New Constructions in Local Quantum Physics

Abstract

Among several ideas which arose as consequences of modular localization there are two proposals which promise to be important for the classification and construction of QFTs. One is based on the observation that wedge-localized algebras under certain conditions have particle-like generators with simple properties and the second one uses the structural simplification of wedge-localized algebras in the holographic lightfront projection. Factorizable d=1+1 models permit to analyse the interplay between particle-like aspects and chiral field properties of lightfront holography.Pacs 11.10.-z, 11.55.-m * Talk presented at the symposium "Rigorous quantum field theory" in honor of J. Bros, Paris, July 2004, submitted for publication in the symposium proceedings.

Two-dimensional models as testing ground for principles and concepts of local quantum physics

“…It also provides an additional strong link between two-dimensional and higher dimensional QFT and admits a rich illustration for chiral theories. For a description of its history and aims, the reader is referred to [77][78][79] It had been known for some time that under very general conditions (for wedge-localized algebras and interval localized algebras of chiral QFT no additional conditions need to be imposed) the localized operator algebras AðOÞ of AQFT are isomorphic to an algebra which belongs to a class which already appeared in the famous classification of factor algebras by Murray and von Neumann and whose special role was highlighted later in mathematical work by Connes and Haagerup. It is somewhat surprising that the full richness of QFT can be encoded into the relative position of a finite number of copies of this ''monade'' 24 within a common Hilbert space [80].…”

“…Let G be a (generally unbounded) operator affiliated with the local algebra AðOÞ. We call such G a vacuum-polarization-free generator (PFG) affiliated with AðOÞ (denoted as GgAðOÞÞ if the state vector GX (with X the vacuum) is a one-particle state without any vacuum polarization admixture [79]. By definition such PFGs are (unbounded) on-shell operators and it is well-known that the existence of a subwedge-localized PFG forces the theory to be interaction-free, i.e., the local algebras possess free field generators.…”

“…The problem of demonstrating the existence of a non-trivial QFT associated with the algebraic structure (49) of the wedge algebra generators is then encoded into a non-triviality statement ðAðDÞ 6 ¼ C1Þ for the double cone intersections; the fact that the Z-F algebra is different from that of free field creation/ annihilation operators has the consequence that the operators in the intersection have infinitely many vacuum polarization components (connected formfactors) involving all particle numbers. In this way the problem of existence of non-trivial QFTs becomes disconnected from the inexorable short-distance problems of the standard approach [79].…”

“…Using concepts of modular theory (modular inclusions and modular intersections of wedge algebras) one can construct the local structure [79] (i.e., the local algebraic net) and identify the subgroup G (LF) of the Poincaré group which is the symmetry group of the holographic lightfront projection. Whereas some of the ambient Poincaré symmetries are evidently lost (in d = 1 + 1 the translation leading away from the lightray), the holographic projection is also symmetry-enhancing in the sense that the rotational symmetry of the Moebius group associated with the compactified lightray (and according to Section 4 also the infinite dimensional Diff (S 1 ) group) becomes geometric.…”

“…Whereas some of the ambient Poincaré symmetries are evidently lost (in d = 1 + 1 the translation leading away from the lightray), the holographic projection is also symmetry-enhancing in the sense that the rotational symmetry of the Moebius group associated with the compactified lightray (and according to Section 4 also the infinite dimensional Diff (S 1 ) group) becomes geometric. These symmetries are already present in the ambient theory, but they are not noticed because they acts there in a non-classical fuzzy manner and hence escapes the standard quantization approach [79].…”