Non-relativistic reduction of spinors, new currents and their algebra

Abstract

A specific mapping is introduced to reduce the Dirac action to the non-relativistic (Pauli -Schrödinger) action for spinors. Using this mapping, the structures of the vector and axial vector currents in the nonrelativistic 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 -Schrödinger 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. 1 rabin@bose.res.in 2

Canonical formulation for a non-relativistic spinning particle coupled to gravity

“…The free spinning nonrelativistic particle in a flat background has been discussed here in details in section 2. From the first two equations of (25) we see that here also the geodesic equation is satisfied by the momenta, although the coordinates apparently do not (see (39)). This happens because the momenta is not aligned with the velocity (see (24)), similar to the last relation of (112).…”

“…It is possible to discuss the coupling of this theory to a background magnetic field as an extension of our approach. By following the same arguments as before, the nonrelativistic limit of the Dirac equation in presence of a magnetic field is obtained [39]. The result is obtainable from (8) by the minimal substitution, p i → p i − A i , where A i is the vector potential,…”

“…One possible way to obtain this is to start from the four component Dirac equation. Expressing the four component Dirac fermion in terms of a pair of two component fermions, taking the nonrelativistic (c → ∞) limit after eliminating the oscillations due to the rest mass of the fermions, eventually yields the following set of coupled equations [38,39],…”

“…where we have put t = τ . Taking a further time derivative and using the form of the momenta given in (24) as well as ṗi = 0, which follows either from (25) or from the Hamiltonian equation of motion, we are immediately led to (39). We reserve further discussion on the implications of the equations of motion (see section 5) till the studies of spinning particle motion in a curved background.…”