How to take non-relativistic limit of the following Lagrangian

[Tan Tixuan]

TL;DR Summary

I want to take the non-relativistic limit of the following Lagrangian.

In https://arxiv.org/pdf/1709.07852.pdf, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part)
$$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$

will yield the following Hamiltonian
$$H=-g\vec{\nabla} a \cdot \vec{\sigma_{\psi}}$$

Where $a$ is the axion field (scalar field), and $\psi$ is a fermion field. g is the interaction strength. $\sigma_{\psi}$ is the spin operator of the fermion field.

Can anyone teach me how to take this limit? How to start from the Lagrangian and obtain the non-relativistic Hamiltonian?

 

[malawi_glenn]

Do you know how to obtain the relativistic hamiltonian?
Do you know the non-relativistic limit of Dirac Spinors?

Here they work it out for Yukawa interacation
https://arxiv.org/pdf/1704.00155.pdf
https://arxiv.org/pdf/1910.01533.pdf