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Parameter of SU(4) and SU(2)

  1. Aug 15, 2016 #1
    1. The problem statement, all variables and given/known data
    Good day,

    From my reading, SU(4) have 15 parameter and SU(2) has 3 paramater that range differently with certain parameter(rotation angle). And all the parameter is linearly independent to each other.

    My question are: 1. What the characteristic of each of the parameter? 2. If I choose two of them, what is the reason behind the theory?

    Can anyone explain to me or suggest any book/paper to me to understand the parameter itself.
     
  2. jcsd
  3. Aug 15, 2016 #2

    fresh_42

    Staff: Mentor

    What you call parameter are either parameters in generators of the groups ##SU(4)## or ##SU(2)##, as an angle of a rotation would be, or the basis vectors in their tangent spaces ##\mathfrak{su}(4) \, , \, \mathfrak{su}(2)## resp., which define the dimensions, and your wording about linearity suggests. You could also mean a parametrized curve on the manifold ##SU(n)## where we may calculate, e.g. tangents at.
    So it's not really clear to me what exactly you are referring to, even though all these are related.

    ##15## and ##3## are the dimensions of these groups over the real numbers, the dimensions of their tangent spaces.
    Linear independency only makes sense on linear structures, which the groups are not. There is no ##0## in sight! (And a rotation of an angle of 0° doesn't count, as it is the identity transformation, i.e. ##1##.) So the term can only be applied to their Lie algebras, their tangent spaces. As basis vectors they are linearly independent. E.g. the three Pauli-matrices build a basis of ##\mathfrak{su}(2)##, the eight Gell-Mann matrices build a basis of ##\mathfrak{su}(3)## and I don't know whether there is a named basis for ##\mathfrak{su}(4)##.

    In general the ##SU(n)## are transformation groups, i.e. groups of transformations, that act on ##\mathbb{C}^n##.
    If one group element acts as a rotation, then the rotation angle is a parameter in the sense that two different angles are two different transformations, although both are still rotations. You could also have different rotation axis with the same angle. So basically we are talking about geometric properties here.

    I don't understand the second of your questions. Maybe you could give an example.
     
  4. Aug 16, 2016 #3
    Thank you so much for reply.

    From the parameter below

    SU(4), it has 15 parameter
    1. 0 ≤ α1,α7,α11 ≤ π
    2. 0 ≤ α3,α5,α9,α13 ≤ 2π
    3. 0 ≤ α2,α4,α6,α8,α10,α12 ≤ π/2
    4. 0 ≤ α14 ≤ √3 π
    5. 0 ≤ α15 ≤ 2π √(2/3)

    and SU(2)
    1. 0 ≤ β1,β3 ≤ π
    2. 0 ≤β2≤ π/2

    What I mean, in SU(4), the parameter is divided into 5 range and in SU(2) in two range respectively to the paramater. MY questions are:
    1.Is it 5 range in SU(4) and 2 range in SU(2) can said a group?
    2. Why the parameter is determined like that?.
    3. If I that for example α2 and α1 to my calculation, what it mean actually? What happen to other?
    4. Is there any rule to me take the parameter or I can simply take it ? And why I can select the certain parameter?
     
  5. Aug 16, 2016 #4

    fresh_42

    Staff: Mentor

    ##SU(4)## are complex ##(4 \times 4)-##matrices which satisfy a certain condition, namely ##U^\dagger U=1## and ##\det(U)=1##.
    How are the ##\alpha_i## defined? Likewise for ##SU(2)##. I suppose, if you write down matrices that contain the ##\alpha_i##, they will generate the group. But I'm not sure.

    If you look at https://en.wikipedia.org/wiki/Special_unitary_group#n_.3D_2, there are other parameters. How do yours fit in?
     
  6. Aug 16, 2016 #5
    thank you. I understand about that but still don't get what happen if I only consider only certain parameter.It is will effect other or not?
     
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