# Pauli Matrices in orthogonal space

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.

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Any chance you can provide a link to some such reference? There is something about $\tau$ matrices for isospin $SU(2)$ on Wikipedia, but I wouldn't know whether that's what you're asking about without some context. –  David Z Dec 8 '11 at 20:37
@DavidZaslavsky I dont have a link but the textbook in which this appears is Georgi's Lie Algebras... , chapters 3 (exercises) and chapter 6 (again, in the exercises) –  yayu Dec 8 '11 at 21:08
That works too. –  David Z Dec 8 '11 at 22:36

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.

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This is correct, of course, but it's a bit of overkill -- to answer the OPs question, the matrix elements of the tau-matrices are exactly the same as the matrix elements of the sigma-matrices. The point is just so that you don't confuse the spin with some other internal SU(2) degree of freedom. –  wsc Dec 9 '11 at 4:07
I see. Thanks! To sum it up: "$\sigma$" $= \sigma \otimes \mathbb{I}$ while "$\tau$" $=\mathbb{I}\otimes \sigma$ –  yayu Dec 9 '11 at 4:26