All Questions
Tagged with differentiation field-theory
15 questions
8
votes
3
answers
3k
views
Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?
The Euler-Lagrange equation for particles is given by
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$
and for fields it is
$$ \partial_{\mu} \frac{\...
- 4,064
6
votes
2
answers
2k
views
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$.
In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
- 27.2k
3
votes
1
answer
347
views
How come $\frac{\partial(\partial_{\beta}A_{\gamma})}{\partial(\partial_{\mu}A_{\nu})} = g_{\beta\mu}g_{\gamma\nu}$?
For context, this equation is used in the following (from Schwartz's QFT 3.44)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 2,022
3
votes
2
answers
2k
views
What does $\partial_{\mu}$ mean?
I've stumbled across the following notation a couple times reading physics articles on wikipedia:
$$\partial_{\mu}$$
But what does it mean? They don't clarify.
Source: https://en.wikipedia.org/wiki/...
2
votes
2
answers
270
views
Does it make sense to speak in a total derivative of a functional? Part II
I am trying to derive the Noether theorem from the following integral action:
\begin{equation}
S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}%
\phi_{r},x\right) , \tag{II.1}\...
- 387
2
votes
2
answers
464
views
Does it make sense to speak in a total derivative of a functional? Part I
I would like to consider the problem of the total derivative of a given functional \begin{equation}
\mathcal{L}\bigg[\phi\big(x,y,z,t\big),\frac{\partial{\phi}}{\partial{x}}\big(x,y,z,t\big),\frac{\...
- 387
0
votes
1
answer
388
views
Scalar Field Theories
The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\...
- 1,229
6
votes
1
answer
1k
views
Covariant derivative of a Dirac spinor and Kosmann lift
In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]:
$$\nabla_{\mu}\psi=\...
- 567
5
votes
1
answer
536
views
What is the definition of $\overleftrightarrow{\partial}$ in Dirac Lagrangian?
In my course, the teacher wrote the Dirac Lagrangian as :
$$ \mathcal{L}=\frac{i}{2} \bar{\psi}\gamma^{\mu}\overleftrightarrow{\partial_\mu} \psi -m \bar{\psi} \psi $$
I just would like to ...
- 1,560
4
votes
1
answer
401
views
Calculating equation of motion in gauge theories: using ordinary derivatives or covariant derivatives?
For general gauge theories, the total Lagrangian density is given as $$L=-\frac{1}{4}F^2+L_M(\psi, D\psi)$$ where $L_M(\psi, D\psi)$ is the matter field with the ordinary derivative replaced by the ...
- 1,706
2
votes
1
answer
877
views
Hermitian conjugate of 4-derivative $\partial_\mu$
I want to find the hermitian conjugate of 4-derivative $\partial_\mu$ for the real scalar Lagrangian defined as
$$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^\dagger(\partial^\mu\phi) - \frac{1}{2}...
- 634
2
votes
3
answers
2k
views
Poincare invariant Lagrangians
The Lagrangian density of a Poincare invariant theory should not depend explicitly on the space-time coordinates. Does this mean
$$
\partial_\mu \mathcal{L}=0~?
$$
If this is the case doesn't the ...
- 1,137
2
votes
2
answers
2k
views
Commutator of scalar field and its spatial derivative
Consider the usual commutation relations of two scalar fields
$$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
- 141
2
votes
1
answer
346
views
Covariant derivative in field theory
I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 equation 7.18 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-$\frac{1}{...
- 432
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862