# The Mystery of Spin

Scientific American has a new article today about the supposedly mysterious fact that electrons have “spin” even though they aren’t classical spinning material objects. The article doesn’t link to it, but it appears that it is discussing this paper by Charles Sebens. There are some big mysteries here (why is Scientific American publishing nonsense like this? why does Sean Carroll say “Sebens is very much on the right track”?, why did a journal publish this?????).

These mysteries are deep, hard to understand, and not worth the effort, but the actual story is worth understanding. Despite what Sebens and Carroll claim, it has nothing to do with quantum field theory. The spin phenomenon is already there in the single particle theory, with the free QFT just providing a consistent multi-particle theory. In addition, while relativity and four-dimensional space-time geometry introduce new aspects to the spin phenomenon, it’s already there in the non-relativistic theory with its three-dimensional spatial geometry.

When one talks about “spin” in physics, it’s a special case of the general story of angular momentum. Angular momentum is by definition the “infinitesimal generator” of the action of spatial rotations on the theory, both classically and quantum mechanically. Classically, the function $q_1p_2-q_2p_1$ is the component $L_3$ of the angular momentum in the $3$-direction because it generates the action of rotations about the $3$-axis on the theory in the sense that
$$\{q_1p_2-q_2p_1, F(\mathbf q,\mathbf p)\}=\frac{d}{d\theta}_{|\theta=0}(g(\theta)\cdot F(\mathbf q,\mathbf p))$$
for any function $F$ of the phase space coordinates. Here $\{\cdot,\cdot\}$ is the Poisson bracket and $g(\theta)\cdot$ is the induced action on functions from the action of a rotation $g(\theta)$ by an angle $\theta$ about the $3$-axis. In a bit more detail
$$g(\theta)\cdot F(\mathbf q,\mathbf p)=F(g^{-1}(\theta)\mathbf q, g^{-1}(\theta)\mathbf p)$$
(the inverses are there to make the action work correctly under composition of not necessarily commutative transformations) and
$$g(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta &\cos\theta &0\\ 0&0&1\end{pmatrix}$$

In quantum mechanics you get much same story, changing functions of position and momentum coordinates to operators, and Poisson bracket to commutator. There are confusing factors of $i$ to keep track of since you get unitary transformations by exponentiating skew-adjoint operators, but the convention for observables is to use self-adjoint operators (which have real eigenvalues). The function $L_3$ becomes the self-adjoint operator (using units where $\hbar=1$)
$$\widehat L_3=Q_1P_2-Q_2P_1$$
which infinitesimally generates not only the rotation action on other operators, but also on states. In the Schrödinger representation this means that the action on wave-functions is that induced from an infinitesimal rotation of the space coordinates:
$$-i\widehat L_3\psi(\mathbf q)=\frac{d}{d\theta}_{|\theta=0}\psi(g^{-1}(\theta)\mathbf q)$$

The above is about the classical or quantum theory of a scalar particle, but one might also want to describe objects with a 3d-vector or tensor degree of freedom. For a vector degree of freedom, in quantum mechanics one could take 3-component wave functions $\vec{\psi}$ which would transform under rotations as
$$\vec{\psi}(\mathbf q)\rightarrow g(\theta)\vec{\psi}(g^{-1}(\theta)\mathbf q)$$
Since $g(\theta)=e^{\theta X_3}$ where
$$X_3=\begin{pmatrix}0&-1&0\\ 1&0&0\\0&0&0\end{pmatrix}$$
when one computes the infinitesimal action of rotations on wave-functions one gets $\widehat L_3 + iX_3$ instead of $\widehat L_3$. $S_3=iX_3$ is called the “spin angular momentum” and the sum is the total angular momentum $J_3=L_3 + S_3$. $S_3$ has eigenvalues $-1,0,1$ so one says that that one has “spin $1$”.

There’s no mystery here about what the spin angular momentum $S_3$ 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.

Where there’s a bit more of a mystery is for half-integral values of the spin, in particular spin $\frac{1}{2}$, where the wave-function takes values in $\mathbf C^2$, transforming under rotations as a spinor. Things work exactly the same as above, except now one finds that one has to think of 3d-geometry in a new way, involving not just vectors and tensors, but also spinors. The group of rotations in this new spinor geometry is $Spin(3)=SU(2)$, a non-trivial double cover of the usual $SO(3)$ rotation group.

For details of this, see my book, and for some ideas about the four-dimensional significance of spinor geometry for fundamental physics, see here.

