Tuesday, November 18, 2014

Why Do Particles Have Spin?

Intrinsic spin is one of those infuriatingly delightful concepts in theoretical physics that leave you with more questions than answers. And it will leave your head spinning. What exactly is it? Is it real or just math? Why do particles have it? And what the heck is spin-1 or even spin-1/2? Spin seems to be built into our universe. Spin and rotational movement are found almost everywhere we look in the cosmos as well as at the sub-microscopic quantum scale. Does quantum spin have anything to do with the cosmic-scale spin and rotation of planets, stars and galaxies? We will explore this last question first.

Does Spin in the Quantum World Translate Into Spin in the Cosmos?

Stars, solar systems and most galaxies have been observed to be rotating in space and there is some recent evidence that the universe itself may be rotating too. Most cosmic objects such as stars, planets and galaxies have angular momentum. They spin about an axis based on the center of mass. The elementary particles that make them up - the electrons and quarks of matter - also possess angular momentum called intrinsic spin, but it is much different in nature. It is quantized and, unlike a spinning planet or galaxy, it is impossible to conceptualize (but we will try).

Once an object, whether it is a collapsing gas cloud, a planet or a galaxy, has angular momentum, it will maintain that momentum because it is a conserved property. In the physics of motion three properties are conserved: momentum, energy and angular momentum. However, this does not explain how these objects attain angular momentum in the first place.

If we look for a mechanism that links this angular momentum with the quantum momenta of the constituent elementary particles that make up the object, we will run into trouble. Much of the challenge in finding such a relationship would be in how to scale up the sum of all the quantized intrinsic spins of the particles to the cosmological scale. First, particles "spin" in a sense but, as we will see, it is not a simple matter of measuring a rate of rotation for that spin. Second, it would be an impossibly monumental task. Third and by far most importantly, we would require the theories of quantum physics, in which quantum particle spin is described, to connect with theories of relativity, where the motion of large-scale cosmic objects is described. Although physicists have been trying to do this for several decades, these two fundamental sets of theories do not match up.

Spin is built into particles. According to current particle theory (gauge theory in particular), all particles of energy and mass were "born" as the result of various energy fields breaking from one another while the universe expanded and cooled. During this process, every type of particle that appeared came with a specific intrinsic angular momentum, or spin. Particles such as electrons and quarks combined to create the first atoms and an additional kind of angular momentum was realized. Electrons orbit nuclei, and in doing so, they exhibit orbital angular momentum in addition to their intrinsic angular momenta. It turns out that neither of these sources of momentum is required to explain why many cosmic objects such as galaxies, stars and solar systems rotate or spin.

According to most current computer modeling theories, the angular momentum of large cosmic objects comes about as a result of torque created when matter begins to collapse together under the attractive force of gravity. The scenario plays out like this: When the universe was very young, it was flooded with particles of mass and these particles were not quite perfectly spread out. There were slight imperfections. This meant that gravitationally unique regions existed, where gravity tugged on denser regions of particles just a bit more than it did on less dense regions. This inhomogeneity meant that gravitational forces on various regions were a bit off-center compared to other regions nearby. Therefore, as regions of matter began to collapse together due to gravitational attraction, they experienced some amount of torque as they did so. Many astronomical websites and Wikipedia do not address this question of spin origin, but this process of how rotation begins during galaxy formation is explained by Astronomy Cast and also answered by The Physics Van. Torque exerted on various tiny scales of matter continued to add up as the matter collapsed together. Differing levels of shear became twists as matter was dragged inward under gravity's attractive pull. Theoretically in a perfectly randomized system there would be torques in all directions as the collapse continued, and overall torque on a perfectly centered mass would even out to zero. There would be no resulting spin, and a star produced in this kind of collapse would have no spin and no planets. In reality, this would happen only extremely rarely if at all because even a very tiny excess of torque in any one direction will generate a significant overall spin once collapse is complete. The reason for this amplification is that angular momentum is conserved. This concept is analogous to a skater spinning faster when she pulls in her arms. The total angular momentum of a vast gas cloud is the same as that for the far smaller radius galaxy or star it evolves into. Galaxies in the universe are observed to rotate in every which way as a result of these early tidal forces. A similar formation process is believed to account for the rotation observed in stars and their solar systems, except that the clouds of dust these bodies form from may often already be very slowly spinning thanks to earlier events in the lives of those clouds.

