# Adjoint transformation of gauge fields

1. ### spookyfish

53
A gauge field $W_\mu$ is known to transform as
$$W_\mu\to W'_\mu=UW_\mu U^{-1} +(\partial_\mu U)U^{-1}$$
under a gauge transformation $U$, where the first term $UW_\mu U^{-1}$ means it transforms under the adjoint representation. Can anyone explain to me why it means a transformation under the adjoint representation? all I know is the definition of the adjoint representation
$$[T_i]_{jk}=-if_{ijk}$$

Last edited: Aug 19, 2014
2. ### ChrisVer

Because the gauge field $W_{\mu}$ belongs to the adjoint representation of the gauge group.

3. ### spookyfish

53
How can you see that?

4. ### ChrisVer

Well in fact you need to check the adjoint representation. The definition of the adj. repr is that of an automorphism.
In particular it's because the transformation $W \rightarrow U^{-1} W U$ preserves the Lie Bracket.
The above transformation in practice means $W^{a}_{\mu} T^{a}_{ij} \rightarrow W_{\mu} (U^{-1} T^{a} U)_{ij}$.

Actually are we talking in particular for SU(2)?
In the SU(2) case you have the (dim) reprs:
$2 \equiv \bar{2}, 2 \otimes \bar{2} = 3 \oplus 1$
the $3$ is where the gauge bosons (spin=1) belong. and that's the adjoint repr.

Last edited: Aug 19, 2014
5. ### spookyfish

53
I understand. But I don't see how this transformation rule is consistent with the definition I know of the adjoint rep: Is it possible to assume that T transforms as $GT_a G^{-1}$ and then prove that it is given by the adjoint representation $[T_a]_{bc}=-if_{abc}$?
where $f_{abc}$ are determined from:
$$[T_a,T_b]=if_{abc}T_c$$
and I don't look specifically for SU(2) reps. also SU(3), and generally any Lie group.

6. ### The_Duck

963
An infinitesimal symmetry transformation can be parametrized by some numbers ##\omega^a##, where ##a## runs over the generators of the symmetry group. Then an object ##A_i## is said to transform in the representation ##R## if, under an infinitesimal transformation,

##A_i \to A_i + i \omega^a (T^a_R)_{ij} A_j##.

where the ##T_R^a##'s are the representations of the generators in the representation ##R##.

Let's look at how the vector potential ##W_\mu^a## transforms under a global gauge transformation. I'll drop the Lorentz index ##\mu## because it's irrelevant. We have

##W \to U W U^{-1}##

where ##U## is the gauge transformation matrix (we will look at a global transformation, so ##\partial_\mu U = 0##). For an infinitesimal gauge transformation ##U## can be written

##U = 1 + i \omega^a T^a_F##

where the ##T^a_F## are the generators in the fundamental representation. Similarly ##W## can be written in terms of the fundamental generators:

##W = W^a T^a_F##.

So we can rewrite the transformation rule, for an infinitesimal gauge transformation, as

##W^a T^a_F \to (1 + i \omega^a T^a_F) W^b T^b_F (1 - i \omega^c T^c_F)##

or, dropping negligible terms of order ##\omega^2##,

##W^a T^a_F \to W^a T^a_F + i \omega^a W^b [T^a_F, T^b_F]##.

But we know from the commutation rules that ##[T^a_F, T^b_F] = i f^{abc} T^c_F##. So the transformation rule becomes

##W^a T^a_F \to W^a T^a_F - \omega^a f^{abc} W^b T^c_F##

By renaming indices this can be rewritten

##W^a T^a_F \to (W^a + \omega^c f^{cab} W^b) T^a_F##

or just

##W^a \to W^a + \omega^c f^{cab} W^b##

Looking back at the first equation, this is the transformation rule for an object that lives in a representation ##R## where the generators are given by

##(T_R^c)^{ab} = -i f^{cab}##.

This is exactly the adjoint representation.

7. ### spookyfish

53
Hi. Thank you very much! This is exactly what I was looking for. the explanation is very clear. I only have one question (that I think does not affect the proof): Why did you assume that $U$ is given in the fundamental representation $$U=1+i\omega^aT_F^a$$

8. ### haushofer

980
I guess because we are talking about matrix Lie groups (their elements are matrices), and their algebra elements correspond to the fundamental representation.

9. ### The_Duck

963
It doesn't matter; you can pick any representation ##R## and think of ##W## as the matrix ##W^a T_R^a## and ##U## as the matrix ##1 + i \omega^a T_R^a##.

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