Finding the 3x3 Matrix Representation of SU(2)

• QuantumLeak
In summary, the conversation is about finding the 3x3 matrix representation of SU(2) for spin 1 particles. The conversation mentions a PDF that shows the isomorphism between SU(2)/Z2 and SO(3), and also discusses the generators of SU(2) for S=1 spin particles. The three dimensional irreducible representation of SU(2) is mentioned, as well as the need for a reference with the explicit form of 3x3 matrices of the generators. Finally, there is a question about the meaning of "symmetric square of standard representation" and a request for a textbook reference.
QuantumLeak
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!

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)

(Si)jk = εijk

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?

1. How is the 3x3 matrix representation of SU(2) found?

The 3x3 matrix representation of SU(2) is found by using the properties of special unitary groups and the Pauli matrices. The Pauli matrices are a set of three 2x2 matrices that represent the three-dimensional rotation group. By combining these matrices with the special unitary group properties, a 3x3 matrix representation of SU(2) can be derived.

2. Why is the 3x3 matrix representation of SU(2) important?

The 3x3 matrix representation of SU(2) is important because it is used in quantum mechanics to describe the behavior of spin-1/2 particles. It is also used in the study of gauge theories and the Standard Model of particle physics.

3. What is the significance of the special unitary group SU(2)?

The special unitary group SU(2) plays a crucial role in many areas of mathematics and physics. It is used to describe the symmetries of physical systems, such as rotations in three-dimensional space. It also has applications in the study of Lie groups, which are used to describe continuous symmetries in mathematics and physics.

4. Can the 3x3 matrix representation of SU(2) be generalized to other groups?

Yes, the 3x3 matrix representation of SU(2) can be generalized to other groups. In fact, the generalization of this representation is an active area of research in mathematics and physics. For example, the 3x3 matrix representation of SU(3) is used in the study of the strong nuclear force.

5. How is the 3x3 matrix representation of SU(2) related to quantum entanglement?

The 3x3 matrix representation of SU(2) is related to quantum entanglement through the concept of spin states. Spin-1/2 particles, such as electrons, can be in a superposition of two states, which are related by a rotation in 3D space. The 3x3 matrix representation of SU(2) describes these rotations and can be used to analyze and manipulate quantum entanglement.

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