How to take non-relativistic limit of the following Lagrangian

In summary, the conversation discusses the non-relativistic limit of a Lagrangian and its corresponding Hamiltonian, which describes the interaction between an axion field and a fermion field. The Lagrangian is given in equation (1) and (2), while the Hamiltonian is given as H. The discussion also includes a request for guidance on how to obtain the non-relativistic Hamiltonian from the Lagrangian and if the conversationalist knows how to obtain the relativistic Hamiltonian and the non-relativistic limit of Dirac spinors. Links to articles on the topic are also provided.
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Tan Tixuan
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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?
 
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Related to How to take non-relativistic limit of the following Lagrangian

1. How do I take the non-relativistic limit of a Lagrangian?

The non-relativistic limit of a Lagrangian involves taking the speed of light, c, to be very large or infinite. This can be done by setting c=∞ in the Lagrangian equations and simplifying the resulting equations. This limit is often used in classical mechanics to approximate the behavior of a system at low speeds.

2. What are the steps to take when taking the non-relativistic limit of a Lagrangian?

The first step is to identify the terms in the Lagrangian that contain the speed of light, c. These terms will typically involve kinetic energy or momentum. The next step is to set c=∞ and simplify the equations. This may involve dropping terms that become negligible when c is large. Finally, the resulting equations should be checked to ensure they are consistent with classical mechanics principles.

3. Can the non-relativistic limit be taken for any Lagrangian?

The non-relativistic limit can only be taken for Lagrangians that describe systems in which the speeds are much smaller than the speed of light. This limit is not applicable for systems that involve high speeds, such as those in particle physics or astrophysics.

4. How does taking the non-relativistic limit affect the Lagrangian equations?

Taking the non-relativistic limit typically simplifies the Lagrangian equations by removing terms that become negligible at low speeds. This can make the equations easier to solve and interpret. However, it is important to note that the resulting equations may not accurately describe the behavior of the system at high speeds.

5. Are there any limitations to using the non-relativistic limit in Lagrangian mechanics?

While the non-relativistic limit can be a useful tool in classical mechanics, it is important to remember that it is an approximation and may not accurately describe the behavior of a system at high speeds. Additionally, this limit is not applicable for systems that involve relativistic effects, such as those in particle physics or astrophysics.

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