arXiv subjects

Klaus Kirsten

Publications and source records attributed to Klaus Kirsten.

At least 55 records · Page 3Linked to original sources

arXiv2012

The Casimir Effect for Generalized Piston Geometries

In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type $I\times_{f}N$ where $I=[a,b]$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at $R\in(a,b)$. By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function $f$ and base manifold $N$.

hep-th↗

arXiv2012

Spectral methods in quantum field theory and quantum cosmology

We review the application of the spectral zeta-function to the 1- loop properties of quantum field theories on manifolds with boundary, with emphasis on Euclidean quantum gravity and quantum cosmology. As was shown in the literature some time ago, the only boundary conditions that are completely invariant under infinitesimal diffeomorphisms on metric perturbations suffer from a drawback, i.e. lack of strong ellipticity of the resulting boundary-value problem. Nevertheless, at least on the Euclidean 4-ball background, it remains possible to evaluate the zeta(0) value, which describes in this case a universe which, in the limit of small 3-geometry, has vanishing probability of approaching the cosmological singularity. An assessment of this result is here performed, discussing its physical and mathematical implications.

hep-th↗

arXiv2011

The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type $I\times_{f} N$

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type $I\times_{f} N$ where $I$ is an interval of the real line and $N$ is a compact, $d$-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on $M$ for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.

math-ph↗

arXiv2011

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a $d$-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly.

math-ph↗

arXiv2011

Pistons modeled by potentials

In this article we consider a piston modelled by a potential in the presence of extra dimensions. We analyze the functional determinant and the Casimir effect for this configuration. In order to compute the determinant and Casimir force we employ the zeta function scheme. Essentially, the computation reduces to the analysis of the zeta function associated with a scalar field living on an interval $[0,L]$ in a background potential. Although, as a model for a piston, it seems reasonable to assume a potential having compact support within $[0,L]$, we provide a formalism that can be applied to any sufficiently smooth potential.

hep-th↗

arXiv2011

Bose-Einstein Condensation on Product Manifolds

We investigate the phenomenon of Bose-Einstein condensation on manifolds constructed as a product of a three-dimensional Euclidian space and a general smooth, compact $d$-dimensional manifold possibly with boundary. By using spectral $ζ$-function methods, we are able to explicitly provide thermodynamical quantities like the critical temperature and the specific heat when the gas of bosons is confined, in the three-dimensional manifold, by the experimentally relevant anisotropic harmonic oscillator potential.

hep-th↗

arXiv2011

Conical Casimir Pistons with Hybrid Boundary Conditions

In this paper we compute the Casimir energy and force for massless scalar fields endowed with hybrid boundary conditions, in the setting of the bounded generalized cone. By using spectral zeta function regularization methods, we obtain explicit expressions for the Casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston. Our general formulas are, subsequently, specialized to the case in which the piston is modelled by a $d$-dimensional sphere. In this particular situation, explicit results are given for $d=2,3,4,5$.

hep-th↗

arXiv2011

The Casimir Effect for Conical Pistons

In this paper we utilize $ζ$-function regularization techniques in order to compute the Casimir force for massless scalar fields subject to Dirichlet and Neumann boundary conditions in the setting of the conical piston. The piston geometry is obtained by dividing the bounded generalized cone into two regions separated by its cross section positioned at $a$ with $a\in(0,b)$ with $b>0$. We obtain expressions for the Casimir force that are valid in any dimension for both Dirichlet and Neumann boundary conditions in terms of the spectral $ζ$-function of the piston. As a particular case, we specify the piston to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.

hep-th↗

arXiv2010

Heat trace asymptotics with singular weight functions II

We study the weighted heat trace asymptotics of an operator of Laplace type with mixed boundary conditions where the weight function exhibits radial blowup. We give formulas for the first three boundary terms in the expansion in terms of geometrical data.

math.AP↗

arXiv2010

Functional determinants in higher dimensions using contour integrals

In this contribution we first summarize how contour integration methods can be used to derive closed formulae for functional determinants of ordinary differential operators. We then generalize our considerations to partial differential operators. Examples are used to show that also in higher dimensions closed answers can be obtained as long as the eigenvalues of the differential operators are determined by transcendental equations. Examples considered comprise of the finite temperature Casimir effect on a ball and the functional determinant of the Laplacian on a two-dimensional torus.

