Relativity symmetry and the structure of quantum theory William H Klink Department of Physics and Astronomy University of Iowa USA; Sujeev Wickramasekara Department of Physics Grinnell College USA ; 1: Galilean quantum theory The history of how quantum mechanics was developed is a fascinating one and underlies the focus of this book; namely given the rules that the founders of quantum mechanics developed is it possible to find principles that lead to the structure of quantum mechanics as it was historically formulated This is the first book in a series of works considering what particular relativity is applicable to a given dynamical theory. The series considers Newton Einstein and de Sitter relativities while this book examines the unitary irreducible representations of the Galilei group and see how they provide the framework for Galilean quantum theory.