- #1
Korybut
- 60
- 2
Hi there!
I am reading textbook "Supergravity" by Freedman and Van Proeyen and got stuck on a simple exercise (Ex 2.4). Usually I would proceed further marking it as a typo but I've checked the errata list on the website and didn't find this exercise there
Exercise 2.4 Show that ## A\bar{\sigma}_\mu A^\dagger=\bar{\sigma}_\nu \Lambda^{-1}{}^\nu{}_\mu## and ##A^\dagger \sigma A=\sigma_\nu \Lambda^\nu{}_\mu## . This gives precise meaning to the statement that the matrices ##\bar{\sigma}_\mu## and ##\sigma_\nu## are 4-vectors
Sigma matrices in this book are defined as
$$\sigma_\mu=\left(-\mathbb{1},\sigma_i\right),\;\;\; \bar{\sigma}_\mu=\sigma^\mu=\left(\mathbb{1},\sigma_i\right).$$
And SL(2,C) transformations is defined as
$$ \mathbf{x}^\prime \equiv A \mathbf{x} A^\dagger$$
The first identity in the exercise is kinda straightforward and it is also easy to see that the second one holds for barred sigma-matrices. But I didn't managed to work out the identity in the form written in the exercise
Is it a typo or I missing something very deep?
I am reading textbook "Supergravity" by Freedman and Van Proeyen and got stuck on a simple exercise (Ex 2.4). Usually I would proceed further marking it as a typo but I've checked the errata list on the website and didn't find this exercise there
Exercise 2.4 Show that ## A\bar{\sigma}_\mu A^\dagger=\bar{\sigma}_\nu \Lambda^{-1}{}^\nu{}_\mu## and ##A^\dagger \sigma A=\sigma_\nu \Lambda^\nu{}_\mu## . This gives precise meaning to the statement that the matrices ##\bar{\sigma}_\mu## and ##\sigma_\nu## are 4-vectors
Sigma matrices in this book are defined as
$$\sigma_\mu=\left(-\mathbb{1},\sigma_i\right),\;\;\; \bar{\sigma}_\mu=\sigma^\mu=\left(\mathbb{1},\sigma_i\right).$$
And SL(2,C) transformations is defined as
$$ \mathbf{x}^\prime \equiv A \mathbf{x} A^\dagger$$
The first identity in the exercise is kinda straightforward and it is also easy to see that the second one holds for barred sigma-matrices. But I didn't managed to work out the identity in the form written in the exercise
Is it a typo or I missing something very deep?