The following 51-minute lecture video by Professor Carolin Crawford explores cosmic spin in great detail for those of you who are curious.

To sum up this section, the spins of particles are not in any known way related to the rotational spins and orbits of cosmic objects.

Intrinsic Spin

All elementary particles have intrinsic spin associated with them, but understanding what that means in a physical common sense way is not just almost impossible, but thoroughly impossible. That being said, we can gain a much deeper understanding of the nature of these particles by treading the difficult territory of spin. Today we will focus once again on the electron, as it is such a thoroughly studied particle, but there will be some very interesting things to say about the spins of quarks, bosons and protons - other particles of matter and energy - as well.

Stern-Gerlach Experiment

In the early 1900's the electron posed a huge puzzle. Quantum theory was in its early developmental stage at this time and physicist Paul Dirac was at work attempting to explain the behaviour of electrons inside atoms. At this time, physicists knew that the electron was a particle with a specific charge and mass and it had a magnetic field associated with it. A moving charge generates a magnetic field. At the time, however, physicists argued that this magnetic field was the result of the electron's orbital movement within the atom, rather than something that originates from the electron itself. Bohr's model, in which the electron rotates around the nucleus in specific circular energy orbitals, had just been introduced as of 1913.

In 1921, Otto Stern and Walter Gerlach developed an ingenious experiment to test this developing picture of the electron's magnetic field.

Walter Gerlach
Otto Stern
They found a way to focus on the behaviour of single electrons, those outside of the atom's influence, by choosing silver atoms for their experiment. These atoms have a single outer electron that is shielded from the positive charge influence of the nucleus by 46 inner electrons. This outer electron therefore behaves almost as if it is a free electron. When silver atoms are shot out in a beam, these valence electrons move in the Coulomb potential created by the 47 positively charged protons.

The researchers sent a beam of silver atoms through a non-uniform magnetic field. A rotating charge in the magnetic field should interact with it. However, they expected that these electrons, now moving in the Coulomb potential, would no longer have any orbital circular movement within the atom. Even though they are moving charges and will create their own (much smaller) magnetic field, they should no longer have orbital angular momentum, so they shouldn't be deflected by the externally applied magnetic field.

Stern and Gerlach were astonished by what they saw. Not only were these "free" electrons deflected by the magnetic field, the pattern of their deflection was itself totally unexpected and mysterious. They found that the beam separated into two distinct parts. The basic experimental set-up is shown below.
This pointed to two things. First, the "free" electrons must have some kind of built-in magnetic moment because they are interacting with the magnetic field. Second, this magnetic moment is no ordinary dipole moment - the electrons are not acting like tiny little magnetic spheres shooting through the magnetic field.

To clarify these terms: The magnetic moment or magnetic dipole moment of an electron, or any magnet, measures the torque it experiences in an external magnetic field. It is a vector force that has magnitude and direction. That vector points from the south pole to the north pole of the magnet. The magnetic field produced by the magnet is proportional to its magnetic moment.

First implication:

The first question this experiment raised was how does this particle produce a magnetic moment? In 1925, Samuel Goudsmit and George Uhlenbeck suggested that the electron must have some intrinsic "built-in" angular momentum that is completely independent of its orbital motion in the atom. In the photograph below right, taken in 1928, Uhlenbeck is on the left and Goudsmit is on the right.

In classical mechanics, a spinning object will generate just the kind of magnetic field observed in the Stern-Gerlach experiment, so they suggested that the electron itself must therefore be spinning.

Second implication:

If each electron is a tiny charged spinning object then it will have two magnetic poles just like a magnet does. A magnetic dipole, as this is called, will experience a force proportional to the magnetic field gradient. If there is a gradient, if the field is uneven in other words, the two poles will be within different fields. If an electron is a tiny spinning sphere, then the dipoles within a beam of electrons should find themselves in all kinds of random orientations as they move through and react to the non-uniform field. The beam will experience a whole range of possible deflections as each electron experiences a force proportional its specific pole-pole orientation. This would result in a continuous smear on the photographic plate that is used to detect the electrons (the classical prediction in the diagram above). But they don't get this. Instead they find two distinct parts, indicating just two possible orientations of the electron's magnetic moment, and therefore only two possible spin orientations for the electron. This was no ordinary tiny spinning sphere! It has just two spin states: spin-up and spin-down (this is where these familiar terms originally came from).

