Variation in action for modified EM Field Action

In summary: This result is significant because it shows that the equations of motion, which are derived from the Lagrangian, are also invariant under this gauge transformation. This means that our theory is consistent and we can use this freedom to simplify calculations. Also, it is interesting to note that this transformation does not change the physical content of the theory, only the mathematical form. This is a fundamental concept in Lagrangian Field Theory and is crucial in understanding the behavior of particles and fields. In summary, the conversation discusses a question and solution related to Lagrangian Field Theory. The question involves determining the variation in the action for a given potential and the solution shows that the variation is equal to zero. This result is significant because it demonstrates gauge invar
  • #1
maverick280857
1,789
4
Hi everyone

I am teaching myself QFT, and am currently learning Lagrangian Field Theory. Here is a question I am trying to solve, and I am not absolutely sure if my solution is correct because I am new to this notation and material. I would be grateful if someone could go over it and let me know if its correct. This isn't homework.

Also, I'd like to know what the significance of this result is, if it is indeed true.

Question: Consider the action [itex]S = \frac{1}{4}\int d^{4}x F_{\mu\nu}F^{\mu\nu}[/itex]. Vary the potential according to [itex]A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}\Phi[/itex] where [itex]\Phi[/itex] is a scalar field. Determine the variation in the action.

Solution:

[tex]\delta S = \frac{1}{4}\int d^{4}x (\delta F_{\mu\nu}F^{\mu\nu} + F_{\mu\nu}\delta F^{\mu\nu})[/tex]

[tex]\delta F_{\mu\nu}F^{\mu\nu} &=& \delta(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})F^{\mu\nu}[/tex]
[tex]\implies = (\partial_{\mu}\partial_{\nu}\Phi - \partial_{\nu}\partial_{\mu}\Phi)F^{\mu\nu} = 0[/tex]

Similarly [itex]F_{\mu\nu}\delta F^{\mu\nu} = 0[/itex]. Therefore [itex]\delta S = 0[/itex].

I have attached a pdf file, which contains this question and my working.
 

Attachments

  • qftquest2.pdf
    34.6 KB · Views: 222
Physics news on Phys.org
  • #2
I did not read your pdf-file, but your conclusion is correct. delta_S is equal to zero. Moreover, F_mu_nu does not vary. It is know as a "gauge" liberty, invariance of transformations. A_mu may be replaced by another A_mu in a certain way, but this does not change fields and particle trajectories. It is similar to the potential energy shifts U -> U+const in CM. The only thing that counts is the exchange, not the absolute value.

Bob.
 
  • #3
Thanks bob. I see, so this is a gauge transformation under which the Lagrangian is invariant.
 

1. What is "Variation in action for modified EM Field Action"?

"Variation in action for modified EM Field Action" refers to the changes in behavior or effects of an electromagnetic (EM) field when it is modified or manipulated in some way. This could include altering its frequency, amplitude, or direction.

2. How does variation in EM field affect its action?

Variation in the EM field can greatly impact its action, as it can change the strength and direction of the field's force, as well as the type of energy it emits. This can have a significant impact on the objects or particles that interact with the field.

3. What are some examples of modified EM field action?

Examples of modified EM field action include using a magnet to change the direction of a compass needle, using a speaker to create sound waves through electromagnetic induction, and using a microwave oven to heat food through the absorption of EM radiation.

4. How is variation in EM field action studied?

Variation in EM field action is studied through various experiments and observations, often using specialized equipment such as EM field meters or oscilloscopes. Scientists also use mathematical models and simulations to understand and predict the behavior of modified EM fields.

5. What are the practical applications of studying variation in EM field action?

Studying variation in EM field action has numerous practical applications, including in the fields of engineering, telecommunications, and medical technology. Understanding how to manipulate and control EM fields allows us to create devices and technologies that improve our daily lives, such as cell phones, MRI machines, and power generators.

Similar threads

Replies
5
Views
289
Replies
24
Views
2K
  • Quantum Physics
Replies
3
Views
1K
Replies
3
Views
889
  • Quantum Physics
Replies
3
Views
3K
  • Quantum Physics
Replies
3
Views
299
Replies
5
Views
757
  • Quantum Physics
Replies
14
Views
2K
  • Quantum Physics
Replies
1
Views
583
Replies
1
Views
816
Back
Top