# On the Cones of <i>α</i>- and Generalized <i>α</i>-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 <i>α</i>- and Generalized <i>α</i>-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