All Questions
10 questions
1
vote
0
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62
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Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
0
votes
1
answer
83
views
What is the relation between gauge field and Levi-Civita connection?
In field theory, covariant derivative is something like
$$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$
while in differential geometry, covariant derivative is something like
$$D_{\mu}V^{\nu}=\partial_{...
- 101
3
votes
0
answers
153
views
d'Alembertian operator in presence of torsion
Consider a Riemann-Cartan 4-dimensional spacetime with torsion. In such a spacetime, I have been asked to compute the d'Alembertian operator acting on a scalar field. Here's what I tried:
$$ g^{\mu\nu}...
- 702
1
vote
0
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170
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?
I know the covariant derivative of a tensor is
$$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$
Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
- 67
2
votes
1
answer
633
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Covariant derivative on $n$-forms
I am new to form notation. I have recently read that one can write (non-abelian) gauge theory in terms of forms. I am stuck and can't derive this equation below:
$$ \nabla_{A} \alpha_p = d \alpha_p + ...
- 882
0
votes
0
answers
83
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Doubt of gauge covariant derivatives: how can I derive it?
In the context of general relativity (GR) it is necessary to introduce the notion of covariant derivatives. From the point of view of a basic introduction, we always start to deal with GR in a highly ...
- 3,159
2
votes
2
answers
164
views
Is there a equivalent theorem for closed form/ exact form for derivative with respect to fields
For a vector (one-form) $A_\mu$, when
\begin{eqnarray}
\partial_{[\mu}A_{\nu]}=0
\end{eqnarray}
then, there exists a scalar $\phi$ such that
\begin{eqnarray}
A_\mu =\partial_\mu\phi
\end{eqnarray}
...
- 91
2
votes
1
answer
807
views
The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative
The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
- 12.3k
0
votes
0
answers
233
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Covariant derivative of a composite field and the chain rule
I have a gauge theory with some rather strange covariant derivatives and I am wondering how they act on a composite field like $\psi= \phi\psi'$. In my setup, the covariant derivative acting on a ...
- 141
2
votes
1
answer
346
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Covariant derivative in field theory
I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 equation 7.18 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-$\frac{1}{...
- 432