Saturday, December 08, 2012

The electron is spinning, after all

First, a brand new movie, Decay (2012), was released by the LHC students 12 seconds ago. If you have 75 minutes for a truly independent (I warned you!) underground bound state of comedy, unexplained quenches, VW New Beetles, unfaithful girlfriends, extra shifts ordered by CERN's Russian Director General (who may be less innocent than you think), LHC waking up spontaneously, radiation alerts, zombies running through the LHC tunnels (horror starts at 24:50 or so and then 26:50) and outside (1:02:00, 1:09:00), and Higgs bosons and Higgs bioentanglement, watch the video above. See also their website, Wikipedia, their Twitter, and HD downloads. To say the least, it's a great opportunity to look into the interiors of the LHC and how the "real people" use it. And let me admit, I was totally terrified when I was watching the film: you don't want to be an experimenter! ;-)

Now, to mention one of them in a less terrifying context, Aidan Randle-Conde of the U.S. LHC blogs is recording some kind of a particle physics advent calendar. For the December 5th, he talks about pentaquarks, and so on. December 7th was all about the spin:
Advent Calendar 2012 December 7th
He thinks the electron is not spinning in any way. Moreover, he thinks that the very term "spin" is just some blunder caused by historical misunderstandings and it perhaps depends on the outdated old Bohr model of the atom.

I disagree with these (widespread) musings and I view them as contributors to the general misunderstanding of quantum mechanics.

This is the comment I originally wrote over there:
It’s an interesting monologue but sorry, Aidan, I completely disagree with your statements that nothing is spinning and/or the word “spin” is caused by some historical inaccuracies or accident.

Spin is a perfectly valid description of the physical concept that doesn’t depend on the old Bohr atom in any way – although people started to observe things related to the spin before the modern quantum mechanics was really established in its current form.

But the electron *is* spinning. It’s rotating. The previous two sentences are mathematically equivalent to the fact that the angular momentum connected with the spin is nonzero. The angular momentum is what measures whether something is spinning. If it’s nonzero, it’s spinning.

A valid comment would be that the spin can’t be described or visualized by classical physics. But no other observable at the microscopic scale – and therefore controlled by quantum mechanics – can be properly described by classical physics, either. It doesn’t prevent us from using observables that become the usual classical observables in the classical limit. The angular momentum is one of them and despite the fact that the spin is tiny and quantized, it is a term in the angular momentum.

The quantization applies not only to the spin but to any other form of the angular momentum, too. It’s a true fact about Nature. For large objects, the spacing between allowed angular momenta (\(\hbar\) or \(\hbar/2\)) is so small relatively to the total angular momentum that the total angular momentum becomes approximately or effectively continuous. But strictly speaking, it’s still quantized, even for the Earth. The spin of a spin-1/2 particle is just the smallest positive contribution to the angular momentum one may have.

When one thinks about Nature correctly in terms of quantum mechanics, it’s clear that the character of various concepts we know from classical physics will be altered, the predictions and behavior will be different. But they’re still the same quantity and the [quantum] spin *is* the [classical] spin subjected to the quantum formalism to treat observables.

It’s true exactly in the same sense as energy is energy and momentum is momentum. In quantum mechanics, momentum may be quantized in the compact space. The energy is quantized for bound states. It also violates the classical intuition of energy as some “continuous substance” that may be “gradually added”, and other things. One may also construct superpositions of different energy eigenstates, just like for the spin. But it’s still energy. It’s the quantity that is conserved because of the time translational symmetry, much like the angular momentum is the quantity that is conserved due to the rotational symmetry, and when these quantities become large and other conditions guarantee that the classical limit is appropriate, these quantities take on their almost exact and usual classical meaning!

So I think it’s misleading to pretend that the spin is something “entirely different” than the spin of the Earth etc. It’s the same thing, it’s just smaller, and what needs to be changed is the thinking about all of physics – all observables. The transition from classical physics (and/or classical rotating objects and other things) to quantum physics (and the spin of a particle) isn’t about adding a new observable analogous to other classical observables but different. Instead, it’s about keeping the same observables and changing the overall logic and “heart” of physics, how we think about statements, measurements, and predictions. It’s about the replacing of the classical framework, realism, and determinism by the probabilistic, positivist postulates of quantum mechanics based on the linear operators on the complex Hilbert space. But the observables are “the same”, just properly treated.
His reply suggests he didn't try to think about my points too much:
Hi Lubos, thanks for the comment! I don’t think you’ll be surprised to find that I disagree with your statements about whether the electron is spinning or not. It can’t be spinning in the classical sense since it is a point-like object...
Well, the electron is not spinning in the classical sense. I would agree with that. But nothing in the microscopic world is behaving in the classical sense – more precisely in the classical way. Particles aren't moving in the classical way, they don't decay in the classical way, they don't oscillate in the classical way, they don't transfer energy in the classical way, they don't carry information in the classical sense. Pick any activity or any verb you want and I may say that the microscopic objects aren't doing it in the classical way or classical sense.

They're doing everything in the quantum way because the world is quantum, stupid.

But that's something else than saying that particles aren't moving, decaying, spinning, or doing other things. They definitely are. When we say just "moving" or "spinning", we should never assume that it means "moving in the classical sense" or "spinning in the classical sense" – which is the huge mistake that Aidan is explicitly and proudly doing – because this huge mistake means that we fool ourselves into thinking that the "classical sense" is the "normal sense" or "default sense" or the "sense we should always assume". But it's not.

