# Pauli matrices in SU(2)

1. Jan 5, 2010

### vertices

Can I check with someone - is the following pauli matrix in SU(2):

0 -i
i 0

Matrices in SU(2) take this form, I think:

a b
-b* a*

(where * represents complex conjugation)

It seems to me that the matrix at the top isn't in SU(2) - if b=-i, (-b*) should be -i...

However, my notes say otherwise (that all pauli matrices are in SU(2)).

2. Jan 8, 2010

### the_house

Pauli matrices are not actually unitary matrices and thus are not actually themselves elements of SU(2). They are the traceless and Hermitian 'generators' of infinitesimal SU(2) transformations. I.e., an arbitrary SU(2) matrix is given by exponentiation of a linear combination of Pauli matrices.

This is if my memory serves me correctly. I'm sure someone will correct me if not.