## Group Theory II – SU(2)

In a previous post, I cursorily touched upon the standard orthogonal and unitary groups. It turns out that in quantum mechanics, the operators of principal interest are either Hermitian or unitary. So it is only natural that we should be interested in transformations from the Special Unitary Group . In particular, we dwell on the case.

First of all, has 3 independent parameters, something that is obvious if one demands that a matrix ,

be unitary, that is

and have a determinant equal to . It follows that any such matrix can be written as

in which there are three independent parameters. If is expanded about the identity, i.e.

then unitarity of demands that be Hermitian, and also **traceless**. For a finite transformation, this generalizes to

That is more generally, implies that for . So, is a traceless matrix.

We can write

From nonrelativistic quantum mechanics, we know that the effect of a rotation through an angle on a spin-1/2 particle about an axis is given by the unitary matrix,

where is the Pauli spin matrix “vector” given by,

Clearly, in this development, we can identify for as the **generators** of the group. The commutation relation for the generators is

This is called the * fundamental representation of *. An -dimensional representation of consists of unitary matrices satisfying such commutation relations.

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