Vacuum Energy as Spectral Geometry

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.Comment: 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

Repulsive Casimir Pistons

Casimir pistons are models in which finite Casimir forces can be calculated without any suspect renormalizations. It has been suggested that such forces are always attractive. We present three scenarios in which that is not true. Two of these depend on mixing two types of boundary conditions. The other, however, is a simple type of quantum graph in which the sign of the force depends upon the number of edges.Comment: 4 pages, 2 figures; RevTeX. Minor additions and correction

Spectral Oscillations, Periodic Orbits, and Scaling

The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.Comment: 19 pages, RevTeX4 with PicTeX. Minor improvements in content, new references, typos correcte

Local Spectral Density and Vacuum Energy near a Quantum Graph Vertex

The delta interaction at a vertex generalizes the Robin (generalized Neumann) boundary condition on an interval. Study of a single vertex with N infinite leads suffices to determine the localized effects of such a vertex on densities of states, etc. For all the standard initial-value problems, such as that for the wave equation, the pertinent integral kernel (Green function) on the graph can be easily constructed from the corresponding elementary Green function on the real line. From the results one obtains the spectral-projection kernel, local spectral density, and local energy density. The energy density, which refers to an interpretation of the graph as the domain of a quantized scalar field, is a coefficient in the asymptotic expansion of the Green function for an elliptic problem involving the graph Hamiltonian; that expansion contains spectral/geometrical information beyond that in the much-studied heat-kernel expansion.Comment: 12 pages. Paper presented at AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications, Snowbird, June, 200

Vacuum energy density and pressure of a massive scalar field

With a view toward application of the Pauli-Villars regularization method to the Casimir energy of boundaries, we calculate the expectation values of the components of the stress tensor of a confined massive field in 1+1 space-time dimensions. Previous papers by Hays and Fulling are bridged and generalized. The Green function for the time-independent Schrodinger equation is constructed from the Green function for the whole line by the method of images; equivalently, the one-dimensional system is solved exactly in terms of closed classical paths and periodic orbits. Terms in the energy density and in the eigenvalue density attributable to the two boundaries individually and those attributable to the confinement of the field to a finite interval are distinguished so that their physical origins are clear. Then the pressure is found similarly from the cylinder kernel, the Green function associated most directly with an exponential frequency cutoff of the Fourier mode expansion. Finally, we discuss how the theory could be rendered finite by the Pauli-Villars method.Comment: 18 pages; v2 and v3 have minor improvement

Surface Vacuum Energy in Cutoff Models: Pressure Anomaly and Distributional Gravitational Limit

Vacuum-energy calculations with ideal reflecting boundaries are plagued by boundary divergences, which presumably correspond to real (but finite) physical effects occurring near the boundary. Our working hypothesis is that the stress tensor for idealized boundary conditions with some finite cutoff should be a reasonable ad hoc model for the true situation. The theory will have a sensible renormalized limit when the cutoff is taken away; this requires making sense of the Einstein equation with a distributional source. Calculations with the standard ultraviolet cutoff reveal an inconsistency between energy and pressure similar to the one that arises in noncovariant regularizations of cosmological vacuum energy. The problem disappears, however, if the cutoff is a spatial point separation in a "neutral" direction parallel to the boundary. Here we demonstrate these claims in detail, first for a single flat reflecting wall intersected by a test boundary, then more rigorously for a region of finite cross section surrounded by four reflecting walls. We also show how the moment-expansion theorem can be applied to the distributional limits of the source and the solution of the Einstein equation, resulting in a mathematically consistent differential equation where cutoff-dependent coefficients have been identified as renormalizations of properties of the boundary. A number of issues surrounding the interpretation of these results are aired.Comment: 22 pages, 2 figures, 1 table; PACS 03.70.+k, 04.20.Cv, 11.10.G

The Dirichlet-to-Robin Transform

A simple transformation converts a solution of a partial differential equation with a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. In the simplest cases this observation enables the exact construction of the Green functions for the wave, heat, and Schrodinger problems with a Robin boundary condition. The resulting physical picture is that the field can exchange energy with the boundary, and a delayed reflection from the boundary results. In more general situations the method allows at least approximate and local construction of the appropriate reflected solutions, and hence a "classical path" analysis of the Green functions and the associated spectral information. By this method we solve the wave equation on an interval with one Robin and one Dirichlet endpoint, and thence derive several variants of a Gutzwiller-type expansion for the density of eigenvalues. The variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighborhood of zero in the frequency variable.Comment: 31 pages, 5 figures; RevTe

Index theorems for quantum graphs

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a "Laplacian") with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe