A brief survey of recent results in the study of boundary integrable quantum field theories, indicating some currently open problems. Based on lectures given at the 2000 Eötvös Summer School in Physics on "Nonperturbative QFT methods and their applications". DCTP/01/13; hep-th/0101174These lectures concerned the properties of quantum field theories in the presence of boundaries. There are many different approaches to this subject. One can begin by studying conformal field theories with boundaries -the principal theme of the lectures at this school by Jean-Bernard Zuber and by Christoph Schweigert -and then, as described in Gérard Watts' lectures, consider their perturbations. In many cases these perturbations result in massive integrable quantum field theories, and it was the direct study of such theories in their own right that formed my main topic. A number of reviews of this subject can be found on the electronic archives, and so in this contribution I shall restrict myself to an outline of the questions touched on in my talks, and a brief list of references to which the interested reader can turn to find at least some of the answers.The focus will be on boundary field theories which are integrable, and if the usual locality conditions are also imposed then, just as for theories without boundaries, the dimensionality of space must be restricted to one. With the time dimension remaining infinite, there are then just two possible 'boundary geometries': the theory can be defined eitherIt can then be studied either as a classical, or directly as a quantum field theory. Key questions that one might ask include the following:(a) For a given classical quantum integrable model on the full line ('in the bulk'), which boundary conditions are compatible with integrability? 1 (b) Given a massive boundary integrable quantum field theory, how do bulk particles scatter off the boundary, and how should the 'exact S matrix' technology be generalised to encompass boundary problems? (c) Can perturbation theory be set up to test any exact predictions? (d) In the presence of a boundary, the spectrum of bulk excitations may, depending on the boundary condition, be augmented by a number of 'boundary bound states'. How is this spectrum encoded in the amplitudes for the scattering of bulk particles off the boundary? (e) What does the spectrum of the theory look like on a finite interval?Before any of these issues can be addressed, the properties of massive integrable quantum field theories on the full line should be understood. The classic reference is the article [1] by Zamolodchikov and Zamolodchikov; a couple of more recent reviews, containing many more references, are [2,3].Having digested this material, we can return to the novel questions which arise when the model has a boundary. Much of the recent interest in this topic can be traced to the pioneering work [4] by Ghoshal and Zamolodchikov. In particular, question (b), concerning the correct generalisation of the ideas of exact S matrix theory to boundary theories, was answered...
