[HN Gopher] The Mystery of Spin
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The Mystery of Spin
Author : cratermoon
Score : 34 points
Date : 2022-11-26 19:36 UTC (1 days ago)
(HTM) web link (www.math.columbia.edu)
(TXT) w3m dump (www.math.columbia.edu)
| BMc2020 wrote:
| _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.
| the__alchemist wrote:
| I'd prefer to see less use of "all one has done", "just",
| "easily", and "trivial".
| marcosdumay wrote:
| That's a joke. The GP is just saying that spin is defined
| this way, so there's nothing strange with it.
| Elpis1 wrote:
| 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.
| whatshisface wrote:
| Why then aren't several bosons in the same state a particle?
| Elpis1 wrote:
| 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!
| howenterprisey wrote:
| >When you construct angular momentum from rotation (which is
| a fascinating discussion in its own right)
|
| 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?
| Elpis1 wrote:
| 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!
| the__alchemist wrote:
| Does it still have rotational symmetry if the elecron has
| n>1? Doesn't this lead to wavefunction shapes that aren't
| spherically symmetric? Thank you. (I'm coincidentally
| running into this conundrum while trying to build a
| chemistry visualizer. Have only attempted for n=1 with
| the potential being a single proton.)
|
| What about an electron in more complicated potentials,
| like the ones you'd see in real life vice textbook
| examples?
| wnoise wrote:
| 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 wrote:
| Magnetic vector potential is actually directly measurable
| with a somewhat esoteric experimental setup.
| [deleted]
| Elpis1 wrote:
| 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 wrote:
| You're missing a "not".
| [deleted]
| Elpis1 wrote:
| Where?
| wnoise wrote:
| 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 wrote:
| Ah, sorry about that! You're right, of course; my brain
| slipped a bit.
| saghm wrote:
| > 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.
| lumost wrote:
| 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.
| [deleted]
| fijiaarone wrote:
| Skipped the impenetrable equations for the hissy fit comments
| section in the fine article and was not disappointed.
| personjerry wrote:
| Anybody have the English translation?
| hgsgm wrote:
| quickthrower2 wrote:
| If I understand the point of the article it is:
|
| a. Not worth understanding the mysteries.
|
| b. Electrons have spin classically, so no need to talk about
| Quantum Field Theory
|
| And then posts some equations about spin.
|
| I presume (a) is because we would get into "God" territory and
| (b) is to make the discussion simpler.
| tinym wrote:
| No, the mysteries that "are deep, hard to understand, and not
| worth the effort" are why Scientific American is publishing
| this junk article. My lay theory is that SA hasn't been worth
| reading in decades and basically nobody can write well about
| quantum mechanics for a casual audience..
| edgyquant wrote:
| That's because quantum mechanics is for making predictions
| not answering theological questions. Casual observers
| generally want to know what this says about our place in the
| universe and quantum mechanics is way too probabilities based
| for the average joe.
| pdonis wrote:
| Not really, no.
|
| Re (a), the mysteries that Woit says are not worth
| understanding are the ones described in the parenthesis at the
| end of the first paragraph. (As far as I can tell from reading
| the actual paper Woit links to, he is being nice about how off
| base the paper actually is.) As he notes in the second
| paragraph, the actual story--i.e., how spin actually works in
| QM-- _is_ worth understanding.
|
| Re (b), Woit is not saying electrons have spin classically,
| he's saying electrons (and other quantum particles) have spin
| in non-relativistic QM, or more precisely that spin can be
| modeled in non-relativistic QM, so the claim made by Sebens and
| Carroll that QFT is needed to understand spin is wrong. (AFAIK
| the key contribution QFT makes is the spin statistics
| connection, which is a different issue that is not discussed in
| the article.)
|
| The equations Woit posts are a basic presentation of _how_ spin
| can be modeled in non-relativistic QM.
| [deleted]
| ajkjk wrote:
| I like Woit because he's skeptical of the same people as me, but
| the fact that he thinks
|
| > Angular momentum is by definition the "infinitesimal generator"
| of the action of spatial rotations on the theory
|
| Is an explanation... is the same as the reason why he hasn't
| succeeded in changing very many people's opinions on this stuff.
| Elpis1 wrote:
| 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!
| ajkjk wrote:
| I'm familiar with all the physics; that's why I think Woit's
| stance is so disappointing! I can't stand physics' tendency
| to be okay with bad explanations. It's fine to not _have_ a
| good explanation, but that doesn't mean you have to be okay
| with bad ones. (also imo the problem with pretty much every
| treatment of Lagrangians, among other things)
| puffoflogic wrote:
| In other words, it is a sequence of words entirely devoid of
| any meaning. If it can only possibly convey an idea to
| someone who already knows that idea and knows that idea is
| the one to be conveyed, then the words carry zero bits of
| information.
| Elpis1 wrote:
| 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.
| andrewflnr wrote:
| No.
|
| Just to elucidate the general principle a bit: sometimes a
| reminder or different perspective of past learning using
| vocabulary you already know can be valuable. Humans aren't
| perfect decoding and recall machines. And that's assuming
| the author and target audience learned the advanced
| vocabulary in the exact same way, which is unlikely.
| Sometimes you need to fill in gaps in some of your
| audience's knowledge, perhaps that they should have learned
| but didn't, maybe because they or their teacher was having
| a rough day in class.
| mhh__ wrote:
| Tomonagas book "the story of spin" is a banger. Really gentle at
| times but very detailed and insightful at other times
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