We present illustrations which show the usefulness of algebraic QFT (quantum field theory). In particular, in low-dimensional QFT, when Lagrangian quantization does not exist or is useless (e.g. in chiral conformal theories), the algebraic method is beginning to reveal its strength.
1997 Academic Press
1. HISTORY AND MOTIVATION
One characteristic feature which distinguishes algebraic QFT from the various quantization approaches to relativistic quantum physics based on classical Lagrangians (canonical, path-integral) is its emphasis on locality and its insistence in separating local from global properties. The former reside in the algebraic structure of local observables, whereas the latter usually enter through states and the suitably constructed representation spaces of local observables.
The idea that the global'' constitutes itself from the local'' is of course the heartpiece of classical electromagnetism as formulated by Faraday, Maxwell, and Einstein.
Algebraic QFT is more faithful to physical principles than to particular formalisms. As such it has more in common with the Kramers Kronig dispersion relations of the 1950's as a test of Einstein causality than with the post-Feynman developments of functional formalisms. Its conceptual and mathematical arsenal is however significantly richer than the QFT underlying the derivation of the Kramers Kronig dispersion relations.
When physicists first analysed the new concepts of quantum theory, they payed little attention to local structure since their main aim was to understand the new mechanics. For example, in von Neumann's early general account of the mathematical framework of quantum physics in terms of observables and their measurement, locality and causality did not enter at all. Even in the subsequent refinements of Wick et al. , concerning limitations on the superposition principle, locality was not used.