The Journey From Classical Spin to Quantum Spin Begins

The intrinsic magnetic moment of any classical object or any fermion such as electrons and quarks depends on its charge, mass and intrinsic angular momentum multiplied by a dimensionless quantity called the spin g-factor. For a classical rotating charged sphere, the g-factor will be 1, meaning that the sphere's mass and charge occupy the same radius (the sphere's density is evenly distributed in other words). The magnetic moment of the electron can be measured using its deflection. If this value is put into the equation, the g-factor is measured to be 2.002319. A g-factor of around 2 rather than 1 suggests that the electron's mass and charge do not occupy the same radius. It is a further hint that the magnetic moment of the electron is a quantum quantity that departs from classical objects. Why it is around 2 and not 4 or 15 or some other number remains a mystery. The g-factor does not offer us any clues about a hidden architecture of the electron or if there is any at all.

However, why it is slightly more than two is due to something called anomalous magnetic moment. This discrepancy is quite interesting. The expected value of exactly 2 can be calculated straight from the Dirac wave equation for the electron, a relativistic and quantum mechanical equation. There is no obvious reason why it shouldn't be exactly 2. However, the addition of 0.002319 can be anticipated as the effects of quantum corrections to the Dirac equation (which can be expressed as Feynman diagrams with loops), giving predictive weight to both the Dirac equation and to quantum field theory.

Is Electron Spin Real?

Electron spin is weird in several ways. It has some properties that you would expect from a physically spinning object, and other properties you do not expect.

There is evidence that suggests that the intrinsic angular momentum (intrinsic spin) of the electron is a physically real phenomenon. The obvious deflection observed in Stern-Gerlach experiment itself can be thought of as evidence: There is a measurable force associated with its angular momentum, as expected. Also, experiments done with light add weight as well. In 1936, physicists showed definitively that light has real angular momentum. This angular momentum can be used to make physical objects rotate and it can be used to make electron spins change state from up to down. This means that momentum is transferred from the photon to the electron's quantum spin. This transferability also strongly implies that the spin of the electron is a physical reality.

Electron Spin Is Quantized

Even though we can think of electron spin as physically real in the sense that it interacts with forces, this spin is not like the spin of a rotating object in the classical world of physics. First, the spin is quantized - only two spin states are allowed, and this right here makes the concept of the electron's angular momentum non-intuitive. A classical spinning object will have angular momentum along its axis of rotation, which is determined by the direction in which it is spinning. If it is spinning clockwise, the angular momentum points down; if it is spinning counterclockwise, the angular momentum points up. Like any classical object, both the direction and the magnitude of the angular momentum can be changed by applying forces to the object. It can be made to point in any direction - up, down, at 45 degrees, etc. Its rotational rate can be increased or decreased.

Units of Intrinsic Quantum Spin

The quantized nature of the electron's spin means that it must be described in a way that is different from classically spinning objects. However, the SI unit for both classical and quantum spin is the same - joule⋅second (not joules per second, that is a watt!). It is expressed as ML2T-1 where M is mass, L is length, and T is time. It is a base measure used to measure either action or angular momentum.

This happens to be exactly the same unit used for Planck's constant. This constant relates the energy in one photon (quantum) of electromagnetic radiation to the frequency of that radiation. This relationship has profound implications. First, it connects frequency, a wave term, with the quantum, a particle term, implying the dual wave-particle nature of particles. Second, we can use the reduced Planck constant, where a factor of 2 pi is absorbed into Planck's constant (it's divided by 2 pi) to get a term for angular frequency (radians per second) from the wavelength frequency of Planck's constant. By doing this we get a measurement for a quantum (smallest possible unit) of angular momentum in quantum physics. All quantum spins are multiples of this value. It does not give us a specific rotational velocity but it does bring home the granularity of spin at the quantum level, in the same way that electron energies are quantized in atoms.

Quantum spin is either written as a multiple of the reduced Planck constant, ћ or as a unit-less number with the ћ omitted. This unit-less number is called the quantum spin number, which parameterizes the intrinsic quantum spin of a particle. It is one of four quantum numbers that describe the unique quantum state of the particle, and it is designated by the letter s.

