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- 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?

$$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$

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?