Update: I realized that I blogged about much this same topic a couple years ago, with more detail, see here. One thing I didn’t write down explicitly either there or here, is the definition of spin in terms of the action of rotations on the theory. It’s very simple: angular momentum is the infinitesimal generator of the action of rotations on the wave-function, spin angular momentum is the part coming from the point-wise action on the values of the wave-function (orbital angular momentum is the part coming from rotating the argument). Using a formula from my older posting, for a rotation about the z-axis, the total angular momentum operator $\widehat J_z$ is by definition
$$\frac{d}{d\theta}\ket{\psi(\theta)}=-i\widehat J_z \ket{\psi(\theta)}$$
The spin operator $\widehat S_z$ is what you get for $\widehat J_z$ when you act just on the wave-function values. For a spin n/2 state particle, the wave-function will take values in $\mathbf C^{n+1}$. For the spin 1/2 case the action of rotations is by 2 x 2 unitary matrices of determinant one (the spinor representation). For a rotation by an angle $\theta$ about the z-axis, this is
$$e^{-i\theta\frac{\sigma_3}{2}}$$
so the spin operator is
$$\widehat S_3=\frac{1}{2}\sigma_3$$

This entry was posted in Quantum Mechanics. Bookmark the permalink.

### 45 Responses to The Mystery of Spin

1. André says:

One more big mystery that you forgot to mention is why Scientific American writes about this three years after its publication (and four and a half years after it first appeared on the arXiv)?

2. Shantanu says:

One more mystery is that this is published in physics section of arxiv and not hep-th etc.

3. Peter Woit says:

Shantanu,
That’s not really a mystery at all…

4. Pascal says:

Peter, can you elaborate on your objections to Seben’s paper? The bulk of the paper seems to be a history of the understanding of spin, from classical ideas to modern ideas based on QFT, and on the obstacles that had to be overcome to get there. Seben also uses this opportunity to advertise his own interpretation. He carefully states that this is just an *interpretation* (perhaps in the sense of “interpretation of quantum mechanics”), not a new theory leading to any new prediction.
Do you think his history is wrong? Or correct, but this material is so well known that writing a historical paper about this is pointless? Did he get the physics wrong? Are you against the discussion of interpretations in general? In summary, to use your own punctuation style: what is wrong with this paper?????

5. Doug McDonald says:

Something that came to mind. Mayybe this is a nonsequitor, but still …

Peter: you’re a mathematician. You are giving a mathematician’s argument. Its based on the Poincare group, which includes spinors. It works. It agrees with the world. But still, it just suggests from the existance of spinors that there might be spin 1/2 particles, or more generally, particles of half-integer spin.

But ….

by the same argument there could be … and is … a “Super-Poincare” group that promotes spinors to generators of a sort of “Supersymmetry” which we should have a look at for particle physics. We HAVE had a look or two or ten thousand … and it doesn’t work.

I see that as a generic problem of going from math to physics. Or to be a bit kinder, of going from math to lots of funding for theoretical physics that does not work.

6. Peter Woit says:

Pascal,

I’m not commenting on the history or “interpretation” in the Sebens paper. The “paradox” Sebens and Becker (and others before them) go on about is based on the idea that “angular momentum” is the usual $\mathbf q \times \mathbf p$ formula from high school physics, a measure of how an extended object is rotating about a point. You can’t understand spin this way, no matter how hard you try to give strained interpretations to terms in the Dirac free field.

The point I was trying to explain in the posting is that you need to start not from high school physics, but from the fundamental fact that angular momentum is the generator of the action of rotations on your theory. Becker and Sebens seem either unaware of or uninterested in this basic fact. From this more fundamental definition, you can not just assume the formula $\mathbf q \times \mathbf p$ you learned in high school, but derive it (by looking at how the theory behaves under infinitesimal rotations of space).

From this point of view, whenever you have fields, wavefunctions or whatever that are not scalars under rotations (a 3d-vector field the simplest example), when you compute the formula for angular momentum, you will get not just a $\mathbf q \times \mathbf p$ orbital angular momentum term, but another term. This is the “spin” angular momentum. It has nothing to do with stuff moving in space.

Another point I was trying to make is that this has nothing to do with the complexities of 4-d space-time, spinors, the Dirac equation, etc. This is a purely 3d story, just look at a vector-valued field in 3d (any dynamics you want, the definition of angular momentum doesn’t depend on dynamics).

7. Peter Woit says:

Doug McDonald,
First of all, my Ph.D. is in physics, not math…
Secondly, as I tried to explain in my previous comment, this has nothing to with 4d symmetries (Poincare), nothing to do with spinors, certainly nothing to do with SUSY. The mathematics is very basic: rotations of physical 3d space (applied to whatever kind of physics you are doing, Hamiltonian, Lagrangian, classical, quantum, field theory or not).

