On the Cones of α- and Generalized α-Positivity for Quantum Field Theories with Indefinite Metric

In order to construct a Krein-space theory (i.e., a *-algebra of (unbounded) operators which are defined on a common, dense, and invariant domain in a Krein space) the cones of α-positivity and generalized α-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 &#981 defined on (C2)&#8855 (tensor algebra over C2) not being α-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 α-positivity (introduced by Jakóbczyk) is provided in the case of tensor algebras.