Now, Aidan says that the electron is not spinning because it's pointlike. Even if we adopt the usual definitions that were developed in the classical framework, this statement by Aidan is wrong at both levels. First, it's not true that the electron is strictly point-like. Before the Planck scale, all particles have to have an internal structure because quantum gravity doesn't allow distances shorter than the Planck length to be resolved. The internal architecture of particles therefore can't be "strictly and sharply point-like". In all perturbative string vacua, the electrons are vibrating strings (occupying space comparable to the string length, a few orders of magnitude longer than the Planck length) and indeed, the spin may be understood as coming from some internal degrees of freedom that are localized at an extended object.

In fact, the stringy picture of the spin tells us exactly where the reasoning that pointlike or tiny objects can't carry a nonzero or rather large angular momentum breaks down. The string is locally a very heavy object – the tension or the linear mass density of the fundamental string in string theory is huge – and the motion of this string just a string length away from the center of mass has a big, Planck's-constant-sized impact on the angular momentum. On the other hand, the overall rest mass of the resulting particle may be small or even zero because of various cancellations, including the fact that the sum of integers is equal to \(-1/12\) which is the primary identity guaranteeing that some stringy vibration modes remain exactly massless (and others' masses only arise from some much smaller corrections).

My broader point is that Aidan's attempt to treat the proton's spin as something qualitatively different from the electron's spin is totally misguided as well. It's absolutely the same property or process, when it comes to the right mathematical description as well as its measurable consequences. The spin carried by the proton is totally the same thing and has the same magnitude as the spin carried by the electron!

The only difference is that we already realize that the proton is a composite, extended particle which is why the angular momentum may be visualized as the motion of the components around the axis, at a nonzero distance from it. The electron is not composite in the Standard Model and most likely, it will always be elementary in any quantum field theory correctly and usefully approximating the phenomena around us. But at the end, it's also an extended object – a string or whatever quantum gravity tells you (quantum gravity doesn't allow you to make things that are strictly point-like) – so it's qualitatively analogous to the proton. And the naive argument that an angular momentum that would be this large would require a very light particle such as the electron to rotate superluminally simply has loopholes because the lightness of the particle may arise – and in string theory (and perhaps many other descriptions) explicitly does arise – from cancellations of large contributions to the mass.

The second level of Aidan's proposition is conceptually misguided, too. He says that a particle isn't allowed to be pointlike for it to be spinning. But that's also false. Check your definition of the rotation or the spin in your closest encyclopedia. The "spin" is the rotation around an axis that goes through the center of mass while the rotation around an external axis is "orbital motion" composed of "revolutions".

It's easy to check that even if the electron were point-like, which it's strictly speaking not in the most complete theory, its type of rotation would agree with the definition of "spin" in the encyclopedia. All the points on the electron remain at a fixed distance from the axis that goes through the center of mass of the electron, so the kind of rotation we see is clearly the spin. If the electron were strictly point-like, we couldn't determine the angular frequency from pure geometry. In particular, it's not true that the angular frequency is apparently zero. The angular frequency is the ratio of the "travelled distance over radius" so for a point-like particle, it is the 0/0 indeterminate form and may be anything.

Fortunately, we don't need to be finding out the angular frequency because the "amount of spin" is measured as the "amount of angular momentum" and we know that relatively to any axis, the electron's spin is \(\pm\hbar/2\). It's nonzero. In fact, the electron's spinning can't be stopped at all.

Again, what I find so deeply misguided about Aidan's (widespread) approach is that he wants to make himself think that instead of the Earth's spin, he is supposed to learn just another type of motion or properties of objects but all these processes will still occur "in the classical sense" because it's his default sense he will attribute to any verb or any idea. He believes that everything that looks unusual about the spin must be due the spin's not being the internal angular momentum. But that's totally wrong because classical physics is wrong both for the spin as well as for everything else in physics, especially physics of the microscopic world. One must adopt quantum reasoning and when he does, he will realize that what we call the spin is nothing else than than the usual concept of the "spin" or "internal rotation" as quantified by the angular momentum that we have always known. The reason why many facts about the spin are surprising is not that it's something else than the internal angular momentum; the actual reason is that while it is the internal angular momentum, the whole reasoning about physics must be replaced by the postulates of quantum mechanics that have an impact on everything, especially in the microscopic world.
The spin manifests as a form of angular momentum because it is the angular momentum required to balance the books, but that’s not same as saying that it’s spinning.
It is the same thing. And electron is spinning is surely a possible – although not the most widespread – way to discuss the existence of electron's spin in Physical Review and elsewhere. When we want to determine whether a particle is doing something or not, we have to find some observable whose value informs us about the answer. The right universal observable by which we measure whether things are rotating around their axis or not – whether they're spinning – is called the angular momentum. Because the value of the angular momentum of the electron is nonzero, it's spinning. Period, end of the story.
In an analogous way there are no magnetic sources, even though it’s useful to speak of them (ie North and South poles). Magnetism is just a relativistic effect that’s required to balance the books and maintain invariance, but it can’t be treated in the same way that electrostatics is treated.
Well, this example has nothing to do with the discussion about the spin but independently of the spin discussion, it is wrong, too. First of all, the magnetic monopoles probably exist although their masses are near the GUT scale. Second, in accessible experiments, we probably won't find one and the total magnetic (monopole) charge is what determines whether an object is magnetically charged. Indeed, we will find out that \(q_m=0\) for the electron or any other object we know, so it means that they're not magnetically (monopole-like) charged. However, the analogous quantity for the spinning is the angular momentum \(\vec J\) and \(J_z\neq 0\) for an electron (similarly for any axis) so the electron is spinning.

