Nuclearity and Thermal States in Conformal Field Theory

Abstract

We introduce a new type of spectral density condition, that we call L ²- nuclearity. One formulation concerns lowest weight unitary representations of $SL(2,\mathbb R)$ and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition ${\rm Tr}(e^{-\beta L_{0}}) < \infty$ for the conformal Hamiltonian L ₀, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L ₀ is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C^*-algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L ²-nuclearity is satisfied for the scalar, massless Klein-Gordon field.