I am only bringing mathematics into this with the claim that to understand what angular momentum is, to derive the formula for it, you have to look at what 3d rotations do to your theory. Not everyone needs to be willing to use this level of math, but if they’re not willing to do so, they shouldn’t be making complicated arguments about supposed paradoxes with angular momentum.

8. anon says:

The field (2nd quantization) gives mass and affects electric charge (vacuum polarization). So why can’t it have an effect on spin, too?

9. Peter Woit says:

anon,
The fundamental definition of angular momentum and spin is the same in QFT as in any other part of physics (generator of infinitesimal rotations) and has nothing to do with the dynamics.

I realized that I actually worked this out very explicitly for the simple cases of spin zero and spin 1/2 in non-relativistic QFT in my book. See sections 38.3.2 and 38.3.3 of
https://www.math.columbia.edu/~woit/QMbook/qmbook-latest.pdf

10. Peter Woit says:

Another comment: here and in the book I’ve emphasized phase space (Hamiltonian) methods. Same thing works in the Lagrangian formalism, where Noether’s theorem first tells you how to define the angular momentum (including the spin term) from the infinitesimal action of rotations on the fields, then tells you that if the Lagrangian is rotation invariant, the angular momentum will be conserved.

11. George says:

The single particle theory is not really a good theory, as it does not give energies that are bounded from below for free particles. Of course, you can have a wavefunction in the spinor representation of rotations and you will get spin quantum numbers indeed. But you do not get the energy spectrum correctly without Dirac sea voodoo. The first time that spin is explained within a logically acceptable framework is in QFT, I would argue.

12. akhmeteli says:

Peter Woit,

Thank you for mentioning the article in the Scientific American.

You wrote: “I’m not commenting on the history or “interpretation” in the Sebens paper.” However, Sebens’ work is specifically about interpretation. He wrote: “What follows is a project of interpretation, not modification.”

Sebens’ work may be good or bad, but I would appreciate your arguments. What you wrote so far sounds to me like “Sebens cannot discuss interpretation of spin unless he uses the same mathematical formalism as I do.” My understanding is your formalism has the same predictions as the traditional one(s), so I am not sure one must exclusively use your formalism to triage interpretations.

If I misunderstood your arguments, I apologize.

13. Peter Woit says:

George,
The problem of negative energies in relativistic quantum mechanics has nothing to do with spin (you have the same problem in the spin-less theory).

Non-relativistically (or in the non-relativistic limit of a relativistic theory), there’s a perfectly well-defined quantum theory of a single spin 1/2 degree of freedom. Sometimes called the Pauli-Schrodinger theory, see chapter 34 of the book I linked to. In this theory you can easily compute the angular momentum and see that it has separate orbital and spin angular momentum terms, and see the simple origin of the spin angular momentum term.

14. Peter Woit says:

akmeteli,
My formalism is exactly the standard one in the textbooks: the angular momentum observable of a theory is the infinitesimal generator of the action of rotations. The most common version of this is the Lagrangian version, and if you compute this by the textbook Noether method in a theory where the values of the fields transform under rotations that gives you the “spin” part of the angular momentum.

Sebens is doing some rather obscure calculations in a relativistic theory and seems to be claiming that these justify “interpreting” spin in terms of some kind of motion in space. This doesn’t explain anything. It’s the exact opposite of an explanation, an attempt to mystify and confuse. In a rough analogy it’s like saying I don’t like potential energy, all energy should be kinetic energy, so I need to come up with a complicated way to “interpret” potential energy as kinetic energy.

15. Blake Stacey says:

The paper’s conclusion, as the last line of the abstract says, is that “The electron’s gyromagnetic ratio is twice the expected value because its charge rotates twice as fast as its mass.” Personally, I’d take that as another objection to the idea that you can think of the electron as “spinning.” If you work very hard to dodge all the objections that physicists raised decades ago, the best case is that you still run into the problem that the charge rotates twice as fast as the mass, which is not how a classical charged body can behave. Your picture of the electron as a classically spinning object is unphysical in a new way.

I feel like Scientific American should have mentioned that Sebens and Carroll were coauthors. It’s not a major thing (that was several years ago), but still. I’d expect that to be included if someone quoted me about work an erstwhile colleague did.

16. Alex says:

I think the reason some people see quantum spin as mysterious is that, even when you correctly define angular momentum in both classical and quantum as the infinitesimal generator of rotations, in the quantum case you get a sum of the usual quantized so-called orbital angular momentum (which has a clear classical counterpart) with the so-called spin angular momentum, which lives in an “internal” finite-dimensional Hilbert space, tensored with the usual L^2(R^3), and which, naively at first sight, doesn’t seem to have a classical analogue.

