  We make a detailed study of the infinite dimensional Galilean Conformal
Algebra (GCA) in the case of two spacetime dimensions. Classically, this
algebra is precisely obtained from a contraction of the generators of the
relativistic conformal symmetry in 2d. Here we find quantum mechanical
realisations of the (centrally extended) GCA by considering scaling limits of
certain 2d CFTs. These parent CFTs are non-unitary and have their left and
right central charges become large in magnitude and opposite in sign. We
therefore develop, in parallel to the usual machinery for 2d CFT, many of the
tools for the analysis of the quantum mechanical GCA. These include the
representation theory based on GCA primaries, Ward identities for their
correlation functions and a nonrelativistic Kac table. In particular, the null
vectors of the GCA lead to differential equations for the four point function.
The solution to these equations in the simplest case is explicitly obtained and
checked to be consistent with various requirements.
