Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Topological Defects on the Lattice I: The Ising model

In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations, cousins of the Yang-Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers-Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.Comment: 45 pages, 9 figure

The kinetic description of vacuum particle creation in the oscillator representation

The oscillator representation is used for the non-perturbative description of vacuum particle creation in a strong time-dependent electric field in the framework of scalar QED. It is shown that the method can be more effective for the derivation of the quantum kinetic equation (KE) in comparison with the Bogoliubov method of time-dependent canonical transformations. This KE is used for the investigation of vacuum creation in periodical linear and circular polarized electric fields and also in the case of the presence of a constant magnetic field, including the back reaction problem. In particular, these examples are applied for a model illustration of some features of vacuum creation of electron-positron plasma within the planned experiments on the X-ray free electron lasers.Comment: 17 pages, 3 figures, v2: a reference added; some changes in tex

Stability and Representation Dependence of the Quantum Skyrmion

A constructive realization of Skyrme's conjecture that an effective pion mass ``may arise as a self consistent quantal effect'' based on an ab initio quantum treatment of the Skyrme model is presented. In this quantum mechanical Skyrme model the spectrum of states with $I=J$, which appears in the collective quantization, terminates without any infinite tower of unphysical states. The termination point depends on the model parameters and the dimension of the SU(2) representation. Representations, in which the nucleon and $\Delta_{33}$ resonance are the only stable states, exist. The model is developed for both irreducible and reducible representations of general dimension. States with spin larger than 1/2 are shown to be deformed. The representation dependence of the baryon observables is illustrated numerically.Comment: 19 pages, Late