Two-dimensional models

Local quantum physics of systems with infinitely many interacting degrees of freedom leads to situations whose understanding often requires new physical intuition and mathematical concepts beyond that acquired in quantum mechanics and perturbative constructions in quantum field theory. In this situation twodimensional soluble models turned out to play an important role. On the one hand they illustrate new concepts and sometimes remove misconceptions in an area where new physical intuition is still in process of being formed. On the other hand rigorously soluble models confirm that the underlying physical postulates are mathematically consistent, a task which for interacting systems with infinite degrees of freedom is mostly beyond the capability of pedestrian methods or brute force application of hard analysis on models whose natural invariances has been mutilated by a cut-off. In order to underline these points and motivate the interest in 2-dimensional QFT, let us briefly look at the history, in particular at the physical significance of the three oldest two-dimensional models of relevance for statistical mechanics and relativistic particle physics, in chronological order: the LenzIsing model, Jordan’s model of bosonization/fermionization and the Schwinger model (QED2). The Lenz-Ising (L-I) model was proposed in 1920 by Wihelm Lenz [1] as the simplest discrete statistical mechanics model with a chance to go beyond the P. Weiss phenomenological Ansatz involving long range forces and instead explain ferromagnetism in terms of non-magnetic short range interactions. Its one-dimensional version was solved 4 years later by his student Ernst Ising. In his 1925 university of Hamburg thesis, Ising [1] not only showed that his chain solution could not account for ferromagnetism, but he also proposed some (as it turned out much later) not entirely correct intuitive arguments to the extend that this situation prevails to the higher dimensional lattice version. His advisor