All Questions
Tagged with differentiation field-theory
73 questions
-1
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0
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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:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
- 213k
1
vote
0
answers
62
views
Adjoint of the covariant derivative of a field?
Let's call $D$ the covariant derivative, $T$ the transposition of a field and $*$ its complex conjugate, so $T*$ is the "adjoint".
Is: $$(D_{\mu}\Phi)^{T*} (D_{\mu}\Phi)=D^{\mu}\Phi^*D_{\mu}\...
- 1,337
0
votes
1
answer
105
views
How to expand $(D_\mu\Phi)^\dagger(D^\mu\Phi)$ in $SU(2)$?
I would like to calculate the following expression:
$$(D_\mu\Phi)^\dagger(D^\mu\Phi)$$ where $$D_\mu\Phi = (\partial_\mu-\frac{ig}{2}\tau^aA_\mu^a)\Phi$$ and $A_\mu^a$ are the components of a real $SU(...
- 213k
6
votes
1
answer
1k
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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)]:
$$\nabla_{\mu}\psi=\...
- 12.9k
0
votes
0
answers
82
views
Taylor expansion of scalar function for a coordinate infinitesimal transformation (Poincaré group)
For a coordinate infinitesimal transformation of the form $x^{\prime \mu} = x^{\mu} + a^{\mu} + \omega^{\mu}_{ \ \nu}x^{\nu}$, we want to derive its effect on a space of scalar functions $f(x)$. This ...
- 213k
2
votes
1
answer
175
views
The Abelian versus the non-Abelian commutator of covariant derivatives in field theory
In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
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0
votes
1
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83
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What is the relation between gauge field and Levi-Civita connection?
In field theory, covariant derivative is something like
$$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$
while in differential geometry, covariant derivative is something like
$$D_{\mu}V^{\nu}=\partial_{...
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0
votes
0
answers
75
views
Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]
Lagrangian for Klein-Gordon equation is given by
$$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$
To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
- 213k
0
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0
answers
117
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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:
$$
\gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
- 60.5k
0
votes
0
answers
90
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How to take the second-order gauge covariant derivative in quantum field theory?
I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field.
(1) The first way is to write the second order gauge ...
- 11
0
votes
0
answers
25
views
What is the partial derivative of a four-covector with respect to its derivative? [duplicate]
If I have a partial derivative for $A_\mu$ of the form $$\frac{\partial A_\mu}{\partial(\partial_\mu A_\upsilon)}$$ where $\partial_\mu A_\upsilon = \frac{\partial A_\upsilon}{\partial x^\mu}$ is the ...
- 213k
1
vote
0
answers
57
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If there is a spin-0 structure and i differentiate it with respect to a space dimension, does it become a spin-1 structure?
It might be a naive question but i was wondering what a derivative can do regarding spin. If there is a Riemann scalar it is clear that its an invariant object under tensor transformation and it does ...
- 213k
0
votes
1
answer
111
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Gauge covariant derivative for fields in tensor representations with multiple indices
In QFT, for fields transforming under some Gauge group, one defines the covariant derivative as
$$
(1)\qquad D_{\mu} \phi = \partial_{\mu}\phi -igA_{\mu}^k \rho(t_k)_{ab}\phi_b
$$
If $dim\rho=dim(\...
- 15.1k
3
votes
1
answer
113
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How does the $\not{\partial}$ work in the Dirac Lagrangian?
The Dirac Lagrangian (Density) is defined in the text "Quantum Field Theory, An Integrated Approach" by Fradkin as:
$$\mathcal{L}=\bar{\Psi}\left(i\not{\partial}-m\right)\Psi\equiv \frac{1}{...
- 213k
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:
$$ g^{\mu\nu}...
- 702
0
votes
1
answer
153
views
Differentiating the index notation
I am always confused with the algebra of differentiating the index notation, and have browsed many other posts but still confused. There must be details I have been missing. It would be really ...
- 6,119
3
votes
0
answers
358
views
Integration by parts of covariant derivative
There already exists posts to discuss this question, but I don't think it's totally done!
We can write the covariant derivative as
$$D_i=\partial_i-igA_i^aT^a \tag{1}$$
There are two kinds of opinions ...
- 1,461
3
votes
2
answers
456
views
Lie derivatives and the tetrad formalism
I have been trying to learn about the tetrad formalism in general relativity and I understand the basic idea, but there is one issue that I can't seem to figure out: Is there a definition of a Lie ...
