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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 \...
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14 votes
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2 votes
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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 ...
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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] ...
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-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. ...
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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 $$G~=~ \partial_{\mu} A^{\mu} .$$ ...
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-1 votes
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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 ...
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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)}{\...
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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 ...
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3 votes
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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 ...
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1 vote
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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=\...
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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 ...
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3 votes
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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 ...
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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 ...
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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}-|\...
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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^...
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2 votes
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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 ...
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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 $A \to pure~gauge$ ...
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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 ...
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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) + \...
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5 votes
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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^{-...
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1 vote
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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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