# How can the $SU(2)$ group have a $3$-dimensional representation?

The group $SU(2)$ consists of $2\times2$ unitary matrices with determinant $1$ can be put in the form: $$U=\begin{pmatrix}a &b \\ c & d\end{pmatrix}$$ By invoking the conditions: $det(U)=1$ and $UU^\dagger=1$, we reduce the matrix $U$ to $$U=\begin{pmatrix}a &b \\ -b^* & a^*\end{pmatrix}$$ This is a 2 dimensional representation of $SU(2)$ group. Now, consider a three dimensional representation: $$U=\begin{pmatrix}a &b&c \\ d & e &f \\ g&h&i\end{pmatrix}$$ and then invoke the same conditions $det(U)=1$ and $UU^\dagger=1$. Isn't this just the same as the 3 dimensional representation for $SU(3)$ group?

I know this in't correct, but I don't know what's wrong.

• Do you know what a representation of a group is? Mar 18, 2017 at 13:06
• I'm very confused by the word representation. I think you are asking how a general matrix of $SU(3)$ looks like. This has nothing to do with representation theory. Mar 18, 2017 at 13:07
• That's very vague and not entirely correct. A group representation is a group morphism $\phi:G\rightarrow \text{GL}(V)$ where $V$ is a vector space. If $V$ is finite-dimensional, we say that $\phi$ is a $\dim(V)$-representation of $G$. Mar 18, 2017 at 13:11
• So suppose $\dim(V)=3$, what is the easiest group morphism you can think of? If you interested in a faithful representation you'll have think some more. Anyways, I would revise the basics on representations, I think you have not understood the theory. Mar 18, 2017 at 13:13
• Here's an example of a group representation. The group $G= \Bbb Z / 3\Bbb Z$ is represented by the three matrices $\begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t\end{bmatrix}$, where $t = 0, 2\pi/3, 4\pi/3$. The group operation in $G$ corresponds to multiplication of matrices. This is a (real) 2-dimensional representation of $G$ because it uses $2 \times 2$ real matrices. Mar 18, 2017 at 13:41

If you are a "physics lover" as your username suggests, then I suggest you think about it like this:

Any $$SU(2)$$ matrix can be written in the form $$U = \exp \left(i \theta_x J_x + i \theta_y J_y + i \theta_z J_z \right),$$ where $$\theta_x, \theta_y, \theta_z$$ are real numbers and $$J_x = \frac 1 2 \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \ \ \ J_y = \frac 1 2 \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) \ \ \ J_z = \frac 1 2 \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).$$ Hopefully you recognise $$J_x, J_y, J_z$$ as the angular momentum operators for spin-$$1/2$$ particles.

To get the three-dimensional representation, you still write $$U$$ in the form$$U = \exp \left(i \theta_x J_x + i \theta_y J_y + i \theta_z J_z \right),$$ using the same $$\theta_x, \theta_y, \theta_z$$. But now, you change $$J_x, J_y, J_z$$ to $$J_x = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{array} \right), \ \ \ J_y = \left( \begin{array}{ccc} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \end{array} \right), \ \ \ J_z = \left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right).$$ These are the angular momentum operators for spin-$$1$$ particles.

To construct the $$n$$-dimensional irreducible representation of $$SU(2)$$, use the spin $$j = (n-1)/2$$ angular momentum matrices as $$J_x, J_y, J_z$$.

• I get it. Thanks!! Also,I'd like to ask what is an $SU(2)$ triplet (like the Higgs triplet)? How does it transform under $SU(2)$?Does it has anything to do with the three dimensional representation? All the particle physics lectures and textbooks I can find just use this terminology directly. Mar 18, 2017 at 16:15
• Particle physics has more than one $SU(2)$ symmetry group. The one that I was talking about is the group of rotational symmetries (well, sort of, up to an important minus sign...). This $SU(2)$ group that you're talking about in connection to the Higgs boson is called a gauge symmetry: it is associated with the electroweak force. And by the way, the Higgs is in a two-dimensional representation of this electroweak $SU(2)$. Mar 18, 2017 at 18:03
• But yeah, if your $SU(2)$ is the spin/rotation group, then an $SU(2)$ triplet is the same thing as a particle with spin one. People call it a triplet because you write the quantum wavefunction is really a column vector with three entries, and the angular momentum operators act on the wavefunction as 3x3 matrices. Mar 18, 2017 at 18:05
• Shouldn't the $J_\alpha$ matrices obey the same algebra as the Pauli matrices? When I try to do that explicitly they don't. Dec 12, 2019 at 9:58
• These are not the usual angular momentum matrices for spin-1 systems (which has dimension 3). For example, $J_z$ is (usually) a diagonal matrix with diagonal elements (1,0,-1). How do we get the matrices in the answer (for arbitrary dimension), then? Sep 2, 2021 at 11:54