Well, Duh.
I once read a comment on how the lorentz invariance "falls out" of the Lagrangian and thus it's not mysterious, this is such a particle theorist perspective (if it's not clear I'm being derisive) who just copied the notes in class but doesn't understand theory or the history of the situation. The Lagrangians you use and then quantize in QFT are chosen to be Lorentz invariant because experiment says physics is Lorentz invariant, not the other way around. Similarly, Zeeman effect and other such things informed Pauli's theory of spin 1/2 and all that followed. And thus, since such facts (constant speed of light, intrinsic spin) are experimental facts, it is natural and fine to wonder if there is a more fundamental reason underneath the theory.
The thing is despite his reputation, I expect better from someone as smart as Peter Woit. This is kind of disappointing.
This is a hyper-correction; in practice physicists apply the term in places adjacent to where it should be all the time.
FLP-II-15–5 The vector potential and quantum mechanics https://www.feynmanlectures.caltech.edu/II_15.html#Ch15-S5
" Precisely this experiment has recently been done. It is a very, very difficult experiment. Because the wavelength of the electrons is so small, the apparatus must be on a tiny scale to observe the interference. The slits must be very close together, and that means that one needs an exceedingly small solenoid. It turns out that in certain circumstances, iron crystals will grow in the form of very long, microscopically thin filaments called whiskers. When these iron whiskers are magnetized they are like a tiny solenoid, and there is no field outside except near the ends. The electron interference experiment was done with such a whisker between two slits, and the predicted displacement in the pattern of electrons was observed.
In our sense then, the A-field is “real.” You may say: “But there was a magnetic field.” There was, but remember our original idea—that a field is “real” if it is what must be specified at the position of the particle in order to get the motion. The B-field in the whisker acts at a distance. If we want to describe its influence not as action-at-a-distance, we must use the vector potential. "
As someone who mostly just coasted in my two required physics courses in college and had no interest to study it further, this isn't really _that_ much more "obviously correct" to me than the first quote about angular momentum. Having never heard the term "generator of translation" or anything like it before, I wouldn't have been able to tell if was a rigorous, well-defined term or made up pseudoscience buzzwords before reading this thread.
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.
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!
I am very fascinated and would like to learn more. Begging your pardon for asking something that's googleable, but assuming at least a few other people reading this care... what are some resources for looking into this further?
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!
What about an electron in more complicated potentials, like the ones you'd see in real life vice textbook examples?
All of the "s" states are spherically symmetric. The "p" (and higher) states aren't spherically symmetric, but are instead a basis for an entire family of states that are related to each other through rotations.
I think you lost something here. Particles are particles — they are not little irreps flying through space. I mean this non-sarcastically: spin-0 particles aren’t scalars flying through space. They are things, the state of which can be described by a function mapping a position to a complex scalar. Similarly, the Dirac equation involves states, and those states involve spinors, but it seems rather odd to say that a particle is a spinor.
(In quantum information, one regularly ignores space: there is a finite set of particles, there is time that is often treated as occurring in discrete steps (just like a computer kind of works in discrete clock cycles), and the particles have states that are, if you squint really hard, complex 2-vectors. (Or spinors! But we’re ignoring space entirely, so there are no rotations, so the transformation laws are irrelevant.) But you have to be squinting pretty hard: the state of the quantum computer is not a pile of 2-vectors or of spinors — it’s more like a map from the set of possible strings of bits, one per particle, to a complex scalar amplitude.)