Wednesday, October 05, 2016

Demystifying Spin 1/2

Theoretical physics is the most math-heavy of disciplines. We don’t use all that math because we like to be intimidating, but because it’s the most useful and accurate description of nature we know.

I am often asked to please explain this or that mathematical description in layman terms – and I try to do my best. But truth is, it’s not possible. The mathematical description is the explanation. The best I can do is to summarize the conclusions we have drawn from all that math. And this is pretty much how popular science accounts of theoretical physics work: By summarizing the consequences of lots of math.

This, however, makes science communication in theoretical physics a victim of its own success. If readers get away thinking they can follow a verbal argument, they’re left to wonder why physicists use all that math to begin with. Sometimes I therefore wish articles reporting on recent progress in theoretical physics would on occasion have an asterisk that notes “It takes several years of lectures to understand how B follows from A.”

One of the best examples for the power of math in theoretical physics – if not the best example to illustrate this – are spin 1/2 particles. They are usually introduced as particles that have to be rotated twice to return to the same initial state. I don’t know if anybody who didn’t know the math already has ever been able to make sense of this explanation – certainly not me when I was a teenager.

But this isn’t the only thing you’ll stumble across if you don’t know the math. Your first question may be: Why have spin 1/2 to begin with?

Well, one answer to this is that we need spin 1/2 particles to describe observations. Such particles are fermionic and therefore won’t occupy the same quantum state. (It takes several years of lectures to understand how B follows from A.) This is why for example electrons – which have spin 1/2 – sit in shells around the atomic nucleus rather than clumping together.

But a better answer is “Why not?” (Why not?, it turns out, is also a good answer to most why-questions that Kindergartners come up with.)

Mathematics allows you to classify everything a quantum state can do under rotations. If you do that you not only find particles that return to their initial state after 1, 1/2, 1/3 and so on of a rotation – corresponding to spin 1, 2, 3... etc – you also find particles that return to their initial state after 2, 2/3, 2/5 and so on of a rotation – corresponding to spin 1/2, 3/2, 5/2 etc. The spin, generally, is the inverse of the fraction of rotations necessary to return the particle to itself. The one exception is spin 0 which doesn’t change at all.

So the math tells you that spin 1/2 is a thing, and it’s there in our theories already. It would be stranger if it nature didn’t make use of it.

But how come that the math gives rise to such strange and non-intuitive particle behaviors? It comes from the way that rotations (or symmetry transformations more generally) act on quantum states, which is different from how they act on non-quantum states. A symmetry transformation acting on a quantum state must be described by a unitary transformation – this is a transformation which, most importantly, ensures that probabilities always add up to one. And the full set of all symmetry transformations must be described by a “unitary representation” of the group.

Symmetry groups, however, can be difficult to handle, and so physicists prefer to instead work with the algebra associated to the group. The algebra can be used to build up the group, much like you can build up a grid from right-left steps and forwards-backwards steps, repeated sufficiently often. But here’s where things get interesting: If you use the algebra of the rotation group to describe how particles transform, you don’t get back merely the rotation group. Instead you get what’s called a “double cover” of the rotation group. It means – guess! – you have to turn the state around twice to get back to the initial state.

I’ve been racking my brain trying to find a good metaphor for “double-cover” to use in the-damned-book I’m writing. Last year, I came across the perfect illustration in real life when we took the kids to a Christmas market. Here it is:

I made a sketch of this for my book:

The little trolley has to make two full rotations to get back to the starting point. And that’s pretty much how the double-cover of the rotation group gives rise to particles with spin 1/2. Though you might have to wrap your head around it twice to understand how it works.

I later decided not to use this illustration in favor of one easier to generalize to higher spin. But you’ll have to buy the-damned-book to see how this works :p


  1. Hi Sabine, thanks.

    I would have thought of a car on a Moebius ring.
    Would it comes a bit closer to the double cover?