But you can actually give a more general Poisson structure on the classical phase space so that you indeed get a new internal space and generators that correspond to a sort of classical spin. The reason for calling this an analogue of the quantum spin is that it’s related to it by a deformation quantization, so it’s indeed its true classical analogue in the sense of quantization theory. For some details see page 31 here: https://arxiv.org/abs/quant-ph/0506082

17. 4gravitons says:

“but one might also want to describe objects with a 3d-vector or tensor degree of freedom”

I would have guessed that this is the crux of the discussion, though. Why would one want to do this (independent of our having observed objects that do indeed have this property)? I would think that when people think of spin as mysterious, they’re asking a question of this form (and as Doug McDonald points out, the principle can’t be something like “if you can describe such objects they exist”, if you don’t expect SUSY to exist).

(Can one say that of the original classical setup too? Kind of? Angular momentum of extended objects is at least derivable from (to speak loosely) linear momentum of point particles. I think it’s a bit silly for a philosopher to decide arbitrarily that motion of point particles is the most fundamental thing in the universe, but eh, I’m skeptical of metaphysics in general, most people aren’t.)

I’d also guess that, to whatever extent Carroll and/or Sebens think the problem is solved in QFT, it’s not because of the mathematics of QFT, but rather because QFT motivates us to think of point particles as not ontologically basic, and thus makes it more acceptable to think of objects as just carrying whatever degrees of freedom we find convenient. Obviously you can do this in QM as well, but you’re less motivated to, in the same way that you’re less motivated to develop classical field theory before Maxwell develops E&M.

From what you describe of Sebens’ “explanation”, it indeed sounds like it’s addressing this in a stupid way, regardless, especially if it’s invoking charge (does he think sterile neutrinos can’t exist?).

18. Low Math, Meekly Interacting says:

99% of this is completely over my head, but I do wonder: Could there be any pedagogical value in abolishing “spin” from the quantum lexicon? I got enough of a physics education to know where the average person gets led astray, and it’s not entirely our fault. “Classically non-describable two-valuedness” is rather a mouthful, but is there any alternative to “spin”? Could the world be rid of the notion that “spin angular momentum is like spin, only it’s not, and nothing’s spinning”? Seems a good place to start would be to swear off entirely the high-school/first-year physics picture asking us to “imagine a little bar magnet spinning”.

Seems the only proper way to popularize QM is to stop putting paradoxical ideas in our heads to begin with, such that we need to spend so much time and energy later disabusing ourselves of misunderstandings rooted in bad classical analogies. Obviously, even professional physicists and philosophers continue to fall into the very same traps the pioneers of QM did, so it seems the first priority is to drill into everyone’s brains that there are no classical analogs worth using. Perhaps then SciAm, etc. would have nothing of general interest to write about.

19. Low Math, Meekly Interacting says:

I think it’s not too hard for a high-school student to absorb the following (assuming I’m not screwing something up myself):

Lorentz invariance and Noether’s Theorem represent two of the deepest and, to the best of our knowledge, inviolate truths of the natural world. It follows directly from the fact we occupy four space-time dimensions that particles like the electron must have four intrinsic degrees of freedom to allow for conservation of energy and momentum. These are fundamentally quantum degrees of freedom with no classical analogs, but bear some resemblance to classical, extended objects in the fact that these degrees of freedom have quantum units of charge and angular momentum.

I.e, if you can handle thinking about charge as an “intrinsic” property of an electron, there’s no justifiable reason to think of “spin” any differently, except that our brains are evolved to conceive of such properties in classical terms. Nevertheless, they allow for no better explanation, and indeed none are necessary.

Start teaching it that way early and ditch “spin”! Many a wayward and frustrated brain would be grateful.

20. Peter Woit says:

Alex,
It’s true that there’s a good way to think of the classical limit of “spin”, as adding a new factor to phase space of a sphere in $\mathbf R^3$. This idea is actually very general: if the values of your field transform under a group G as an irreducible representation of G, take as new factor in phase space the corresponding co-adjoint orbit in the dual of the Lie algebra of G. If you want, you can then use a phase-space path integral, and think of your classical spinning particle as moving along a trajectory in the usual phase space times the sphere (or co-adjoint orbit in general).

The problem with this is that a curved, finite volume phase space like the sphere behaves rather differently than the conventional phase space. Additionally, it’s only in the limit of infinite spin that you recover a classical limit. Things like spin 1/2 and spin 1 are at the opposite, truly quantum limit.

