I SU(2) representations

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Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a vector space. Also you can have any n x n representation of SU(2) (here I am a bit confused why is it called SU(2) if you can represent it as any n x n matrix).
So I assumed that this means they act (in the 2 x 2 case) on 2x1 column matrices. But, from what i read in the attached screenshots, I understand that the Hilbert space where these matrices act is the one that undergoes a representation, as they say "to completely reduce the Hilbert space of the world to block diagonal form". And also the (j, m) notation I thought it was for eigenvectors, but they say that they use that notation for the representations of the Hilbert space. From what I read (I just started all this stuff with representation theory so I really need help to understand this), I understood that the matrices that act on the Hilbert space can have different representations, but the Hilbert space itself (formed of vectors) stays the same.
So, to resume, my question is why you need to have a representation of the Hilbert space itself and not just of the operators (the Pauli matrices in this case), or in case I misunderstood what I read, what does the author means by the parts I attached? Thank you!
 

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Maybe the question can be boiled down to "Why do we need abstract groups and can we describe everything about a group with a specific set of generator transformations?" Some points
  • There may be multiple choices of the field type (##\mathbb{R}## or ##\mathbb{C}##) and dimension of linear transformations which obey the abstract group algebra. As you mention ##SU(2)## can be represented by ##n\times n## matrices as long as ##n\geq2##.It may be useful to see matrix representation as a separate notion from the group itself.
  • Pauli matrices have an element of arbitrary choice. Even with field and dimension fixed there is an element of arbitrary parameter-ization which doesn't reflect any new group structure.
  • Not all Lie groups are simply connected and can be expressed as exponential expansion. The Lorentz group is a common example of a Lie group that is not simply connected.
As for the statement about ##(j,m)## - they are looking at representations that consist of operators on a subspace of the ##\mid j,m\rangle## states. They label the representations by ##j##, which is the highest ##J_3## value in that subspace. The dimension of a subspace of fixed ##j## is ##2j+1##.

So a ##j=\frac{1}{2}## representation would be ##2\frac{1}{2} + 1 = 2## dimensional and thus we would use ##2\times2## matrices.
 
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A good place to look for stuff on representations of SU(2) is discussions of quantum-mechanical angular momentum. Each possible value of angular momentum j corresponds to a representation (rep) of SU(2), an irreducible representation (irrep) with dimension 2j+1. Each of the dimensions in it corresponds to a value of the projected angular value or magnetic quantum number m. The matrices of the rep are given by the Wigner D-matrices, matrices for doing 3D rotations in that rep. Addition of angular-momentum values corresponds to a product representation constructed from each value's rep. The resulting angular-momentum values correspond to irreps in that rep.
 

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