  2. akidbelle,

    I actually started with sketches of a Moebius ring. The problem with this is that it's damned hard to draw. Whatever you do, there's some part of the ring's surface which you don't see. So I started putting it in front of a mirror and that became really messy and ugly... Ah, try drawing a Moebius ring and you'll see what I mean!

  3. well, there is always the example in Misner, Thorne & Wheeler page 1149 - but it is overly complicated, and all the strings can be replaced with a single tape, which one can attach to a desk.

  4. "Probabilities always add up to on" Just sayin. .

  5. Hi Sabine,

    I agree, that's funny. I would probably cut a band of paper, put a small car on it, and then make a photograph.. But you're right, by definition there is a part that cannot be seen.

    I found a good one on Youtube:

    There's also this one where you can put a car:


  6. If "spin 1/2" means "twice around" what does "spin 2" mean?

  7. Hi Sabine,
    Here is another example that you need to turn 720 degrees to make it come back to its initial state, the Geared 5x5x5 cube by Oskar van Deventer:

  8. Dr Benway, Phillip,

    Thanks, I've fixed that!


    It means half round.

  9. Hi Sabine,

    I will have to buy "the damn book" but not until I have finished the Susskind books I am very slowly working through (very handy to have his Stanford lectures to watch while I do so).

    Have I found a typo? In your paragraph listing all the half integer spins you have the following 4 series: 1, 1/2, 1/3 : 1, 2, 3 : 2, 3/2, 5/2 : 1/2, 3/2, 5/2 . Should the 3rd series read 1/2, 3/2, 5/2 ? The lack of symmetry is worrying me but if I am correct then why have the last 2 series as they are then identical which is also worrying me.

  10. Bob,

    Sorry, that was a typo! I hope it's correct now. I wrote this post in an editor that insisted on automatically replacing some of the fractions with symbols that then disappeared when I copied them. Hence I had to put them all in again!

    I don't think my book will come out until 2018, so no need to hurry. Best,


  11. "particles that have to be rotated twice to return to the same initial state" occupy non-orientable surfaces? Physical helices are chiral. Is a mathematical Möbius band chiral? Physics postulates universal mirror symmetry for elegance and simplicity despite observation (but anti-symmetric Berry curvature and cyclic adiabatic evolution).
    LPOF bar

    Commercial high purity and perfection single crystal quartz, right-handed and left-handed, then a geometric Eötvös experiment. buy the-damned-book to see how this works Do the damned experiment to see how everything works. “8^>)

  12. "It [2] means half round" ha! I suppose there is a good mathematical reason for that convention! Thanks Sabine!

  13. Matthew,

    Well, yes, the spin * rotation must give a multiple of 2*Pi to map the state back onto itself (because exp(I*2*Pi*n)=1 for all integer n). So if the spin is 1 you turn by 2*Pi, ie once, if the spin is 1/2, you turn by 4*Pi and turn twice, if the spin is 2, you turn by Pi, which is half round. And so on. A spin zero particle doesn't change to begin with. Best,


  14. Bee;

    "If readers get away thinking they can follow a verbal argument, they’re left to wonder why physicists use all that math to begin with. "the obvious reason, which i assume the readers f your blog must realize is that using math enables us to make quantitative predictions of behavior. and that's the purpose of theory - to make experimentally testable predictions.


  15. Sabine - there is a simple illustration. Take a cup of tea, and put it on a hand, up. Then try to rotate it, not pouring a tea, out off the cup. You have to take care about vertical axis of the cup, while trying to put it below Your elbow and do on. During that process ( rake look on the cup handle!) Cup is rotated by 720 degrees in order to return to original position. 360 degrees is only a strange Han position where You have to continue rotation of Your hand.
    Try it.

  16. @Uncle Al,

    the Moebius is of no importance, the car round trip on it is chiral and spin 1/2.


  17. Lots of lay-persons are familiar with computer concepts, so you could try using bits as an analogy for spin-1/2 particles. It doesn't really generalize to higher spins, but the idea of a simple two-state thing is pretty common.

