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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


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    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


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