hep-th↗

arXiv2010

Basic zeta functions and some applications in physics

It is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the Casimir effect and Bose-Einstein condensation. A brief introduction into these areas is given in the respective sections. We will consider exclusively spectral zeta functions, that is zeta functions arising from the eigenvalue spectrum of suitable differential operators. There is a set of technical tools that are at the very heart of understanding analytical properties of essentially every spectral zeta function. Those tools are introduced using the well-studied examples of the Hurwitz, Epstein and Barnes zeta function. It is explained how these different examples of zeta functions can all be thought of as being generated by the same mechanism, namely they all result from eigenvalues of suitable (partial) differential operators. It is this relation with partial differential operators that provides the motivation for analyzing the zeta functions considered in these lectures. Motivations come for example from the questions "Can one hear the shape of a drum?" and "What does the Casimir effect know about a boundary?". Finally "What does a Bose gas know about its container?"

hep-th↗

arXiv2009

Small Mass Expansion of Functional Determinants on the Generalized Cone

In this paper we compute the small mass expansion for the functional determinant of a scalar Laplacian defined on the bounded, generalized cone. In the framework of zeta function regularization, we obtain an expression for the functional determinant valid in any dimension for both Dirichlet and Robin boundary conditions in terms of the spectral zeta function of the base manifold. Moreover, as a particular case, we specify the base to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.

hep-th↗

arXiv2009

Casimir Effect for a Semitransparent Wedge and an Annular Piston

We consider the Casimir energy due to a massless scalar field in a geometry of an infinite wedge closed by a Dirichlet circular cylinder, where the wedge is formed by $δ$-function potentials, so-called semitransparent boundaries. A finite expression for the Casimir energy corresponding to the arc and the presence of both semitransparent potentials is obtained, from which the torque on the sidewalls can be derived. The most interesting part of the calculation is the nontrivial nature of the angular mode functions. Numerical results are obtained which are closely analogous to those recently found for a magnetodielectric wedge, with the same speed of light on both sides of the wedge boundaries. Alternative methods are developed for annular regions with radial semitransparent potentials, based on reduced Green's functions for the angular dependence, which allows calculations using the multiple-scattering formalism. Numerical results corresponding to the torque on the radial plates are likewise computed, which generalize those for the wedge geometry. Generally useful formulas for calculating Casimir energies in separable geometries are derived.

hep-th↗

arXiv2009

Kaluza-Klein models as pistons

We consider the influence of extra dimensions on the force in Casimir pistons. Suitable analytical expressions are provided for the Casimir force in the range where the plate distance is small, and that where it is large, compared to the size of the extra dimensions. We show that the Casimir force tends to move the center plate toward the closer wall; this result is true independently of the cross-section of the piston and the geometry or topology of the additional Kaluza--Klein dimensions. The statement also remains true at finite temperature. In the limit where one wall of the piston is moved to infinity, the result for parallel plates is recovered. If only one chamber is considered, a criterion for the occurrence of Lukosz-type repulsion, as opposed to the occurrence of renormalization ambiguities, is given; we comment on why no repulsion has been noted in some previous cosmological calculations that consider only two plates.

hep-th↗

arXiv2008

Simplified Vacuum Energy Expressions for Radial Backgrounds and Domain Walls

We extend our previous results of simplified expressions for functional determinants for radial Schrödinger operators to the computation of vacuum energy, or mass corrections, for static but spatially radial backgrounds, and for domain wall configurations. Our method is based on the zeta function approach to the Gel'fand-Yaglom theorem, suitably extended to higher dimensional systems on separable manifolds. We find new expressions that are easy to implement numerically, for both zero and nonzero temperature.

hep-th↗

arXiv2008

The ubiquitous $ζ$-function and some of its "usual" and "unusual" meromorphic properties

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions of these $\z$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

math-ph↗

arXiv2008

General moment theorems for non-distinct unrestricted partitions

A well-known result from Hardy and Ramanujan gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers. In this vein, we consider the general partitioning problem of writing an integer $n$ as a sum of summands from a given sequence $\gL$ of non-decreasing integers. Under suitable assumptions on the sequence $\gL$, we obtain results using associated zeta-functions and saddle-point techniques. We also calculate higher moments of the sequence $\gL$ as well as the expected number of summands. Applications are made to various sequences, including those of Barnes and Epstein types. These results are of potential interest in statistical mechanics in the context of Bose-Einstein condensation.

math-ph↗

arXiv2008

Elliptic aspects of statistical mechanics on spheres

Our earlier results on the temperature inversion properties and the ellipticisation of the finite temperature internal energy on odd spheres are extended to orbifold factors of odd spheres and then to other thermodynamic quantities, in particular to the specific heat. The behaviour under modular transformations is facilitated by the introduction of a modular covariant derivative and it is shown that the specific heat on any odd sphere can be expressed in terms of just three functions. It is also shown that the free energy on the circle can be written elliptically.

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