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]
relate to lattice statistical models
[statistical mechanics]
(x)
interrelation of the program
to other areas
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.