# A Tensor algebra

1. Apr 7, 2017

### spaghetti3451

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!

Last edited: Apr 7, 2017
2. Apr 12, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Apr 14, 2017

### haushofer

I don't get a trace either. Can you give some references where the trace is written? Maybe they use the standard choice trace(t_a t_b)~ delta_{ab} or some similar rewriting.