Pauli Matrices in orthogonal space

Question

In some literature there is reference to $\tau$ matrices which are the same pauli matrices in an orthogonal space. I have not seen any explicit constructions of this anywhere. Could someone tell me or point to literature on how to find the matrix elements of these $\tau$ matrices.

Answer 1

Georgi is in Exercises 3D, 3E and 6C using the word orthogonal in a non-standard sense. Basically, he means independent copies of sigma matrices that act in different spaces. In detail, first let us define the $gl(2,\mathbb{C})$ Lie algebra as the span of the sigma matrices and the unit matrix $\sigma_0:={\bf 1}_{2\times 2}$,

$$ gl(2,\mathbb{C})~:=~ {\rm span}_{\mathbb{C}}\{\sigma_0, \sigma_1, \sigma_2, \sigma_3 \}. $$

Then Georgi is considering another 'orthogonal' copy of the Lie algebra, call it

$$ gl(2,\mathbb{C})^{\prime}~:=~ {\rm span}_{\mathbb{C}}\{\tau_0, \tau_1, \tau_2, \tau_3 \}. $$

And then he is basically considering the tensor product

$$ gl(2,\mathbb{C})\otimes gl(2,\mathbb{C})^{\prime}$$

as a new $4 \times 4=16$ dimensional Lie algebra with Lie bracket

$$ [\sigma_i\otimes\tau_a, \sigma_j\otimes\tau_b]~:=~\sigma_i\sigma_j\otimes\tau_a\tau_b - \sigma_j\sigma_i\otimes\tau_b\tau_a. \qquad (1)$$

There exist well-known formulas to reduce products $\sigma_i\sigma_j=\sum_k f_{ij}^k\sigma_k$ of sigma matrices, etc., so that the rhs. of eq. (1) again belong to the Lie algebra. In this sense, the $\sigma$'s and the $\tau$'s commute.