Quantum Spin States

For any quantum system, including elementary particles, angular momentum (intrinsic spin) is quantized so it can only take on certain values. These allowed states happen to be integer or half-integer multiples of reduced Planck's constant up to a maximum value and down to a minimum allowable value. It's perfectly logical to think that a smallest quantum unit of spin should be 1, so why is there a 1/2 spin? This is a very good question and one I will attempt to answer shortly.

A theoretical quantum particle might have one the following possible spin states +3, +5/2, +2, +3/2, +1, +1/2, 0, -1/2, -1, -3/2. -2, -5/2 and -3 (where +3 is spin-up and -3 is spin-down for example). Any particle, no matter what its quantum spin number is (1 for example) has only two possible spin states (+1 and -1 in this case). The spin number of a particle cannot be changed by any known mechanism. A spin-1 particle is always a spin-1 particle; that quantum spin of 1 is built into it. However, that particle's +/-1 spin state (recall there are two allowable states) can be changed through the application of a force, from spin-up to spin-down and vice versa. The term spin state can be confusing when reading online and in printed literature because both the quantum spin number (0, 1/2, 1, 2 etc.) and the +/- state of the particle are sometimes called spin state.

An electron (or quark) can never have zero angular momentum - it is always spinning in one orientation or the other and its spin is always at the same rate - it never slows down. Intrinsic spin is built in. Every electron has the same spin rate as every other electron, and it has exactly the same intrinsic angular momentum.

Spin-up and Spin-down

Spin-up and spin-down used for the +/- state, is also confusing because it is tempting and incorrect to visualize the particle as physically spinning in an upward or downward direction. Here I offer an explanation of spin state that I am paraphrasing from an exchange on physics.stackexchange.com, an excellent place to snoop around and see how grad-level and above students tackle the hard stuff. I think it describes the situation in the most understandable non-technical way: You might measure a spin-up electron for example. If you could then measure its left-right spin you would assume it is zero since it is spin-up, right? However, you would find that it is either left or right as well, with a 50/50 chance of being either one. This is not intuitive in any way. It has to do with the 2-dimensional vector space that is used to describe spin states. Choosing up and down spin states is like choosing basis vectors in this space. This peculiarity comes about when you consider that your measurement induces a collapse of the particle's quantum state. For example, let's say that you choose the z-axis in 2-dimensional space for your measurement basis. No matter what alignment the electron might have actually been in, it will come out as either up or down along the z-axis, with 100% probability. A left or right state also exists and obviously it tilts neither up nor down. However, it must be represented in the 2-dimensional space, so this leaves the possibility of left versus right as completely up to chance as there is no specific vector in this space to accompany it - it is always a 50% probability. This argument emphasizes the important point that spin state is a mathematical construct rather than a physical spin direction.

Quantum Spin Numbers For Real and Theorized Particles

Only some of the possible spin values listed above represent known particle quantum spin numbers. Boson particles have whole-integer spin numbers (0,1,2). Bosons such as photons, W and Z bosons and gluons are all known to be spin-1 particles so they have two possible spins: +1 and -1 (not zero). Some gravity theories such as string theory suggest a spin-2 graviton boson. The Higgs boson is thought to be a spin-0 particle. This particular particle is interesting for many reasons but its zero spin is especially so. The Higgs boson mediates, or gives rise to, the Higgs field. This means is that the Higgs field, which pervades the universe and "gives" mass to some particles, is a spin-zero field. While an electric field or magnetic field have both magnitude and direction, the Higgs field has only magnitude at any given location in space. It's a scalar field in other words. The Higgs boson itself, with zero spin, does not have any rotation-like behaviour whatsoever (with the caveat here that thinking about spin simply as rotation is always going to get you in trouble because it is technically not accurate).

All known elementary fermion particles have a spin number of +/- 1/2. This includes electrons, quarks and neutrinos. There are no known elementary or composite fermion particles with a spin state of 5/2. However, unstable delta baryons, made of three quarks, have a spin of 3/2. Mesons, unstable particles made up of a quark and an antiquark, have a spin state of 1. Though they are composed of fermionic quarks, mesons are bosonic composite particles, which act like bosons rather than fermions. Both mesons and baryons (neutrons and protons) are hadrons. These are composite particles made up of quarks.

