  A specific mapping is introduced to reduce the Dirac action to the
non-relativistic (Pauli - Schr\"odinger) action for spinors. Using this
mapping, the structures of the vector and axial vector currents in the
non-relativistic theory are obtained. The implications of the relativistic Ward
identities in the non-relativistic limit are discussed. A new non-abelian type
of current in the Pauli - Schr\"odinger theory is obtained. As we show, this is
essential for the closure of the algebra among the usual currents. The role of
parity in the non-relativistic theory is also discussed.