  18. Hi Sabine - Good start! How's about the Dirac belt trick to illustrate the 720 degree rotation thing? Much easier to follow than the illustration in MTW. I do the Balinese cup dance version in my classes for lay audiences, mumbling something about how the 720 degree rotation is needed to return to the original orientation accounting for connections with other things.

  19. Correct me if I'm wrong but the way I think of a spinor is as a particular class of eigenvector of the gamma matrices. In particular a spinor A is an eigenvector of the double equation:

    G_1A = (cos(a)G_1 + sin(a)G_2)A = A

    With gamma matrices G_1 and G_2 and the quantity in brackets is a unit vector. It then follows:

    [(1+ cos(a))G_1 + sin(a)G_2]A = 0

    So that (cos(a/2)G_1 + sin(a/2)G_2)A=0

    Which means A must transform by half an angle to match the half angle transformation of the object it multiplies.

  20. Kakaz,

    I know this trick. I did it once in a lecture. I don't think anybody understood what it was meant to explain. And in any case, it works very badly as an illustration.

  21. Seemingly, all these mjetaphors do not sufficiently convey the most peculiar characteristic, that kind of inherent "orientational relativity" which is ultimately remote, not tied up to the Spin's carrier itself but to some external "boundary" - as if they could be exchanged somehow!

  22. Thanks Sabine, that is the best explanation of spin 1/2 I've come across in the 35 years since I first came across it in a popular science book as a teenager. As an enzymologist I have an appreciation of the power of maths but never having gone beyond the absolute basics of calculus and matrix algebra I cannot follow the technical arguments. Perhaps when I retire I'll have the time to learn maths more fully...maybe the mental exercise and stimulation will stave off senility!

  23. scott,

    I have no idea what that angle is that you are using and what it has to do with rotations. You can indeed obtain the transformation behavior of spinors from that of gamma matrices, but that merely moves the task to figuring out how the gamma matrices transform under rotations. Best,


    1. Hi Sabine it's the angle representing the directional cosines in that particular gamma representation. The formula I presented is missing a minus sign but this should be obvious.

      The point is that a spinor can be constructed for a particular vector as being an eigenvector of that vector and at the same time an eigenvector of one of the chosen gamma matrices.

  24. "The mathematical description is the explanation."

    I've finally figured out why so many people believe this. Nobody thinks that Humean philosophy *just is* English prose, and that's because there are other natural languages besides English where people have made strong contributions to analytical philosophy. But the endeavor of creating a non-verbal system with moving parts which mirror the moving parts of reality has thus far resulted in only one major "language", despite the large variety of "dialects" which do exist. Look at a Japanese physics textbook, and even though the prose is in Japanese, the physics is in "math"--the same math that we use in the West. Seeing only one possible way to encapsulate the information of empirical experience leads people to think that the way they're looking at is itself the information.

    The mathematical description is often the most precise explanation we have, but using that as an excuse to refuse to show one's hand and simply lay bare what the mathematics is referring to is a deep failing. Physics talks about the movement of 3D objects in space. Surely you can simply *show* what it's talking about. The equations are better encapsulations, since they remove extraneous information and contain a huge number of causal connections in their small formulation. But ultimately what they're describing is real movement of real things. The equation is a language act, not a thing to behold as the ultimate piece of data. But of course understanding this and taking it seriously would mean giving up the ruse, and showing the world that much of modern physics is designed first and foremost not to benefit engineering, but to dazzle the monkey brain with shapes and words which will produce in our minds the much sought-after feelings of fascination, like eating a classy cheesecake to dazzle the taste buds.

  25. Ian,

    It is trivially possible to express math in any language you want, basically by reading it out. The problem with this is that the words in spoken languages are already occupied with meaning. Hence, if you allow me to redefine words - English, Japanese, or otherwise - sure I can explain every equation using these words. If not, I can't - if the definitions are missing, it'll just be ambiguous.

