What does the $N$ in $SU(N)$ mean?

Question

So I know this is a very basic question, but I can't really wrap my head around it.

I was told $N$ is the number of dimensions in the rotations of the group theory that we are considering, so I assumed that the reason why we used $SU(2)$ in 1/2 spin particles, but I also was told that for particles with spin 1 we still use $SU(2)$ even though there are 3 basis (1 0 0), (0 1 0) and (0 0 1) and the Pauli matrices are 3x3. Then what's the difference between $SU(2)$ and $SU(3)$?

Another thing that I heard is that the reason why we use $SU(2)$ for photons is because $SU(2)$ is equivalent to $SO(3)$ and the 3 comes from the 3 dimensions of space, but if that's the case why do we use $SU(3)$ for strong force and $U(1)$ for the electromagnetic force? I assumed we use $U(1)$ for EM because there's one quantity conserved (charge) and $SU(3)$ for strong force because there are 3 quantities conserved (colors)

Please don't go too deep within the theory, I don't understand it too deeply yet, I just want to know what is the meaning of $N$ in simple terms.

Answer 1

There's a few things going on here.

Global vs Gauge Symmetries

First, it's true that $SU(2) \sim SO(3)$ if you only worry about small rotations. The more precise statement is that $SU(2)$ is the "double cover" of $SO(3)$, so you can think of $SU(2)$ as two copies of $SO(3)$ that are glued together in a particular way. The "number of states" you're referring to in your post is the number of spin states. Every particle has a spin because every particle exists in space. Spin captures what happens when you rotate a particle in space. If you know what happens to a particle for small rotations, you know what happens for big rotations, because a big rotation is just a lot of small rotations put together. Therefore, information about spin is captured by $SU(2)$ transformations.

A completely unrelated phenomenon is that a particle can have a charge under a force. It turns out that charge is deeply related to group theory (see the next section) and hoe particles transform under different group transformations. So particles not only need to know how they transform under physical rotations, they need to know how they transform under the more abstract "gauge transformations" associated with the various forces of nature. These are two separate pieces of information, and particles need to have the complete list for all the relevant forces of nature.

In our universe, the relevant gauge groups are $SU(3),SU(2), U(1)$ for the strong, weak, and electromagnetic forces. The $SU(2)$ for the weak force is completely unrelated to the $SU(2)$ for spin.

Representation Theory

There's a difference between a symmetry group and the matrices we use to apply that symmetry to various vector spaces. In other words, there's a difference between abstract rotations and rotation matrices. The study of the difference between the two is called representation theory.

The $N$ is $SU(N)$ tells you the smallest dimensional vector space that $SU(N)$ can act on in a non-trivial way. This "smallest non-trivial vector space" is called the fundamental representation.

To explain why the photon still uses $SU(2)$ for rotations despite having three states, we first need an example of a different representation. This will be a little technical, but bear with me if you want to understand that point. Recall that any unitary matrix has eigenvalues that are a pure phase. This implies we can rewrite them as $$U = \exp(i \alpha_a T^a)$$ where $\alpha_a$ is a list of numbers and $T^a$ is a different set of matrices that, when exponentiated, always give the usual $N \times N$ unitary matrices. They are basically the Pauli matrices for more general groups. How many $T^a$s are there? $N^2-1$, because that's the dimension of the group. So you can think of $\alpha_a$ as being a vector in a $N^2-1$ dimensional space. And this space has an action of $SU(N)$ because if you conjugate $U$ by a different unitary $V$, $$ U' = VU V^\dagger = \exp(i\alpha'_a T^a)$$ so the $N^2-1 \times N^2-1$ matrix which transforms $\alpha \to \alpha'$ is also a representation of $SU(N)$, not $SU(N^2-1)$. If it was $SU(N^2-1)$, then you should be able to act by ANY unitary. But you can't: only a $N$ dimensional subspace of them. Finally, notice that $2^2-1=3$, so photons transform under precisely this representation of $SU(2)$, which is called the adjoint representation.

The last thing I'll say is that "how a particle transforms under a group" is the same thing as "what representation of the group is the vector space the particle lives in transforming under". So for a gauge symmetry, the charge of a particle is the representation it transforms under. I strongly recommend learning more about representation theory if you want to learn particle physics more deeply, because it's kind of the language that everyone uses and is really useful in describing stuff.

Answer 2

$N$ denotes the dimension of the so-called defining representation.

