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I've looked a lot of other places for an explanation for why $\rm SU$ groups are so ubiquitous in particle physics, but they all look like I'd need to teach myself lie algebra and group theory just to understand them, and they usually just describe when they're useful, such as:

The $\rm SU(n)$ groups find wide application in the Standard Model of particle physics, especially $\rm SU(2)$ in the electroweak interaction and $\rm SU(3)$ in quantum chromodynamics.

from Wikipedia. But I want to know what they do that makes them useful. The answer to this question is written for someone who fully understands lie groups and how to use them. I don't care how much you dumb it down as long as I can still get the general idea.

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The framework for this question is quantum field theory. The main thing that you need to know about quantum field theory, for this answer, is that we describe a physical theory using fields, which are functions of space and time. The advantage of using fields is that we can write theories that are manifestly local -- two or more fields interact when we include a term in the energy where those fields are multiplied together at a single point. However, we want to use these fields to describe particles, which are the entities we actually detect in particle physics experiments -- and this turns out to lead to a world of mathematical trouble (or wonder, if you are into this kind of thing).

Electromagnetism

Let's briefly describe electromagnetism, at a very high level, in a language that can be generalized to the strong and weak forces. Electromagnetism is a theory of a spin-1 particle (the photon), which is coupled to other matter with electric charge (like the electron and proton). However, the photon itself does not have a charge.

Now, for deep reasons that are very difficult to explain without math, it is very difficult to write down a theory that is simultaneously consistent with the laws of special relativity (in particular Lorentz invariance) and quantum mechanics (particularly unitarity). In fact, it is so difficult, that just knowing that we want to write down a relativistic, quantum mechanical theory of a spin-1 photon coupled to charged particles, uniquely determines the form of the theory at "low energies." (You can take "low energy," here, to mean "at all energies probed by experiments to date.").

At a very high, rough level, what happens is that a generic theory of a spin-1 particle will include terms that violate either Lorentz invariance or unitarity. What is needed is a special symmetry that guarantees these offending terms are cancelled.

Why does a symmetry guarantee such a cancellation? As I will say a lot in this answer, to really explain this requires some math. One high-level answer is that we need some special principle to prevent the "bad" terms from appearing, and this special principle takes the form of a symmetry. Maybe more to the point is to give a (way over-simplified) example of how a symmetry can prevent a term from occurring. Let's say we have a quantity $x$, which is made of a sum of terms that can be positive or negative. For example, $x=1-2+3$. In general, $x$ can be positive, negative, or zero. Let's say that our theory will break down if $x$ is nonzero. We can write $x = 0 + \delta$, where $\delta$ is some "symmetry breaking parameter" -- in general we expect $\delta \neq 0$ (and therefore our theory will not work). However, if we impose the requirement that every time we add a term $a$ to $x$, we also add $-a$, then $\delta = 0$, because the terms that make up $x$ will always cancel in pairs (example: $x = 1 - 1 + 2 - 2 = 0$). Every new term has to be added with a symmetrical partner with the opposite sign, and this guarantees $x$ will be zero. (In fact this "symmetry" underlies double-entry bookkeeping). The point of this example is just to illustrate how a symmetry can guarantee that certain unwanted terms in a mathematical expression are zero.

Anyway, the symmetry used by electromagnetism is given the obscure name $U(1)$. The symmetry also has to be "local" or a "gauge symmetry," again for reasons that are hard to explain (although I could try if you are interested).

Mathematically, a $U(1)$ symmetry refers to the fact that you can multiply all charged fields by a factor $e^{i q \theta}$, where $q$ is the charge of the field, and $\theta$ is a parameter. The "gauge" symmetry means we can choose a different value of $\theta$ for every point in spacetime.

However, the mathematical details are not important at this high level of description; the key idea is that this symmetry "protects" the theory from offending terms which would violate special relativity or quantum mechanics.

Weak and strong interactions

Now, the weak and strong interactions are also described spin-1 particles. But, in these cases, the spin-1 particles themselves are charged.

As in the case of the photon, there are stringent consistency conditions that fix the form of the interactions of such particles. In this case, however, we have to consider a more complicated symmetry group than for the photon. The reason is that there are multiple charged spin-1 particles associated with the weak force (and also for the strong force), and we have to allow for transformations where these spin-1 particles are exchanged with each other. The larger symmetry that allows for these exchanges, and is consistent with special relativity and quantum mechanics, has to be a so-called "non-Abelian Lie group," which are types of symmetries that have been cataloged and are well-understood. For reasons we don't understand, out of this infinite catalog of options, Nature has chosen to use two particular Lie groups, $SU(2)$ for the weak interactions, and $SU(3)$ for the strong interactions.

Mathematically, these groups allow you to multiply fields by unitary matrices with determinant equal to 1 -- that is the meaning of the $SU$ part. The $2$ and $3$ are an additional requirement on the types of matrices that are allowed. Very roughly speaking, these numbers have to do with the size of the symmetry group. This part is very hard to explain precisely without more math, in particular representation theory. However, keeping to a high level, probably the most concrete consequence of this parameter is that $SU(2)$ requires us to have 3 spin-1 particles (the $W^+$, $W^-$, and $Z$ particles), and $SU(3)$ requires us to have 8 (which are the 8 gluons). To dig into the mathematical details is where you need a course on Lie groups.

The high-level takeaway is that, just like with electromagnetism, the weak and strong interactions need a symmetry to protect the theory against terms that would make the theory inconsistent with either special relativity or quantum mechanics. But, this symmetry needs to be more complicated, because the spin-1 particles for the weak and strong interactions are themselves charged. $SU(2)$ and $SU(3)$ are two examples of symmetries that meet these requirements, and Nature has chosen to use these two for the weak and strong interactions.

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  • $\begingroup$ So what are the members of the groups? Are they particles like quarks and leptons, are they gauge bosons, or something else? $\endgroup$
    – zucculent
    Nov 3, 2021 at 4:17
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    $\begingroup$ @zucculent The members of the group (to be precise, the group representation) are mathematical operations applied the fields. In the electromagnetic case, "multiplication by $e^{i q \theta}$" is a member of the group representation. The symmetry is a mathematical operation that acts on the actual physical fields (quarks, leptons, etc). $\endgroup$
    – Andrew
    Nov 3, 2021 at 4:20
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    $\begingroup$ The reason I'm differentiating "group" from "group representation" is because a "group" is actually a completely abstract, mathematical construct that doesn't need to be applied to anything physical. In physics, we are interested in representations where we instantiate this abstract group into a specific operation that actually acts on fields. $\endgroup$
    – Andrew
    Nov 3, 2021 at 4:21
  • $\begingroup$ So in summary, a unitary group is a group "representation" of different mathematical operators to make sure the math comes out correctly, and a bigger $N$ for $SU(N)$ means it's more complicated? $\endgroup$
    – zucculent
    Nov 3, 2021 at 4:25
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    $\begingroup$ @zucculent Yep if you want to know about particle physics you need to know all of those things :) Specifically for group theory, James Grime on youtube has some fun videos: youtube.com/watch?v=ylAXYqgbp4M. If you are looking for like, a list of textbooks to study from, that could quickly become a huge list. 't Hooft has made a huge list like this, I don't know most of the books on it but it's got all the topics you need goodtheorist.science $\endgroup$
    – Andrew
    Nov 3, 2021 at 20:07

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