A remark on the cluster theorem
type: Item
instance of: scholarly article
title: A remark on the cluster theorem
zbMATH Open document ID: 0582.46068
DOI: 10.1007/BF01213409
author: Klaus Fredenhagen
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: The cluster theorem states that in a massive relativistic quantum field theory the vacuum correlations of local observables decay exponentially fast with their spacelike separation. Proofs of the cluster theorem have been given by \textit{D. Ruelle} [Helv. Phys. Acta 35, 147--163 (1962; Zbl 0158.45702)] (decay faster than every power, functional analytic techniques), \textit{H. Araki}, \textit{K. Hepp} and \textit{D. Ruelle} [Helv. Phys. Acta 35, 164--174 (1962; Zbl 0161.23503)] (correct exponential decay with power corrections, only simple geometric situations, techniques of the Jost-Lehmann-Dyson representation in the theory of analytic functions of several variables) and \textit{R. Haag} and \textit{J. A. Swieca} [Commun. Math. Phys. 1, 308--320 (1965; Zbl 0149.23803)] (exponential decay with a smaller parameter, functional analytic techniques). In this paper the following improved version of the cluster theorem is proved. Theorem. Let \(H\) be a positive selfadjoint operator in some Hilbert space \({\mathcal H}\) with \(\text{sp}\;H\subset \{0\}\cup [m,\infty)\), \(m>0\) and with a nondegenerate eigenvalue 0. Let \(A\) and \(B\) be bounded operators in \({\mathcal H}\) with \([A,e^{itH} Be^{-itH}]=0\) for all \(t\) with \(| t| <\tau\), \(\tau >0\). Then \[ | (\Omega,AB\Omega)- (\Omega,A\Omega)(\Omega,B\Omega)| \leq \{\| A\Omega \| \| A^*\Omega \| \| B\Omega \| \| B^*\Omega \| \}^{1/2}e^{-m\tau}. \] The proof relies on techniques of the theory of analytic functions. Due to the assumptions on \(A\), \(B\) and \(H\) the function \[ f(z)=(\Omega,Ae^{izH} B\Omega)- (\Omega,A\Omega)(\Omega,B\Omega),\quad \text{Im}\;z>0 \] can be analytically extended to the cut plane \[ {\mathbb C}_{\tau}=\{z\in {\mathbb C}, \text{Im}\;z\neq 0 \text{ or } | \text{Re}\;z| <\tau \}. \] A holomorphic mapping from the unit disc onto \({\mathbb C}_{\tau}\) makes \(f\) into an analytic function on the unit disc. An application of Jensen's inequality then leads to the desired result.
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 47N50
zbMATH DE Number: 3931800
zbMATH Keywords: cluster theorem
zbMATH Keywords: massive relativistic quantum field theory
zbMATH Keywords: the vacuum correlations of local observables decay exponentially fast with their spacelike separation
zbMATH Keywords: Jensen's inequality
MaRDI profile type: Publication
cites work: Q5543456
cites work: Q5542029
cites work: When does a quantum field theory describe particles?
cites work: Q5597066
description: scientific article; zbMATH DE number 3931800