What are the commutation relations for the electroweak gauge bosons?

[Safinaz]

Hi all,

I have the following exercise about the The electroweak gauge bosons commutations relations:

Homework Statement

If $[ \tau_i ,\tau_k] = 2 i \epsilon_{ikl} \tau_l$ and
$\{ \tau_i ,\tau_k\} = 2 \delta_{ik}$

where $\bar{\tau}$ are the Pauli matrices,

Then prove that:
(1) $\bar{ \tau} \bar{A_\alpha} . \bar{ \tau} \bar{A^\alpha} = ( A_\alpha^1 + i A_\alpha^2) ( A^{\alpha, 1} - i A^{\alpha, 2} ) + A_\alpha^3 A^ {\alpha,3}$

The Attempt at a Solution

I said that
$\tau_i \tau_k = \delta_{ik} + i \epsilon_{ikl} \tau_l$, then

$\bar{ \tau} \bar{A_\alpha} . \bar{ \tau} \bar{A^\alpha} = \bar{A_\alpha} \bar{A^\alpha} + i \epsilon_{ikl} \tau_l \bar{A_\alpha} \bar{A^\alpha}$

But I can't complete for the next step to prove the enquiry ..

Bests,
Safinaz

 

[dauto]

That notation is confusing. The left side of equation (1) seems to be a 2x2 matrix while the right side isn't. That only makes sense if we assume there is implicit unit matrix on the right side. But the equation still doesn't seem right. Doesn't it simplify to A_μ.A^μ ?