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General context and motivation

My main research expertise is on quantum field theory (QFT), which has developed over the last century into an extremely impressive and successful branch of theoretical physics, culminating so far with the formulation of the standard model. Success of the theory is to be understood in the very classical sense of physics, that is agreement of the theory with the experiment, exemplified by quantum electrodynamics which is very often quoted as the theory confirmed to the highest degree by experiments. The predictions of the standard model for the anomalous magnetic moment of the electron agrees with the experiment up to a precision of 11 digits (e.g. [ 120 ]). Nonetheless, despite these remarkable results, there exist well known phenomena which can not be explained within the framework of the standard model (e.g. neutrinos seems to have masses [ 114 ] and the predictions for the magnetic moment of the muon show deviations between theory and experiment after 6 digits [ 42 ]), not to mention the conceptually more difficult and still outstanding unification with gravity. Most troubling, however, and surely related to the open issues, is the status of the current understanding of the mathematical structure underlying QFT, which is still fairly poor. Essentially all concrete results rely upon perturbative calculations in the weak coupling regime around the only analytic solution in higher dimensions, that is the free, that is non-interacting, solution. Since there are no free quarks this still constitutes the main technical problem in quantum electrodynamics. This apparent need to go beyond non-perturbative considerations was already expressed almost sixty years ago by Heisenberg [ 88 ], who stressed the importance of studying analytic continuations of scattering amplitudes into the complex momentum plane. These ideas turned out to be very fruitful and lead to interesting results on-shell [ 44 ], e.g. for the scattering matrix, as well as off-shell [ 7 ], that is in particular for the two-particle form factors. Moreover, in the seventies it was realized, that once one restricts ones attention to integrable quantum field theories (IQFTs) in 1+1-dimensional space-time these concepts reveal their full strength. The main reason for this is that the Coleman-Mandula [37 ] theorem is less constraining than in higher dimensions, where it states that symmetries additional to the Poincaré invariance and internal gauge symmetries lead to non-interacting theories. In 1+1 dimensions it allows the existence of non trivial theories with additional symmetries, possibly infinite in which case they are often called integrable, a terminology borrowed from classical dynamical systems. As a consequence these symmetries lead to conserved charges and in fact the existence of one of them is enough to guarantee the crucial property of these theories, namely that the multi-particle scattering matrix factorizes into two-particle scattering matrices. Remarkably, as will be explained below, these two-particle amplitudes may be determined non-perturbatively in an exact manner. Due to this success these IQFTs attracted some attention and were originally viewed as excellent laboratories in which many general ideas and concepts of QFT could be tested and understood better than in higher dimensions. Pioneered to a very large extend by the theory group of the Freie Universität Berlin [ 112 ] these investigations have turned into the formulation of an entire program, the so-called ``bootstrap (form factor) program" (for reviews see [ 92 ]). Meanwhile the program has been extended in several aspects, partly motivated by advances in string (brane) theory and partly by the progress in experimental techniques of condensed matter physics. The latter point will be more important for this proposal as it has turned this program from a purely theoretical and mathematical playground into a real physical scheme which allows to compute experimentally measurable quantities. Including some new insights, in particular those which came about with the breakthrough in conformal field theory (CFT) in the middle of the eighties [ 9 ] (for a CFT-review see [41 ]), the program consists of the following principle steps:
(o)
classical foreplay [dynamical systems]
(i)
determination of the S-matrix
(ii)
construction of the form factors
(iii)
consistency checks [ultraviolet limit, perturbation theory (in g, 1/N, OPE), TBA]
(iv)
calculation of correlation (Wightman) functions
(v)
identification and classification of the (local) operator content
(vi)
organize and extend the zoo of models [algebraic structures]
(vii)
add boundaries and impurities
(viii)
compute measurable quantities [condensed matter physics]
(ix)
relate to lattice statistical models [statistical mechanics]
(x)
interrelation of the program to other areas
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Of course this is only a schematic view, which helps to clarify the concepts. In general the individual points do not have to be carried out in the particular order stated, as only some off them are subject to a strict order. The original formulation for instance consists merely of the points (i), (ii) and (iv), which have to be carried out consecutively. The scheme should help orientation and provide a guide in particular when the order is not strict. Quantities and abbreviations will be explained in the adequate sections. For instance one sees that one may add boundaries or defects (point (vii)) once the bulk S-matrix is known and before any knowledge of the points (ii)-(vi). One may also adopt the point of view that one can drop point (o) completely from the list as it is well known that a consistent quantum field theory does not need a classical Lagrangian counterpart. The points (iii) and (x) are essentially related to all other points. In my research so far I have contributed to all of these points and I will now go through them in more detail, discuss my contributions and highlight the open issues at the end of each point, which constitute part of my proposal.



Next: (o) Classical foreplay Up: Quantum field theory Previous: Quantum field theory