The 1/2 Spin of Protons and Neutrons is a Mystery

You might think that a 3-quark fermion such as a proton would have a quantum spin number of 3/2. You just add up the spins. However, quarks come in several different kinds including up and down, and these kinds, or flavours, have nothing to do with intrinsic spin. A proton is made of two up quarks and one down quark. A neutron is made of two down quarks and one up quark. Until recently physicists thought that the two up quarks must align in opposite directions. Because they are fermions they should obey the Pauli exclusion principle (no two fermion particles can occupy the same quantum state). Their spins should cancel, leaving just the spin of the single down quark to contribute to the protons overall quantum spin of 1/2. This makes perfect theoretical sense but the proton spin crisis proved it to be wrong.

To hopefully clarify, up and down quarks are not the same as spin-up and spin-down quarks. The quark is a fermion like the electron. The quark has a spin, s, of 1/2, and spin state of +/- 1/2. An up quark, for example, can be spin-up OR spin-down. Likewise, a composite particle like a proton or neutron can also be in a spin-up or spin-down state just like an electron. For example, a spin-up (+1/2) proton is made of two spin-up up quarks and one spin-down down quark. A spin-down proton is made of two spin-down up quarks and one spin-up down quark and will have a spin state of -1/2.

The proton spin crisis proved that straightforward quark spin cancelling is not the reason why the spin number is 1/2. This crisis stemmed from a 1987 experiment that showed that quarks account for only a small fraction, at most 25%, of the proton's spin. Now scientists think that gluons, the particles that "glue" the quarks together inside a proton and mediate the strong force, account for a significant amount of the proton's spin, and there may be far more of them than first thought. Gluons are bosons, each with a spin of 1. Recent work shows that gluons might be responsible for the rest of the proton's spin but uncertainty remains. This evidence comes from high-energy proton-proton collisions carried out at the Large Hadron Collider. Internal orbital angular momentum resulting from quarks and gluons swarming around inside the proton is likely to contribute significantly to the proton's overall spin. Quarks and gluons are never found outside of hadrons. They are always confined (why they are is a mystery), and the dynamics of their confinement could affect the direction of the spins of the quarks and gluons inside hadrons, and thus have an effect on their spin contributions. It is also possible that even ghostlier transient and virtual quark-antiquark pairs inside the proton, called sea quarks, contribute to the proton's spin.

To learn more about the proton's internal structure, I highly recommend physicist Matt Strassier's website called Of Particular Significance. You will notice three links to his recent posts on proton structure in the article "Following up on the Proton's Structure." They are all excellent reads for the layman.

This tells us that at first glance the proton seemed to be fairly simple. Recent evidence shows that it is anything but. The proton, and neutron by extension, is a writhing tangle of far more particles than anyone would have guessed and even virtual particles may contribute to its spin. These contributions to spin are orbital spin contributions rather than intrinsic spin contributions. The 1/2 spin of the proton is therefore orbital rather than intrinsic like the electron spin. The question of why such a complex structure would have a spin of exactly +/- 1/2, exactly the same as the electron, remains utterly mysterious.

Why and What is Spin 1/2?

Why are all possible spins not simply whole numbers? This is actually a fairly deep question. The experimental evidence that fermions have a fractional spin comes once again from the Stern-Gerlach experiment. In general, when a beam of atoms is run through an uneven magnetic field, the beam splits into N parts along a particular axis, with N depending on the angular momentum of the atoms. The smallest whole integer N is 1, but for an atom or particle to have this smallest possible whole-integer momentum, the beam would be split into three parts, corresponding to spin states (along the axis) of -1, 0 and +1. W and Z bosons as well as mesons have these three possible spin states (which does not mean that these bosons or mesons physically exist in a spin-less spin-0 state).

Remember that the silver atom was used because it has a very handy lone, and very shielded, 5s electron and this is what the researchers were focused on. It vastly simplifies the experiment by largely eliminating the very complex electric and magnetic goings-on inside the large atom. The silver atom therefore ends up acting like a massive neutrally charged object flying through the field (no magnetic deflection) so that the two part beam can be attributed just to this lone 5s electron. The "electron" beam along the single axis is composed of less than three parts. This means it must have a spin of less that N =1. It must have a fractional spin.

