TABLE OF CONTENTS In this very short chapter we shall briefly review several approximate methods which are used in the theory of one-dimensional quantum chains. Many recent excellent review articles and books considered these approximate methods, and the purpose of the introduction of this short chapter is only for the "completeness" of the impression of the reader.
The first (and the simplest) class of methods used in the approximate description of quantum correlated chains is connected with the renormalization group approach. Such a study shows that exponents for characteristics of low-dimensional systems are non-integer in general, in contrast to simple perturbation theories, which of course are not legitimate. An application of scaling relations provides a simple tool to understand some essential aspects of the behaviour of quantum correlated chains under a relevant perturbation. To remind, the response of a classical Helmholtz free energy f cl and the correlation function ? cl of a classical critical d-dimensional system perturbed by a relevant operator 
with the renormalization group eigenvalue y ?1 > 0 near a critical point is
where d is the space dimension. A quantum critical d-dimensional system formally behaves in a scaling regime equivalently to a ( d + z)-dimensional classical system, where z is the dynamical critical exponent. To remind, the divergence of correlation functions for quantum critical points implies divergencies not only in space, but also in time, because the real space in which one has to...
