- #1
Safinaz
- 259
- 8
Hi all,
I have the following exercise about the The electroweak gauge bosons commutations relations:
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} ##
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
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