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Group and the Principle of Local Stability in Algebraic Quantum Field Theory

In the present contribution we report on a new approach to the structural analysis of the short distance behaviour of quantum field theories in the operator-algebraic formulation which has recently been proposed in [1] (see also [2,3,4]), and the extension of this framework to quantum field theory in curved spacetimes as set out in [5]. I shall begin by giving a few indications as to why one is interested in having a model-independent framework for the analysis of the short-distance properties of general quantum field theories. By general agreement, quantum field theory is so far the best description of elementary particle physics available. In elementary particle physics there is experimental evidence that the hadrons (e.g., protons and neutrons) are built up from particle-like constituents, the quarks and gluons which, however, are never observed as free particles but are “confined” in their bound states. Only when hadrons participate in collision processes at very high energies/short distances in particle accelerators, traces of those particle-like sub-structures become indirectly visible. In the — up to now mainly perturbative — quantum field theoretical treatment of strong interaction processes (quantum chromodynamics) the phenomenon of confinement, and the related notion of asymptotic freedom, can be explained by the method of renormalization group analysis of the short distance behaviour in the corresponding quantum field models. (Standard references include e.g. [6].) In its conventional form, the renormalization group analysis depends on the use of Wightman-type quantum fields. This is quite unsatisfactory from the point of view of structural analysis since the description of a theory in terms of quantum fields is not intrinsic. The field operators (operator-valued distributions) merely serve as a “coordinatization” of the fieldor observable-algebras, and there is some arbitrariness in their choice [7]. Therefore, to have e.g. an intrinsic criterion as to whether a given quantum field theory obeys a dynamics where confinement occurs, one needs a formulation of renormalization group analysis in local algebraic quantum field theory which uses only the intrinsic