Incidentally, when the GUT-scale magnetic monopole charges are correctly added to the theory, magnetic phenomena may be treated exactly by the same mathematical tools as electric phenomena, in a strike contrast with Aidan's opinions. This fact is guaranteed by the electromagnetic duality of Maxwell's equations – a rigorous way to exchange the electric and magnetic field (a symmetry transformation) that we know in many important field theories as the S-duality. Even when we assume that there are no magnetic monopoles, which we often do for analyses of all low-energy physics situations we know of, the electromagnetic duality is a very potent weapon to solve many questions.
The comparison between spin and energy/momentum is not a fair one to make. They remain continuous in quantum mechanics (although in some circumstances they are, of course, quantized) and there is no “extra” momentum or energy that must be expressed in the same way as spin.
The claim that energy is continuous in quantum mechanics is a remarkably stupid statement to be made. The fact that energy – namely the energy carried by the electromagnetic waves (separated to \(E=\hbar\omega\) quanta, photons) in 1900 or 1905 and, later, in the 1910s and 1920s, the energy of the electron in the Hydrogen atom and other atoms – has a discrete spectrum was the original reason why people invented the name "quantum theory" and later "quantum mechanics" in the first place. The momentum on compact spaces is discrete, too. In other situations, energy may have a continuous or mixed spectrum but even when it does, the classical tools to analyze its behavior are as wrong as they are in the context of the discrete spectrum! It's not true that the discreteness of the spectrum is the only deviation from classical physics that quantum mechanics introduces. Far from it.

The situation and transformation of the concept of spin is absolutely equivalent to the situation and transformation of the concepts of energy and momentum. After all, in some situations – e.g. electron in the magnetic field – the energy is the same thing as the spin, up to an overall coefficient. With the arrival of quantum mechanics, absolutely the same thing has happened to all these observables. They had to be reinterpreted and reanalyzed in a completely new framework for physics. All of them became Hermitian operators on a Hilbert space. All of them reduce to the old classical concepts whenever the classical limit is applicable (usually for macroscopic, decohering objects). All of them have totally new properties in any microscopic situation.

It's just totally wrong and stupid to suggest that the spin is any different from other observables in this respect.
The axioms of quantum mechanics make it very clear that energy and momentum are to be treated in the same way as they are classically (although subject to the uncertainty principle), that is they are continuous and the result of the spatial and temporal translational invariance.
Huh!? Quantum mechanics says that the energy and momentum are to be treated in the same way as they are classically? Sorry, Aidan, but this is too much stupidity to be tolerated for an experimental particle physicist. Nothing is kept about the properties of momentum and especially energy, especially its allowed values, the ways how the values are calculated and predicted, and everything else that makes sense to be discussed. Nothing from classical physics is left intact. The concept of energy becomes the Hamiltonian and it is the operator that dictates all of time evolution. If it were treated in the "same way" as in classical physics, we would treat time evolution in the classical way. The whole physics would be classical.

But the correct physics isn't classical. It's quantum and it's an entirely different thing. All observables – isospin, spin, orbital angular momentum, energy, momentum, number of particles, you name it – are Hermitian operators and predicted probabilistically. Most of them have a discrete (or mixed) spectrum in most situations. The reason for the discreteness is always the same, it's the basic postulates of quantum mechanics, it's the discreteness of the solutions to an eigenvalue equation.

The most general definition of the angular momentum has a totally analogous origin to that of momentum or energy. Via Noether's theorem, energy and momentum are conserved because of Nature's symmetry under the temporal and spatial translational symmetry. Totally analogously, the angular momentum is conserved due to Nature's rotational symmetry. The philosophy behind all these concepts is exactly the same, and so is the character of the transformation that these observables underwent during the quantum revolution.

It's really incredibly idiotic to suggest that energy is what it was and only the spin is new.
All quantum mechanics does is add the requirement that momentum is proportional to the gradient of the wavefunction, and quantum effects follow from there.
In the very same way, quantum mechanics "only adds" that the angular momentum \(J_z\) is the derivative of the wave function with respect to the polar or axial coordinate \(\phi\), a gradient of a sort, and quantum effects (including the quantized values) follow from there. But in both situations, it's wrong to say that "everything in quantum mechanics" follows from these mathematical ideas. The "gradient thing" is an extremely ill-advised starting point to be treated as as a postulate of quantum mechanics. It's just some technicality – a particular form of a particular operator in a particular basis. It's just not a way to learn quantum mechanics and it's clear that Aidan has never understood quantum mechanics.
If you want your electron to physically spin then you need to find a way to show that a single point has enough structure to do so. Imagine you’re sitting next to the electron- how would you see if it’s spinning if you only have access to a single point in space?
There are millions of ways to measure the angular momentum of the electron. Send it to a magnetic field. If it is spinning, the running electric currents inside the electron – using a classical explanation that Aidan needs – give the particle a nonzero magnetic moment, too. This magnetic moment will drive the electron in one direction or another, depending on the value of the spin, in the magnetic field. The spinning electron is a little magnet, indeed. Every charged spinning particle – and every spinning particle with charged components – is a little magnet.

I may also shoot the electron to other particles and try to induce reactions that only occur if the electron is spinning up or if the electron is spinning down (or in another way).

Also, if you give me one trillion of electrons or other particles with the same spin, I may just shoot them to a thin foil so that they are absorbed. The foil will start to macroscopically spin and its totally classical spin may be divided by one trillion to determine the spin of each electron. The spin is the spin. It is the internal rotation and every experiment able to check whether it's there will say Yes.
(Ask the same question of the proton and things get easier. Just watch the valence quarks move around each other!)
This is bullshit. It's not possible to measure the spin of the proton by measuring the velocity of quarks – a simple reason is that the velocity of a single quark doesn't commute with the overall spin and, even more seriously, the position and velocity of a quark inside the proton can't be measured accurately enough at the same moment due to the uncertainty principle. At most, one may get some correlation between the velocities and the spin, but not a sharp, reliable answer. The measurement of the proton's spin is actually done by the same methods as the measurement of the electron's spin because it's exactly the same thing! The only reason why Aidan and others think that it's totally different is that they don't understand quantum mechanics at all.