That is the experimental evidence. The theory behind why fermions have a fractional spin of exactly 1/2 is quite mathematical and I think it is best explained here as a proof for those of us who are more mathematically inclined. In general, it starts with the statements that according to spin statistics theorem, quantum fields of integral spins commute, which means you don't change the result when you change the order of the operands. These integral spins must be bosons. Quantum fields of half-integral spins anticommute (the order of the operands does make a difference in the result). These spins must be fermions. The proof of these statements is worked out in four dimensions using quantum field theory. In three or more dimensions of space, only fermion and boson solutions work. The professor who wrote this proof went further to explain that in two spatial dimensions the mathematics of spin statistics theorem allow for an anyon particle, which is neither boson nor fermion and its spin number can be any fractional or even irrational number. In condensed matter physics, anyons exist as quasiparticles in thin layers of semiconductors in magnetic fields, where they play an important role in the quantum Hall effect.

We might be able to better appreciate (but not visualize unfortunately) the nature of spin-1/2 if we look into what spinors are. Electron (and quark) spin is a spinor, and this makes it very hard to visualize. I suspect most theorists would tell you that any attempt to visualize it as a real object is misguided. A spinor is not a physical description, but instead it is a purely mathematical construct. What makes this construct so useful is that it takes complex space and uses it to expand on the idea of a vector in ordinary space. Complex space is built from both real and imaginary (such as the square root of -1) parts or dimensions. Don't even try to get a mental picture. In ordinary three-dimensional space, you take vectors and build them up into multidimensional tensors. The space of spinors does not build up in this natural way. While a spatial vector or tensor will transform spatially (you can rotate it around in three-dimensional space and you will be right back at the starting point), spinors do not transform well. A 360-degree rotation turns it into its negative and it takes a 720-degree rotation to bring it back to its starting state. A spinor in three dimensions is used to describe the spin of all 1/2 spin particles.

For a classical spinning object (in ordinary vector space), you can change the direction of angular momentum through 360 degrees, something that makes sense and is expected. All whole-integer particles such as bosons operate exactly the same way. You can start with a +1 direction or state, for example, and change it to 0, then to -1, then back to 0 and then to +1 once again. It is analogous to a 360-degree rotation. The +1 state you end up with is identical to the +1 state you started out with. This transformation operation, called Bose-Einstein statistics, works when you deal with whole integer particles and bosons such as photons, W and Z bosons and gluons (I want to mention here again that you don't ever sit at spin state 0 - bosons ONLY have +/- spin states; their spin is quantum and it never "slows down" to zero. The zero here is only used to help describe the full rotation using a classical analogy).

For fermions, such as electrons and quarks, with half integer spins, this doesn't work. When you change the direction of angular momentum from spin-up to spin-down and back again to spin-up you get a state that is not quite what you started out with. The spin is pointing the same direction as it did before but the overall wave function of the electron is multiplied by -1. If you continue to transform the direction of the angular momentum you go back around 360 degrees again and end up, after a 720 degree rotation, at a state identical to the first one.

What does this -1 mean? It has to do with the wave in the fermion's wave function. By multiplying the wave function by -1, you are shifting the phase of the particle's de Broglie wave by 180 degrees. This shift in phase, a delay of 1/2 wavelength, actually does nothing to a singular electron's spin. It looks just the same. However, just as when you delay a light beam by half a wavelength, you encounter negative interference. By itself the delayed beam of light has just the same intensity and so on as before, but if you add this to a second beam of light that is not delayed, negative interference reduces the overall light intensity. When we take this analogy to electrons and quarks, we have Fermi-Dirac statistics and the Pauli exclusion principle, which states that no two fermions can be in the exact same quantum state at the same time.

Though the mathematics behind why we have spin-1/2 particles is no less than totally esoteric, the rules that this spin follows have huge consequences for our very real universe. If the de Broglie waves of electrons did not experience negative interference (if they were not spinors in other words), matter would hardly take up any space at all, as there would be no need for the separation of electrons into larger and larger energy shells, and all atoms would be the same size as the hydrogen atom. Chemistry as we know it would not exist and stars would not exist. We would not be here.

Bosons, such as photons have no problem occupying the same state. For example, a laser beam is a collection of photons all occupying the same quantum state. The de Broglie waves of these particles experience no interference.

