Problem with Maxwell Lagrangian Density

In summary, the given term needs to be expanded to the form of ##\dfrac{1}{2}\left(\partial_{\mu}A_{\nu}\right) \left(\partial^{\mu}A^{\nu}\right) - \dfrac{1}{2}\left(\partial_{\mu}A^{\mu}\right)^2##. This can be achieved by using the hint ##{\partial}_{\mu}A_{\nu}{\partial}^{\nu}A^{\mu}={\partial}_{\nu}A_{\mu}{\partial}^{\mu}A^{\nu}=({\partial}_{\mu}A^{\mu})^{2}
  • #1
Strangelet
4
0

Homework Statement


I have to expand the following term:

$$\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu} = \dfrac{1}{4} \left(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}\right) \left(\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}\right)$$

to get in the end this form:

$$\dfrac{1}{2}\left(\partial_{\mu}A_{\nu}\right) \left(\partial^{\mu}A^{\nu}\right) - \dfrac{1}{2}\left(\partial_{\mu}A^{\mu}\right)^2$$

Homework Equations



I really don't know how to make the calculation. I tried to multiply terms but I think I din't get some rule about index.. sigh!

The Attempt at a Solution

 
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  • #2
Here's a hint, ##{\partial}_{\mu}A_{\nu}{\partial}^{\nu}A^{\mu}={\partial}_{\nu}A_{\mu}{\partial}^{\mu}A^{\nu}=({\partial}_{\mu}A^{\mu})^{2}## . I'm not near a computer and have a hard time writing TeX from my phone, but foiling it out and using this relation should get you the right answer.
 
  • #3
Strangelet said:
I have to expand the following term:

$$\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu} = \dfrac{1}{4} \left(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}\right) \left(\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}\right)$$

to get in the end this form:

$$\dfrac{1}{2}\left(\partial_{\mu}A_{\nu}\right) \left(\partial^{\mu}A^{\nu}\right) - \dfrac{1}{2}\left(\partial_{\mu}A^{\mu}\right)^2$$

I don't believe the expression ##\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu} ## is equal to the form you want to get. However, recall that two Lagrangian densities lead to the same equations of motion if they differ by the divergence of some expression. So, try to show that the initial and final forms differ only by a divergence of some expression.
 

FAQ: Problem with Maxwell Lagrangian Density

1. What is the Maxwell Lagrangian density?

The Maxwell Lagrangian density is a mathematical quantity that describes the dynamics of electromagnetic fields. It is derived from the Lagrangian formalism, which is a way of expressing the laws of physics in terms of the energy of a system.

2. What is the problem with the Maxwell Lagrangian density?

The problem with the Maxwell Lagrangian density is that it does not take into account the quantum nature of electromagnetic fields. This means that it cannot fully describe the behavior of these fields on a microscopic level.

3. How does this problem affect our understanding of physics?

The problem with the Maxwell Lagrangian density means that our current understanding of physics is incomplete. It does not fully explain the behavior of electromagnetic fields at the quantum level, which is crucial for understanding many phenomena in the universe.

4. Is there a solution to this problem?

There have been attempts to modify the Maxwell Lagrangian density to include quantum effects, but a complete and satisfactory solution has not yet been found. This is an active area of research in theoretical physics.

5. How does the Maxwell Lagrangian density fit into the larger context of physics?

The Maxwell Lagrangian density is an important part of the Standard Model of particle physics, which describes the fundamental particles and forces in the universe. It is also a key component of classical electrodynamics, which is used to describe the behavior of electromagnetic fields in macroscopic systems.

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