Elpis1

Well, one of course has to make a distinction between the physical thing and the mathematical representation of said thing. The set of states which model a type of elementary particle furnish an irreducible representation of the appropriate group. Folks usually (somewhat imprecisely, but physicists aren't generally known for their pedantry) say that this is 'defining a particle'. The idea gets across, but for pedagogical purposes, being more precise is better, I'd reckon; thank you for bringing this to our attention.

Good question! What's important is that the Hamiltonian or Lagrangian has these symmetries. Particular solutions having rotational invariance may signify some nice properties of that solution, but it's quite irrelevant to Noether's theorem.

Ah, sorry about that! You're right, of course; my brain slipped a bit.

Where?

Of course! What you're looking for is Noether's theorem; this tells us that for every (continuous) symmetry of a system one may construct a conserved quantity. There are subtleties and exceptions, of course, but that's the gist of it. This is generally how we define things like angular momentum (Woit is referring to this song and dance when he says "Angular momentum is by definition the “infinitesimal generator” of the action of spatial rotations on the theory, both classically and quantum mechanically.")

As a quick example, a hydrogen atom has rotational symmetry, and this corresponds to conserved angular momentum. In turn, this leads to the structure behind the periodic table!

Not at all. To the audience the blog is written for, the article is very sensible. When two folks who know computers quite well discuss some esoteric issue, they will use technical language and assume a certain level of competency and background knowledge; it's the same in physics.

The system would be described by a multi-particle state, which would be reducible.

Of course, Wigner's classification is just for classifying (most) elementary particles. A hydrogen atom can be considered a particle in some contexts, as can waves of spin in a magnet; I am specifically talking about elementary particles!

It's not quite measurable; the magnetic potential is gauge invariant, which means, among a great deal many other things, that it has no well-defined measurable value. However, it is certainly physical; things like the Aharonov–Bohm effect prove that.

Dr. Woit's blog is directed towards physicists, for the most part. This sort of thing is covered very early on in a graduate education in physics; it's old hat for that crew, but incomprehensible to anybody else!

This sort of business is covered in any textbook on quantum mechanics worth its salt.

The idea goes something like this: physics should certainly not depend on where you are at or the orientation of your measuring device. If we change this (called 'changing the frame of reference'), observable quantities should remain the same. Essentially, the point is that if I move an electron from Asia to the United States or spin it around, it remains an electron.

So, we want to encode this mathematically. Quantum mechanically, we describe systems (like this electron) with a mathematical object, the state vector (mathematical physicists, this is good enough). We need some sort of way to describe what it means to move or spin this state. Well, we can construct operators that do such things (rotation operators, translation operators, etc.); the real insight is that translation and rotation can be mapped to objects called groups. A group is a set with an operation that takes two members of the group and outputs a third (with some qualifications on the structure of the operation). Translations can be described with a group; if I drag an object three meters north and then four meters east, this is the same as dragging it five meters in a northeastern direction. Likewise, rotations also form a group.

So, say we have a state that describes an electron. When we act an operator corresponding to a rotation or translation on the state, the resulting state should also describe an electron. Mathematically, we define the description of electrons this way; they're described by the set of states that mix among themselves when acted on by operations from a specific group (in the nonrelativistic case, this is the Galilean group; in the relativistic one, this is the Poincare group).

A set of objects that transform among themselves under group operations are called a group representation. We add a couple other reasonable stipulations: there shouldn't be a subgroup of electron states that only transforms among itself; given any electron state, I should be able to move it or rotate it into any configuration I'd like. Thus, our representation is a so-called irreducible representation. Furthermore, when I rotate or translate my state, the observable predictions should remain the same (a scattering process does not care if it is done in China or Germany), which, due to the structure of quantum mechanics, imposes an additional constraint: unitarity. Thus, particles are defined as irreducible unitary representations of the Galilean/Poincare group. Particles are distinguished from one another by their quantum numbers (mass, charge, and yes, spin, among others). This is known as Wigner's classification.

Now, this imposes incredible restraints on what sort of states you can have. In relativistic and non-relativistic theory, particles have to remain particles after rotation in plain old three-dimensional space: this translates to, in technical terms, as being an irreducible unitary representation of the group SU(2), which encodes rotations in three-dimensional space (it is a subgroup of both the Galilean and Poincare groups). The "irreducible unitary" part enforces stringent qualifications on the states; you get different possible families of states, each (traditionally) labeled by half-integers: j=0,1/2,1,3/2,...

This is spin. States of non-zero j have internal degrees of freedom that mix among themselves when mathematically rotated (this is what Woit means by "in this case rotations also act on the vector values"). When you construct angular momentum from rotation (which is a fascinating discussion in its own right), this corresponds to intrinsic angular momentum.