21. Peter Woit says:

LMMI,
The word “spin” is part of the problem, but just part of the more general problem of trying to explain what a quantum particle is, starting from a classical particle. This just can’t work, it inherently makes quantum theory seem like incoherent, incomprehensible voodoo.

To understand angular momentum you just need to think about 3d space and 3d rotations of this space, invoking four dimensions and special relativity only makes this harder to understand. There is a difference between spin and purely internal degrees of freedom like charge. In both cases in quantum theory you are using the fact that the values of your wave-function (or field) transform under a group acting point-wise on the values. In the internal symmetry case (charge), your group doesn’t act on space. But in the spin case, the group is the rotation group, acting at the same time moving around points in space. You can separately see the action on values and the action on spatial points, and this separation is the separation between spin angular momentum and orbital angular momentum.

22. Andrew Thomas says:

Could the world be rid of the notion that “spin angular momentum is like spin, only it’s not, and nothing’s spinning”? Seems a good place to start would be to swear off entirely the high-school/first-year physics picture asking us to “imagine a little bar magnet spinning”.
Well, I would suggest exactly the opposite. I think we need to get away from this idea that a particle’s spin is “nothing like actual spin and nothing’s spinning”. The particle has angular momentum: if you shine a beam of electrons with each having the same spin value onto a object (like a brick) then that object will start to spin – for real. I don’t think this is generally appreciated. That sounds a lot like real spin to me.

Here’s Frank Wilczek (Nobel prize particle physics) on the subject from his book Fundamentals: “If you’ve ever played with a gyroscope, you’ll have a head start on understanding the spin of elementary particles. The basic idea of spin is that elementary particles are ideal, frictionless gyroscopes which never run down. Spin changed my life. I always liked math and puzzles, and as a child I loved to play with tops”. Argue with Frank at your peril.

Nobody is suggesting that quantum spin is the same as classical spin, but it is much closer in nature than is generally realised, certainly closer than is usually suggested in popular science books.

23. Peter Woit says:

Andrew Thomas,
Yes, spin is angular momentum, and if you want to play with metaphorical descriptions of it as Wilczek does that’s fine. But you should keep reading in the text (page 75-6) that you quote. In the rest of the paragraph he goes on to explain that the way “spin changed my life” for Wilczek is that he took Peter Freund’s advanced course on “the application of mathematical symmetry” (e.g. Lie groups and representations) to physics. In this course
“Professor Freund showed us how some extremely beautiful mathematics, building on the idea of symmetry, leads directly to concrete predictions about observable physical behavior… To me, the most impressive example of this connection was – and still is – the quantum theory of angular momentum, which he showed us… To experience the deep harmony between two different universes – the universe of beautiful ideas and the universe of physical behavior – was for me a kind of spiritual awakening. It became my vocation. I haven’t been disappointed.”

To me what Wilczek is saying is that if you want to talk about physics at the level of vague metaphor, fine to think of spin as something spinning. If you want to understand what spin actually is, you need to understand it as a contribution to angular momentum, where angular momentum is defined mathematically in terms of rotational symmetry.

24. Lars says:

The spin of elementary particles like electrons is one thing.

But the spin of composite particles would seem to be a different beast.

For example, the experimental evidence indicates that the spin of the proton IS at least partly the result of “motion in space”, depending on the “orbital” angular momentum of the component quarks.

See
“What we know and what we don’t know about the proton spin after 30 years” (Y. Zhao, 2020, Brookhaven National Laboratory)

“Results from Jlab 6 GeV experiments and HERMES data suggest a substantial quark orbital contribution”

25. Peter Woit says:

Lars,
When you have a composite bound state like the proton which does not correspond to an elementary field, you can assign it a spin, by looking at the lowest energy state and seeing how it behaves under rotations. For a proton, you find it transforms as the spinor representation, so has spin 1/2. The relation between the spin of the proton and the spin of its constituents is complicated and as far as I know not completely understood. The problem is that the proton is a strongly coupled system of both quarks and gluons, and we lack good calculational methods for such systems. For a composite system of free particles you can understand the total angular momentum in terms of the spins of the components and orbital angular momentum, but a strongly coupled system is something different.

I think what is going on with some people is that they don’t like intrinsic spin of an elementary particle because it doesn’t correspond to something they can visualize in a model of reality which consists of scalar elementary particles moving in space. They want to explain the spin 1/2 of an electron in terms of this model of reality. But when you do this, you run into an intractable problem: if you have spinless particles and only orbital angular momentum, all states will have integral total angular momentum, no possible way to get 1/2 integral values.

