# Irreducible representation of su(2)

• Kara386
In summary, the conversation discusses the use of the irreducible representation of ##su(2)## with ##j=\frac{5}{2}## to calculate ##J_z##, ##exp(itJ_z)##, and ##J_x##. The "ladder" operator method is suggested and it is mentioned that ladder operators are not elements of ##su(2)##. It is then suggested to look into a physics book on quantum mechanics and the use of ##\mathfrak{sl}(2,\mathbb{R})## to find representations. The conversation also mentions the definition of j values and their associated increase in dimension of the representation.
Kara386

## Homework Statement

Using the irreducible representation of ##su(2)##, with ##j=\frac{5}{2}##, calculate ##J_z##, ##exp(itJ_z)## and ##J_x##.

## The Attempt at a Solution

There seem to be loads of irreducible representations of ##su(2)## online, but no reference at all to a specific irreducible representation in my lecture notes. It will be a matrix representation, I suspect, involving something from physics because that's the context we're working in, so maybe the Pauli matrices? I'm completely stuck and any guidance or thoughts on what my lecturer might mean would be very much appreciated! :)

Last edited:
Look into ladder operator methods. One defines ##J_\pm = J_x\pm i J_y## and gets commutation relations ##[J_z,J_\pm]=\pm J_\pm## where the signs I just wrote are likely all screwed up.

Paul Colby said:
Look into ladder operator methods. One defines ##J_\pm = J_x\pm i J_y## and gets commutation relations ##[J_z,J_\pm]=\pm J_\pm## where the signs I just wrote are likely all screwed up.
But the ladder operators aren't even elements of ##su(2)##... and they aren't generators. How can they be a representation?

Okay, best looked up in a book. In a nut shell one starts with ##J_-\vert -5/2\rangle = 0##. By applying ##J_+## to this "ground" state one generates all the eigen states in the rep. From these follow all operators in matrix form. It's work, that's why it's homework. It's also extremely elegant.

Kara386 said:
But the ladder operators aren't even elements of su(2)...

Ah, best looked up in a Physics book on quantum mechanics. Any intro text will do.

##\dim \mathfrak{su}(2) = 3## and therefore (both are simple) ##\mathfrak{su}(2) \cong \mathfrak{sl}(2,\mathbb{R})##.
Therefore you can get all representations as representations of ##\mathfrak{sl}(2,\mathbb{R})## which has a basis ##\{Y,H,X\}## with ##[H,X]=2X\, , \,[H,Y]=-2Y\, , \,[X,Y]=H## which can be represented by the matrices
$$H=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\, , \,X=\begin{bmatrix}0&1\\0&0\end{bmatrix}\, , \,Y=\begin{bmatrix}0&0\\1&0\end{bmatrix}$$
This makes it easier to find the representations as those of ##\mathfrak{sl}(2,\mathbb{R})## and easier to see the "ladder", as ##X=J_+\, , \,Y=J_-\, , \,H=J_z\,.## By the way, is ##j=\frac{5}{2}## meant to be the highest weight?

You can also look up the Wikipedia entry, which is not bad:
https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)

fresh_42 said:
##\dim \mathfrak{su}(2) = 3## and therefore (both are simple) ##\mathfrak{su}(2) \cong \mathfrak{sl}(2,\mathbb{R})##.
Therefore you can get all representations as representations of ##\mathfrak{sl}(2,\mathbb{R})## which has a basis ##\{Y,H,X\}## with ##[H,X]=2X\, , \,[H,Y]=-2Y\, , \,[X,Y]=H## which can be represented by the matrices
$$H=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\, , \,X=\begin{bmatrix}0&1\\0&0\end{bmatrix}\, , \,Y=\begin{bmatrix}0&0\\1&0\end{bmatrix}$$
This makes it easier to find the representations as those of ##\mathfrak{sl}(2,\mathbb{R})## and easier to see the "ladder", as ##X=J_+\, , \,Y=J_-\, , \,H=J_z\,.## By the way, is ##j=\frac{5}{2}## meant to be the highest weight?

You can also look up the Wikipedia entry, which is not bad:
https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)
The j values were defined as j=0 are scalars, ##j=\frac{1}{2}## are spinors, and so on. Every half integer increase in j seems to be associated with an increase in 1 of the dimension of the representation.

## 1. What is an irreducible representation of su(2)?

An irreducible representation of su(2) is a mathematical concept used in the study of group theory, specifically in the context of Lie algebras. It refers to a representation of the special unitary group of degree two (su(2)) that cannot be broken down into smaller, independent representations.

## 2. How is an irreducible representation of su(2) different from a reducible representation?

An irreducible representation cannot be decomposed into smaller, independent representations, whereas a reducible representation can be broken down into smaller, independent representations. Essentially, an irreducible representation is the simplest and most basic form of a representation in su(2).

## 3. What are the applications of irreducible representations of su(2)?

Irreducible representations of su(2) have numerous applications in physics, particularly in the fields of quantum mechanics and particle physics. They are also used in the study of symmetry in mathematical and physical systems.

## 4. How are irreducible representations of su(2) related to angular momentum in quantum mechanics?

In quantum mechanics, angular momentum is a fundamental concept that describes the rotational motion of particles. The irreducible representations of su(2) are closely related to angular momentum in that they provide a mathematical framework for understanding the possible values and states of angular momentum in a system.

## 5. Are irreducible representations of su(2) unique?

No, there can be multiple irreducible representations of su(2) with different dimensions and characteristics. The uniqueness of an irreducible representation depends on the specific Lie algebra and group being studied.

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