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.

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 |

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.

JohnCD;Wikipedia |

*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 ML^{2}T^{-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.
Conclusion

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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