# Transformation of a 2x2 matrix with Pauli matrices

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## Homework Statement

Suppose the vector ##\phi## transforms under SU(2) as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where ## (t_j)_{kl} = −i \epsilon_{jkl}## and ##j, k, l \in \left\{1, 2, 3\right\}.##

Based on ##\phi,## we define the ##2 \times 2## matrix ##\sigma = \tau \cdot \phi## where ##\tau## are the Pauli matrices.

1) Assume that the real parameters ##\alpha = (\alpha_1, \alpha_2, \alpha_3)## are small and show, by expanding up to first order in ##\alpha##, that the transformation law for the ##2 \times 2## matrix ##\sigma## is given by ##\sigma \rightarrow \sigma' = U\sigma U^{\dagger}## where ##U = \exp(i\alpha \cdot \tau/2)##

(b) By construction, ##\sigma## is hermitian and traceless. Verify that ##\sigma'## is also hermitian and traceless, and that det ##\sigma'## = det ##\sigma##. Show that ##\phi'## and ##\phi## are related by a rotation.

## Homework Equations

The ##(t_j)_{kl}## are 3 matrices each of which are 3x3. These correspond to matrices in the adjoint representation of SU(2). I think I can write ##\tau_i \rightarrow u\tau_i u^{\dagger}##, where ##u \approx 1 + i \beta^a \tau^b ## with ##u \in SU(2)## - i.e can decompose a general group element in terms of the lie algebra and since the taus are 2x2 matrices the corresponding representation of the generators are also pauli matrices. (I think?)

## The Attempt at a Solution

##\sigma_{ij} \rightarrow (\tau_k' \phi_k')_{ij}##. Using the transformation of the taus above and the algebra ##[\tau_a/2, \tau_b/2] = i/2 \epsilon_{abc} \tau_c## I get that $$\tau_k' = \tau_k + 2i \beta_a \epsilon_{akm}\tau_m$$

The transformation of ##\phi## is given in the question and, as far as I understand, it is transforming in the adjoint representation of SU(2).

$$\phi_k' = (1-i\alpha^b(t^b)_{kl})\phi_l = (\delta_{kl} - i \alpha^b (-i \epsilon_{bkl}))\phi_l = \phi_k - \alpha^b \epsilon_{bkl} \phi_l$$

Putting these two transformations together (that of ##\phi## and that of the ##\tau_k## ) I don't get something resembling the answer. Did I go wrong somewhere?

Thanks!