Positivity of Wightman functionals and the existence of local nets (Q752425)

In the frame of Wightman quantum field theory, a Wightman field \(\Phi\) is said to be associated to a local net of von Neumann algebras if each field operator \(\Phi\) (f) has an extension to a closed operator affiliated with the von Neumann algebra \({\mathcal A}(K)\) (associated to the double cone K in Minkowski space) if the support of the test function f is contained in the interior of K. The problem of associating \(\Phi\) to a local net is particularly simple if, for all K, the algebra \({\mathcal P}(K)\) of field operators with Supp \(f\subset K\) has a weak commutant \({\mathcal P}(K)^ w\) which is an algebra (it is the case if \(\Phi (f)^{**}e^{-H^{\alpha}}\) is bounded for all f where H is the Hamiltonian and \(0\leq \alpha <1)\). In fact, in this case, it is shown that this problem has a positive answer if and only if the corresponding Wightman state W over the tensor algebra of test functions is, for any test function f, centrally positive with respect to f when restricted to the subalgebra generated by f and the test functions with support in \(K^ c\) (the causal complement of K). The central positivity of W with respect to f is equivalent to the central positivity, in the sense of Powers, of the operator representing f in the GNS representation associated to W.