Basic zeta functions and some applications in physics

Abstract

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?"

To appear in "A Window into Zeta and Modular Physics", Mathematical Sciences Research Institute Publications, Vol. 57, 2010, Cambridge University Press

References in corpus (22)

• Theory of Bose-Einstein condensation in trapped gases
• Precision Measurement of the Casimir Force from 0.1 to 0.9 microns
• Measurement of the Casimir force between parallel metallic surfaces
• Nonlinear Micromechanical Casimir Oscillator
• The Casimir effect: Recent controversies and progress
• Precise comparison of theory and new experiment for the Casimir force leads to stronger constraints on thermal quantum effects and long-range interactions
• Casimir forces between arbitrary compact objects
• Casimir interaction between a plate and a cylinder
• Casimir force on a piston
• Attractive Casimir Forces in a Closed Geometry
• Bose-Einstein condensation of atomic gases in a harmonic oscillator confining potential trap
• Experimental test for the conductivity properties from the Casimir force between metal and semiconductor
• Towards a precision measurement of the Casimir force in a cylinder-plane geometry
• Casimir interaction of two plates inside a cylinder
• A more accurate analysis of Bose-Einstein condensation in harmonic traps
• Kaluza-Klein models as pistons
• Bose-Einstein Condensation in Anisotropic Harmonic Traps
• Bose-Einstein condensation in arbitrarily shaped cavities
• Precise Quark Mass Dependence of Instanton Determinant
• Finite temperature Casimir effect in spacetime with extra compactified dimensions
• Finite Size Effects from General Covariance and Weyl Anomaly
• Vacuum Stress and Closed Paths in Rectangles, Pistons, and Pistols

Cited by in corpus (7)

• Will a physicist prove the Riemann Hypothesis?
• Geometrically confined thermal field theory: Finite size corrections and phase transitions
• Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics
• The zeta function for circular graphs
• Casimir effect in 2+1 Horava gravity
• Zeta function regularization technique in the electrostatics context for discrete charge distributions
• Vacuum energy, temperature corrections and heat kernel coefficients in $(D+1)$-dimensional spacetimes with nontrivial topology