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2
votes
0
answers
61
views
Ostrogradsky instability and fractional derivatives
Are fractional derivatives (or even more generally differentegrals) also under the scope of the Ostrogradsky instability theorem?
- 213k
-2
votes
1
answer
73
views
How do I keep track of what to differentiate in a Dirac Hamiltonian/Lagrangian?
Suppose we have the dirac Hamiltonian:
$$
H = \int d^3y\bar\psi(y)_b(-i\gamma^k\partial_k+m)_{bc}\psi(y)_c.
$$
My question is should I think the derivative operator $\partial_k$ is acting on the ...
- 213k
4
votes
1
answer
230
views
Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 213k
1
vote
2
answers
78
views
What is the meaning of the differential in Doran and Lasenby's discussion of Noether's theorem for spacetime transformations?
Doran and Lasenby (Geometric Algebra for Physicists, pg. 450) state that if a transformation involves spacetime dependence (this brings to my mind common examples: translation and rotation), then ...
0
votes
1
answer
28
views
Clarification for derivatives under a change of variables
In Special Relativity and Classical Field Theory by Susskind, he says that we can imagine a function of $(x+ct)$, then he says that we can consider its derivatives and easily see that $$\frac{\...
- 213k
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
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
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0
answers
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 $\Phi(x)$...
- 556
0
votes
1
answer
155
views
Finding the Euler-Lagrange equation for a scalar field
Consider a scalar field with the following lagrangian density:
$$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi).$$
I want to find the corresponding Euler-Lagrange equation, ...
- 16.4k
0
votes
1
answer
110
views
Taking the second time derivative of a scalar field
Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get:
$$\frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
- 213k
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 1,253
1
vote
1
answer
246
views
Four-vector differentiation (E-M Euler-Lagrange eq.)
$$\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})}{\...
- 467
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)
$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\...
- 467
2
votes
1
answer
159
views
Notation and Terminology Questions from Schwartz' QFT Book
I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing.
First off, on page 34 he defines a translation of a field to first order as
$$...
- 213k
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:
$$ \nabla_{A} \alpha_p = d \alpha_p + ...
- 13.4k
0
votes
1
answer
51
views
Derivative with respect a partial derivative in lagrangian density [duplicate]
I am reading about field theory and lagrangian densities, and I found the following lagrangian density in my book:
$$
\mathcal{L} = \dfrac{1}{2} (\partial_\mu \phi)^2 - \dfrac{1}{2} m^2 \phi^2\ ,
$$
...
- 391
2
votes
1
answer
877
views
Hermitian conjugate of 4-derivative $\partial_\mu$
I want to find the hermitian conjugate of 4-derivative $\partial_\mu$ for the real scalar Lagrangian defined as
$$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^\dagger(\partial^\mu\phi) - \frac{1}{2}...
- 213k
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} \...
- 2,820
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 ...
- 55.5k
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 (...
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0
votes
1
answer
196
views
Derivative of a complex potential for the $\lambda \Phi^{4}$-model
A charged scalar particle is described by a complex field $\Phi(x) = \phi_{1}(x)+i\phi_{2}(x)$. Consider a Lagrangian of the $\lambda \Phi^{4}$-model whose potential in the Euclidean action is given ...
- 213k
0
votes
0
answers
83
views
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 ...
- 213k
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}{...
- 363k
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 ...
- 11.5k
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}
...
- 213k
4
votes
1
answer
402
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 ...
- 213k
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.
- 822
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}(\...
- 3,013
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
$$\int d^dx\,\phi(x)V(x-y)\phi(y).$$
...
- 213k
0
votes
2
answers
244
views
Why partial w/respect to $\phi$ of d'Alembertian of $\phi$ = 0?
Why is $\frac{\partial }{\partial \phi}(\frac{\partial^2 \phi}{\partial x^2})$ = 0?
I'm happy to read an article, but I don't know what keywords to search on.
Background:
Apologies if this is a math ...
- 213k
0
votes
1
answer
403
views
Form of the Lagrangian for 1D String Dynamics
I am currently reading part of Franz Gross' Relativistic Quantum Mechanics and Field Theory, and am getting stuck on a particular point in chapter 1 where he attempts to motivate field theory by ...
- 213k
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 $\...
- 829