If you ignore the half-integrality problem and try and build a model of electrons as bound spinless particles anyway, you run into the problem that electrons exhibit pointlike behave on scales much smaller than their Compton radius. Then when you try to explain spin 1/2 in terms of orbital angular momentum you get into problems with your hoped-for model, since it would imply faster than the speed of light motion of the component particles.

26. Peter Orland says:

Hi Peter,

Feel free to delete this, if it seems off-topic, but spin (the physical phenomenon) can be visualized as mechanical revolution. Wess-Zumino actions for spin are well-known (even for Dirac particles, which I wrote a paper about ages ago. There is a close relation with twistors).

Now these Wess-Zumino actions are equivalent to (Fock-Tamm) charge-monopole systems, even if one does not accept the charge and monopole as existing in physical space. This picture is actually useful for understanding Skyrmion spin.

Of course, this way of understanding spin has nothing to do with the topic of the Scientific American article.

27. Peter Woit says:

Peter,
That’s related to what Alex above brought up. If you enlarge phase space by adding a factor of a co-adjoint orbit (for 3d rotations, the orbit is a sphere in 3d space) and quantize this, you get wave-functions with values in a non-trvial representation of the rotation group. The phase-space path integral for the sphere factor will be given by integrating over paths on the sphere with action a Wess-Zumino term, one you could think of as the phase factor picked up from the vector potential of a monopole at the center of the sphere. But, the monopole is unphysical, as is the sphere you’re moving on.

This whole story leads to really beautiful mathematics, it’s the simplest case of the Borel-Weil-Bott geometric constructions of representations of compact Lie groups. I’m a huge fan of this way of thinking, but I don’t think those who want to get more sophisticated mathematics out of physics really want to go down this path…

28. Davide Castelvecchi says:

Peter,

This is a very interesting discussion! You point out that there is a well-understood machinery for understanding spin, based on group representations. But the very fact that there is Dirac’s correspondence between Poisson brackets and commutators and that all of this works is at least a bit mysterious, no? (Also, I thought the quantization procedure wasn’t even uniquely defined?)

Two: the thing that’s measurable, spin along a certain axis, is often described as telling us that the spin is “up” or “down” — even though it doesn’t tell us that the angular momentum points in a certain direction, only that it has a certain angle with the axis; it ostensibly measures a component of a vector, but that vector is not even well-defined (because you can’t measure all of its components).

Three: the ontology of a particle’s spin. If are to believe that the wavefunction has physical meaning, then each electron in a (unpolarized) beam has a well-defined spin state — in the sense that at any given time and place there exists a direction along which it will have 100% probability of giving a spin “up”. But there doesn’t seem to be a way to do anything with this because we can only ever deal with mixed states and the properties of the ensemble. (Ok I guess this is a mystery about the quantum measurement problem, not specifically spin.)

Four: The Hilbert space where the spin state lives is some abstract three-dimensional sphere outside of the physical world. And if I understand correctly, when you associate an operator L_z to measuring spin in a certain direction, the correspondence between the operators L_x and L_y and the other directions of space is partly arbitrary.

All of this sounds pretty mysterious to me!

29. Peter Woit says:

Davide,
I’m not claiming there are no mysteries here, just that trying to understand what spin is in terms of stuff moving in space is a huge mistake. Also, that trying to do this in the 4d relativistic (Dirac equation) context is another sort of mistake, since the situation there is quite tricky, and the phenomenon you are trying to understand is already there in a much simpler context in the 3d non-relativistic theory. Maybe you can shed some light on the mystery of why this article was published by SciAm…

I just took a look at various sources to see how they explain “spin”. Found everything from the completely dreadful (Wikipedia) to some places that get things right. As you might expect, one of the latter is Weinberg, who in his QM book (chapter 4) starts off with “we should think of the existence of both spin and orbital momentum as consequences of a symmetry principle” and an explanation of how to get from rotations to angular momentum. He also correctly states “there is nothing about spin that requires relativity to be taken into account”.

About your four mysteries, the first three involve the basic mystery of the relation of classical and quantum mechanics, including the mysteries of measurement theory. The spin 1/2 system doesn’t explain any of these mysteries, but it does provide the simplest examples of non-classical behavior so a good place to start in thinking about this basic mystery.

