Lecture 2: Massless scalar field partition functions & real analytic Eisenstein series

Glenn Barnich (ULB, Brussels)

Sep 14. 2021, 11:00 — 12:00

We start by reviewing the computation of the partition function of a massless scalar field in the large volume limit, which corresponds to a spacetime manifold of the form S^1\times R^3, using 3 different methods: functional integration with zeta function regularization, Hamiltonian operator quantization, and heat-kernel/proper time/worldline methods. We then discuss the modifications of this computation for a spacetime manifold of the form T^2\times R^{d-1}, which is the case relevant for Casimir physics when d=2. The result is simply expressed in terms of a real analytic Eisensten series after turning on a chemical potential for linear momentum in the compact spatial direction. How to recover the standard conformal field theory result in d=1 is briefly discussed. Temperature inversion symmetry and modular covariance are derived. As a consequence, the entropy for photons in a Casimir box is shown to scale with the area of the plates at low temperature.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Geometry for Higher Spin Gravity: Conformal Structures, PDEs, and Q-manifolds (Thematic Programme)
Xavier Bekaert (U of Tours)
Andreas Cap (U of Vienna)
Stefan Fredenhagen (U of Vienna)
Maxim Grigoriev (U of Mons)
Alexei Kotov (U Hradec Králové)