An Algebraic Approach to Quantum Field Theory

Abstract

It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors ℌ1 and ℌ2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.

Froissart bound on inelastic cross section without unknown constants

“…Fortunately, [3] it was possible to prove this bound rigorously in the much more general frame work of Wightman's [4] axiomatic local field theory as applied to hadrons. Later, the needed analyticity properties, and polynomial boundedness at fixed momentum transfer squared t, were obtained by Epstein, Glaser and Martin [5] in the even more general framework of the theory of local observables of Haag, Kastler and Ruelle [6]. It has nevertheless been questioned [7] if these properties apply to hadrons made of quarks and gluons.…”

Untitled

“…(But, at the same time, this last fact may be criticized, because we loose contact with Dirac's original theory of constraints [6].) We hope, nevertheless, that the present examples will also serve to clarify the relation of the theory of quantum constraints with the algebraic approach to QFTà la Haag and Kastler ( [7,8]). …”

Representations of Baer ∗-semigroups

“…For this reason, representations of Baer »-semigroups may be physically interpreted as representations of the set of operations for a quantum system. One then obtains a representation theory which is more fundamental than the usual representation theory for the C*-algebra of bounded observables for a quantum system [4], [9]. …”

“…Representations of Baer »-semigroups are not only of interest in their own right, they can be important in the study of quantum logics [7], [11], [13], [18] and operational quantum mechanics [2], [3], [7], [9]. Because of the intimate connection between Baer »-semigroups and orthomodular lattices [5], [6], representations of the former can provide embeddings of the latter [8].…”