Some Clarification OR NOT of the Pauli Exclusion Principle

The Pauli exclusion principle states exactly what we've been talking about: two identical fermions cannot occupy the same quantum state simultaneously. However the mechanism by which fermions are excluded from identical states is not clearly stated by Wikipedia beyond saying that it is due to the antisymmetric states of the fermions. Is Pauli exclusion a repulsive type of force, and if so which of the four fundamental forces is it? Is there a fifth force? Some sources in textbooks and on the internet say that Pauli exclusion originates from spin-spin interaction, implying that the magnetic dipoles of two nearby electrons repulse each other and prevent them from occupying the same location. Others claim that destructive interference of the two de Broglie waves is the more accurate explanation, and is based on spin statistics theorem. As far as I can tell from online sleuthing, there is no agreed upon mechanism for Pauli exclusion. This question was raised on physics.stackexchange.com. The second answer offers a mechanism as a possible resonant boundary condition (where the force is more a matter of inertia in an accelerating frame than a true force). My personal preference is for this last possibility, probably because I still have inertia on my mind from writing the previous article.

The Sizes of Electrons, Quarks and Protons

Angular momentum can be fairly easily visualized when we think of classical objects such as spinning spheres. The quantum nature of particles, however, makes such a visualization impossible. If we strictly adhere to the mathematics of quantum mechanics, elementary particles such as electrons and quarks, the two particles that makes up atoms, have no physical size about which to spin. However, their sizes can be estimated using classical methods but these are estimates that depend upon the mathematics used and have no claim to be the "real" physical size. Protons and neutrons DO have a physical size but this is based on the interactions of forces within these particles rather than on their constituent quarks taking up space. Likewise, atoms have physical size for a similar reason. Within, they are almost completely empty. This zero size not only makes particles non-intuitive in terms seeing them as physical objects; it begs the question, where does their angular momentum come from?

Electron Size

We can say that quantum physics uses a zero-size particle out of mathematical necessity, but there is also experimental evidence that deals a fatal blow to any notion that the electron is a tiny spinning sphere: high-energy electron scattering experiments also indicate that the electron has no physical size, down to a resolution of about 10-18 m. In accelerators these particles scatter in the same manner that points, not spinning spheres, would scatter.

Right about now the uncertainty principle unfortunately interjects into our neat zero-size wrap-up, however. According to quantum theory, electrons are both points and not points (which we will get into) and that means that various "classical" radius measurements still play an important role in many physics applications.

First, just to drive home a point, let's estimate a largest possible classical size for the electron - a radius of about 10-15 m, and then calculate how fast that sphere would need to rotate in order to produce its observed magnetic moment, which is very precisely known to be about -929 x 10-26 J⋅T-1.

But first, a few notes: Where does this measurement of a radius come from? It is calculated as the size an electron would need to have in order for its mass to be completely converted to its electrostatic potential energy, a purely mass-energy equivalency situation using classical electrostatics and a relativistic model of the electron. This use of "classical" should come with a warning because while we usually associate "classical" measurements with real-life measurements, in this case the classical radius of the electron bears no relationship to any physical radius. It is the Thomson scattering length of the electron and this length serves only to offer a "biggest possible" electron "size" to use to make my point:

It turns out that a point on the equatorial surface of a sphere this size would have to be rotating over 100 times the speed of light (page 5 in reference) to account for the strength of its magnetic moment, something prohibited by special relativity. Here, we might be tempted to ignore special relativity for a moment and imagine a spinning point simply spinning all that much faster (approaching infinity) but the framework itself breaks down. A particle with no radius, a point particle, will not lend itself to any mathematical notion of an infinitely tiny spinning charged sphere.

Physicists know that even if we imagine the electron as a point particle we are not quite accurate because we must take quantum field theory into account to properly describe the electron on such a small scale. The electron as a point particle is also described in quantum mechanics as a wave function. The de Broglie wave associated with the electron cannot be spatially localized because of the Heisenberg uncertainty principle. The electron's quantum state instead forms a three-dimensional pattern. This wave function, however, is not the particle. It is the superposition of all possible quantum states of the particle, where the particle itself is considered to be exactly localized somewhere within this "cloud of probability." To clarify further, measuring or colliding an electron collapses the wave function to a single point particle.