    There are some examples where certain systems can indeed be perfectly mapped onto each other, so you can use one system to explain the other. But this is fairly rare. Almost all cases where you can use some 'real world' example for what's actually a math equation will fail. I'm sure you've noted, for example, that the above image relies on there being a vertical direction which removes the ambiguity. Well, you could ask, which direction is "vertical" when it comes to a spin 1/2 particle? That's where this analogy ends.

    If you think it's possible, why don't you explain us plain English what a spinor is?



  26. I like this simplified picture which is derived from the Dirac belt trick.

    It appears in a post explaining quaternions as an alternative view of the double cover of the space of rotations.

  27. Great article all around I really enjoyed it. Regarding the rotation; is it a classical (physical) rotation, or is it a worded description that also doesn't truly translate the mathematics?

  28. The problem is that natural language works fairly well describing the macroscopic world. When we use nouns, they refer to a realm of familiar things, concrete or otherwise. When we use verbs, they refer to a realm of imaginable actions. That makes describing classical physics in natural language fairly straightforward. There may be a formal definition for acceleration, but we have all seen and felt acceleration. Even if we are talking about objects too small to see or too large to comprehend easily, we can talk about their acceleration by simple analogy.

    The problem with quantum theory is that the basic entities are nothing like the things we encounter in daily life. They aren't like anything we are ever likely to encounter at our human scale, and our brains are designed to work at our human scale. We can talk about an electron being accelerated, as in a CRT or cyclotron, but it does not accelerate like a ball or a car. To start with, an electron doesn't really have a location until we contrive to observe it. What does it even mean for something that doesn't have a location to have a velocity or to accelerate? This can be described mathematically, but any natural language description would be lacking.

  29. Good try, Sabine. It's still amazes me that the particles that make up the observable universe are described mathematically by the spinor formalism, which something like 99.999% of humans (and probably 100% of us Americans) will never understand. Your trolley example is essentially the same as Dirac's twisted belt or the rotated palmed cup, but it's one my two-year-old grandson (who's currently infatuated with Mr. Rogers' trolley) certainly appreciates.

  30. How about isotropic vector language?

  31. Kaleberg, In answering my question by embellishing what Sabine said in her article I'm inferring the answer to my question is; the rotation is mathematical and not a physically observable one.

  32. Kaleberg, Louis,

    Not sure what you mean. That a particle has a 'spin' doesn't mean a rotation takes place. You calculate the spin by asking 'How would the particle behave if it was rotated?' Best,


  33. Sabine, That's what I was asking, if particle spin is a calculation and not a physical one; i.e it doesn't spin like a top or the earth on it's axis. Maybe it's a bad question?

  34. Wasn't it Feynman who said that if you think you understand quantum mechanics, then you don't understand quantum mechanics?

    In any case, modern physics is unprecedented in the modern sciences, including the science of physics in general, in how it cannot explain, but merely describes. It's better to be honest and own up to that lamentable shortfall, than to obfuscate it and conflate description with explanation.

    Explanation happens when we ask why the sun rises and sets, why the seasons change, why the stars and planets move as they do, and answer with the theory of gravity and the solar system. Note that this theory, while very hard won and difficult to arrive at (it took longer than QM has existed!), is understandable in general terms to the educated layman. It's likewise with so many of our successful scientific theories, such as the theories of evolution, genetics, electricity, the periodic table, etc.

    But with mere description, you get the Ptolemaic model, which (like QM) would baffle the layman. It's important to recognize this history and its lesson, because it might pave a way to a theory of the quantum that actually does explain.

  35. Dear Miss Hossenfelder,

    Thank you for you blog and your posts!

    I have a question. But since I am a layman, chances are reasonably high that my question is not very well formulated and/or ill-defined. But I give my best.

    I know that if one measures the spin of a spin 1/2 along a particular axis, the spin can be either up or down. Before I measure it, the spin-state of a particle can be in a superposition of states. If I now measure the particle´s spin along a particular axis, e.g. the z-axis, the wavefunction collapses, and the spin is either up or down. From now on, as long as I continue to measure the particle´s spin along the z-axis, I will always obtain the same result of my initial spin measurement.

