An Algebraic Approach to Quantum Field Theory

TLDR

The paper discusses an algebraic formulation of quantum field theory and contrasts it with the conventional operator approach. It investigates how to separate global from local features in this algebraic framework. The authors prove that two quantum theories in distinct Hilbert spaces are physically equivalent when they share a faithful representation of the same abstract algebra of observables, allowing a purely algebraic formulation; they attribute unitarily inequivalent representations to irrelevant asymptotic behavior and demonstrate that, in electrodynamics, the zero‑charge Hilbert space already contains all relevant physical information.

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.