The matrix representation ##U## for the group ##SU(2)## is given by(adsbygoogle = window.adsbygoogle || []).push({});

##U = \begin{bmatrix}

\alpha & -\beta^{*} \\

\beta & \alpha^{*} \\

\end{bmatrix}##

where ##\alpha## and ##\beta## are complex numbers and ##|\alpha|^{2}+|\beta|^{2}=1##.

This can be derived using the unitary of ##U## and the fact that ##\text{det}\ U=1##.

Is any complex ##2\times 2## matrix with unit determinant necessarily unitary?

Consider the following argument:

##\text{det}\ (U) = 1##

##(\text{det}\ U)(\text{det}\ U) = 1##

##(\text{det}\ U^{\dagger})(\text{det}\ U) = 1##

##\text{det}\ (U^{\dagger}U) = 1##

##\text{det}\ (U^{\dagger}U) = \text{det}\ (U)##

##U^{\dagger}U = U##

##U^{\dagger}= 1##

Where's my mistake in this argument?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I SU(2) matrices

Have something to add?

Draft saved
Draft deleted

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**