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There’s no mystery here about what the spin angular momentum is: all one has done is used the proper definition of the angular momentum as infinitesimal genera

by BMc2020 4y ago

There’s no mystery here about what the spin angular momentum is: all one has done is used the proper definition of the angular momentum as infinitesimal generator of rotations and taken into account the fact that in this case rotations also act on the vector values, not just on space. One can easily generalize this to tensor-valued wave-functions by using the matrices for rotations on them, getting higher integral values of the spin. Well, Duh.

deleted 4y ago
[deleted]
the__alchemist 4y ago
I'd prefer to see less use of "all one has done", "just", "easily", and "trivial".
- marcosdumay 4y ago
  That's a joke. The GP is just saying that spin is defined this way, so there's nothing strange with it.
wnoise 4y ago
I know you're being sarcastic, but this is actually just table-stakes for any sort of research in fundamental physics. This is quite analogous to linear momentum being the generator of translation, yet having mysterious components in E&M that aren't a particle moving, but charge interacting with the somewhat inscrutable "vector potential".
- jojobas 4y ago
  Magnetic vector potential is actually directly measurable with a somewhat esoteric experimental setup.
  - deleted 4y ago
    [deleted]
  - Elpis1 4y ago
    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.
    - wnoise 4y ago
      You're missing a "not".
      - Elpis1 4y ago
        Where?
        wnoise 4y ago
        in "the magnetic potential is gauge invariant". It differs for different gauges, so is not gauge-invariant, but gauge-dependent. A choice of A _is_ a choice of gauge. The theory using it (i.e. E&M Lagrangian) is what is gauge-invariant. This is a hyper-correction; in practice physicists apply the term in places adjacent to where it should be all the time.
        Elpis1 4y ago
        Ah, sorry about that! You're right, of course; my brain slipped a bit.
      - deleted 4y ago
        [deleted]
  - mikewarot 4y ago
    Yes, I'd like to own a few SQUIDs to play with. I've got some old Tesla patent drawings (bifilar coils) I'd like to understand better.
- lumost 4y ago
  For a particular set of formalisms which some may find esoteric. It’s not wrong to wonder if the above is a true, but ultimately I insightful statement.
- saghm 4y ago
  > This is quite analogous to linear momentum being the generator of translation, yet having mysterious components in E&M that aren't a particle moving, but charge interacting with the somewhat inscrutable "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.
Elpis1 4y ago
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.
noobermin 4y ago
I read Peter Woit's reply to an objection, and it really didn't help either. It feels circular, basically, "angular momentum mathematically looks like this if you start from the infinitesimal generators"...okay, so that means the spinor components of the dirac field transform like they're spinning...but why man, why must that be the case? God didn't give you the dirac states and the gamma matrices and say "all fermions behave as so! " Arguably, you can say the opposite: people use spinor valued fields because they describe spin 1/2 particles, which electrons and positrons are observed to be experimentally. 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.

deleted 4y ago

[deleted]

the__alchemist 4y ago

I'd prefer to see less use of "all one has done", "just", "easily", and "trivial".

marcosdumay 4y ago
That's a joke. The GP is just saying that spin is defined this way, so there's nothing strange with it.

wnoise 4y ago

I know you're being sarcastic, but this is actually just table-stakes for any sort of research in fundamental physics. This is quite analogous to linear momentum being the generator of translation, yet having mysterious components in E&M that aren't a particle moving, but charge interacting with the somewhat inscrutable "vector potential".

jojobas 4y ago
Magnetic vector potential is actually directly measurable with a somewhat esoteric experimental setup.
- deleted 4y ago
  [deleted]
- Elpis1 4y ago
  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.
  - wnoise 4y ago
    You're missing a "not".
    - Elpis1 4y ago
      Where?
      - wnoise 4y ago
        in "the magnetic potential is gauge invariant". It differs for different gauges, so is not gauge-invariant, but gauge-dependent. A choice of A _is_ a choice of gauge. The theory using it (i.e. E&M Lagrangian) is what is gauge-invariant. This is a hyper-correction; in practice physicists apply the term in places adjacent to where it should be all the time.
        Elpis1 4y ago
        Ah, sorry about that! You're right, of course; my brain slipped a bit.
    - deleted 4y ago
      [deleted]
- mikewarot 4y ago
  Yes, I'd like to own a few SQUIDs to play with. I've got some old Tesla patent drawings (bifilar coils) I'd like to understand better.
lumost 4y ago
For a particular set of formalisms which some may find esoteric. It’s not wrong to wonder if the above is a true, but ultimately I insightful statement.
saghm 4y ago
> This is quite analogous to linear momentum being the generator of translation, yet having mysterious components in E&M that aren't a particle moving, but charge interacting with the somewhat inscrutable "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.

Elpis1 4y ago

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.

noobermin 4y ago

I read Peter Woit's reply to an objection, and it really didn't help either. It feels circular, basically, "angular momentum mathematically looks like this if you start from the infinitesimal generators"...okay, so that means the spinor components of the dirac field transform like they're spinning...but why man, why must that be the case? God didn't give you the dirac states and the gamma matrices and say "all fermions behave as so! " Arguably, you can say the opposite: people use spinor valued fields because they describe spin 1/2 particles, which electrons and positrons are observed to be experimentally. 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.