https://www.math.columbia.edu/~woit/wordpress/?p=13152 Not Even Wrong [cropped-blackhole] Skip to content * Home * Euclidean Twistor Unification * Frequently Asked Questions - Math Job Rumors The Mystery of Spin Posted on November 22, 2022 by woit 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 Schrodinger 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. This entry was posted in Quantum Mechanics. Bookmark the permalink. - Math Job Rumors 26 Responses to The Mystery of Spin 1. Andre says: November 23, 2022 at 4:11 am 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: November 23, 2022 at 11:10 am One more mystery is that this is published in physics section of arxiv and not hep-th etc. 3. Peter Woit says: November 23, 2022 at 12:27 pm Shantanu, That's not really a mystery at all... 4. Pascal says: November 23, 2022 at 2:30 pm 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: November 23, 2022 at 3:00 pm 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: November 23, 2022 at 3:12 pm 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: November 23, 2022 at 3:29 pm 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: November 23, 2022 at 3:29 pm 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: November 23, 2022 at 3:40 pm 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: November 23, 2022 at 3:45 pm 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: November 23, 2022 at 4:57 pm 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: November 23, 2022 at 5:58 pm 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: November 23, 2022 at 10:16 pm 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: November 23, 2022 at 10:31 pm 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: November 24, 2022 at 12:06 am 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: November 25, 2022 at 7:30 am 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: November 25, 2022 at 8:19 am "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: November 25, 2022 at 9:21 am 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: November 25, 2022 at 11:14 am 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: November 25, 2022 at 12:30 pm 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: November 25, 2022 at 12:44 pm 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: November 25, 2022 at 2:56 pm 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: November 25, 2022 at 3:23 pm 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: November 27, 2022 at 4:24 pm 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: November 27, 2022 at 4:55 pm 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: November 27, 2022 at 5:56 pm 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. Leave a Reply Informed comments relevant to the posting are very welcome and strongly encouraged. Comments that just add noise and/or hostility are not. Off-topic comments better be interesting... In addition, remember that this is not a general physics discussion board, or a place for people to promote their favorite ideas about fundamental physics. Your email address will not be published. Required fields are marked * [ ] [ ] [ ] [ ] [ ] [ ] [ ] Comment * [ ] Name * [ ] Email * [ ] Website [ ] [ ] Save my name, email, and website in this browser for the next time I comment. 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