# Tag Info

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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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### 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
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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 $A \to pure~gauge$ ...

### 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 ...
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^{-...
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)...