Transformation of a 2x2 matrix with Pauli matrices

[CAF123]

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!

 

[mark726]

Your approach seems to be correct, but there are a few errors in your calculations.

First, the transformation of $\tau_k$ should be written as $$\tau_k' = \tau_k + \frac{i}{2}\alpha_a\epsilon_{akm}\tau_m,$$ as the $\alpha$'s are real parameters, not matrices. This will change the sign of the imaginary term in your expression for $\tau_k'$.

Second, the transformation of $\phi_k$ should be written as $$\phi_k' = \phi_k - \frac{1}{2}\alpha_a\epsilon_{akl}\phi_l,$$ as the $\tau_k$'s are matrices, not numbers. This will change the sign of the imaginary term in your expression for $\phi_k'$.

Finally, when putting the two transformations together, you should get $\tau_k' \phi_k' = (\tau_k + \frac{i}{2}\alpha_a\epsilon_{akm}\tau_m)(\phi_k - \frac{1}{2}\alpha_a\epsilon_{akl}\phi_l)$. Expanding this out and keeping terms up to first order in $\alpha$ will give you the desired result.