    In your post, you write: "Spin 1/2 particles are usually introduced as particles that have to be rotated twice to return to the same initial state"

    And now, my question. Let´s say I have measured the spin of a particle e.g. along the z-axis, and end up with the result "spin up", does your statement mean that I have to rotate my measurement apparatus in the lab twice to obtain the same measurement result, that is "spin-up"?

    Or does your statement refer to the wavefunction? I believe I read that one needs two complex numbers to describe the particle`s wavefunction in terms of spin, thus two rotations (SU2) are needed to return to the identical state.

    So, my question could also be: In a laboratory measurement, how can one measure the consequences of your statement that "spin 1/2 particles have to be rotated twice to return to the same initial state"? Because i can do two rotations in the abstract space that the wavefunction lives in, but how is this manifested in real measurements?

    I am sorry for my long question. If it is totally ill-defined, then just do not answer it! And I wish you good luck and a lot of energy for you damned book!

    Best regards,

  36. Well, orbital ("classical") angular momentum by itself is not conserved. The conserved angular momentum is the combination of orbital & spin AMs. In the hydrogen atom, when the 2 forms of AM couple (due to a weak external magnetic field), they combine like a vector & the effect is called "Anomalous Zeeman". When the external field is strong, they decouple & the resulting effect is called Paschen-Back. I take it that conservation of the combination of AM together with spin is, as I understand it, an experimentally verified fact.

  37. Shayne,

    As I said, mathematics *is* the explanation. You have it all backwards.

  38. Louis,

    No, a spin 1/2 particle doesn't spin like a top or the Earth on its axis because these wouldn't have spin 1/2. I think I still don't know what you mean. Can you actually rotate a spin 1/2 particle? Yes, you can. You can actively perform the operation I was referring to in a merely mathematical sense. Best,


  39. Armin,

    The statement refers to the wavefunction. Referring to the image with the trolley on the tracks, you don't measure the difference between the first and second level (it's a -1 that squares to +1). Hence, you don't have to turn your apparatus twice. Best,


  40. Shayne, Sabine,

    OK, right now mathematics "is" the explanation.

    Let us parallel: in chemistry, before atom models, unbreakable atoms "were" the explanation - now it is explained by quantum chemistry using atom models.

    Right now, a classification of "elementary" particles exist; it suggests a lower level of reality (not only ad-hoc quantum numbers like strangeness). So "those" mathematics may not be the ultimate level of explanation.

    What I mean is that if you want an explanation at level N, you need at least a description at level N-1. This is true in all domains of science. The point is that this blog discusses the bottom line of the time, the "current" level 0, by definition unexplained. And if it was explained, I bet this blog would discuss the level -1.


  41. akidbelle,

    > What I mean is that if you want an explanation at level N, you need at least a description at level N-1.

    Indeed, and that was my point -- an explanation *needs* a description, it *isn't itself* a description. Stipulation of unbreakable atoms is not by itself an explanation of anything, you also have to have a line of reasoning that takes you from those hypothetically really existing unbreakable atoms to the observed physical effects they cause, e.g. "when they move around a lot then things are hot." When you use mathematics, it's just to have a high fidelity link between the causes and effects you're talking about; it's not the explanation itself, it's a tool of precision.

    Mathematics can be an explanation -- of mathematics. We can show how one side of an equation is equal to another. But there is no mathematics in the really existing world causing things and therefore mathematics is not, by itself, an explanation of anything in the real world.

    So I more or less agree with your post, except the first sentence -- overall the post is non-sequitur as in you haven't shown that your metaphor to chemistry actually works. Or we might say that you have *described* what you think the situation is, but you haven't *explained* (and likewise regarding Sabine's earlier response to me).

