All Questions
33
questions
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 ...
- 161
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
$$
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.
$$
...
- 35
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 ...
- 838
1
vote
1
answer
226
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 139
0
votes
0
answers
88
views
How compute the expression of electromagnetic tensor explicitly as given here?
I am trying to understand how the second line arrives at the last line of this expression.
For $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$
And $F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu ...
- 13
3
votes
2
answers
622
views
Gauge Invariant terms of Lagrangian for Electromagnetism
Besides the usual EM Lagrangian $\mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$, we can add an additional term $\mathcal{L'} = \epsilon_{\mu \nu \rho \sigma }F^{\mu \nu}F^{\rho \sigma} = -8 \vec{E} ...
- 413
-3
votes
1
answer
204
views
What does “Integrating out field” mean?
In Schwartz’s QFT book, there is a couple of exercise problems of particle polarization in chapter 3. I have trouble with finding interaction terms from the given Lagrangians. Is it just okay to ...
- 1
-1
votes
1
answer
132
views
Gauge invariance of a Lagrangian
How do I check whether or not the Lagrangian is a gauge invariant? A Lagrangian is
$$
\mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}m^2A_\mu A^\mu
$$
- 21
1
vote
1
answer
87
views
Deriving Lagrangian density in field theory
While reading a field theory book, there's a (rather simple) equation derivation part that I can't quite understand.
Apparently from $({\partial}^2 + m^{2})A_{\mu} = 0$ (for the vector field carrying ...
- 11
4
votes
1
answer
1k
views
Canonical conjugate momenta of EM Field Lagrangian density
I have the EM Field Lagrangian density given as
$
\mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
$
where $F^{\mu \nu}$ is the Field strength tensor defined as $F^{\mu \nu} = \partial^\mu A^\nu- \...
- 624
1
vote
1
answer
569
views
How to evaluate the Euler-Lagrange equation for the electromagnetic Lagrangian? [duplicate]
I'm fascinated with field theories, but have little knowledge about them, so excuse Me if this is a dumb question.
We all know, that if we have a Lagrangian in terms of a field $\Phi $, we can just ...
- 163
1
vote
0
answers
235
views
How to derive some part of the Proca lagrangian for a Vector (spin-1)? [closed]
I'm trying to derive Eq. (10.17) & Eq. (10.18) from the textbook. Where does the term
-1/(4*pi) come from, and how do I cancel out the rest of the term (see my text, second picture).
- 11
3
votes
2
answers
767
views
Hamiltonian formalism of the massive vector field
I am currently working through a problem concerning the massive vector field. Amongst other things I have already calculated the equations of motion from the Lagrangian density $$\mathcal{L} = - \frac{...
- 157
0
votes
1
answer
4k
views
Given the Lagrangian density, how do I find the equations of motions for fields? [closed]
Given Lagrangian densities, for example:
$ L = \partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2\phi^2 +\lambda \phi(x)$,
the Euler-Lagrange equation yields
$\partial^2 \phi + m^2 \phi = \lambda ...
- 32
1
vote
0
answers
44
views
MCS Lagrangian and Euler-Lagrange
I'm trying to solve the Euler-Lagrange equation for the MCS Lagrangian density as given by Kharzeev in this article (Eqn. 7):
$$ \mathcal{L}_{\textrm{MCS}} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^{\...
- 71
2
votes
0
answers
406
views
Derivation of Coulomb's law from classical field theory
In the section on Coulomb's law in QFT by Schwartz, he expands $-\frac{1}{4}F_{\mu\nu}^{2}$ to get $-\frac{1}{2}(\partial_{\mu}A_{\nu})^{2} + \frac{1}{2}(\partial_{\mu}A_{\mu})^{2}$, can someone ...
- 43
0
votes
2
answers
372
views
Derivative of $\nabla\times(\nabla\times A)$ by A
I'm trying to find out how to quantize EM field. It seems like $\vec{A}$ and $\vec{E}$ are it's canonical coordinates. For example:
$$\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