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Thread: Fun in the SU(N)

  1. Top | #11
    Elder Contributor barbos's Avatar
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    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by barbos View Post
    What do you mean by that? Unitary matrix consists of rows which are ortho-normal to each other.
    In other words, any ortho-normal basis corresponds to unitary matrix.
    That is a constraint on the matrices, not a specification of the matrices in terms of some parameters.

    For the SU(2) matrices, one can find them in terms of the unit quaternions:

    D = x0*I - i*(x1*s1 + x2*s2 + x3*s3)

    where 4-vector x = (x0,x1,x2,x3) has unit length, x.x = 1, and obeys the quaternion multiplication law. The matrices s1, s2, s3 are the Pauli matrices: {{0,1},{1,0}}, {{0,-i},{i,0}}, {{1,0},{0,-1}}. The matrix I is the identity matrix: {{1,0},{0,1}}.

    There is a similar expression for the SO(3) matrices, but it is more complicated, and it contains a sort of square of the quaternion vector. Rotation matrix

    Quaternions are often used in computer 3D graphics, because it is relatively easy to smoothly change them -- it is much easier to keep a vector normalized than a matrix orthonormalized.
    Still don't understand your complain about not having an expression when you yourself gave said expression, not to mention me giving it earlier.

  2. Top | #12
    Administrator lpetrich's Avatar
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    barbos, what you describe are constraints. I will be more explicit:

    Orthogonality:
    M.MT = MT.M = I
    MT = transpose of M

    Unitarity:
    M.M+ = M+.M = I
    M+ = Hermitian conjugate of M = transpose complex conjugate of M

    Another kind: symplecticity:
    M.J.MT = J
    where
    J = {{0, I}, {-I, 0}}
    where J has size 2n * 2n and the I's and 0's in it have size n * n.

    Special matrices: determinant of them = 1

  3. Top | #13
    Administrator lpetrich's Avatar
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    Here are some group and algebra interrelationships.

    U(n) ~ SU(n) * U(1)

    Group O(n) has subgroup SO(n) with quotient group Z2 {1, -1}

    O(n) ~ SO(n) * Z2 only for odd n

    • SO(1) ~ empty algebra, identity group
    • SO(2) ~ U(1)
    • SO(3) ~ SU(2) ~ Sp(2)
    • SO(4) ~ SU(2) * SU(2)
    • SO(5) ~ Sp(4)
    • SO(6) ~ SU(4)

    Sp = symplectic group / algebra

  4. Top | #14
    Elder Contributor barbos's Avatar
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    Quote Originally Posted by lpetrich View Post
    barbos, what you describe are constraints. I will be more explicit:
    Nope, I wrote the expression A=exp(i*H) where H is any hermitian operator which itself is a linear combination of what is called generator operators.

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