All Questions
33
questions
7
votes
2
answers
2k
views
Derivation of the quadratic form of the Dirac equation
I am asked to derive the quadratic form of the Dirac equation in an electromagnetic field,
$\left[\left(i\hbar \partial - \frac{e}{c}A\right)^2 - \frac{\hbar e}{2c} \sigma^{\mu\nu} F_{\mu\nu} - m^2c^...
- 475
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
5
votes
3
answers
3k
views
Energy-Momentum Tensor for Electromagnetism in Curved Space
$\newcommand{\l}{\mathcal L} \newcommand{\g}{\sqrt{-g}}$$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$I want to calculate the energy-momentum tensor in curved free space by functional ...
- 4,021
4
votes
1
answer
225
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 838
4
votes
1
answer
1k
views
Canonical conjugate momenta of EM Field Lagrangian density
I have the EM Field Lagrangian density given as
$
\mathcal{L} =- \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
$
where $F^{\mu \nu}$ is the Field strength tensor defined as $F^{\mu \nu} = \partial^\mu A^\nu- \...
- 624
3
votes
2
answers
622
views
Gauge Invariant terms of Lagrangian for Electromagnetism
Besides the usual EM Lagrangian $\mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$, we can add an additional term $\mathcal{L'} = \epsilon_{\mu \nu \rho \sigma }F^{\mu \nu}F^{\rho \sigma} = -8 \vec{E} ...
- 413
3
votes
2
answers
767
views
Hamiltonian formalism of the massive vector field
I am currently working through a problem concerning the massive vector field. Amongst other things I have already calculated the equations of motion from the Lagrangian density $$\mathcal{L} = - \frac{...
- 157
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
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):
$$L=-...
- 385
3
votes
1
answer
155
views
Non-linearities in Lagrangian of a scalar field coupled to point-like source
I have an exercise where I did not manage to understand the questions. Basically, I have this Lagrangian
\begin{equation}
\mathcal{L}=\frac{1}{2}(\partial \pi)^2-\frac{1}{\Lambda^3}(\partial \pi)^2\...
2
votes
1
answer
351
views
Hamilton's equations of motion on Dirac's formalism
I'm having several doubts about the procedure proposed by the Dirac-Bergmann algorithm in order to get the correct equations of motion of electrodynamics (Maxwell's equations).
Suppose I've already ...
- 1,742
2
votes
0
answers
406
views
Derivation of Coulomb's law from classical field theory
In the section on Coulomb's law in QFT by Schwartz, he expands $-\frac{1}{4}F_{\mu\nu}^{2}$ to get $-\frac{1}{2}(\partial_{\mu}A_{\nu})^{2} + \frac{1}{2}(\partial_{\mu}A_{\mu})^{2}$, can someone ...
- 43
1
vote
1
answer
226
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\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
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
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