The group SU(N) is the group of $N\times N$ special unitary matrices (i.e. determinant=1).

Similarly, the group $U(N)$ is the group of $N\times N$ unitary matrices (determinant not necessarily =1). The group $SO(N)$ is the group of $N\times N$ special orthogonal (i.e. determinant=1), and $Sp(2N)$ is the group of $2N\times 2N$ symplectic matrices.

It may (or may not) be possible to find matrices in dimension $r$ with r greater, equal or smaller than $N$ for which the basic multiplication law of the corresponding $N\times N$ matrices still holds. One then speaks of an $r\times r$ representation of $SU(N)$, $U(N)$, $SO(N)$ whatever.

Answer 3

$SU(N)$ is a mathematical object (a group), and the $N$ means different things in different contexts. Asking what the $N$ means is a bit like asking what the variable $x$ means in $x^2 + x + 1$; it has different meanings in different contexts.

One of the meanings (as you bring up) is associated with rotations. As you may know, the group $SO(3)$ represents rotations in our three-dimensional world. As it turns out (this is just a math fact) the group $SU(2)$ is a "universal cover" of $SO(3)$. Technically what this means is that if we want to find out all the different ways that objects can transform under rotations, which is important to be able to classify all the different kinds of particles in the world, we should study representations of $SU(2)$ and not representations of $SO(3)$.

Another place $SU(N)$ shows up is in the $SU(3)$ gauge theory called QCD, which describes quarks and gluons (the constituent pieces of protons, neturons, etc.) $N = 3$ here has nothing to do with the fact that we live in three dimensions, it's just a happenstance that quarks come in 3 different 'colors'. You could imagine a different world, still with three spatial dimensions, that had four different colors of quarks, so that the gauge theory is an $SU(4)$ gauge theory. In this world, there wouldn't be protons or neutrons, but rather different composite particles made up of four quarks bound together.

Answer 4

What does the N in SU(N) mean?

This $N$ value means the size of the "special" (i.e., unit determinant) unitary matrices of the defining representation.

For example, the group $SU(2)$ is defined by the properties of $2\times2$ unitary matrices with determinant 1. But this does not mean that all SU(2) representations have to be $2\times2$ matrices. The $2\times2$ special unitary matrices are used to define the group properties, but the group can have many different representations. (N.b., the word "representation" here means the set of matrices that "represent" the abstract group elements, where "represent" means that they behave in the same way under the group operations as the abstract group elements that they represent.)

For example, the group $SU(3)$ is defined by the properties of $3\times3$ unitary matrices with determinant 1. But this does not mean that all SU(3) representations have to be $3\times3$ matrices. The $3\times3$ special unitary matrices are used to define the group properties, but the group can have many different representations.

For example, a $1\times 1$ representation of $SU(2)$ is provided by: $$ e^{triv}_i = 1 $$
for every element $e_i$ of the group. This is called the "trivial" representation. This is also called the "spin 0" representation since $1 = e^{-i\times 0}$ is generated by the generators $$ g^{triv}_i = 0 \;, $$ which, of course, (trivially) satisfy the usual commutation relations of SU(2) generators: $$ 0=0\times 0 - 0\times 0 = [g_1, g_2] = i g_3 = 0\;. $$

You can have a representation of the group SU(2) that consists of matrices of any size: $1\times 1$, $2\times 2$, $3\times 3$, and so on. The generators of each of these representations have to obey the fundamental rules established by considering the defining representation (the $2\times 2$ unit-determinant unitary matrices), but that does not mean every representation has to be 2 dimensional. For example, we describe the spin angular momentum of spin-1/2 particles using the $2\times 2$ representation of the $SU(2)$ group. For example, we describe the spin angular momentum of spin-1 particles using the $3\times 3$ representation of the $SU(2)$ group. This relation between spin and representation size (size equals twice the spin plus one) also holds for composite systems. E.g., Manganese(II) has spin 5/2 and so its spin angular momentum is described by the $6\times 6$ representation of SU(2).

You can read more about groups and their representations in the excellent book by Tinkham.

Answer 5

$SU(N)$ is the group of $N \times N$ unitary matrices with determinant $1$. Hence, $N$ literally tells you the size of the matrices.

Your remarks in your question are misunderstandings related to representation theory. It might benefit to ask a separate question because an answer to the question you ask in this post will not resolve all said confusions.