All Questions
Tagged with gauge-theory variational-calculus
19
questions
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71
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How to do Variational Principle in QFT? ($SU(2)$-Yang-Mills)
I am currently familiarizing myself with QFT and have a question about this article. My understanding is that the Lagrangian density in (2) couples my gauge fields to the Higgs field. And with ...
- 43
3
votes
1
answer
78
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What does it mean when the EOM of a field is trivially satisfied if other EOMs are satisfied?
If a Lagrangian has the fields $a$, $b$ and $c$ whose equations of motion (EOM) are denoted by $E_a=0, E_b=0$ and $E_c=0$ respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c\tag{1}
...
- 392
3
votes
1
answer
109
views
Change in number of gauge symmetries after adding auxiliary fields to the action
As per part (c) of Ex. (3.17) in Ref. 1, the number of gauge symmetries of an action does not change after adding auxiliary fields to it. But we know that a Stueckelberg field is an auxiliary field, ...
- 392
3
votes
1
answer
147
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How to derive the gauge transformation of a Lagrangian with auxiliary fields?
Suppose Lagrangian $L_1(y_1,y_2)$ is a functional of fields $y_1$ and $y_2$, and Lagrangian $L_2(y_1,y_2,z_1,z_2)$ is a functional of the fields $y_1,y_2$ and the auxiliary fields $z_1$ and $z_2$. If ...
- 392
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61
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If two different gauge transformations of an action commute, does it imply anything?
If two different gauge transformations of a Lagrangian commute with each other, does it imply anything?
- 392
2
votes
1
answer
102
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How to distinguish a trivial gauge transformation from a non-trivial one?
Two days ago I posted a post that discusses a very generic gauge transformation. I repeat it here. Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We ...
- 392
2
votes
2
answers
213
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Question about Trivial Gauge Transformation
Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We denote the variation of $S$ wrt to a given field, say $a$, i.e. $\frac{\delta S}{\delta a}$, by $E_a$....
- 392
2
votes
1
answer
105
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BRST variation of $\delta_{\alpha}F^A$ in $S_3$ in BRST section of Polchinski
The Faddeev-Popov action reads
$$S_3=b_Ac^{\alpha}\delta_{\alpha}F^A(\phi).\tag{4.2.5}$$
I want to find the BRST variation of the gauge variation of $F^A$ in $S_3$ i.e. $$b_Ac^{\alpha}\color{red}{\...
- 3,043
1
vote
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answers
341
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Equations of motion for electromagnetic field coupled with axion-like particle in curved spacetime
I've been reading the article "Geometric optics in the presence of axion-like particles in curved space-time" by Dominik J. Schwarz, Jishnu Goswami and Aritra Basu. They introduce the action
...
- 11
2
votes
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answers
143
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EOM, spin and mass of a particle described by a given Lagrangian
Consider the Lagrangian density
$$L=\frac{1}{12}A^{\alpha \beta \gamma}A_{\alpha \beta \gamma}$$
and $B_{\alpha \beta}$, an antisymmetric two-indices, 4 dimensional, free field; moreover $A_{\alpha \...
- 21
2
votes
0
answers
74
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Wilson action equations of motion
Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$:
\begin{equation*}
S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right),
\end{equation*}
where $\beta$ ...
- 2,613
0
votes
2
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46
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Variation of QED gauge missing step
I have several questions about this problem. I have been given a non-linear gauge condition for a QED theory:
$$F = \partial_{\mu}A^{\mu} + \frac{\lambda}{2}A_{\mu}A^{\mu}.$$
I have found online that ...
- 41
3
votes
2
answers
400
views
Functional derivative in Faddeev Popov method (Lorenz Gauge)
When applying Faddeev and Popov method (am using Peskin and Schroeder as reference), we use the identity:
$$1=\int \mathcal{D}\alpha(x)\delta(G(A^\alpha)) \det\left(\frac{\delta G(A^\alpha)}{\delta\...
- 980
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85
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Equations of motion involving terms with four vectors
So I am trying to find equations of motion for the Lagrangian associated with a non-Abelian Gauge theory for $SU(N)$, and while I was doing the math, I was a bit confused the indices.
So I have $\...
- 157
4
votes
1
answer
344
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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 ...
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