Nonrelativistic Limit of Dirac Theory From Effective Field Theory

TL;DR

This paper derives the nonrelativistic limit of Dirac theory using effective field theory, showing how the low energy effective action relates to the Pauli-Schrödinger equation and analyzing the significance of various terms.

Contribution

It provides a systematic derivation of the nonrelativistic limit of Dirac theory within the effective field theory framework, clarifying the role of different terms in the effective action.

Findings

Derived the low energy effective action for Dirac theory

Connected the effective action to the Pauli-Schrödinger equation

Analyzed the significance of terms using anisotropic dimensional analysis

Abstract

In this work we analyze the low energy nonrelativistic limit of Dirac theory in the framework of effective field theory. By integrating out the high energy modes of Dirac field, given in terms of a combination of the two-components Weyl spinors, we obtain a low energy effective action for the remaining components, whose equation of motion can then be compared to the Pauli-Schr\"odinger equation after demanding normalization of the wave function. We then discuss the relevance of the terms in the effective action in the context of an anisotropic dimensional analysis which is suitable for nonrelativistic theories.