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

### Gerald Hofmann

HTWK Leipzig, Germany

## Abstract

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 $ϕ$ defined on $(C_{2})_{⊗}$ (tensor algebra over $C_{2}$) 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.

## Cite this article

Gerald Hofmann, On the Cones of $α$- and Generalized $α$-Positivity for Quantum Field Theories with Indefinite Metric. Publ. Res. Inst. Math. Sci. 30 (1994), no. 4, pp. 641–670

DOI 10.2977/PRIMS/1195165793