  42. Shayne,

    It is correct of course, as I said previously (many more times than is good for my health), that to have physical theory you need both the mathematics and a map to reality. I don't think I ever said that math ALONE can be an explanation. In fact I'm basically writing a whole book about how that is a fallacious attitude. Neither, for that matter, did I ever say that math will remain the best explanation (indeed I have extensively argued that this is a naive position to hold). In any case, I get the strong impression you're not interested in my opinion to begin with, you just want to urgently voice your opinion that you think something must be wrong with what I explained (if you allow the word). Best,


  43. Sabine,

    I agree with you that it's not fair to ask a physicist to explain something in layman terms and then expect the explanation to be fully correct, that there are certain critical steps that require intensive mathematics. Furthermore it's a good criticism of journalism that it should qualify as in “It takes several years of lectures to understand how B follows from A.”

    Also, if someone has misunderstood you it's more productive to find out why, before imputing bad motives. Not only does this avoid unfair rudeness, it might be that what you said could have been more clear.

  44. Sabine,

    "It is trivially possible to express math in any language you want, basically by reading it out."

    That would be akin to fingerspelling in American Sign Language (ASL). Most people are unaware that sign languages are their own systems, and that ASL is no more related to English than musical notation is related to calculus. If we fingerspell English words in ASL, we're not really speaking ASL. We're using ASL to represent English.

    With that said, I'm not saying we should push one language to the side (mathematics) and use a different language instead (English). What I'm suggesting is that we always make sure that we're talking with a meaning. I'd like to periodically push aside *both* languages (mathematics and English), and show concrete pictures and videos to really make clear what aspect of reality we're talking about. I'm concerned that modern physics has moved away from engineering and switched to something much more similar to chess (i.e., a fascinating game for smart people).

    "There are some examples where certain systems can indeed be perfectly mapped onto each other, so you can use one system to explain the other. But this is fairly rare."

    Mathematics is a system that humans have built over the course of centuries of trial and error to be just that: a system where the moving parts, and the patterns by which those parts move, can be mapped onto certain sections of reality. An equation describing the trajectory of a projectile given certain factors is supposed to be a set of rules for moving around symbols on a page, where naming each of those symbols ("weight", "speed", and so forth) will reveal a man-made system of lines on a page where the rules governing 'legal' and 'illegal' operations is isomorphic to the 'rules' of certain aspects of physical reality.

    "Almost all cases where you can use some 'real world' example for what's actually a math equation will fail."

    This will only fail when the physicist hasn't made clear in their mind what it is that they're talking about.

    This is okay in certain cases, since there's a division of labor in physics and mathematics where, for example, some people focus more on the elegance of the mathematical machinery and others more on the engineering applications. But we need to make sure we're on the same page when it comes to what exactly is happening here.

    "If you think it's possible, why don't you explain us plain English what a spinor is?"

    I'm not well-versed on that topic, so I'll let others deal with that. But if you want to read a book by a physicist whose views match mine on the epistemology of physics and mathematics, check out The Science of Mechanics by Ernst Mach.

  45. Ian,

    I agree on most of what you say, except that you still insist it is of advantage for a physicist to use analogies rather than mathematics. Not only is this wrong, it's a problematic belief. Much time is being wasted because some people urgently want to find a 'better explanation' for quantum mechanics, while quantum mechanics is a perfectly fine explanation. This isn't to say that it can't be improved or shouldn't be improved. I'm just saying that the argument for why that should done is wrong. Best,


  46. Hi Prof. Hossenfelder - You are wrong or unclear or I misunderstand you. Humans do not think in mathematics, not physicists, not anyone (excluding savants etc. It's commonly said but what people and probably yourself included are referring to is the common instinctual process in which regular usage of a thing - anything - will see a degree of support emerge in instinct for use of that thing. It doesn't mean you're thinking in mathematics. That is impossible because mathematics involves complex multistage operations and transformations. What is possible is that you are familiar with certain ideas and equations, and also the common parlance with and between those you have to interact with. But it's just the common machinery of instinct.

  47. piein,

    You didn't only misunderstand me, you seem to have entirely invented something I never said.

