  Using the notion of Galilean conformal algebra (GCA) in arbitrary space
dimension d, we introduce for d=3 quantized nonrelativistic counterpart of
twistors as the spinorial representation of O(2,1){\oplus}SO(3) which is the
maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant
quantization of such nonrelativistic spinors, which shall be called also
Galilean twistors, is presented. We consider for d=3 the general spinorial
matrix realizations of GCA, which are further promoted to quantum-mechanical
operator representations, expressed as bilinears in quantized Galilean twistors
components. For arbitrary Hermitian quantum-mechanical Galilean twistor
realizations we obtain the result that the representations of GCA with
positive-definite Hamiltonian do not exist. For non-positive H we construct for
N{\geq}2 the Hermitian Galilean N-twistor realizations of GCA; for N=2 such
realization is provided explicitly.
