EM Lagrangian: Question on $(\partial_\mu A^\mu)^2$ Term

In summary, the EM Lagrangian in QFT notes from Tong is written in the form of a Lagrangian density, with a term involving the square of the derivative of the vector potential. This term is obtained through integration by parts and is relevant for the action.
  • #1
-marko-
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The EM Lagrangian is
$$\mathcal{L} = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A_\nu)(\partial^\nu A^\mu)]$$

In the QFT notes from Tong the EM Lagrangian is written in the form
$$\mathcal{L} = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A^\mu)^2]$$

I don't see how did he get ##(\partial_\mu A^\mu)^2## term? Thanks :)
 
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  • #2
Integration by parts. What is relevant is not the Lagrangian density, but the action.
 
  • #3
Orodruin said:
Integration by parts. What is relevant is not the Lagrangian density, but the action.
Many thanks, it's clear now.
 

Related to EM Lagrangian: Question on $(\partial_\mu A^\mu)^2$ Term

1. What is the $(\partial_\mu A^\mu)^2$ term in the EM Lagrangian?

The $(\partial_\mu A^\mu)^2$ term in the EM Lagrangian represents the kinetic energy of the electromagnetic field. It is a gauge-invariant term that describes the self-interaction of the electromagnetic field.

2. How does the $(\partial_\mu A^\mu)^2$ term contribute to the equations of motion?

The $(\partial_\mu A^\mu)^2$ term contributes to the equations of motion by providing a term that relates the second derivatives of the electromagnetic field to the field itself. This is important for understanding how the field evolves over time.

3. Why is the $(\partial_\mu A^\mu)^2$ term important in the EM Lagrangian?

The $(\partial_\mu A^\mu)^2$ term is important because it helps to describe the behavior of the electromagnetic field and how it interacts with itself. Without this term, the EM Lagrangian would not accurately reflect the dynamics of the electromagnetic field.

4. Is the $(\partial_\mu A^\mu)^2$ term present in other Lagrangians?

Yes, the $(\partial_\mu A^\mu)^2$ term is present in other Lagrangians, such as the Yang-Mills Lagrangian which describes the interactions of non-abelian gauge fields.

5. How does the $(\partial_\mu A^\mu)^2$ term relate to the gauge symmetry of the EM Lagrangian?

The $(\partial_\mu A^\mu)^2$ term is gauge-invariant, meaning that it does not change under a gauge transformation. This is important for maintaining the gauge symmetry of the EM Lagrangian, which is a fundamental principle in understanding the behavior of the electromagnetic field.

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