Bert Schroer

Superselection sectors with braid group statistics and exchange algebras. I. General theory

The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.

Einstein causality and Artin braids

Abstract Within the restricted context of conformal QFTh 2 we present a systematic analysis of the exchange algebras of light-cone fields which result from the previously studied global decomposition theory of Einstein-causal fields. Although certain aspects of the representation theory of exchange algebras with their Artin braid structure-constants appear in our illustrative examples (minimal and WZW models), our main interest are algebraic aspects. We view the present work as a new non-lagrangian (non-hamiltonian) approach to non-perturbative QFTh.

Superselection sectors with braid group statistics and exchange algebras. 2. Geometric aspects and conformal covariance

The general theory of superselection sectors is shown to provide almost all the structure observed in two-dimensional conformal field theories. Its application to two-dimensional conformally covariant and three-dimensional Poincare covariant theories yields a general spin-statistics connection previously encountered in more special situations. CPT symmetry can be shown also in the absence of local (anti-) commutation relations, if the braid group statistics is expressed in the form of an exchange algebra.

Axial anomaly and Atiyah-Singer theorem

Abstract The fermion integration for gauge theories requires the study of the Euclidean c number Dirac equation in arbitrary external gauge fields. Winding of Aμ leads to a trapping for the Euclidean ψs. We complete our previous discussion by relating this problem to the Atiyah-Singer index theory. For gauge theories a suitably modified axial anomaly relation provides a new proof of this theorem.

Polarization-Free Generators and the S-Matrix

Abstract: Polarization-free generators, i.e. “interacting” Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.

String-localized quantum fields and modular localization

We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representations and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.

Modular Wedge Localization and the d=1+1 Formfactor Program

Abstract In this paper I continue the study of the new framework of modular localization and its constructive use in the nonperturbative d =1+1 Karowski–Weisz–Smirnov formfactor program. Particular attention is focussed on the existence of semilocal generators of the wedge-localized algebra without vacuum polarization (FWG-operators), which are closely related to objects fulfilling the Zamolodchikov–Faddeev algebraic structure. They generate a “thermal Hilbert space” and allow us to understand the equivalence of the KMS conditions with the so-called cyclicity equation for formfactors, which are known to be closely related to crossing symmetry properties. The modular setting gives rise to interesting new ideas on “free” d =2+1 anyons and plektons.

The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit

Abstract We show that the sin/cos functions of the axial potential of the free massive Dirac field appropriately describe the correlation functions of the ( T > T c order/disorder variable of two independent superimposed Ising systems taken in the continuum limit. The well known Kramers-Wannier transformation has a simple implementation in this description. It is also demonstrated that the square relationship at the level of correlation functions between the doubled and the single versions of the Ising model originates from an operator relationship between Dirac and Majorana spinor fields in two-dimensional space-time. Finally a connection is made between the Wightman functions of these local field operators and the statistical correlation functions obtained by McCoy, Tracy and Wu. The use of the doubled version of the Ising model based on the Dirac field allows a simple description of the scale invariant limit and therefore bridges the gap between the rigorous work and that of Kadanoff and Ceva as well as that of Luther and Peschel.

Topological fluctuations and breaking of chiral symmetry in gauge theories involving massless fermions

Abstract Using the method of Euclidean functional integrals, we show the occurrence of a certain types of chiral symmetry breaking as a result of local fluctuations in the winding number. We mainly restrict our discussion to QED 2 in order to have an independent check of our methods from the known solution of this model. We do not use instantons (pseudo-particles), and we also avoid functional integration over fields with global winding (Pontryagin) number different from zero.

Towards an explicit construction of the sine-Gordon field theory

Abstract We sketch a program for the explicit construction of the sine-Gordon and the massive Thirring-model fields. This construction only works in the phase of the model in which the infinite set of “soliton conservation laws” are valid. The procedure entails two steps of which we only indicate explicitly the first, namely the determination of the S -matrix leading to the sine-Gordon spectrum.