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.
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".
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.
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.
"
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.
"
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.
> 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.
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.
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!
>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?
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!
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?
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.
Symmetry of boundary conditions cashes out in symmetry of solutions in two ways. The obvious one is solutions that are themselves symmetrical. The less-obvious one is families of solutions, where the symmetry maps one solution to another.
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.
> Thus, particles are defined as irreducible unitary representations of the Galilean/Poincare group.
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.)
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.
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.
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..
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.
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.
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!
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.
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.
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.
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)
It’s interesting the reply from John Baez (another mathematical physicist) to Woit’s “The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here.”:
“And perhaps no physics here, either, unless we say how the state of the world is described by that vector: that is, how we can use the vector to make predictions of experimental results.”
Physics uses fields to describe physical quantities that have a value for each point in space. For example; the temperature in a room is given by a field, the breeze in a room is given by a field. These two fields are not of the same kind, though. The temperature assigns a single value to each point. For example, over the radiator the temperature field says 30 C (86 F in freedom units) and next to an open window 8 C. While the breeze field assigns a vector (a little arrow) to every point in space. Over the radiator, a small arrow pointing up indicating air is flowing with a small speed upwards as the hot air raises. Next to the window a bigger arrow pointing inwards into the room, indicating the breeze is stronger and entering the room.
Different kinds of fields do not transform the same as you change your point of view in the room. If you rotate your point of view, the temperature field remains invariant. For the same point in space you assign the same temperature. But the breeze field changes as you change your point of view with a rotation. If you look directly to the window the breeze in the window points against you, if you face in the opposite direction it points in the same direction you are looking. A rotation changes the breeze vector but leaves the temperature vector unchanged.
How these different fields change as you rotate your point of view is what spin describes. It's a property of the field. The temperature field has spin 0, the breeze field has spin 1.
Fundamental particles are defined by fields too. The electron is defined by a field, the photon is a field, the Higgs boson is a field, the graviton is a field... As fields they have this property characterizing their behavior under rotations.
The Higgs boson is given by a spin 0 field (like the temperature example above). Rotations leave the field invariant.
The photon is given by a spin 1 field (like the temperature example). Rotations make the field changes such that a complete 360 rotation leaves the field invariant.
The graviton is given by a spin 2 field. Rotations make the field change such that a half rotation leaves the field invariant.
The electron is given by a spin 1/2 field. Rotations make the field changes such that you need to do 2 full rotations to leave the field invariant.
Spin characterizes their behavior under rotations. As physics is invariant under rotations this property is a conserved quantity.
Note that even for a spin-1/2 field, nothing weird-looking happens at a macro level. This is because the wave-function enters twice into the expectation values of observables, leaving a spin-0 incoherent portion, and spin-1 vector portion.
You don't need angular momentum to define spin. Spin is a concept that naturally arises just from the geometry of our space. You can define spin of a human body. I wrote a blog post about this https://dandanua.github.io/posts/the-spin-of-a-human-body/
51 comments
[ 4.1 ms ] story [ 109 ms ] threadWell, Duh.
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.)
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.
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.
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.
> 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.
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.
https://www.math.columbia.edu/~woit/wordpress/?p=10533
It’s interesting the reply from John Baez (another mathematical physicist) to Woit’s “The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here.”:
“And perhaps no physics here, either, unless we say how the state of the world is described by that vector: that is, how we can use the vector to make predictions of experimental results.”
Different kinds of fields do not transform the same as you change your point of view in the room. If you rotate your point of view, the temperature field remains invariant. For the same point in space you assign the same temperature. But the breeze field changes as you change your point of view with a rotation. If you look directly to the window the breeze in the window points against you, if you face in the opposite direction it points in the same direction you are looking. A rotation changes the breeze vector but leaves the temperature vector unchanged.
How these different fields change as you rotate your point of view is what spin describes. It's a property of the field. The temperature field has spin 0, the breeze field has spin 1.
Fundamental particles are defined by fields too. The electron is defined by a field, the photon is a field, the Higgs boson is a field, the graviton is a field... As fields they have this property characterizing their behavior under rotations.
The Higgs boson is given by a spin 0 field (like the temperature example above). Rotations leave the field invariant.
The photon is given by a spin 1 field (like the temperature example). Rotations make the field changes such that a complete 360 rotation leaves the field invariant.
The graviton is given by a spin 2 field. Rotations make the field change such that a half rotation leaves the field invariant.
The electron is given by a spin 1/2 field. Rotations make the field changes such that you need to do 2 full rotations to leave the field invariant.
Spin characterizes their behavior under rotations. As physics is invariant under rotations this property is a conserved quantity.
Of all of this, the only mind-bending stuff is that a spin 1/2 field requires 2 rotations to leave it invariant. So here is a visualization: https://upload.wikimedia.org/wikipedia/commons/a/a9/Belt_Tri...