Field algebras in quantum theory with indefinite metric. II. Formulation of a modular theory in Pontryagin space
The authors investigate \(WJ^*\) algebra M in a Pontryagin space \({\mathcal H}\) of type \(\Pi_{\kappa}\) possessing a bicyclic vector \(\xi_ 0\) as the field algebra in a rigorous nonperturbative formulation of quantum gauge field theory. They study the generalized J-Hilbert algebras \({\mathcal U}=M\xi_ 0\) and \({\mathcal U}'=M'\xi_ 0\) corresponding to the algebra M. Associated canonically with the algebra M there are closed involution operators S and F satisfying \(S^ 0=F\) and \(S^ 2=F^ 2=1\). Here \(S^ 0\) means the conjugation of S with respect to the indefinite form \(<\cdot,\cdot >\) on \({\mathcal H}\). The operators S and F have the domains \({\mathcal D}^{\#}=dom S\) and \({\mathcal D}^ b=dom F\), and give a J- selfadjoint operator \(\Delta =FS\) such that \(\Delta^{-1}=SF\). Associated with \(\eta\in {\mathcal D}^ b\) there is a closed and, in general, unbounded operator \(\pi\) '(\(\eta)\). The algebra \({\mathcal U}'\) is also the set of all \(\pi\) '-bounded vectors in \({\mathcal D}^ b\). The authors prove that \({\mathcal U}'\) is dense in the space \({\mathcal H}\) and \({\mathcal D}^ b\). Their aim is to generalize the construction of the Tomita-Takesaki spectral algebra for \(W^*\) algebras. In order to do this, they study the properties of J-selfadjoint operators contained in M and M'', boundedness of J-selfadjoint operator \(AA^ 0\) by a closed operator A in J space and the spectrum of \(\Delta\) and \(\Delta^{-1}\). Furthermore they study the spectral algebra \({\mathcal U}'\!_ 0(\nu)=\{f(A_{\nu})\xi_ 0,\quad f\in K\},\) where \(A_{\nu}=\pi '(\nu)\pi '(\nu)^ 0\) for \(\nu\in {\mathcal D}^ b\) and K is the space of continuous complex-valued functions of \(R^ 1\) with compact support not containing the critical point of \(A_{\nu}\). Finally, they prove the first resolvent property \((\Delta -\omega)^{-1}{\mathcal U}'\subset {\mathcal U}''\) and the second resolvent property \((\Delta^{-1}-\omega)^{- 1}{\mathcal U}\subset {\mathcal U}''\) for all \(\omega \in C^ 1\setminus [0,\infty]\setminus \Gamma\), where \(\Gamma =\sigma (\Delta)\cap (- \infty,0)\) or \(\sigma (\Delta^{-1})\cap (-\infty,0).\) [For part I see 54, No.1, 57-77 (1983; Zbl 0519.47031).]
nonperturbative formulation of quantum gauge field theory
Tomita-Takesaki spectral algebra for \(W^*\) algebras
scientific article; zbMATH DE number 3911109