For the fourth, it is an observed fact of nature that fundamental particles are best described by a $\mathbf C^2$ valued wave-function, transforming as a spinor under rotations. This is mysterious in the sense that it is highly non-classical and not in accord with our classical intuitions at all. It’s also telling you that fundamental physics is described not in terms of usual geometry, but in terms of spinor geometry. To me that’s a big hint about what to look for in a more fundamental theory. It should most naturally be formulated in terms of spinors, with a compelling argument for twistors that in twistor theory a point in space time is tautologically a spinor $\mathbf C^2$. Yes, this is an argument from mathematical beauty.

What’s not mysterious here is how you get from “particles are described by spinor-valued wavefunctions” to the description of what spin is. This is nothing other than the simple fact about 2 by 2 matrices that you rotate spinors by exponentiating Pauli matrices.

Was thinking of writing a part II of this posting that explains this more, maybe later today…

30. Peter,

”transforming as a spinor under rotations. This is mysterious in the sense that it is highly non-classical and not in accord with our classical intuitions at all.”

In fact, everything about spin 1/2 was already discussed by Stokes in 1852, long before quantum mechanics was born. He had no problems with intuition, though it was different from the usual one!
See my article ”A Classical View of the Qubit” at
https://www.physicsforums.com/insights/a-classical-view-of-the-qubit/

31. Levy-Leblond says:

Levy-Leblond explained how spin can arise without relativity more than 50 years ago:

32. Peter Woit says:

Levy-Leblond,
This goes back much further than that, to more than 95 years ago, and Pauli’s 1927 paper

What Levy-Leblond is working out is going from the usual story about elementary particles being irreducible representations of the Poincare group given by solutions of relativistic wave-equations to instead the non-relativistic case of the Galilean group and its representations. But this is still more complicated than you need, the Galilean boosts have nothing to do with spin. All you need is spatial translations and rotations (the Euclidean group in 3d). Quantized these give momentum and angular momentum operators. If you look at one component wave functions you just get orbital angular momentum, if you take multiple component wave-functions, with rotations acting non-trivially on the values at a point, the angular momentum acquires the additional spin angular momentum term.

Again, I worked out the details of all of this explicitly in chapter 34 of my book, but there’s nothing there that Pauli (with some help from Weyl) wouldn’t have already understood in the late 1920s.

33. martibal says:

Is there any conceptual explanation of why the multicomponent wave function transforms under the double cover of the rotation group, and not the rotation group itself ? (I know this is a fact of nature, is there any understanding of why it should be so ?)

34. martibal,

not every multicomponent wave function transforms nontrivially under the double cover of the rotation group, , and not the rotation group itself; only (and precisely) those for fermions do. This is explained by the spin-statistics theorem from relativistic quntum field theory.

35. Peter Woit says:

Martibal,
To me the conceptual explanation is just that the geometry of physics is inherently spinorial, not surprising since spinor geometry is more fundamental than the usual geometry in terms of vectors. In arbitrary dimensions you have a non-trivial double-cover of the rotation group and spinor representations. The spinor representations are more fundamental than the usual vectors (and tensors built out of them) since you can build vectors out of spinors but not spinors out of vectors.

In three dimensions there is a very simple way of seeing this using the quaternions. If you identify space with imaginary quaternions, rotations are given by conjugation by unit quaternions q. Unit quaternions form a 3-sphere, isomorphic to group SU(2). The double cover comes from the action of q and -q being the same rotation. The spinor representation and its fundamental nature comes from the fact that SU(2) acts on $\mathbf C^2$, and all representations of SU(2) can be build out of this one.

36. vmarko says:

I can see that the discussion has covered a wide range of points of view and properties of spin, but apparently not all, so maybe I can chip in a bit. 🙂

Maybe it should be emphasized that the notion of spin does not really have much to do with quantum mechanics. Rather, it can be regarded as a field theory phenomenon, completely classically.

For example, in Maxwell’s electrodynamics, the electromagnetic wave has nonzero (integer) spin. Likewise for a massive Proca theory, and for general relativity. And likewise for the half-integer spin of the Dirac, Weyl, Majorana, Rarita-Schwinger fields. All of these can be understood as classical field theories. And as Peter noted, even the nonrelativistic Schrodinger-Pauli equation (regarded as a classical field theory) describes nonzero spin.

In general, in field theory one can talk about the action of a rotation operator on the spacetime point (the “argument”) and on the field components (the “function”). The former gives rise to orbital angular momentum, while the latter gives rise to spin. And there is nothing quantum-mechanical in any of that.