Incidentally, if you adopt some "nonreliable" measurement of the spin from the constituents' positions and velocities (which must be approximate), I must say that exactly the same thing applies to the measurement of the electron's spin from positions and velocities of pieces of the underlying string. It's undoable in practice today but the same comment applies to the "measure positions and velocities of quarks" to find the total spin, too.
Hello again! I should also point out that certain atomic transitions are forbidden precisely because of spin being an act of balancing the books, and not an actual spinning of the electron. If the electron was actually spinning it should be possible to change its spin by radiating a photon, flipping its spin and jumping down to a lower energy orbital.
An electron at rest can't emit a photon because of the energy conservation law. In the initial electron's rest frame, the total energy is \(E=m_0c^2\) while the total final energy in this frame is the higher kinetic energy of a moving electron (already too much) plus another positive energy of the photon (even higher, even worse). This reason has nothing whatsoever to do with Aidan's bizarre claim that the spin isn't a spin.
(If this would violate some symmetry then it can exchange a virtual photon with the nucleus and everything is happy again.) Since an electron cannot do that, then it cannot just change the direction of its spin spontaneously, implying it is not spinning in the first place.
What a breathtaking amount of ignorance. Of course that when the energy conservation obstacle is circumvented by considering a more complicated process, the electron will become able to emit a photon and change its spin accordingly to guarantee the angular momentum conservation law. Ever heard of Bremsstrahlung? Even ordinary radiation of the atoms also becomes sensitive to the spin – because it is the angular momentum, after all – above some accuracy (and in the real world).

The process of an electron spontaneously falling to a lower energy level surely happens – that's why 2s may fall to 1s and emit the corresponding photon – but most of such emissions preserve the electrons' spins for a simple reason: the leading non-relativistic Hamiltonian commutes with the spin.

But this ain't the case for the exact Hamiltonian. The leading relativistic corrections include the spin-orbit interaction and be sure that this term in the Hamiltonian is responsible for processes that Aidan considers impossible or blasphemous. The only ultimate, exact conservation law of this kind is the total angular momentum conservation law. If it isn't violated, the process is allowed.

(The orbital angular momentum can change by \(0\) or \(\pm 1\) as well but I don't want to discuss all the selection rules here. The only fact I need and stress is that \(\Delta S = 0\) isn't a valid selection law.)

The electron's spin is a spin and in most situations, quantum mechanics allows more processes to occur than classical physics. Be sure that we observe tons of atomic radiation emissions in which an electron's spin is changed. Aidan's understanding of all these things is upside down. It's wrong at every level. Unfortunately, he's very far from being the only one.

And that's the memo.


  1. OMG, is Aidan a particle physics student?

    Some time ago, I often read the US LHC blog. Flip used to write quite nice articles .

    But I amost cant believe that Aidan writes all this nonsense and confused things while being a particle physics student (?). How did he manage to pass any exams? If one can be that confused and still study (experimental) particle physics, I could do it as well :-P (joking).

    I always thought that all particle physicists (and students thereof) have the same good fundamental understanding of things, apply the same cute and clear way of thinking and approach to look at problems and physics questions, which I admire and appreciate of Lumo, Lenny Susskind, Prof. Strassler, etc ... (?)

    And by continuously pointing out that the electron should be a point particle etc I can guess Aidan probably does not like ST too much ...

    You have written quite a nice clarification in the comments, worth an US LHC blog article by it self :-D.

    I like best your stringy explanation of the electon spin :-).

    Thanks for this nice article Lumo, it made me happy again and detracted me from other less funny things.

  2. There is an article of H. Ohanian about spin:

    However I think it is tautological to a great extent.

  3. Can an electron acquire higher values of the spin? If not why?

  4. anyname not takenDec 8, 2012, 2:37:00 PM

    with regard to "continuous" position, momentum, energy, spin etc, i can't remember the details but i would have thought that there is a basis where the spin hilbert space is continuous. it is just that when one applies the Sz operator, it forces the basis to be discrete, either up or down. completely analogously for position there is in general a continuous hilbert space basis, but if one constructs a two slit experiment and observes both slits, at the same time, then the hilbert space basis at the time interval where the slits are positioned, is discrete, either slit a or slit b. Would you agree Lubos?

  5. Lubos, you are wrong on so many levels and misrepresent me so much that I do not have time to address them all (I do have a day-job, after all.)

    Let's start with where we agree. The electron has spin. Spin is a form of angular momentum. We can perform many tests to demonstrate that the electron has spin and therefore angular momentum. Angular momentum is conserved in a given frame of reference. We should not use classical concepts and analogies for quantum effects if we expect to get reliable results.

    There are some models such as GUT and superstrings that make predictions about things like electronic substructure or magnetic monopoles. If you want to believe these theories and entertain their predictions then go ahead. They have no evidence to support them and they are not part of the Standard Model, so I do not take them seriously. In other words: "Proof, or it didn't happen!"

    If the electron is spinning about an axis as a quantum object then let's a apply a quantum model to it. We can use the rigid rotator Hamiltonian. When we do that we expect to see a spectrum of spin values, but we clearly do not. If the electron really is spinning about an axis we would expect it to be allowed to have spin 3/2, 5/2 etc. Note that other quantum states can be described in this manner, such as the Ds system, leading to well studied spectra. The same cannot be said of the electron, hence it is not actually spinning. At that point we can probably agree to a semantic difference.

