All Questions
Tagged with differentiation field-theory
73 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
$$ \partial_{\mu} \frac{\...
- 4,064
6
votes
2
answers
2k
views
Variation of Lagrangian density $\mathcal{L}$ w.r.t $x^{\mu}$
If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$.
In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
- 27.2k
6
votes
1
answer
1k
views
Covariant derivative of a Dirac spinor and Kosmann lift
In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]:
$$\nabla_{\mu}\psi=\...
- 567
5
votes
2
answers
2k
views
Is the derivative with respect to a fermion field Grassmann-odd?
Fermion fields anticommute because they are Grassmann numbers, that is,
\begin{equation}
\psi \chi = - \chi \psi.
\end{equation}
I was wondering whether derivatives with respect to Grassmann numbers ...
- 382
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
4
votes
1
answer
230
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 862
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
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
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)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 2,022
3
votes
1
answer
113
views
How does the $\not{\partial}$ work in the Dirac Lagrangian?
The Dirac Lagrangian (Density) is defined in the text "Quantum Field Theory, An Integrated Approach" by Fradkin as:
$$\mathcal{L}=\bar{\Psi}\left(i\not{\partial}-m\right)\Psi\equiv \frac{1}{...
- 179
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/...
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:
$$
\left[i\partial_\mu-eA_\mu,i\partial_\nu-eA_\nu\right]~=~-e i[\partial_\mu A_{\nu}-\partial_\nu A_{\mu}]~=~
-e i F_{\mu \nu}. \tag{...
- 2,134
3
votes
2
answers
456
views
Lie derivatives and the tetrad formalism
I have been trying to learn about the tetrad formalism in general relativity and I understand the basic idea, but there is one issue that I can't seem to figure out: Is there a definition of a Lie ...
- 1,113
3
votes
3
answers
578
views
Infinite derivatives, Locality and Lagrangian
Providing the derivative of a single valued function $f(x)$ is like providing its value at two infinitesimally close points.
My question consists of two parts:
Can Higher derivatives be thought of as ...
- 1,202
3
votes
1
answer
519
views
Prove $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ [Question closed, the statement was not true]
In gauge transformation, $D_\mu$ was defined to be $\partial_\mu-igA_\mu$.
However, I have hard time to see that $D_\mu\phi^\dagger=(D_\mu\phi)^\dagger$ without ambiguity.
(A comparable example in QED ...
- 3,306
3
votes
0
answers
153
views
d'Alembertian operator in presence of torsion
Consider a Riemann-Cartan 4-dimensional spacetime with torsion. In such a spacetime, I have been asked to compute the d'Alembertian operator acting on a scalar field. Here's what I tried:
$$ g^{\mu\nu}...
- 702
3
votes
0
answers
358
views
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
votes
0
answers
68
views
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
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
$$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^\dagger(\partial^\mu\phi) - \frac{1}{2}...
- 634
2
votes
2
answers
2k
views
Commutator of scalar field and its spatial derivative
Consider the usual commutation relations of two scalar fields
$$\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
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
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 $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
- 12.3k
2
votes
1
answer
346
views
Covariant derivative in field theory
I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 equation 7.18 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-$\frac{1}{...
- 432
2
votes
1
answer
175
views
The Abelian versus the non-Abelian commutator of covariant derivatives in field theory
In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
- 290
2
votes
1
answer
159
views
Notation and Terminology Questions from Schwartz' QFT Book
I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing.
First off, on page 34 he defines a translation of a field to first order as
$$...
- 221
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
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
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
1
answer
388
views
How to take derivative with respect to Lagrangian of complex field?
Basics: The Lagrangian in field theory was written as
$$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$.
Question 1:
Is $\...
- 3,306
2
votes
0
answers
61
views
Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
- 6,727
2
votes
1
answer
634
views
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:
$$ \nabla_{A} \alpha_p = d \alpha_p + ...
- 882
1
vote
2
answers
305
views
Is $ \partial_{\mu} \partial^{\mu} $ the second derivative or derivative squared?
This might be a silly question, but I'm just getting my feet wet with field theories.
So far I have assumed that $ \partial_{\mu} \Phi\partial^{\mu}\Phi $ means $ (\Phi_t)^2-(\Phi_x)^2-...$ . I ...
- 163
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862
1
vote
2
answers
819
views
Showing the form of the covariant derivative of $\phi$, if $\phi$ transforms as the adjoint representation of $SU(n)$
I want to show that if $\phi$ transforms as the adjoint representation of SU(n), its covariant derivative is given by $\textbf{D}_\mu \phi = \partial_\mu \phi + i [\textbf{A}_\mu, \phi]$. (Exercise in ...
- 1,055
1
vote
2
answers
70
views
Mandl & Shaw QFT chapter 1 question [closed]
Page 3 of Mandl & Shaw claims that, given a vector $\pmb{A}(\pmb{x},t)=\pmb{A}_{0}e^{i(\pmb{k}\pmb{\cdot} \pmb{x} - \omega t)}$, $\pmb{\nabla} \pmb{\cdot} \pmb{A} = 0$ (eq. 1.6) implies $\pmb{k} \...
- 21
1
vote
2
answers
815
views
Field momentum of Klein-Gordon Lagrangian
Given the Lagrangian $L$ of the field $\phi$ the field momentum $\Pi$ reads:
$$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
1
vote
2
answers
78
views
What is the meaning of the differential in Doran and Lasenby's discussion of Noether's theorem for spacetime transformations?
Doran and Lasenby (Geometric Algebra for Physicists, pg. 450) state that if a transformation involves spacetime dependence (this brings to my mind common examples: translation and rotation), then ...
- 163
1
vote
1
answer
246
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
- 149
1
vote
1
answer
602
views
SUSY chiral covariant derivatives under change of coordinates
Reading Martin's SUSY Primer, section 4.4 on Chiral Superfields, he makes the statement that the SUSY chiral covariant derivatives
$$D_\alpha=\dfrac{\partial}{\partial\theta^\alpha}-i(\sigma^\mu\...
- 1,389
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
$$
\mathcal{L}(x) = \frac{1}{2} \partial^\mu\phi \partial_\...
- 1,259
1
vote
1
answer
113
views
Calculating the variation of an operator in two different ways
Let
$$
H_{T}=\dot{x}^{I}\frac{\partial}{\partial \psi^{I}}T(x,\psi)
$$
and consider the transformation:
$$
x^{I}\mapsto x^{I}+i\epsilon\psi^{I}
\\
\psi^{I}\mapsto\psi^{I}-2\epsilon\dot{x}^{I}
$$
where ...
- 587
1
vote
0
answers
62
views
Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
1
vote
0
answers
57
views
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
vote
0
answers
170
views
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
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
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
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
0
votes
2
answers
274
views
Tensor Question (Klein–Gordon equation) [closed]
I have a question following the derivation of the Klein-Gordon equation from a lagrangian. From
Eq. (13d), where does $\delta^\mu_\nu$ come from? I guess it's a conversion factor of some sort.
- 11