  In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local
densities and currents from canonical commutation relations in nonrelativistic
quantum field theory. The corresponding Lie group is infinite dimensional: the
natural semidirect product of an additive group of scalar functions with a
group of diffeomorphisms. Unitary representations of this group describe a wide
variety of quantum systems, and have predicted previously unsuspected
possibilities; notably, anyons and nonabelian anyons in two space dimensions.
We present here foundational reasons why this semidirect product group serves
as a universal kinematical group for quantum mechanics. We obtain thus a
unified account of all possible quantum kinematics for systems with mass in an
arbitrary physical space, and clarify the role played by topology in quantum
mechanics. Our development does not require quantization of classical phase
space; rather, the classical limit follows from the quantum mechanics. We also
consider the relationship of our development to Heisenberg quantization.
