Understanding Group Theory in QFT

Question

Recently it was asked what the reason for Pauli's Exclusion Principle, and the most well-recieved response looks like hieroglyphics to me:

I think that while these "explanations" are all dancing around the same pole, they aren't created equal. I think the meat is in the fact that nature has a local Lorentz symmetry, so we expect to be able to decompose things into representations of the group SO(3,1). It's a mathematical fact that this group (or it's algebra, rather) has integer and half-integer representations.

Once you have this structure, then a few meagre assumptions about causality and unitarity lead to the Spin-statistics theorem. In order to understand the proof you'll need to first dig deeper into the representations of the Lorentz group, and how they label single-particle states.

I am fairly familiar with second quantization but don't have much QFT knowledge beyond that. Group Theory consistently comes up even in regular quantum mechanics...but it's always come across as being complete nonsense. A professor writes down a matrix or some system of equations that are completely understandable, and then says it's really important to understand that this is a "group" and writes SU(2) on the board. Then we proceed to do completely normal math without ever using that knowledge. Sure he writes down some rules for what exactly you need to have to have a "group", but what's the point?

It's obvious that there is something here that's important...but I've never managed to uncover it. Any attempt I get to learn about it, I just get overwhelmed with even more of the formalism (here for example). What is this stuff and what is the point of it?

Answer 1

Groups are tied intimately with symmetries. I looked up once what the connection between the was, and the results were hieroglyphics. But they were useful hieroglyphics... after a lot of wikidiving into category theory!

At the highest level, groups are mathematical objects which describe how symmetries compose. Often just seeing how they compose provides great insight into the possibilities that can occur.

For example, one of the classic rules in physics is time-symmetry. This means that you may not know what's going to happen at t=1, but you're pretty sure that if you started the experiment 10 seconds later, whatever would have happens at t=1 happens at t=11. This behavior is associated with an $R$ group. The similar behavior in 3-space is associated with an $R^3$ group, so a property which is the same at any time or place exhibits $R\times R^3$ symmetry.

The SU class of groups is associated with $n\times n$ unitary matrix multiplication, so things with SU(2) symmetry behave the same when multiplied at the right place by a 2x2 unitary matrix.

What's great about this is that it empowers all of the other math we might do later. It lets us ignore any possible answer which violates said symmetry. And, as it turns out, much of that can be done in group-theory land without actually having to operate on the real equations. We can often predict some behaviors, such as the existence of half spin solutions, simply from what must be true to keep that symmetry functioning.

Symmetries are a powerful thing. There is a famous derivation by Emmy Nother which proved that, in Lagrangian mechanics, every continuous symmetry yields a conserved value. Time symmetry, as mentioned above, is actually associated with conservation of energy. Spatial symmetries result in conservation of momentum. And in the quantum world, the SU(3) symmetry found in the equations for color charge yield conservation of color. Group theory is simply the power-tool used to leverage these symmetries to yield results!

As to your specific example, consider whether you can have a value represented by thirds. You could work the math and prove that such a value will result in a Lorenz symmetry failure. But what about quarters? Nope. Breaks the symmetry. What about irrational values? We could be here all night. But a little trip into group theory and lie algebras immediately tells us that integers and half integers are all we need to consider. Any other solution will not exhibit Lorenz symmetry.

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A quick edit (which I may properly incorporate later).

In category theory, a symmetry of some function $\phi$ is a pair of functors $(\alpha, \beta)$ such that $\alpha\circ\phi = \phi\circ\beta$. The class of all symmetries for some $\phi$ forms a group (it has the properties that a group hash). Now that's basically gibberish if you ask me, but it is meaningful. Let's put it in function notation: $\alpha(\phi(X))$ = $\phi(\beta(X))$. It's a bit easier to see the power. Now look at the right side. I'm modifying X, the input to $\phi$ with some $\beta$ function. On the right hand side, I'm modifying $\phi(X)$, the output of $\phi$ with some $\alpha$ function.