    Energy and momentum are variables which are conserved under spatial and temporal symmetries, and hence their intrinsic nature depends on the nature of the space-time. Quantum field theory is embedded in Minkowski space (ignoring the oft-failed attempts to include gravity) so energy and momentum must obey the constraints of this space. They are still continuous for the following reason. Construct an infinite square well potential (or really any potential you like that has energy eigenstates.) Label this W1. Can we construct another potential that has energy levels between two adjacent energy levels in W1? Of course, so call this system W2. Repeat this indefinitely and you will find that we can always obtain an energy level at an arbitrary point between two other energy levels. Hence energy remains continuous. The difference of two energies in a given system may not be continuous, but that does not stop us from defining energy as a continuous variable. The energy levels are properties of a system, and not energy itself.

    The only point I was making was that we can talk of things spinning in the classical sense, but that we should not apply that to the electron, or any other fundamental particle. Now stop misrepresenting me (this is not the first time you have done so). I've taken seven courses at undergraduate and graduate level in quantum, atomic, nuclear and particle physics, and quantum field theory at both Oxford and UCL, and passed every one of them. I have a good grasp of quantum mechanics and I know what I'm talking about. If you want to think of electrons as literally spinning around as a convenient shorthand then I won't stop you. It's a useful analogy, but there are times when it simply isn't true. You're the only person claiming to be a physicist I've ever met who literally thinks that electrons spin about an axis.

    As for the spin of the proton, well this took me by surprise too, but recent studies show that the quarks and gluons in a proton are responsible for no more than 70% of its total spin. See, for example

  6. By a certain transformation, the electron can spin down to a selectron ... :-P

  7. Dear Dilaton, the selectron's spin is 0 and it's a good question whether we may consider electron and selectron to be "states" of the same particle. The terminology is not designed in this way. However, the electron and the selectron belong to the same multiplet - the supermultiplet - and the multiplet terminology is analogous to the multiplet terminology for the nucleon (multiplet consisting of proton and neutron) or any other particle.

    Rather Strange: one may derive that the angular momentum vector for an electron always obeys J^2 = j(j+1) hbar^2 for j=1/2. It's because an electron is an excitation of a spin-1/2 field. But the same thing applies to the proton, too. Despite the proton's being demonstrably composite, there's no proton state with j=3/2 or higher (and of course no integer-valued j is possible because there couldn't be 3 valence quarks in it). You may derive similar constraints for much more complicated composite states, too.

    At the end, the proofs are not qualitatively different from the proofs that the spin of the Earth can't be adjusted to some insanely high value because the Earth would be torn apart to pieces by the centrifugal force. In physics, it's just not true that every object may rotate arbitrarily quickly. The limitations on the magnitude of the angular momentum often seem to be far away and some of us are forgetting about them altogether. But such constraints become absolutely immediate in the microscopic world. At the same moment, they're not new features of the microscopic world because similar constraints on the angular momentum of large objects exist, too.

    The constraints derivable from quantum mechanics are finer than those in classical physics. In classical physics, we are sending hbar to zero which means that the spacing of the angular momentum, hbar/2, becomes infinitely fine. We're no longer able to distinguish "adjacent" levels because in the classical limit, the spectrum becomes effectively continuous. But in quantum mechanics, it is discrete and the discreteness is very important if we study characteristic quantum phenomena. So quantum mechanics may say many things about the angular momentum that strongly or qualitatively depend on J being changed just by hbar or hbar/2 etc. even though this change of the angular momentum would be considered "infinitesimal" in the classical limit and that's why it would have to be inconsequential.

  8. Hi, if one considers bound objects whose angular momentum J^2 is smaller than a maximum value, then there only exists a finite number of states - basis vectors of the Hilbert space. And finite-dimensional Hilbert spaces simply don't admit any operators with a continuous spectrum.

    If J^2 is unbounded from above, of course you may have infinite-dimensional spaces and infinite-dimensional spaces admit operators with continuum bases. So if your Hilbert space is the direct sum of multiplets with values of L=0,1,2,3,4..... and each of them is taken once, then it's already guaranteed that for each L, the number M may be an integer between -L and L, and up to the value of LMAX-1, you get LMAX^2 basis vectors which becomes infinite if the upper bound is lifted.

    This Hilbert space is isomorphic to the Hilbert space of functions on the two-sphere, f(theta,phi), which may be expanded into the spherical harmonics Y_{LM}. One may also define operators such as cos(phi) or cos(theta) acting on functions f(theta,phi) whose spectrum is continuous.

  9. Dear Lubos, This is an ignorant question, but since I am a layman I will ask it anyway: Angular momentum is always conserved, yet when you start a gyroscope spinning by pulling a string wrapped around the axis it seems as though you are creating a lot angular momentum which wasn't there before. Is a reverse angular momentum somehow set up in the atoms of the string in the process of pulling? This has bothered me for a long time. thanks,

  10. Dear Luke, first start with the momentum. It is conserved but it may be transferred from one body to another by something called the force F.

    When A acts on B by force F, it means that the change of the momentum of B per unit time is dp/dt=F. The momentum of A is changing in the opposite way, dp/dt=-F.

    Now, it's totally equivalent with angular momentum. The total angular momentum is also conserved but it may be transferred from A to B. The derivative of the angular momentum per unit time is called the torque, dJ/dt = tau, for B. The object A is changing its angular momentum in the opposite way, dJ/dt=-tau, so the total J of A and B is conserved.

