A remark on the cluster theorem
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.
massive relativistic quantum field theory
the vacuum correlations of local observables decay exponentially fast with their spacelike separation
scientific article; zbMATH DE number 3931800