Now look how powerful that is. This says that if $(\alpha, \beta)$ is in the symmetry group of $\phi$, and I solve $\phi(X)$ once, I can use $\alpha$ to modifying the output to account for any $\beta$ in that group. I've moved the modification from the input side of the process to the output side. To answer your question in the comments, it's not always linear, but it is defined by the symmetry group.

Group theory lets you make statements based on the existence of these symmetric pairs without having to actually identify the functions. Thus you can either take the time to prove that a particular system with a Lorentz symmetry can be represented with integers and half integers (and that you can't simplify it further), or you can just say "All systems with Lorentz symmetry have an associated SO(3,1) group, and SO(3,1) groups can always be represented by integers and half integers, so this system must be representable in the same way.". Now you may come across a far more complex system, with far more details to work with. But if you look at it from a symmetry perspective, and find that the system has a symmetry group of U(1)SO(3,1)SU(2), then you know it has all the charactaristics of any system with SO(3,1) symmetry, without having to run any extra math.

Answer 2

Symmetries are things you can do to a system that keep its behavior the same, such as rotating it. You can compose two such operations and invert them, so the set of symmetries forms a group.

Next, you can consider how these symmetry operations act on the states of your system. For example, you know that in the hydrogen atom (ignoring spin), rotating the $1s$ state doesn't do anything, rotating a $2p$ state moves you around in the subspace of three $2p$ states, rotating a $3d$ state moves you around in the subspace of five $3d$ states, and rotating a $3p$ state moves you around in the subspace of three $3p$ states, in just the same way that $2p$ states are rotated. Because rotation is a symmetry of this system, each of these individual sets of states are at the same energy.

You can show this by performing laborious manual calculations on the wavefunctions, but group theory provides a lot of information that saves you work. All you have to know is that the group of rotational symmetries is $SU(2)$. Then the general tool of group theory tells you that:

• the basic subspaces of states related by rotations (like the set of $3d$ states) are $1,3,5,\ldots$ dimensional (group theory tells you that you can't get an even number)
• for each of these dimensions, there's exactly one way the rotations can act on the states (group theory tells you the $2p$ and $3p$ states transform identically among themselves under rotations)
• all of this applies when the system is more complicated but still has rotational symmetry (group theory lets you apply these results to, e.g. many-electron atoms)
• all of this applies when you have a symmetry described by $SU(2)$, even if those symmetries have nothing to do with physical rotations (group theory lets you apply your intuition about rotation to, e.g. isospin)

The reason group theory might currently look useless to you is that you're still learning about the first example, $SU(2)$. Suppose you were teaching somebody how differentiation is useful for finding the slopes of curves, and you started by showing them the derivative of $ax+b$, because that's the simplest possible example. They might protest that differentiation is useless, because they already knew the slope was $a$. But differentiation isn't useless, because it can handle far more than a line. Similarly, when you're starting with $SU(2)$ you basically do everything manually, to build intuition. It starts paying off once you get to reuse these results in other contexts, or consider more complicated groups.

Answer 3

I don't know about groups in general but in physics Lie groups usually correspond to symmetry transformations. The useful thing about Lie groups is the Lie algebra and exponential map ( you can think of these as the general basis for constructing usually any possible element of the Lie group).

So it's kind of like the language of underlying mathematical structure and enables you to express the stuff in the "foundational" building blocks. I don't know much about this yet though and i would recommend you to read the usual recommended books about group theory and look at examples where it is used in physics like the Lorentz group.

I personally read a pretty mathsy book before and a part of it was about Lie groups and I also thought "this is great and beautiful and all but what does this mean in physics" and then I read the special relativity chapter in Jackson about the Lorentz group and it's applications and was really impressed, I'm sure it is similarly useful in quantum mechanics etc..

All in all it just seems to be the natural language of symmetries in physics.