jet10 said:
Let's say we have S= n1.sx + n2.sy +n3.sz showing us the magnitude in each component (sx, sy, sz are the spin matrices/vectors). What does it show us? How should we interprete the imaginary no.?
You seem not to take the difference between matrices and vector not too seriously. This is a
big issue. Consider say a 3-vector (ordinary vector). How do you define such an object ? Usually by giving its transformation law under the rotations. This is equivalent to saying you deal with a certain representation of SO(3).
Alternatively, one can represent such a three vector using a hermitean, traceless 2x2 matrix. This is related to the fact that SU(2) is the covering group of SO(3). The usual rotation of such a matrix is a unitary transformation. Now the reason for the imaginary numbers is the following : when passing from a Lie group to its algebra, the usual convention is g=\exp^{\imath G}=\exp^{\imath \alpha^iG_i} where g is in the group and G is in the algebra. The \alpha^is are the coordinates in the G_i basis. The fact that we actually obtain an algebra can be seen as \exp^{A}\exp^{B}=\exp^{A+B+\frac{1}{2}[A,B]+\cdots} where \cdots contain commutators of commutators, commutators of commutators of commutators...
so by the very definition of what generators are, we can infer the fact that
the commutator of two generators is a linear combination of the generators : the algebra "closes". In the case of SO(3) we have antisymmetric structure constants : [G_i,G_j]=-\imath\epsilon_{ij}^kG_k. Now the punchline : we actually can make the \imath disappear by absorbing it into the generator :
If G\rightarrow -\imath G then g=\exp^{G} and [G_i,G_j]=\epsilon_{ij}^kG_k.
But then the generators are antihermitean !
For instance, for translation, we can set P_i=\partial_i and for rotations J_{ij}=x_{[i}\partial_{j]} since anyway the \imath is doomed to be canceled eventually.
I think there is not much more into the \imath than convenience for manipulating expressions. In the case of spacetime symmetries it is legitimate to eliminate the \imath from the definition and use antihermitean generators. In the case of internal symmetries, it seems more convenient to keep hermitean generators. Also, complex Lie algebra are nice to use, whereas real Lie algebras require more care (just as real polynomials are not as nice to factorize as complex polynomials).
Again about the vector/matrix "duality" : the \gamma_\mu matrices of Dirac are both spin matrices and (one single) Lorentz vector (the set of Dirac matrices is a vector. Each component of the vector is a matrix).
They are the link between Lorentz and spin indices.
Also in the case of Pauli matrices for the fundamental SU(2), the 3-vector "duality" previously mentionned is explicitely
(V)_{\alpha\dot{\beta}}=\left( \begin{array}{cc}V_+,V_1^*\\V_1,V_-\end{array} \right) = \frac{1}{\sqrt{2}} \left( \begin{array}{cc}V_0+V_1,V_2-\imath V_3\\V_2+\imath V_3,V_0-V_1\end{array} \right) =V_a (\sigma^a)_{\alpha\dot{\beta}}
and in that case also, the Pauli \sigma matrices are (the three components of) a 3-vector (even though each component is
a matrix).
EDIT : in the last formula, one should set V_0 = 0 for a three vector. This form displayed also works with 4-vectors.
