New answers tagged gauge-theory
1
vote
Accepted
Why render the kinetic part of a vector field (other than the photon) $U(1)$ symmetry?
The most general first-order kinetic term you could write down for a vector field $A_\mu$ is
$$ S_\text{kin}[A] = \int \left( c_1 \partial_\mu A_\nu \partial^\mu A^\nu + c_2 \partial_\mu A_\nu \...
14
votes
Accepted
Gauge Theory determined by Gauge Group and Representation: What about specifying the bundle?
You have correctly diagnosed at the end of your question why many texts never bother with bundles: As long as we're only doing physics on $\mathbb{R}^4$ or are only interested in local phenomena that ...
3
votes
Local gauge transformations and Noether Current
A global/$n$-parameter quasisymmetry is the realm of Noether's 1st theorem.
Example: Global gauge symmetry in E&M with matter sectors implies electric charge conservation, cf. e.g. this Phys.SE ...
1
vote
Local gauge transformations and Noether Current
Funnily enough this has only really been understood in recent years in the connection to large gauge transformations and soft theorems. See https://arxiv.org/abs/2112.05289 for a recent in depth ...
1
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Local gauge transformations and Noether Current
Noether's theorem is about global symmetries. Gauge "symmetries" are not physical symmetries. They are local redundancies. In quantum field theory, a gauge transformation is a do-nothing ...
0
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Extending Wightman axioms to gauge theories
As @DanielC Pointed out, Krein spaces are the main ingredient to generalise Wightman axioms to free gauge theories. There is a framework, which you might be looking for: the Strocchi Wightman ...
2
votes
Accepted
On the Ward Identity in QED
Using translation operators the position space amplitude can be written $$\mathcal{M}^\mu(x)\equiv \langle f|j^{\mu}(x)|i\rangle=\langle f|j^{\mu}(0)|i\rangle e^{i(\sum p_f-p_i)x}$$
After Fourier ...
0
votes
Nonlinear symmetry realization: what is it for and caveats?
$\newcommand{\ex}[1]{\mathrm{e}^{#1}}\newcommand{\i}{\mathrm{i}}$Let me add a couple of points to Cosmas' answer
A crucial observation for your first question is that
\begin{align}
[X_\alpha,X_\beta] ...
-1
votes
What is $U(1)$ symmetry?
You can intuitively see that U(1) corresponds geometrically to a circle, and multiplication among elements is equivalent to adding the parameter θ, that is rotating with some angle around the circle. ...
0
votes
Accepted
What is 't Hooft-Veltman gauge? What are the interactions in SM in 't Hooft-Veltman gauge?
For what it's worth, the 't Hooft-Veltman gauge
$$ G~=~ \partial_{\mu} A^{\mu} +\frac{\lambda}{2} A_{\mu} A^{\mu} $$
is a non-linear deformation of the Lorenz gauge
$$G~=~ \partial_{\mu} A^{\mu} .$$
...
-1
votes
Accepted
Nonlinear symmetry realization: what is it for and caveats?
I am not sure with what text I'd be shadow-boxing; I suggested a pretty good summary in the comments. It is understood you are completely, and hands-on familiar with the SO(4)→SO(3) σ model, which ...
2
votes
Accepted
The Faddeev-Popov generating functional and its independence on the gauge-fixing function
In a nutshell the independence of the gauge-fixing function in the path integral/partition function $Z$ is a generalization of the fact that
$$ \int_{\Omega}\! d^nx ~\left|\det\frac{\partial f(x)}{\...
0
votes
Anomaly of the $\text{U}(1)$-$\text{SU}(2)$-$\text{SU}(3)$ triangle diagram
It is easier to understand when you try to write the amplitude of the triangle diagram for U(1)$\times$ SU(2)$\times$ SU(3). Just to remember, we need the following Feynman rule for non-abelian gauge ...
3
votes
Accepted
Commutator between covariant derivative and a field
It may help to evaluate the commutator on a function, i.e.
$$[\partial_\mu,\Phi]f=\partial_\mu(\Phi f)-\Phi\partial_\mu f=(\partial_\mu \Phi)f+\Phi \partial_\mu f-\Phi \partial_\mu f.$$
The last step ...
