On the Cones of $\alpha$- and Generalized $\alpha$-Positivity for Quantum Field Theories with Indefinite Metric

Abstract

In order to construct a Krein-space theory (i.e., a $\ast$-algebra of (unbounded) operators which are defined on a common, dense, and invariant domain in a Krein space) the cones of $\alpha$-positivity and generalized $\alpha$-positivity are considered in tensor algebras. The relations between these cones, algebraic $\#$-cones, and involutive cones are investigated in detail.

Furthermore, an example of a $P$-functional $\varphi$ defined on $({\mathbf{C}}^2{)}_{\otimes }$ (tensor algebra over ${\mathbf{C}}^2$) not being $\alpha$-positive and yielding a non-trivial Krein-space theory is explicitely constructed. Thus, an affirmative answer to the question whether or not the method of $P$-functionals (introduced by Ôta) is more general than the one of $\alpha$-positivity (introduced by Jakóbczyk) is provided in the case of tensor algebras.