Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Why does SU(2) have 3 parameters/generators like the SO(3)?

  1. May 8, 2017 #1
    From "Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson", it states that 'SU(2) matrix has one of the real parameters fixed,leaving three real parameters'.

    I don't really get this part and hope someone can clear my doubt.
    _____________________________________________________

    To ensure that the matrix is unitary, c=-b* & d=a* while a=a and b=b. To ensure that it is special, ad-bc=1

    Together, $$ ad-bc=a.a^*-b.(-b^*)=|a|^2 + |b|^2 = 1 $$

    $$a=Re(a)+i Im(a)$$ & $$b=Re(b)+i Im(b)$$ ← From here I got 4 parameters.

    $$ |a|^2 + |b|^2 = (Re(a))^2 +(Im(a))^2 + (Re(b))^2 +(Im(b))^2 = 1 $$← I still see 4 parameters thus I am still not sure how the unit determinant fix one of the parameter.

    ___________________________________________________________________________________
    Can I think of it another way as in just like SO(3) rotation, in SU(2) we also have a 3D space except instead of z axis, we have the i(imaginary) axis. So is no different from rotation in 3D space other than just changing the one of the axis to i-axis.
     
  2. jcsd
  3. May 8, 2017 #2

    fresh_42

    Staff: Mentor

    You can write ##a=\pm \sqrt{1-b^2-c^2-d^2}## which leaves you with three free variables ##b,c,d##.

    Or you can pass over to its Lie algebra ##\mathfrak{su}(2)##, which is of the same dimension and has the ##i-##multiples of the Pauli-matrices as real basis.

    Or you can identify an element ##\begin{bmatrix}x_1\mathbf{1}+x_2\mathbf{i} & -x_3\mathbf{1}+x_4\mathbf{i} \\x_3\mathbf{1}+x_4\mathbf{i} & x_1\mathbf{1}-x_2\mathbf{i} \end{bmatrix}## of ##SU(2)## with a point ##x_1\mathbf{1}+x_2\mathbf{i}+x_3\mathbf{j}+x_4\mathbf{k}## in ##\mathbb{R}^4##. Now the determinant condition reduces these point to the sphere ##\mathbb{S}^3##, which is a three-dimensional manifold.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Why does SU(2) have 3 parameters/generators like the SO(3)?
  1. SU(2) in 3-d (Replies: 6)

Loading...