Finding the 3x3 Matrix Representation of SU(2)

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Discussion Overview

The discussion revolves around finding the 3x3 matrix representation of SU(2), specifically focusing on rotation matrices for particles with spin 1. Participants are seeking explicit forms of the generators of SU(2) in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests the 3x3 matrix representation of SU(2) for spin 1 particles.
  • Another participant provides a link to a PDF that discusses the isomorphism between SU(2)/Z2 and SO(3).
  • A participant expresses interest in the generators of SU(2) specifically for a spin 1 particle, asking for explicit forms of the 3x3 matrices.
  • One participant mentions that the three-dimensional irreducible representation of SU(2) can be realized as the symmetric square of the standard representation, but seeks clarification on this point.
  • Another participant acknowledges the need for commutation relations and requests a reference for the explicit form of the matrices used for rotations around the z-axis for a spin 1 particle.
  • A participant seeks clarification on the meaning of the symmetric square of the standard representation and requests a reference for it.

Areas of Agreement / Disagreement

Participants do not seem to reach a consensus on the specific form of the 3x3 matrices or the references needed, indicating that multiple views and uncertainties remain in the discussion.

Contextual Notes

There are limitations regarding the explicit forms of the matrices and the references to the representations being discussed. The discussion also reflects a dependence on definitions and unresolved mathematical steps related to the generators of SU(2).

QuantumLeak
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Hi all,
do you know where i can find the 3x3 matrix representation of SU(2)? Which means basically rotation matrices for particles of spin 1.

Thanks!
 
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That pdf shows the isomorphism between SU(2)/Z2 and SO(3). What I would like to find is the generators of SU(2) for a S=1 spin particle, i.e. 3x3 generators of SU(2)
 
The three dimensional irreducible representation of SU(2) can be realized as the symmetric square of the standard representation. But I am not sure what exactly you are looking for. May be you are asking about the representation of the Lie algebra and the matrices by which the standard basis elements act.
 
Yes Bill I know the commutation relation that they have to satisfy. What I need is a reference with the explicit form of 3x3 matrices of the generators. In other words, if I have to make a rotation of an angle \alpha around the z axis of a spin 1 particle, which matrix I have to use to model this rotation?

martin what do you mean by symmetric square of standard representation? Do you have a reference?

Thanks!
 
Oh, sorry Bill now I see. you mean that the jk element of the matrix of the i-th commutator has to satisfy that relation. Ok thanks. Do you have a textbook reference for that?
 

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