All Questions
33
questions
0
votes
1
answer
197
views
Finding the resonant frequency of a rectangular resonator filled with a magnetic material
The prompt is to find the resonant frequency $f_r$ of a rectangular resonator which is filled with a magnetic material rather than standard air or vacuum. I'm confused as how the resonance frequency ...
CommunityBot
- 1
0
votes
1
answer
39
views
Covariant derivative property
I am trying to demonstrate this propertie
$$
\not{D}^2= \mathcal{D}^\mu \mathcal{D}_\mu-\frac{i}{4}\left[\gamma^\mu, \gamma^\nu\right] F_{\mu \nu}
$$
where $\not{}~$ is the Feynmann slash, and $D_\mu ...
- 26.6k
1
vote
0
answers
40
views
Detailed derivation of the energy-momentum tensor from the Maxwell Lagrangian [duplicate]
I have started studying QFT, and I am currently reviewing briefly on the classical field theory. I have come across the Maxwell Lagrangian given by
$$
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.
$$
...
- 207k
3
votes
2
answers
430
views
How to expand Maxwell Lagrangian?
I am given $$L=-\frac{1}{4}F^2_{\mu\nu}-A_{\mu\nu}J_\mu$$ to calculate equations of motion I have to expand the terms in the Lagrangian as following (note this is from Schwartz QFT book page 37):
$$L=-...
- 4,902
4
votes
1
answer
225
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 207k
1
vote
1
answer
226
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 457
0
votes
0
answers
88
views
How compute the expression of electromagnetic tensor explicitly as given here?
I am trying to understand how the second line arrives at the last line of this expression.
For $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$
And $F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu ...
- 16k
3
votes
2
answers
622
views
Gauge Invariant terms of Lagrangian for Electromagnetism
Besides the usual EM Lagrangian $\mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$, we can add an additional term $\mathcal{L'} = \epsilon_{\mu \nu \rho \sigma }F^{\mu \nu}F^{\rho \sigma} = -8 \vec{E} ...
- 16k
-3
votes
1
answer
204
views
What does “Integrating out field” mean?
In Schwartz’s QFT book, there is a couple of exercise problems of particle polarization in chapter 3. I have trouble with finding interaction terms from the given Lagrangians. Is it just okay to ...
- 1
-1
votes
1
answer
132
views
Gauge invariance of a Lagrangian
How do I check whether or not the Lagrangian is a gauge invariant? A Lagrangian is
$$
\mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}m^2A_\mu A^\mu
$$
- 207k
1
vote
1
answer
87
views
Deriving Lagrangian density in field theory
While reading a field theory book, there's a (rather simple) equation derivation part that I can't quite understand.
Apparently from $({\partial}^2 + m^{2})A_{\mu} = 0$ (for the vector field carrying ...
- 4,361
2
votes
0
answers
406
views
Derivation of Coulomb's law from classical field theory
In the section on Coulomb's law in QFT by Schwartz, he expands $-\frac{1}{4}F_{\mu\nu}^{2}$ to get $-\frac{1}{2}(\partial_{\mu}A_{\nu})^{2} + \frac{1}{2}(\partial_{\mu}A_{\mu})^{2}$, can someone ...
4
votes
1
answer
1k
views
Canonical conjugate momenta of EM Field Lagrangian density
I have the EM Field Lagrangian density given as
$
\mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
$
where $F^{\mu \nu}$ is the Field strength tensor defined as $F^{\mu \nu} = \partial^\mu A^\nu- \...
3
votes
1
answer
155
views
Non-linearities in Lagrangian of a scalar field coupled to point-like source
I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian
\begin{equation}
\mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\...
CommunityBot
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1
vote
1
answer
569
views
How to evaluate the Euler-Lagrange equation for the electromagnetic Lagrangian? [duplicate]
I'm fascinated with field theories, but have little knowledge about them, so excuse Me if this is a dumb question.
We all know, that if we have a Lagrangian in terms of a field $\Phi $, we can just ...
- 207k