- #1
spaghetti3451
- 1,311
- 31
Consider the expression
$$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$
where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers.
How can I extract the term ##\text{Tr}(\phi^{c}\phi^{d}[T^{a},T^{c}][T^{b},T^{d}])A^{a}_{\mu}A^{b\mu}## from this expression?I suppose you have to square the third term, but I do not get a trace!
$$\left(T^{a}\partial_{\mu}\varphi^{a} + A_{\mu}^{a}\varphi^{b}[T^{a},T^{b}] + A_{\mu}^{a}\phi^{b}[T^{a},T^{b}]\right)^{2},$$
where ##T^{a}## are generators of the ##\textbf{su}(N)## Lie algebra, and ##\varphi^{a}##, ##\phi^{a}## and ##A_{\mu}^{a}## are numbers.
How can I extract the term ##\text{Tr}(\phi^{c}\phi^{d}[T^{a},T^{c}][T^{b},T^{d}])A^{a}_{\mu}A^{b\mu}## from this expression?I suppose you have to square the third term, but I do not get a trace!
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