All Questions
Tagged with differentiation field-theory
15 questions with no upvoted or accepted answers
3
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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:
$$ g^{\mu\nu}...
- 702
3
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0
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358
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Integration by parts of covariant derivative
There already exists posts to discuss this question, but I don't think it's totally done!
We can write the covariant derivative as
$$D_i=\partial_i-igA_i^aT^a \tag{1}$$
There are two kinds of opinions ...
- 1,461
3
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0
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68
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Lattice differentiation and Locality
Assume we define the locality of a theory in the following way:
Assume we have a theory of real scalars, so this theory is non local if the action has terms like
$$\int d^dx\,\phi(x)V(x-y)\phi(y).$$
...
- 1,429
2
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0
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61
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Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
- 6,727
1
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0
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62
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Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
1
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0
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57
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If there is a spin-0 structure and i differentiate it with respect to a space dimension, does it become a spin-1 structure?
It might be a naive question but i was wondering what a derivative can do regarding spin. If there is a Riemann scalar it is clear that its an invariant object under tensor transformation and it does ...
- 153
1
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0
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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
1
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0
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583
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Partial derivative vs Total derivative
This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives.
Consider a Lagrangian density
$$\mathcal{...
- 1,674
0
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0
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82
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Taylor expansion of scalar function for a coordinate infinitesimal transformation (Poincaré group)
For a coordinate infinitesimal transformation of the form $x^{\prime \mu} = x^{\mu} + a^{\mu} + \omega^{\mu}_{ \ \nu}x^{\nu}$, we want to derive its effect on a space of scalar functions $f(x)$. This ...
- 11
0
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Are eigenvalues of slashed covariant derivative real?
I am trying to demonstrate that the slashed covariant derivative
$$
\gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu)
$$
has real eigenvalues:
$$
\gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
- 161
0
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90
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How to take the second-order gauge covariant derivative in quantum field theory?
I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field.
(1) The first way is to write the second order gauge ...
- 11
0
votes
1
answer
111
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Gauge covariant derivative for fields in tensor representations with multiple indices
In QFT, for fields transforming under some Gauge group, one defines the covariant derivative as
$$
(1)\qquad D_{\mu} \phi = \partial_{\mu}\phi -igA_{\mu}^k \rho(t_k)_{ab}\phi_b
$$
If $dim\rho=dim(\...
- 23
0
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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
0
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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
-1
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63
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
- 141