All Questions
33
questions
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 ...
- 43
0
votes
2
answers
372
views
Derivative of $\nabla\times(\nabla\times A)$ by A
I'm trying to find out how to quantize EM field. It seems like $\vec{A}$ and $\vec{E}$ are it's canonical coordinates. For example:
$$\mathfrak{H} = \frac12E^2 + \frac12(\nabla\times A)^2$$
$$H = \int ...
3
votes
2
answers
2k
views
How is solving Proca equation equivalent to scalar field equation?
My prof. told me that using differential forms proca equation reduces to solving for scalar field equation. How is that? I can’t see how does one relate to Scalar equation using differential forms.
...
user183683
0
votes
2
answers
57
views
Magnetic moment of a radially symmetric current
In my latest assignment I'm tasked with finding a magnetic moment $\mu$ of a hydrogen atom, whose current distribution $\mathbf{j}(\mathbf{r})$ looks like
$$\mathbf{j}(\mathbf{r})=\frac{e\hbar}{3^8 \...
-1
votes
2
answers
235
views
Relativistic EM Lagrangian and the derivation of equations of motion
As mentioned in my other post, I am attempting to learn from Gross'"Relativistic quantum mechanics and field theory", and I have a question concerning the manipulation of the antisymmetric 4x4 tensors ...
- 255
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\...
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 ...
5
votes
3
answers
666
views
How to see $\mathbf{E}\cdot\mathbf{B}$ is a total derivative?
Since $\mathbf{E}\cdot\mathbf{B}$ is a Lorentz invariant of the electromagnetic fields it seems like an interesting thing to plug into a Lagrangian to see what happens. However, this ends up ...
- 1,289
0
votes
3
answers
2k
views
Square of the Maxwell Field Tensor
I want to prove that the square of the Maxwell field tensor
$$F_{\mu\nu}F^{\mu\nu}=2(B^2-E^2),$$
but I got $F_{\mu\nu}F^{\mu\nu}=2(-B^2+E^2)$ instead.
Here's what I did:
$$F_{\mu\nu}F^{\mu\nu}=F_{0\nu}...
- 1,161
1
vote
0
answers
222
views
Index notation with four-gradient
Reading Schwarz's textbook on quantum field theory, early on he gives the Lagrangian
$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_{\mu}J_{\mu}.$$
With $F^{\mu\nu}=(\partial_{\mu}A^{\nu}-\partial_{\...
user77629
1
vote
0
answers
67
views
EM Lagrangian in terms of gauge fields [duplicate]
I have a question that may be very simple and potentially for that very reason I can't find a sensible answer to it - everyone just skips over it. I have a EM Lagrangian given by:
$L -\frac{1}{4} F^{\...
- 521
3
votes
2
answers
430
views
How to expand Maxwell Lagrangian?
I am given $$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=-...
- 385
2
votes
1
answer
351
views
Hamilton's equations of motion on Dirac's formalism
I'm having several doubts about the procedure proposed by the Dirac-Bergmann algorithm in order to get the correct equations of motion of electrodynamics (Maxwell's equations).
Suppose I've already ...
- 1,742
5
votes
3
answers
3k
views
Energy-Momentum Tensor for Electromagnetism in Curved Space
$\newcommand{\l}{\mathcal L} \newcommand{\g}{\sqrt{-g}}$$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$I want to calculate the energy-momentum tensor in curved free space by functional ...
- 4,021
0
votes
1
answer
505
views
How do you take the derivative with respect to a rank two tensor?
I am learning classical field theory and am trying to find the momentum density of the electromagnetic lagrangian as part of an example of Noether's Theorem. The derivative I am encountering is:
$$
\...
- 935