    The torque is proportional to the product of the force F and the distance of the point where the force F acts, r, from the origin. It's the cross product but I suspect that any piece of maths like that would scare you. At any rate, the pulling of the string away from the axis acts by some nonzero force and nonzero torque and that's what's changing the angular momentum of the gyroscope. The angular momentum of the rest of the Earth (or just your body, whatever is attached) is changing in the opposite direction but because the Earth is heavy, the same absolute change of J is manifested by much smaller - unobservable - change of the angular frequency, the axis and frequency of spinning.

  11. Right, the field may also carry the angular momentum, and when it's the angular momentum of a localized configuration of quantum fields that is mostly at rest, the angular momentum should be called the spin.

    One can't really get J=1/2 without having some building blocks - fields or anything else - that have a half-integral J to start with. Nevertheless, strictly speaking, the total J surely does include contributions from the field as well. Only the total J is reasonably observable in practice for tiny enough systems.

  12. Great article, Lubos. A lot of people get confused because they assume the electron to be point like. String theory should remove all confusion.
    A little bit off topic: You talk about the relativistic corrections of quantum mechanics. The correct theory here is quantum field theory. But most quantum field theory books study cross sections and decay rates and do not deal with the relavistic corrections for the hydrogen atom. It looks like one reason is that cross sections are studied at infinity and therefore there is no issue of introducing a preferred time slice. Therefore I find this relation between non relativistic quantum mechanics and quantum field theory also conceptually interesting.

  13. Thanks. So I was basically right. That math is not beyond me. I've learned it many times. Unfortunately I forget it the very next day!

  14. most experimental particle physicists probably know much less than Aidan on the subject. whether experimental particle physicists should know more about the theory behind what they are doing and why they are doing it is not something i know much to comment on.

    i know for sure in what i worked on in astronomy the students should know more about the theory but the professors are most to blame for not showing very simple, basic things on purpose.

  15. If you are right it is shoking how little they know about it. But it probably explains why many experimental (even particle) physicists are so hostil against beyond the standard model physics...
    See for example Tommaso Dorigo, I have stopped clicking him for exactly that reason.

  16. Sorry Lubos your " there's no proton state with j=3/2 or higher " does not count the resonances? Delta is 3/2 ,3/2 and is a baryon resonance.,

  17. Dear Anna, it may be mostly due to terminology but the Delta baryon simply form a different multiplet than the nucleon, so they're not counted as the nucleon states.

  18. The main subject of this post reminds me of Galileo , because at this time we have an advert on greek tv where Galileo says "Yes it is spinning", an advertisement for bets on football :)

    I watched snippets of the video because I do not like horror movies, and have to tell you that shifts are not taken by students in the beam tunnels. Only the detector areas are monitored, and mostly remotely. If something is wrong with the beam one calls the experts. So it is not so hard being an experimentalist on that account :).

  19. Great post indeed.
    Would Aidan be damaged by positivism I wonder ?

  20. Doesn't the Einstein-deHaas effect demonstrate that electron spin is fundamentally the same thing as classical angular momentum (though as you say a bit smaller than usual and subject to quantum rules)?

  21. Marcel van VelzenDec 9, 2012, 3:17:00 PM

    Dear Lubos,

    I want to thank you for all your exhibitions on the right interpretation of quantum mechanics, it surely helped to get me on the
    right track. I now understand that we should take the quantum rules
    for what they are, learn to apply them
    correctly, check that they don't lead to contradictions and show that the rules agree with experiment and so agree with classical
    physics for macroscopic bodies. But we should NOT use our
    macroscopic (or statistically averaged) picture, shown to follow from the rules of quantum mechanics, to THEN try to
    interpret the rules of quantum mechanics! That would be silly. Do
    you agree with this interpretation of what you are trying to teach
    us over and over again?

    I totally agree with your view that IF you want to picture spin (in
    correspondence of how we picture mass, energy and momentum for a point
    particle, which is also a highly idealized picture), a
    truly spinning electron is the only right way. I could try to
    explain why,
    but you already did!

  22. Hi Lubos, I added a comment on Aidan's blog that I hope help clarifies the issue, let me repeat it here and let me know what you think:
    "I think these particles “really” are spinning, in some sense. For the electron, the amount of spin is always just hbar/2, so its hard to picture or take any classical limit. So it may be useful to think of a spin-1 photon instead. By Aidan’s reasoning it is not spinning either, which I also disagree with. Its spin is always hbar, but we can make it large. Because it is a boson we can prepare a macroscopic collection of photons all in the same state, each with the same spin. This is known as a “circularly polarized wave”. The E and B fields thst make up this wave are certainly spinning round and round, it is macroscopic and measurable, and it is all connected to each individual photon’s spin, nothing else. The electron is harder to picture because it is a fermion, but its still true that its spinning."

  23. Exactly, it does.

    The angular momentum in quantum mechanics can do all kinds of fun things because of quantum mechanics but fundamentally speaking, it can do everything that the angular momentum does in classical physics because it is still the angular momentum. In particular, a changed spin of many electrons in the ferromagnet indeed leads to a macroscopic rotation of the ferromagnet - angular momentum we know from classical physics.

    Spin is the part of the angular momentum that is "intrinsic" to an object or a particle and that doesn't depend on the motion of the center of mass. That's true for the Earth's just just like the electron's spin or proton's spin. The mathematics is the same. To imagine they're qualitatively different things means to imagine some differences that don't really exist.

  24. Exactly, a classical circularly polarized electromagnetic wave is a collection of many photons with some spin, but it's also a classical object with a macroscopic angular momentum that's actually able to rotate a foil that absorbs the radiation. One may divide the angular momentum among the individual photons and find out that each of them has hbar.

