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I SU(2) matrices

  1. Nov 17, 2016 #1
    The matrix representation ##U## for the group ##SU(2)## is given by

    ##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?
     
  2. jcsd
  3. Nov 17, 2016 #2

    TeethWhitener

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    ##\det(A) = \det(B)## does not imply ##A=B##.
     
  4. Nov 17, 2016 #3
    Thanks!

    I was wondering what is the most general form of the complex ##2 \times 2 ## matrix with unit determinant.

    My first hunch was that it is the ##2\times 2## matrix representation of the ##SU(2)## group, but then, a complex ##2 \times 2 ## matrix with unit determinant is not necessarily ##SU(2)##.

    Can you help me with finding the most general form of the complex ##2 \times 2 ## matrix with unit determinant?
     
  5. Nov 17, 2016 #4

    TeethWhitener

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    I don't think there's anything special about it. As far as I know, it's just:
    $$U= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$
    where ##a,b,c,d \in \mathbb{C}## and ##ad-bc = 1##.
     
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