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
225
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 139
0
votes
0
answers
88
views
How compute the expression of electromagnetic tensor explicitly as given here?
I am trying to understand how the second line arrives at the last line of this expression.
For $F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu$
And $F^{\mu\nu} = \partial^\mu A^\nu -\partial^\nu ...
- 13
3
votes
2
answers
619
views
Gauge Invariant terms of Lagrangian for Electromagnetism
Besides the usual EM Lagrangian $\mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$, we can add an additional term $\mathcal{L'} = \epsilon_{\mu \nu \rho \sigma }F^{\mu \nu}F^{\rho \sigma} = -8 \vec{E} ...
- 413
-3
votes
1
answer
199
views
What does “Integrating out field” mean?
In Schwartz’s QFT book, there is a couple of exercise problems of particle polarization in chapter 3. I have trouble with finding interaction terms from the given Lagrangians. Is it just okay to ...
- 1
-1
votes
1
answer
132
views
Gauge invariance of a Lagrangian
How do I check whether or not the Lagrangian is a gauge invariant? A Lagrangian is
$$
\mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}m^2A_\mu A^\mu
$$
- 21
1
vote
1
answer
87
views
Deriving Lagrangian density in field theory
While reading a field theory book, there's a (rather simple) equation derivation part that I can't quite understand.
Apparently from $({\partial}^2 + m^{2})A_{\mu} = 0$ (for the vector field carrying ...
- 11
4
votes
1
answer
1k
views
Canonical conjugate momenta of EM Field Lagrangian density
I have the EM Field Lagrangian density given as
$
\mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
$
where $F^{\mu \nu}$ is the Field strength tensor defined as $F^{\mu \nu} = \partial^\mu A^\nu- \...
- 634
1
vote
1
answer
564
views
How to evaluate the Euler-Lagrange equation for the electromagnetic Lagrangian? [duplicate]
I'm fascinated with field theories, but have little knowledge about them, so excuse Me if this is a dumb question.
We all know, that if we have a Lagrangian in terms of a field $\Phi $, we can just ...
- 163
1
vote
0
answers
233
views
How to derive some part of the Proca lagrangian for a Vector (spin-1)? [closed]
I'm trying to derive Eq. (10.17) & Eq. (10.18) from the textbook. Where does the term
-1/(4*pi) come from, and how do I cancel out the rest of the term (see my text, second picture).
- 11
3
votes
2
answers
758
views
Hamiltonian formalism of the massive vector field
I am currently working through a problem concerning the massive vector field. Amongst other things I have already calculated the equations of motion from the Lagrangian density $$\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