Nuclearity and Thermal States in Conformal Field Theory

TL;DR

This paper introduces L^2-nuclearity, a spectral density condition in conformal field theory, linking representation theory with local algebra inclusions, and explores implications for thermal states and higher-dimensional spacetimes.

Contribution

It establishes the equivalence of spectral and algebraic formulations of nuclearity in conformal QFT and derives conditions for thermal states and split properties.

Findings

L^2-nuclearity is equivalent to the existence of characters in certain representations.

The trace class condition implies Buchholz-Wichmann nuclearity and split property.

L^2-nuclearity holds for the massless Klein-Gordon field.

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,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 for the semigroup generated by 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 beta-KMS state for the translation dynamics on the net of C*-algebras for every inverse temperature beta>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.