  In this work, we develop conformal bootstrap for Galilean conformal field
theory (GCFT). In a GCFT, the Hilbert space could be decomposed into
quasiprimary states and its global descendants. Different from the usual
conformal field theory, the quasi-primary states in a GCFT constitute
multiplets, which are block-diagonized under the Galilean boost operator. More
importantly the multiplets include the states of negative norms, indicating the
theory is not unitary. We compute global blocks of the multiplets, and discuss
the expansion of four-point functions in terms of the global blocks of the
multiplets. Furthermore we do the harmonic analysis for the Galilean conformal
symmetry and obtain an inversion formula. As the first step to apply the
Galilean conformal bootstrap, we construct generalized Galilean free theory
(GGFT) explicitly. We read the data of GGFT by using Taylor series expansion of
four-point function and the inversion formula independently, and find exact
agreement. We discuss some novel features in the Galilean conformal bootstrap,
due to the non-semisimpleness of the Galilean conformal algebra and the
non-unitarity of the GCFTs.
