##SU(2)## generators in ##1##, ##2## and ##3## dimensions

In summary, the Lie algebra in a representation ##\bf R## isdefined by##[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},##where ##T^{a}_{\bf R}## are the ##3## generators of the algebra. The generators in ##1## dimension are the identity matrix and the two matrices that commute with it.
  • #1
spaghetti3451
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The ##{\bf su}(2)## Lie algebra in a representation ##\bf R## is defined by

##[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},##

where ##T^{a}_{\bf R}## are the ##3## generators of the algebra.

In ##2## dimensions, these generators are the Pauli matrices

##T^{1}_{\bf 1} = \frac{1}{2}\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}, \qquad T^{2}_{\bf 1} = \frac{1}{2}\begin{pmatrix}0 & -i\\ i & 0 \end{pmatrix}, \qquad
T^{3}_{\bf 1} = \frac{1}{2}\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}.##

In ##3## dimensions, these generators are

##T^{1}_{\bf 2} = \frac{1}{\sqrt{2}}\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}, \qquad T^{2}_{\bf 2} = \frac{1}{\sqrt{2}}\begin{pmatrix}0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0 \end{pmatrix}, \qquad
T^{3}_{\bf 2} = \begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}.##

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1. How can you derive the generators in ##2## and ##3## dimensions?

2. What are the generators in ##1## dimension?
 
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  • #2
1. The two-dimensional representation is the fundamental representation. You can find out what the matrices J in its Lie algebra are by writing ##e^{-itJ}## and requiring that this matrix is part of SU(2). This will give you a set of matrices of which you can pick a basis.

For the three-dimensional irrep, it is the symmetric part of the tensor product representation. You can construct the generators explicitly by constructing a basis for the tensor product space and checking how the representations act on this basis.

2. I will give you a hint: All 1x1 matrices commute so [A,B]=0 for any A and B.
 
  • #3
Orodruin said:
2. I will give you a hint: All 1x1 matrices commute so [A,B]=0 for any A and B.

So, are the matrices all equal to (0) for the one-dimensional representation?
 
  • #4
spaghetti3451 said:
So, are the matrices all equal to (0) for the one-dimensional representation?
Yes. It is the trivial representation.
 
  • #5
It is also called non-faithful. Is this because every element of ##SU(2)## is being mapped to the same element, that is, ##1##?
 
  • #6
The three-dimensional representation is not faithful either - but it is not trivial. It maps ##A## and ##-A## to the same matrix.

The trivial representation by definition maps all group elements to the identity.

Also, you have a mistake it the third three-dimensional generator above - it should not be the identity.
 
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Related to ##SU(2)## generators in ##1##, ##2## and ##3## dimensions

1. What is the significance of SU(2) generators in 1, 2, and 3 dimensions?

SU(2) generators are mathematical objects that represent the symmetries of a system in 1, 2, and 3 dimensions. They are used to describe the behavior of fundamental particles, such as quarks and leptons, and play a crucial role in understanding the dynamics of quantum field theories.

2. How do SU(2) generators act on physical states in 1, 2, and 3 dimensions?

SU(2) generators act on physical states by transforming them into different states that are related by a unitary transformation. In 1 dimension, the generators are represented by Pauli matrices and act on spin states. In 2 dimensions, they are represented by 2x2 matrices and act on isospin states. In 3 dimensions, they are represented by 3x3 matrices and act on color states.

3. What are the commutation relations of SU(2) generators in 1, 2, and 3 dimensions?

The commutation relations of SU(2) generators differ in 1, 2, and 3 dimensions. In 1 dimension, the Pauli matrices commute with each other, while in 2 dimensions, they satisfy the SU(2) algebra. In 3 dimensions, the SU(2) generators satisfy the SU(3) algebra, which is a generalization of the SU(2) algebra.

4. How are SU(2) generators related to the concept of angular momentum in 1, 2, and 3 dimensions?

In 1 dimension, the Pauli matrices are used to represent the spin angular momentum of particles. In 2 dimensions, the SU(2) generators are related to the isospin of particles, which is a type of internal angular momentum. In 3 dimensions, the generators are related to the total angular momentum, which includes both spin and orbital angular momentum.

5. What are some applications of SU(2) generators in 1, 2, and 3 dimensions?

SU(2) generators have numerous applications in theoretical physics, particularly in the study of quantum field theories and the Standard Model of particle physics. They are also used in mathematical models of crystal structures and in the description of symmetries in condensed matter systems.

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