How to find 3D representation of SU(2)

In summary, the conversation is about finding the 3D representation of the commutators [T_a, T_b] for the SU(2) group. The generators in SU(2) are the 3 Pauli matrices and the equation exp^(i*alpha_i*X_i) is used to determine the elements of the group. The commutator is given by [T_a, T_b) = T_a*T_b - T_b*T_a and since there are 3 generators, there should be three matrices in a 3x3 representation. The resource provided on page 13 of the given link has helped clarify any confusion.
  • #1
lonewolf219
186
2

Homework Statement


Find the 3D representation of what I think are the commutators [T_a,T_b] for the SU(2) group

Homework Equations



I think the generators(X_i) in SU(2) group are the 3 Pauli matrices, which are 2X2 matrices... I assume I need to find the matrices for these generators as 3x3?

The Attempt at a Solution



I think the equation exp^(i*alpha_i*X_i) determines what the elements of the group are...
[T_a,T_b)=T_a*T_b - T_b*T_a is the commutator
Since there are 3 generators, does that also mean there should be three matrices in 3X3 representation?

Do I sound confused? :cry:
 
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  • #2
http://www.physics.rutgers.edu/~steves/502/Lectures_Final/Lec03_SU(2).pdf

page 13
 
  • #3
A huge thanks to you, sgd37! No more tears!
 

Related to How to find 3D representation of SU(2)

1. How can I find a 3D representation of SU(2)?

To find a 3D representation of SU(2), you will need to use mathematical tools such as matrix representations and group theory. Specifically, you can use the Pauli matrices and unitary transformations to construct the 3D representation. This representation is known as the spinor representation and is commonly used in quantum mechanics.

2. What is the significance of a 3D representation of SU(2)?

A 3D representation of SU(2) is significant because it allows us to visualize and understand the properties and symmetries of SU(2) in a tangible way. This representation is also important in quantum mechanics and particle physics, as SU(2) is a fundamental symmetry group in these fields.

3. Are there any real-world applications of a 3D representation of SU(2)?

Yes, there are several real-world applications of a 3D representation of SU(2). One example is in the study of atomic and molecular structures, where the spinor representation is used to describe the electron spin state. It is also used in particle physics to describe the properties of fundamental particles and their interactions.

4. Can I use software to find a 3D representation of SU(2)?

Yes, there are software programs available that can help you find a 3D representation of SU(2). Some popular options include Mathematica, MATLAB, and Python libraries such as NumPy and SymPy. These programs have built-in functions and tools for working with matrices and group theory, making it easier to construct the 3D representation.

5. Is there a specific method or algorithm for finding a 3D representation of SU(2)?

Yes, there are several methods and algorithms for finding a 3D representation of SU(2), such as the Gel'fand-Tsetlin algorithm and the Wigner-Eckart theorem. These methods involve using matrix representations and group theory concepts to construct the 3D representation. Depending on your specific goals and needs, some methods may be more suitable than others.

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