    The Einstein de Haas effect - rotation of a ferromagnet that was remagnetized (whose spins were changed) is an example for the electrons.

  25. Thanks for your comments. I think my comment is still waiting to be moderated on his blog.

  26. Thanks for your kind words!

    But we should NOT use our macroscopic (or statistically averaged) picture, shown to follow from the rules of quantum mechanics, to THEN try to interpret the rules of quantum mechanics!

    Right, I totally agree. And that's one way to describe the wrong approach that Aidan is taking. He is deciding whether the electron is rotating by using the classical rules of how things should be rotating, and because the electron – much like everything else, especially in the microscopic world – refuses to obey the laws of classical physics, he decides that the electron isn't spinning. But that's exactly the fallacy you mentioned. To properly decide whether "electron is spinning" is right, one must convert the statement to a projection operator on the Hilbert space. The right projection operator is one projecting on all the states with nonzero eigenvalues of J_z etc. And indeed, all electron states are preserved by this projection operator – which means that the answer is Yes. The operator measuring how much things are spinning is the angular momentum and because the electron's angular momentum eigenvalue is nonzero with 100% probability, the electron is spinning, the question is answered.

    Quantum mechanics implies that the angular momentum - of electron, proton, atom, gyroscope, or Earth - is quantized in multiples of hbar/2. It has many other interesting implications, nonzero commutator, and so on. But the phrase "an object is spinning" can't possibly mean anything else than the claim that even in its reference frame where p=0, it has a nonzero angular momentum. It's the only right interpretation of the verb "is spinning" in our quantum world. So the answer is clearly Yes for the electron.

    Even if we need a visual model and even if we imagine a point-like electron, the electron is still spinning and whether we like it or not, we must imagine it's spinning with an infinite angular frequency. Whether this infinite value is allowed is a subtle question, the answer is Yes, so the calculation of the angular momentum is really about the evaluation of the indeterminate form 0.infinity of a sort and it's just not true that it must be 0. For the electron, the 0.infinity indefinite form for J_z yields hbar/2, a nonzero value, as the right result. The very fact that we encountered an indeterminate form means that our approximation broke down, it was singular, and a better theory - full-fledged quantum mechanics - is needed. But it doesn't allow us to determine that the particle can't be spinning i.e. can't have a nonzero angular momentum which would indeed be a wrong conclusion for the electron. Such a conclusion means to trust a limiting approximate theory – classical physics – outside the range of its validity.

  27. Both the circular photon and the Einstein de Haas effect are solid enough to trump the subject. Thanks to both. I hope I remember the arguments if I ever need them.

  28. I agree Dilaton, it's shocking that Aidan has such silly ideas about quantum mechanics. Lubos explains all the misconceptions so clearly here - a really good article.

  29. It is really, really hard to abandon the classical view of the world, especially for people who are not particularly bright. Thus, I do have a bit of sympathy for Aiden.

  30. If he were to really adopt positivism he would be a lot wiser, Shannon.

  31. Hi Lubos,

    I read that engineers are working on a system to double the amount of data that can be transmitted through an optic fiber cable by separating and packaging the photons transmitted according to spin direction. Now, that opened a whole new door to me. Photons can be generated by electrons, and the photon seems to inherent the spin direction of it parent electron? And they are already looking at incorporating photon spin detection in fiber optic cable transmission technology. Yet some people are questioning the spin of an electron. What am I missing in this argument?

  32. I thought the production values were pretty good on the movie. I'll reserve comment on some of the science in the plot, since liberties are somewhat expected in any zombie movie, but I have seen a lot of b-movie zombie flicks, and this was at least competitive with those. Definitely good location for a movie.

    The best pop culture ref in the movie is to "sector 7 G" which is well known as Homer Simpson's workplace. So very appropriate.

    About this spin business. I haven't read all the notes, but the point particle stuff is merely a limit for computation, and its the fact the electron has internal structure at a classical limit where there should be none which is the important part. It is indeed spinning, and it is intrinsic to the fundamental particles like the electron, and it is a perfectly appropriate name.

    This to me seems to be another case where one of the more shocking discoveries about our universe is total misunderstood and glossed over as being some quaint artifact of 20th century reasoning. The story that was presented in the video suggests that it was some feeble minded professors that proposed to take the Bohr model and claim spin was like spinning planets. The whole point was that these guys were actually really sharp, used the same reasoning as the creator of the video, said that there should be no intrinsic spin for a point particle, and were not expecting to see any. The fact that it was present was not some early victory for the Bohr model, but the death of classical physics. Sure the discovery of spin seems trivial since it could be easily incorporated and understood by analogy to a planetary system, but it shouldn't have been there based on classical lines of reasoning.

    This was huge! (and still is)

    I am not surprised this is hard for a lot of people. I think the relative ease of understanding the spin property by using the Bohr model makes people think, "Oh that's obvious". However, this is because people think that the Bohr model with spinning objects is classical when they first encounter it. It isn't.

    The fact that spin is easily measurable is perhaps even more shocking. Spin is a strong effect, so strong that degeneracy pressure is a major factor in understanding life cycles of the stars themselves. This again is an example of how fundamental misunderstandings between classical and quantum mechanics leads to understatement and relevancy of key discoveries. Stars simply can not be understood as classical objects, because any classical approach to understanding their lifecycle would lead one to readily observable contradictions.

  33. Is the electron a trapped photon?

  34. I was excited about similar ideas when I was 14 but sometime during the following year, I understood what was wrong with them.

  35. Off-topic: Dilaton (and other Germanophone TRF readers) what is your opinion of this three-part introduction to the Special Theory of Relativity (part 1 here)? Too simplified? Mistake-free?