Also, regarding the distinction between non-relativistic and relativistic theories — while both can describe spin, there are a few subtle differences. First, the Dirac equation predicts the electron magnetic moment, whereas the Schrodinger-Pauli equation makes no such prediction (magnetic moment is a free parameter). Second, in relativistic theory the action of a rotation group element has to have the same parameter for both the orbital and spin parts (since relativistically there is just one notion, the “total” angular momentum operator, as a generator of the Lorentz group), while in non-relativistic theory the actions of the orbital and spin parts are independent, with a priori different parameters. This has been known to lead to incorrect predictions in nonrelativistic theory, since one is falsely allowed to “rotate” the orbital and spin angular momenta independently of each other. And finally, in QFT there is the spin-statistics theorem, which is inherently a relativistic (and also quantum) statement, with no non-relativistic counterpart.

Best, 🙂
Marko

37. Jeffery says:

Hi Peter,

I came across an old paper with similar claim published in 1985. Does this make any sense at all?

https://physics.mcmaster.ca/phys3mm3/notes/whatisspin.pdf

38. Peter Woit says:

Jeffery,
Sebens references that one, likely used some of it in whatever he is doing. Very explicitly same motivation: the only “physical explanation” of spin angular momentum is to try and interpret it as some kind of orbital angular momentum. Zero mention of or any interest in what angular momentum really is (generator of action of rotations on the theory). The 1985 author quotes 1928 and 1939 papers with similar claims. Another mystery is why SciAm is selling this as something new.

39. martibal says:

If you think classically (i.e. non quantum) then why the need of going to the double cover of SO(3) ? Why is nature not happy just with rotations and needs to go to the double cover ? And why does it stop to double cover and do not consider other covering group ?

40. Peter Woit says:

Martibal,
The only interesting covering group of SO(n) is the spin double cover, other covers are trivial. Why should nature be happy with vectors and SO(3) and not exploit the more fundamental spinor representation and the double cover?

The classical situation is a little complicated to state, but here are some comments:

You can make a classical theory of a particle with spin in two ways:

1. Extend phase space by a sphere. You’ll then get an extra “spin” contribution to angular momentum, which can take any non-negative value (determined by the radius of the sphere). No integrality or half-integrality of spin until you quantize.

2. Extend phase space by an anti-commuting coordinate. This is “pseudo-classical” mechanics and gives you a spin-1/2 term in the theory.

If you work with classical fields, you can write down classical field theories where the field transforms under any representation (including spinor) of the rotation group that you want. These can only be half-integral.

Spinors are not inherently quantum. If you go from 3 to 4 dimensions, the story of 4d spinors is very much a basic part of the theory of twistors. Twistor theory is not a quantum theory, almost all the work in twistor theory has been based on applying the twistor geometrical setup to classical field theories.

41. tulpoeid says:

I might be naive in my knowledge of QM (honestly), but having to “take into account the fact that in this case rotations also act on the vector values, not just on space” is or at least results in a very deep mystery. And definitely not the case that “When one talks about “spin” in physics, it’s a special case of the general story of angular momentum”. Spin is fundamentally about more than rotation and angular momentum, while at the same time surprisingly connected to them.
If I’m misunderstanding something obvious, I’d be grateful for a useful nudge.

42. Peter Woit says:

tulpoeid,
There is no mystery. All I’m saying is that if you write down a theory of fields (classical, quantum, whatever) and decide to use fields that take 3d vector values, then when rotations act on your theory, you have to take into account that the values of the fields rotate (giving spin angular momentum, “spin 1” in this case) not just the arguments (giving orbital angular momentum).

43. Giorgio Pastore says:

Peter,

I think that your judgment of Seben’s work is too drastic. Your considerations about angular momentum as the generator of the rotations are correct, of course, but I think they do not exhaust the subject. Indeed, they do not directly address the content of Seben’s paper.

I understand that the central issue is not to reduce the angular momentum to some cross-product of position and momentum. I think that Seben’s work is on the same track as previous work by Belinfante and other people, one of the last being Ohanian.
Work that was not, and it is not an alternative to the view of angular momentum as the generator of rotations, and it is not a nonsense either. It is a work that helps to address a different issue: why, in the case of charged particles, such orbital and angular momenta carry a magnetic moment?

Electromagnetism requires the presence of charged currents to get magnetic moments. Seben’s point of view has the advantage of clarifying the relation between such currents and the properties of the wave field. How would you get the presence of magnetic moment from the generators of the rotations?

44. Giorgio Pastore,

”How would you get the presence of magnetic moment from the generators of the rotations?”

It simply comes from the current $j(x)=:\psi(x)^*\sigma\psi(x):$

45. Peter Woit says:

Giorgio Pastore,

From my reading of Sebens (and Ohanian), their main arguments are not about the magnetic moment issue. The way spin couples to EM is very interesting and non-trivial, but I don’t see how anything they do elucidates that question at all.