# Non-relativistic Reduction of Spinors, New Currents and their Algebra

## TL;DR

This paper introduces a mapping to derive non-relativistic spinor actions and currents from the Dirac theory, revealing new currents and algebraic structures crucial for understanding non-relativistic limits of relativistic symmetries.

## Contribution

It presents a novel mapping from relativistic to non-relativistic spinor actions and uncovers new non-abelian currents necessary for algebra closure.

## Key findings

- Derived non-relativistic vector and axial vector currents
- Identified a new non-abelian current in the Pauli-Sch"odinger theory
- Discussed the impact of parity on non-relativistic currents

## Abstract

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.