  48. Sabine,

    I actually don't believe it's necessarily to the benefit of a physicist to use analogies rather than mathematics.

    An analogy is, by definition, not the thing being talked about. It's another thing in reality which bears some isomorphism to the original thing. For example, we could liken the power balance in a relationship to a seesaw: when one party's power increases the other's decreases. Mathematics also works by isomorphism, but rather than finding a physical system in the wild which just happens to allow isomorphism, we create a system of rules for writing lines on a page which is specifically designed such that its moving parts map onto the moving parts of the original thing we're discussing.

    Therefore, I think mathematics works perfectly fine, and I now think I see what you mean when you say that using an analogy will never capture everything the mathematics captures. An analogy can help, but it's a crutch. Learning the mathematics is where you develop a full understanding. An analogy will only rarely be more than a partial analogy, whereas mathematics is the full 'analogy', if you will.

    With that out of the way, I want to make more clear what I mean.

    I'm not suggesting we find analogies to replace the mathematics. I'm asking to step away from the mathematics, the analogies, even the English words... and point directly at what we're discussing. Sure, we can't see the objects of quantum mechanics by the naked eye. But there must be something in reality that we're talking about, even if we have to resort to pointing to indirect effects in the macroscopic world.

    Again, I'm concerned that much of modern physics, including quantum mechanics, has become nothing more than a signaling game for analytical people. Creating a norm where we constantly re-orient ourselves by pointing at the actual thing we're discussing will prevent us from going down long and winding paths which are divorced from reality.

  49. "I agree with you that it's not fair to ask a physicist to explain something in layman terms and then expect the explanation to be fully correct, that there are certain critical steps that require intensive mathematics."

    As Feynman said (surely in his Far Rockaway accent) to a journalist after it was announced that he would receive the Nobel Prize: "Listen, buddy, if I could tell you in a minute what I did, it wouldn't be worth the Nobel Prize."

  50. I often say that when you can measure what you are speaking about, and express
    it in numbers, you know something about it; but when you cannot measure it,
    when you cannot express it in numbers, your knowledge is of a meagre and
    unsatisfactory kind.

    ---William Thomson, Lord Kelvin

    (Fun fact: Neither Lord Kelvin nor J.J. has a "p" in his name.)

  51. Ian,

    do you mean that QM needs some sort of substrate to live in?

    And that this thing has some undiscovered properties from which QM emerges?


  52. My thoughts, maybe useless:

    There is this youtube video of Feynman explaining QED to some people at some sort of meditation retreat. I think the explanation is basically the same one as he gives in his short book on QED. Anyway, as preliminary material he provides a good illustration of the limitations of using what amounts to non-mathematical language to describe models that are built to calculate something precisely.

    The example refers to how in the Mayan culture, the more sophisticated priest/scientists had worked out ways of predicting the motion of Venus, but this involved arithmetic which could not be explained to the "normal" people. So what they said instead was "here is a bowl of beans. Every day take one bean out, and when the bowl is empty Venus will be visible." So they had computed some appropriate constant period for the motion, and just found and easy, but shallow, way to convey this information to the people who would not know the "whole" answer.

    He then goes on to explain various QED-related phenomena with a clever mechanical model involving rotating arrows, always being careful to note that the audience now understands *some* of the answer but not *all* of the answer. His arrow explanation of QED gives a bit of the flavor of various things that are going on (path integrals, wave functions, the role of phase in quantum behaviors, etc) but he never gives the audience the tools that they need to actually use the theory to predict anything, because to do that you have to use the math.

    In this sense I don't completely understand the various calls here to "step away from the math" or "figure out what we are really talking about". The way I see it what we are really talking about it pretty clear: we are trying to explain the results of experiments, and the results of experiments in quantum mechanics are pretty odd. The mathematics is the deepest and most precise model that we have for showing why the experiments come out this way. You can try to frame less technical and more "intuitive" explanations for these things, but they will inevitably lack the precision needed to make actual predictions unless they fit into the mathematical structures that people can use to make calculations. That's just the reality of it. It may seem unsatisfyingly abstract, but it's the best answer we have.

