CFTs are conformally invariant
field theories, primarily in two spacetime dimensions. There are several
motivations to study them:
IQFTs are theories which classically have infinitely many conserved
charges that are preserved at the quantum level, mostly in two spacetime
dimensions. In most cases they can be considered as perturbations of an
ultraviolet fixed point CFT by some relevant operator. This is the UV (ultraviolet,
or Lagrangian) description.
On the other hand, using the bootstrap approach one can conjecture their
spectra and S-matrices. These are called the IR (infrared) data of the
theory. Using these data, one can calculate exact matrix elements of local
operators, the so-called form factors which can be used to give an approximation
for the large-distance behaviour of correlation functions.
IQFTs have many interesting dynamical symmetries, some of
which are generated by nonlocal charges and correspond to algebras of the
Hopf / quantum group type. These symmetries play an extremely important
role in the determination of the spectrum and the S-matrix. They can be
used to determine S-matrices of massive perturbations of rational conformal
field theories (RCFT) and play a fundamental role in calculating the scattering
amplitudes in affine Toda field theories.
There is always the question whether the S-matrices determined (or rather,
conjectured) using these methods actually correspond to what is called
the Lagrangian of the theory. To demonstrate this, one needs a link
between the UV and IR descriptions which is provided by
Finite size effects mean the dependence of the spectrum of the QFT
on a finite spatial volume. They can be studied using the following methods:
TCS is a numerical approach (an implementation of the well-known
variational approach in quantum mechanics), while the TBA and NLIE approaches
are analytic methods, both based on Bethe Ansatz.
Using these methods one can establish the correctness of conjectured S-matrices, compute relations between the physical masses of particles and Lagrangian parameters (i.e. matching IR and UV data) and calculate the Casimir energy for the vacuum and the excited states of an IQFT.
The scaling functions describing the finite size effects contain a great
deal of information about the QFT system and have some beautiful analytic
properties which play a significant role in recent developments.
Some of the techniques developed for studying CFT & IQFT, together
with the insight gained in course, can be applied to the study of nonintegrable
quantum field theories. From the point of view of QFT, they are much more
realistic models than their integrable counterparts since
Currently, one of the most active areas of development is
motivated partially by issues in nonperturbative string physics
and partially by condensed matter problems. In this case, the field theory
problem is set up in a spacetime with a boundary on which certain boundary
conditions and interactions are prescribed. The focus is on the effect
of the boundary on physical phenomena. Finite size effects, once again,
provide a very efficient method to approach these problems.
From a field theorist's point of of view, ultimately, all these efforts
serve to clarify issues in
Nonperturbative effects are extremely important for our understanding
of nature. The most well-known particle physics example is the description
of hadrons in QCD, but they play a significant role in many other areas
of high energy physics, cosmology and condensed matter physics. The
hope is that studying CFT and IQFT can lead us closer to the understanding
of these problems. There are cases when the results are directly relevant
to phenomenology, e.g. quantum Hall effect or the Kondo problem.