  Low-dimensional systems are an important field of current theoretical and
experimental research. Recent technological developments provide many possible
realizations of effectively one-dimensional systems. These devices promise to
give us access to a new range of phenomena. It is therefore very interesting to
develop theoretical methods specific for such systems to model their behavior
and calculate the correlators of the resulting theory. Incidentally, one such
method exists and is known as Bosonization. It can be applied to
one-dimensional systems and effectively describes low energy excitations in a
universal way. We use the example of a correlator known as the Emptiness
Formation Probability to show that Bosonization fails to describe some long
range correlators corresponding to large disturbances (the EFP measures the
probability for the ground state of the system to develop a region without
particles). We trace this failure to the fact that Bosonization is constructed
as a linear approximation of the full theory and we set up to develop a
collective description with the required non-linearity. The resulting scheme is
essentially a Hydrodynamic paradigm for quantum systems. We show how to
construct such a hydrodynamic description for a variety of exactly integrable
models and illustrate how it can be used to make new predictions. For the
special case of the spin-1/2 XY model we take advantage of the structure of the
model to express the EFP as a determinant of a very special type of matrix,
known as Toeplitz Matrix. We use the theory of Toeplitz determinants to
calculate the asymptotic behavior of the EFP in the XY model and discuss its
relation with the criticality of the theory.