    Apologies to our Esteemed Host, please let me know if I am posting too many off-topic comments so that I will know to stop.

  36. " I've taken seven courses at undergraduate and graduate level in quantum, atomic, nuclear and particle physics, and quantum field theory at both Oxford and UCL, and passed every one of them. I have a good grasp of quantum mechanic and I know what I'm talking about. "

    II don't think this has much bearing on whether you are wrong or right....passing exams is one thing, actually understanding a subject on a deep level something else entirely. I passed exams in QFT - but I would not say I understand it. Its easy to pass if you are good at maths because you just do the calculations. To actually understand it properly takes much more work IMO....and I know what I understand about QM came from reading well beyond what we were taught (which is why a I know I do not understand QFT because I never found time to do so in this case)

  37. The electron's spin *can* be stopped! Only you need some non-standard interactions to do that. What you get is a spin-zero object that goes under the name of selectron and the type of theories that let you do this are called supersymmetric. You might object that a selectron is not just a electron robbed of its spin because we know its mass should be significantly larger, but this is just due to the fact that SUSY is spontaneously broken. The selectron really is just a different face of the electron: one without its spin

  38. Right :-) I was just reading Steven Weinberg essay "Against Philosophy" the other day and this is what lead me to this conclusion.

  39. Absolutely. Whenever we use words to describe the maths of QM we lose precision. But when the math works out as quantized angular momentum and the best minds like Motl can extend the concept to teach others it is the best words and analogies we have to grasp the wonder of nature.
    Unlike some I do not know enough to know what is right or wrong. I do know that what I learned long ago to pass my exams and what I studied since because it interests me tells me that Lubos is s great teacher.
    Btw my bird was scared when the zombies screamed so my wife and I watched the movie at low volume after he went to bed. Pretty good acting by especially the lead actress, we thought. The zombies reminded us of a British flick Shaun of the Dead.

  40. Dear Robert, thanks for your excessive compliments but not again. ;-)

    Whenever we use normal life's words in advanced physics and we're assuming that the everyday life meaning is the primary one, we are losing precision.

    The alternative - and a necessary one, to a large extent - is to redefine the words a little bit so that they mean something related to the everyday life but more accurate, so that the concepts are optimized for physics, quantum or otherwise. When it's done in this way, we don't lose any precision.

    But I don't even think we really need any redefinition of "is spinning" here. The electron is spinning. The only reason to say it's not is to imagine that lots of other not-quite-directly-related and unrelated properties of the situation that we're used to (from the Earth etc.) are not satisfied for the electron. But it's just wrong to "encapsulate" all these vaguely related and unrelated conditions to the words. We are not saying that the electron is the Earth. We are saying it is spinning. The latter statement is a weaker one and it just doesn't assume tons of special properties of the Earth's spin which are related to the large magnitude of the planets' angular momentum.

    You're not the first one comparing Decay to Shaun of the Dead. But the latter was a professional movie, right?

  41. Luboš, instead of piecemeal, reactive corrections to the world's misunderstandings of QM why don't you write _the_ book? Your explanatory powers are brilliant and I really believe that if you were to start at the beginning and move coherently through the subject from your understanding of the world such that each statement was fully supported by what you said prior (with no gaps for people to argue about) then that world would be done a great service and many arguments could be brought to a close.

    Stop short of strings, though, if you would. :-)

  42. Thanks, Don, but the main reason why I don't is that my experience indicates it would be a frustrating fight against the windmills. The reaction would be most likely dominated by a combination of disinterest and downright hostile reactions and I am just not attracted to such things.

    Moreover, even if I imagine that such a book of mine would have an extra "X factor" relatively to any book on the market, and I am willing to imagine that, it's still untrue that it would be "unprecedented". There are some great books – including books that were written by the founding fathers of QM – but they're just unpopular for certain reasons.

    Various extra constraints that many people would like to impose – such as "stop short of strings" – are another reason why I am not attracted by such an idea anymore. Why would I stop short of strings? Even when one talks about the foundations of quantum mechanics only, there exist various modern aspects for which it is appropriate to discuss quantum field theory or even string theory. At the end, I think that almost all readers refuse to search for the truth, by their own research or in the books. They want to be told that their preconceptions have been right from the beginning and your comment "stop short of strings" shows that you're no different, sorry.

  43. Dear Lumo,

    Of course I would just love such a book written by you that includes strings, M-/ and F-theory ;-)

    But I agree that it would most probably lead to too much annoying hassle for you etc ...

    The sourballs and trolls would attack it just because you have written it ... :-(.

    I appreciate it a lot when you and Matt Strassler are correcting things people say wrong in blogs, in the news, and elsewhere, but already this looks much like a fight against windmills to me.

    Sabine Hossenfelder has just reported about a potentially cool application of the AdS/CFT business in an invormative and non negative way. And as sure as eggs is eggs, she gets immetiadely attacked by a troll called Eric for it :-/

  44. Hi Eugene,

    thanks for these links, I'll look at it at the weekend :-)
    (I am quite busy at the moment)


  45. I don't buy your excuses and I'm sorry my attempt at humor didn't make it into text.

  46. Dear Lumo,

    I like your comment at Sabine Hossenfelder's blog a lot, that will pick the sourballs :-)

    In some papers about renormalization group analysics of turbulence I have just read that the UV limit should corresponds to QCD.

    That scares me now ... ;-P

    Should then something like AdS/Turbulence (written as QFT) exist ...?

    I am actually quite seriously interested in applications of QFT methods in turbulence theory.


  47. Positivism is itself a philosophical position.

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