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

Parametrization of su(2) group

  1. Jul 27, 2010 #1
    all elements of su(2) can be written as


    with H being a traceless hermitian matrix

    thus H can be written as the sum of \sigma_x,\sigma_y,\sigma_z

    H=\theta (n_x \sigma_x + n_y \sigma_y+ n_z \sigma_z).

    Here (n_x,n_y,n_z) is a unit vector in R^3.

    we can take \theta in the interval (-\pi,\pi]

    Thus the su(2) group has the same parameter space as SO(3) group.

    what is wrong?
  2. jcsd
  3. Jul 27, 2010 #2


    User Avatar
    Science Advisor

    Not quite. A rotation about the angle phi is obtained with theta=phi/2, hence all rotations
    in the interval phi in ]-pi,pi] are already obtained with theta in ]-pi/2, pi/2].
    E.g. a 180 deg rotation of around x is obtained by exp(i pi/2 sigma_x)=i sigma_x which is not equal to a -180 deg rotation exp(-i pi/2 sigma_x)=-i sigma_x. Especially, a rotation of 360 deg is equal to -1 and not to 1.
  4. Jul 27, 2010 #3
    but at which step i am wrong?
  5. Jul 27, 2010 #4
    i now see one traphole

    for \theta=\pi, \exp(iH) is -1 regardless of the direction of the vector n

    so in contrast to so(3), where on the surface of the pi-radius sphere, two antipodal points are identified, for su(2), the whole surface is identified.
  6. Jul 27, 2010 #5


    User Avatar
    Science Advisor

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook