All Questions
Tagged with differentiation field-theory
73 questions
1
vote
2
answers
815
views
Field momentum of Klein-Gordon Lagrangian
Given the Lagrangian $L$ of the field $\phi$ the field momentum $\Pi$ reads:
$$L_{KG}=-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2$$
$$\Pi=\frac{\partial L}{\partial(\partial_\...
- 43
2
votes
1
answer
346
views
Covariant derivative in field theory
I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 equation 7.18 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-$\frac{1}{...
- 432
5
votes
1
answer
3k
views
Lorentz transformation of the Klein-Gordon equation
In the Lorentz transformation of the field $\partial_\mu\phi(x)$ (Peskin, p.36)
\begin{eqnarray}
\partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\phantom{\nu}\mu}(\...
- 497
5
votes
1
answer
536
views
What is the definition of $\overleftrightarrow{\partial}$ in Dirac Lagrangian?
In my course, the teacher wrote the Dirac Lagrangian as :
$$ \mathcal{L}=\frac{i}{2} \bar{\psi}\gamma^{\mu}\overleftrightarrow{\partial_\mu} \psi -m \bar{\psi} \psi $$
I just would like to ...
- 1,560
2
votes
2
answers
442
views
How do I find the function derivative $(\delta/\delta \phi) (\partial_\mu \phi)$?
The question is simple: How do I find the function derivative of $$(\delta/\delta \phi(x)) (\partial_\mu \phi(x))~?$$ As far as I can tell, I cannot use any of the standard computational rules for the ...
- 1,420
6
votes
1
answer
1k
views
Covariant derivative of a Dirac spinor and Kosmann lift
In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]:
$$\nabla_{\mu}\psi=\...
- 567
3
votes
3
answers
578
views
Infinite derivatives, Locality and Lagrangian
Providing the derivative of a single valued function $f(x)$ is like providing its value at two infinitesimally close points.
My question consists of two parts:
Can Higher derivatives be thought of as ...
- 1,202
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)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 2,022
3
votes
1
answer
165
views
Schwartz QFT coupling to the photon
I am reading Schwartz's QFT textbook. In Eq. (10.104) he writes:
$$
\left[i\partial_\mu-eA_\mu,i\partial_\nu-eA_\nu\right]~=~-e i[\partial_\mu A_{\nu}-\partial_\nu A_{\mu}]~=~
-e i F_{\mu \nu}. \tag{...
- 2,134
1
vote
1
answer
602
views
SUSY chiral covariant derivatives under change of coordinates
Reading Martin's SUSY Primer, section 4.4 on Chiral Superfields, he makes the statement that the SUSY chiral covariant derivatives
$$D_\alpha=\dfrac{\partial}{\partial\theta^\alpha}-i(\sigma^\mu\...
- 1,389
2
votes
2
answers
2k
views
Commutator of scalar field and its spatial derivative
Consider the usual commutation relations of two scalar fields
$$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
- 141
1
vote
1
answer
1k
views
Lagrange density for massless scalar field [duplicate]
I am reading a book on QFT which is stating the following.
For a massless scalar field $\phi$ the simplest possible Lagrangian is given by
$$
\mathcal{L}(x) = \frac{1}{2} \partial^\mu\phi \partial_\...
- 1,259
0
votes
1
answer
309
views
What does index $\mu$ in $\partial_{\mu}$ mean? [duplicate]
I am a beginner in QFT, and am reading it from Quantum Field Theory Demystified by David McGowan, a Tata McGraw-Hill publication.
Here, in this book, the author at one point, while explaining ...
- 2,910
3
votes
2
answers
2k
views
What does $\partial_{\mu}$ mean?
I've stumbled across the following notation a couple times reading physics articles on wikipedia:
$$\partial_{\mu}$$
But what does it mean? They don't clarify.
Source: https://en.wikipedia.org/wiki/...
0
votes
1
answer
388
views
Scalar Field Theories
The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\...
- 1,229
8
votes
3
answers
3k
views
Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?
The Euler-Lagrange equation for particles is given by
$$ \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
0
votes
1
answer
142
views
What's the generator of spinor field shifts?
The shift of a scalar field $\Phi$:
$$ \Phi \rightarrow \Phi'=\Phi - i \epsilon $$
is generated by
$$ G = -i \frac{d}{d\Phi},$$
because
$$ \mathrm{e}^{-i \epsilon \frac{d}{d\Phi} } \Phi = (1-i\...
- 10.3k
0
votes
2
answers
678
views
Derivative with respect to the spacetime derivative of a field $\phi$
I've encountered the following notation several times (for example, when discussing Noether's Theorem):
$$\frac{\partial L}{\partial(\partial_\mu \phi)}$$
And it's not immediately clear to me what ...
- 2,135
1
vote
1
answer
449
views
Covariant and contravariant derivatives in Klein-Gordon equation
Whilst exposing how a scalar product for the solutions of the Klein-Gordon equation (written as $(\Box + m^2)\varphi(x)=0$) can be derived, my textbook starts from the following system
\begin{cases} \...
- 245
6
votes
2
answers
2k
views
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$.
In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
- 27.2k
1
vote
0
answers
583
views
Partial derivative vs Total derivative
This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives.
Consider a Lagrangian density
$$\mathcal{...
- 1,674
4
votes
2
answers
653
views
What does it mean to differentiate a spinor-valued field?
Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense (...
- 576
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