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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!
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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