Phase space properties and the short distance structure in quantum field theory

The paper investigates relations between the phase space structure of a quantum field theory ("nuclearity") and the concept of pointlike localized fields. Given a net of local observable algebras, a phase space condition is introduced that allows a very detailed description of the theory's field content. An appendix discusses noninteracting models as examples.Comment: v3: minor changes, as to appear in J. Math. Phys.; 15 page

Characterization of local observables in integrable quantum field theories

Integrable quantum field theories in 1+1 dimensions have recently become amenable to a rigorous construction, but many questions about the structure of their local observables remain open. Our goal is to characterize these local observables in terms of their expansion coefficients in a series expansion by interacting annihilators and creators, similar to form factors. We establish a rigorous one-to-one characterization, where locality of an observable is reflected in analyticity properties of its expansion coefficients; this includes detailed information about the high-energy behaviour of the observable and the growth properties of the analytic functions. Our results hold for generic observables, not only smeared pointlike fields, and the characterizing conditions depend only on the localization region - we consider wedges and double cones - and on the permissible high energy behaviour.Comment: minor changes, as to appear in Commun. Math. Phys.; 39 pages, 4 figures, 1 vide

Automated assessment in a programming course for mathematicians

The paper reports on a programming course for undergraduate Mathematics students in their 2nd year, with some parts compulsory for single-subject students. Assessment takes the form of several programming projects. Formative feedback as well as summative assessment is aided by automated unit tests, which allow for rapid and consistent marking, while focussing marker’s time on students who require the most help

Relativistische Materie in zwei Raum-Zeit-Dimensionen

Quantum field theory unifies concepts from quantum theory and from special relativity. Its mathematically rigorous description is quite intricate and is only partially understood; this is particularly true for the construction of operators corresponding to measurements at a space-time point or in a bounded space-time region. We explain this fact in simplified examples on 1+1 dimensional space-time, namely in so-called integrable models. We give a characterization of local operators by means of the analyticity properties of their coefficients in a certain series expansion. This also allows us to explicitly construct examples of local operators.Comment: in German; 12 pages, 5 figures; as appeared in the proceedings of the symposium "Raum und Materie", Villigst, October 2012; references update

Towards an explicit construction of local observables in integrable quantum field theories

We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh-Schlieder property, and in which sense they generate the net of algebras. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model

Fermionic integrable models and graded Borchers triples

We provide an operator-algebraic construction of integrable models of quantum field theory on 1+1 dimensional Minkowski space with fermionic scattering states. These are obtained by a grading of the wedge-local fields or, alternatively, of the underlying Borchers triple defining the theory. This leads to a net of graded-local field algebras, of which the even part can be considered observable, although it is lacking Haag duality. Importantly, the nuclearity condition implying nontriviality of the local field algebras is independent of the grading, so that existing results on this technical question can be utilized. Application of Haag-Ruelle scattering theory confirms that the asymptotic particles are indeed fermionic. We also discuss connections with the form factor programme.Comment: form factors of disorder operators added, minor amendments throughout the text; 21 page

Operator product expansions as a consequence of phase space properties

The paper presents a model-independent, nonperturbative proof of operator product expansions in quantum field theory. As an input, a recently proposed phase space condition is used that allows a precise description of point field structures. Based on the product expansions, we also define and analyze normal products (in the sense of Zimmermann).Comment: v3: minor wording changes, as to appear in J. Math. Phys.; 12 page