a sum over all closed paths.…”