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MP: Theoretische und Mathematische Grundlagen der Physik

MP 16: Quantenfeldtheorie 2

MP 16.6: Vortrag

Donnerstag, 23. März 2000, 17:30–17:45, W A317

Spektralfunktionen in Mathematik und Physik — •Klaus Kirsten — University of Manchester, Department of Physics and Astronomy, Oxford Road, Manchester M13 9PL

Several functions of the spectrum of second order elliptic differential operators play a central role in the analysis of properties of physical systems. E.g. in statistical mechanics relevant spectral functions comprise of various partition sums for the evaluation of thermodynamical quantities as critical temperatures or fluctuations of the ground state occupation. In quantum field theory under external conditions relevant quantities are effective actions (closely related with functional determinants) and groundstate or vacuum energies, which describe e.g. the influence of external fields or of boundaries on the properties of the vacuum. In this context, results are generically divergent and need a renormalization to give a physical meaning to them. The renormalization procedure at one-loop is completely determined by the heat-kernel coefficients, central objects of spectral geometry. All mentioned spectral functions can be related to an associated zeta function. In recent years an analysis of zeta functions in spherically symmetric situations has become available. These results (and techniques involved) allow the analysis of vacuum properties in the presence of spherically symmetric boundaries or background fields as well as the determination of thermodynamical properties of ideal gases in magnetic traps. Furthermore, when combined with other methods, it provides an effective scheme for the calculation of heat-kernel coefficients on arbitrary smooth Riemannian manifolds with smooth boundaries.