A tricky part to this (and I have to laugh here because this whole article is tricky is it not?) is that this cloud of probability theoretically extends forever in all directions (though the probability drops off very rapidly), raising the question of where do you draw the boundary for the electron? You can get at least a partial answer from Compton wavelength. For the electron and any particle, there is a minimum wave function wavelength possible, called Compton wavelength. If you try to localize the electron within a smaller region than this wavelength, the energy of the electron (its momentum) will be so high that pair production will result. Two electrons will annihilate into gamma rays. This gives you the smallest possible space in which a single electron state can exist. The Compton wavelength of the electron is about 3 x 10-12 m (radius of 1.5 x 10-12 m).

Likewise, the quark, which is also a fermion and follows Fermi-Dirac statistics, has no measurable size. Using Compton wavelength, however, you can obtain a smallest possible radius estimate for the quark at 1.6 x 10-19 m according to this recent paper published by a collaboration of authors at CMS (Compact Muon Solenoid Experiment at CERN). The comparison between quark size and electron size based on Compton wavelength (quarks have a smaller Compton radius), we should realize, is more of a statement about differences in the frequencies of their wave functions than any useful statement about physical size.

Dirac's precise and experimentally predictive quantum mechanical model of the electron treats the electron as a point particle. High-energy electron scattering experiments indicate that there is no local physical dimension to electrons (the wave function is collapsed). These experiments don't prove that the electron is a point particle, however. They tell us more specifically that the electron's charge has no spatial extension and shape (not to be confused with the charge cloud of an electron which does have a spatial extension and shape).

The Guts of the Electron: Hello, Is Anyone In There?

There is no evidence that the electron has any internal structure. Current colliders can smash electrons together or other particles with forces as strong as the strong force, and they remain intact, meaning that if a force binds the electron together, it must be stronger than the strong force itself. Furthermore, using even the largest calculated size of the electron - the upper limit of classical electron size - makes the electron so tiny that its bound state would require far more mass (mass/energy) than the electron's measured mass to keep it bound. Therefore, the intrinsic spin or intrinsic angular momentum of the electron is truly intrinsic, just as its charge and mass are. How it is built in, at this point, is just one of those things we are left to fidget over and wonder about.

A point in space with no size doesn't even remotely satisfy our common sense. The electron seems to be more of a quantum field solution than a particle, as we think of particles. Research in quantum mechanics seems to be turning toward looking at the structure of space itself to better understand the puzzle of elementary particles such as electrons and quarks. One example is the work of Werner Hofer, which offers a model of an extended electron, in which the charge of the electron has a physically real density distribution. He suggests that the high-energy scattering experiments indicating point particles could be re-interpreted. I leave it to you to explore this and possibly other options that focus on the electron in terms of a quantum field within space-time rather than as a point or wave/particle.

Unlike elementary particles, high-energy scatterings of neutrons and protons, both composite particles with internal structure, show that these particles do have a physical size.

Proton Size

Unlike the electron and quark, the proton has a physical size, about 9 x 10-16 m. However, this being said the proton is believed to have a fuzzy boundary because it is "defined by the influence of forces that do not come to an abrupt end" as Wikipedia puts it. This size comes from two measurements - measuring the proton's energy level using hydrogen spectroscopy as well as measuring the way electrons scatter off protons when fired at them at great velocities.


The intrinsic spin of particles is far from cut and dried in theoretical physics. The question of exactly what contributes to the (orbital) spin within a proton or neutron is open, as the internal contents of these particles are far more complex than previously thought. Intrinsic spin can be modeled and measured for elementary particles, such as electrons and quarks, but these models are challenging and often the only answers to the many questions we have about spin come in the form of non-intuitive mathematical formulae and proofs.

The even more pressing question of where this intrinsic spin actually originates from remains absolutely open. The best answer I can offer is a negative one - elementary particles are not tiny spinning spheres. They are point particles with no shape and no extension into space and yet they aren't, at the same time. Spin as well as charge and mass are intrinsically built into to them but how? As mentioned earlier and as hinted at in other articles, my personal guess is that physics must find a way to look into the quantum nature of space-time in order to solve some of these mysteries. If there will ever be an intuitively satisfying answer to the question of what gives elementary particles quantum spin, it will come from a more thorough understanding of how quantum fields operate in space-time itself.

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