1
vote
Accepted
Abelian flat connection maps to zero
Hint: Given an abelian flat connection $F=\mathrm{d}A=0$, from Poincare Lemma we know that there exists a locally$^1$ defined $0$-form gauge transformation $\alpha$, so that 1-form gauge potential $A=\...
1
vote
Harmonic Gauge in linearized GR and meaning of coordinate system
The reason we want the harmonic gauge is that it greatly simplifies the linearized Einstein tensor (I believe it knocks out 3/4 of the terms). So just by inspection you can see (though this is by no ...
3
votes
Accepted
Gauge invariance of the Abelian Chern-Simons term
To explain the problem in details, I will start from the most generic (non-Abelian) case of the Chern-Simons theory.
Attention: if you are only interested in the answer for Abelian Chern-Simons ...
3
votes
Accepted
Equivalence of quantization in Coulomb gauge and Lorenz Gauge
One may show that gauge-invariant physical observables of a gauge theory (in this case QED) does not depend on the specific gauge-fixing condition (e.g. Lorenz gauge, Coulomb gauge, etc). See e.g. my ...
0
votes
Does an electromagnetic gauge transform induce a $U(1)$ transform on the field?
The Lagrangian of a scalar QED is given by
\begin{align}
\mathcal{L}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)^{\ast}(D^{\mu}\phi)-m^{2}|\phi|^{2} \\
&=\frac{1}{2}(|\mathbf{E}|^{2}-|\...
2
votes
Why is quantizing the free electromagnetic field in the Lorenz gauge more subtle than in the Coulomb gauge?
The thing about the Coulomb gauge is that, in vacuum, you get both $A_0 = 0$ and $\nabla \cdot \vec A = 0$, so it eliminates the $(A_0,\pi^0)$ d.o.f. right out of the gate - we just have $A_0 = 0,\pi^...
2
votes
Accepted
Are there "physical fields" in non-abelian gauge theories?
In confining theories you only observe gauge singlets, as in QCD (hadrons), and you might think of the gauge fields (gluons) as "merely" brilliant indirect calculational devices, leading up ...
1
vote
Why is it that in gauge theories the assumption "all fields decay sufficiently rapidly at infinity" not justified anymore?
Usually, one uses some argument related to finiteness energy/angular momentum/charges, etc. to show that $A \to 0$ at infinity. However, all of these arguments remain unchanged if $A \to pure~gauge$ ...
0
votes
Why is it that in gauge theories the assumption "all fields decay sufficiently rapidly at infinity" not justified anymore?
One idea would be that even at infinity your gauge symmetry still has to hold and so for example even if you set out with a gauge field $A_\mu \rightarrow 0$ at infinity through a gauge transformation ...
2
votes
How are these Covariant Derivative Identities found?
By Leibnitz' rule
$$
e^{-ikx} \partial_x \{e^{ikx} f(x)\} = e^{-ikx}\{f(x)(\partial_x e^{ikx})+ e^{ikx}(\partial_x f)\}\\
= e^{-ikx}\{ f(x) (ik e^{ikx})+ e^{ikx}(\partial_x f)\}\\
= ik f(x) + \...
5
votes
Accepted
How are these Covariant Derivative Identities found?
Hint for eq. (2):
$$e^{-ik\cdot x} f(D) e^{ik\cdot x}~=~f\left(e^{-ik\cdot x} D e^{ik\cdot x}\right)$$
and
$$\begin{align} e^{-ik\cdot x} D_{\mu} e^{ik\cdot x}~\stackrel{\text{Hadamard}}{=}&~e^{-...
1
vote
Accepted
Constructing gauge invariants
I am using this reference,
Spin Multiplicities, T Curtright, T van Kortryk, and C Zachos, Phys Lett A381 (2017) 422-427.
The character of a spin j, dimension 2j+1 irrep of SU(2) is
$$
\chi_j (\theta)...
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