All Questions
8 questions
3
votes
1
answer
67
views
"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
1
vote
1
answer
67
views
How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
- 467
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862
1
vote
1
answer
95
views
How to show the equivalence between Lagrangians?
I have a Lagrangian of a form:
$$\mathcal{L}=\frac{1}{2}\left (A_\mu g^{\mu\nu}\partial^2 A_\nu-A_\mu \partial^\mu \partial^\nu A_\nu\right ) $$
And I want to show that it is equivalent to the ...
- 47
0
votes
2
answers
310
views
Canonical momentum of a 4-vector field
In a four-vector field theory,
we have a given Lagrangian:
$$\mathscr{L} = C_{1} (\partial_{\nu} A_{\mu}) (\partial^{\nu} A^{\mu}) + C_2 (\partial_{\nu} A_{\mu}) (\partial^{\mu} A^{\nu}) + C_3 A_{\mu} ...
- 112
1
vote
1
answer
288
views
Help with relativistic notation (Derivative of Lagrangian)
I am trying to learn QFT, but I haven't taken a course in general relativity so the relativistic notation stuff is taking me a bit to get used to. I do not understand how to do the following.
For a ...
- 1,001
2
votes
1
answer
2k
views
Derivatives with upper and lower indices
I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate
$$\...
- 187
2
votes
2
answers
4k
views
Total time derivative of magnetic vector potential $A$
I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why:
$$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
- 604