    The other bad analogy that I was going to bring up goes like this: this is like trying to figure out what's really going on in a computer program that you use. You can use its user interface to model some behaviors. You can even instrument it in various ways to watch what it does w.r.t the rest of your system. But the only way to know precisely is to find someone with the source code and check (and even that might not be enough, if the program uses libraries for which you don't have the source).

    You might decry this and say "well, that makes computer programs too hard to understand" ... and you'd be right. But that's just how it is. Not exactly math, but similar.

    The relationship between "plain language" explanations of science and the actual theories is similar in some ways, and different in others. Analogies and natural language only go so far.

  53. Sabine said, "No, a spin 1/2 particle doesn't spin like a top or the Earth on its axis because these wouldn't have spin 1/2. I think I still don't know what you mean. Can you actually rotate a spin 1/2 particle? Yes, you can. You can actively perform the operation I was referring to in a merely mathematical sense. Best,"

    Because of the picture where a toy car had to rotate twice to get back to it's original position; it made me question if the spin of a particle had some sort of physically observable rotation (observable states other than start and end). I now think that is not the case; the rotation is mathematical, and in between start and end the process (rotation) is not observable. I understand much of QM is mathematical probability of an observable state. I wondered if spin could be observed in between states due to the example of the toy car around the track analogy.

  54. @ Louis Tagliaferro,

    in my opinion, the question is what would you use to observe it?
    Another particle? Then you will observe a statistical result consistent with QM.

    Anyway, if "something" rotates there it does not look like 3D. It rather look like 4D (check the wikipedia page on 4D rotations and the taurus and tesseract figure:


  55. "You didn't only misunderstand me, you seem to have entirely invented something I never said."

    sorry about that

  56. A model I consider simple is two interlocked gears, one twice the diameter of the other. The smaller one needs to rotate twice to get the whole system back to zero.

    Of course a similar model would work for spin 1/3 particles, which don't exist and don't fit the math, so the only model that really works is mathematical.

  57. @ Louis Tagliaferro
    Electrons have spin 1/2, exquisitely characterized by resonance spin-flip spectrometries. Protons and and neutrons are also spin 1/2, observed in kind. Summed proton and neutrons, atomic nuclei, can have very large net spins (Ta-180m decay is a spin-8 conversion), integral or half-integral. The sign of the spin indicates a prolate (American football), or oblate (M&M), or triaxial (chiral; rattlebacks) shape. Of course, 2S+1 spin transitions.

    Given what we know about molecular rotational spectrometry, a physical centrifugal model is not explanative.

  58. Thank you all for the replies. What I was trying to pin down with my question is; do Physicists know with absolute certainty a physical rotation of spin takes place, or can it only be said with certainty it is a calculable quantity that results in an accurate predictable observable outcome?

  59. Bee!
    “Virtual Think Tanks: Physicists Who Blog” APSNEWS 25(9) 3 (2016) [October 2016] Sabine “Bee” Hossenfelder” Backreaction

    “occasionally offers unadulterated opinions” When everybody thinks the same thoughts, nobody thinks at all. Mediocrity is a vice of the doomed.
    Classic Superman pose re article.


  61. "What I was trying to pin down with my question is; do Physicists know with absolute certainty a physical rotation of spin takes place, or can it only be said with certainty it is a calculable quantity that results in an accurate predictable observable outcome?"

    The former. This has been known for more than 100 years.

  62. @Phillip Helbig, Thank you, that's what I wanted to know. From the linked article, "the Einstein–de Haas effect demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics."

  63. Perhaps you could compare "double cover" to a key ring. You pry apart one end of the key ring, rotate it through the hole in the end of the key, and voila, the key is on the ring. To get the key off, back to its original state of detachment, you have to do it again.

  64. Just like snagging a flipped coin out of the air for probability wave collapse to H or T.


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