## Monday, December 15, 2014

### Quantum Chromodynamics (QCD): An Introduction for Laypeople

This is the theory of the strong force. It is probably the most firmly established quantum field theory there is, backed up a great amount of success in predicting and explaining phenomena associated with quarks and gluons - how these fundamental particles bind to create protons and neutrons (together called nucleons) as well as how nucleons bind to create atomic nuclei. This force, one of the four fundamental forces in physics, is unique in that it is by far the strongest of the forces and yet it acts only on the minutest scale - less than 3 x 10-15 m, just a bit larger than the size of an atomic nucleus. The strong force makes atoms, and therefore matter, stable and it contributes to the majority of the mass-energy of atoms. This means that most of the mass of the universe owes its origin to the strong force.

The strong force is technically a theory defined by a quantum field called the gluon field. It is part of the Standard Model of particle physics, which mathematically is a quantum field theory incorporating the fermion field of all matter particles - quarks and electrons, the electroweak boson fields, the gluon field and the Higgs field. In this article we will explore what this quantum field theory is about. It is very difficult to introduce QCD without getting into some technical jargon. There will be a lot of unfamiliar words but I hope to explain why they are there and why they are useful.

QCD, Mathematically Speaking

Wikipedia describes what a quantum field is quite well. For a classical field, a system can be described by a small set of variables; the dynamics of the system is the time evolution of these variables. For a quantum field, the variables are turned into operators (functions over the space of physical states) and the system itself is encoded into a wavefunction, a description of the quantum state of a system of one or more particles. There is a subtle point here: the field itself is not a wavefunction but the equations that govern its time evolution look a lot like wavefunction formulations. Here, unlike in a classical field, there is no variation in strength of the field in different points in spacetime. The variation is instead a difference in something called phase factor, a unit complex number commonly used in quantum physics.

The quantum field theory behind the strong force is described by a surprisingly simple-looking equation, a Lagrangian. The Lagrangian of a dynamical system is the mathematical function that summarizes the dynamics of that system. If you would like a more thorough understanding of the Lagrangian, try this Harvard University physics course (for real enthusiasts or for those who want to go on to a graduate degree).

Although at first it appears reasonably simple, once one begins to explore this Lagrangian in detail, the mathematics involved become very complex and difficult to solve (even for the experts!) and QCD very quickly becomes one of the most difficult theories to understand. There are different versions of QCD (versions of the Lagrangian) that work best for particular problems. For example, lattice QCD examines the thermal behaviour of QCD very well. This is the version physicists use to probe the nature of quark-gluon plasma, where at extremely high energies, quarks and gluons are thought to exist in a free soup-like plasma state, which they believe existed shortly after the Big Bang. A similar physical state, called quark or QCD matter, may be found deep inside intensely dense  neutron stars.

What the mathematics of QCD predicts can be tested inside high-energy particle colliders and there is a great deal of research in this field currently going on. Here, as newcomers to QCD, we will focus on how the theory enhances our intuitive understanding of how gluons and quarks interact inside the nucleon.

QCD Is Part of the Gauge Theory Behind the Standard Model

We have explored the proton in recent articles (for example, look up the subheading "The 1/2 Spin of Protons and Neutrons is a Mystery" in the last article, Why Do Particles Have Spin?). Over the last 30 years our concept of the proton has become much more complex than the simple picture of three quarks and a number of binding gluons interacting together to form a little sphere. Today it "looks" more like a seething fuzzy-edged ball filled to brim not only by three quarks but with many more gluons than first thought and possibly with virtual quark-antiquark pairs as well. This new picture comes from many recent high-energy collision studies where protons are smashed into protons at almost the speed of light. As collider velocity (energy) increases, so do the numbers of particles created, offering an increasingly detailed picture of what is going on inside protons.

QCD, in addition to being a quantum field theory, is a gauge theory. Gauge theory is an important part of quantum field theory. It forms the basis for the Standard Model of particles. These particles are shown below. (MissMJ;Wikipedia)
You might remember gauge theory from a previous article I devoted to it. A gauge theory is a field theory that can be described technically as having redundant degrees of freedom in the Lagrangian. What this technical bit means is that you can have transformations between possible gauges, which in turn form a symmetry group. The following diagram gives you an idea of what a symmetry group means. In this case, the rotational symmetry group of a tetrahedron is shown, where 12 distinct positions formed by rotation create the group.

From this symmetry group (called a Lie group) you get a vector field (the assignment of a vector to each point in spacetime), and when you quantize this vector field you come up with a quantum of each gauge field and this quantum is a gauge boson particle, something you are probably familiar with by now. Photons, W and Z bosons and gluons are all gauge bosons. These gauge field theories describe the dynamics of particles, how they act and interact with each other. QCD is a gauge theory with the symmetry group SU(3) (special unitary transformation on complex 3D vectors). It describes the strong field and its eight associated bosons called gluons.

In this article we will explore how quarks and gluons interact with one another (QCD) by comparing QCD to its closely related and much simpler analogue theory - quantum electrodynamics, or QED. QED may seem familiar to you if you have read past articles, as I have talked about it several times. Like QCD, QED is one of quantum field theory's most important foundation theories. It is a gauge theory with the symmetry group SU(1), the simplest possible transformation of the gauge at a point in spacetime, which transforms via its use of complex numbers. It has one gauge field, the electromagnetic field, and one gauge boson, the photon. To get our bearings, the Standard Model is a gauge theory with the symmetry group U(1) x SU(2) x SU(3). SU(2) is the symmetry group for the weak force or electroweak field and it is associated W and Z gauge bosons.

In 1926, Paul Dirac laid down the foundation for QED with a relativistic quantum theory describing the motion and spin of electrons. The theory was refined in the 1940's by Richard Feynman and others. The basic idea is that electrically charged electrons interact by emitting and absorbing virtual photons, the boson carriers of the electromagnetic force. These virtual photons cannot be isolated or observed but (non-virtual) photons can be emitted in a free state in which they can be observed (the photons of EM radiation). QCD began its development in the 1950's and 1960's with the advent of bubble and spark chambers and the plethora of new particles called hadrons (composite particles made up of quarks) that showed up in them. QCD theory was formulated as an analogy to the earlier QED theory. QCD had to incorporate not just one kind of charge but three (the colour charges). Here, quarks interact through the exchange of boson strong force carriers called gluons. Unlike photons and electrons, gluons and quarks are never observed in a free state with the exception of theorized quark-gluon plasma. These differences from QED make QCD more complex and much more difficult to solve, but the similarities between them can help us to define QCD in terms of how it is structured, and we can't really appreciate the beauty of QCD without looking at this theory's mathematical structure.

Comparing QCD With QED

First, A Look at the Lagrangian

As just mentioned, the dynamics of quarks and gluons can be described by just one fairly simple (in the world of theoretical physics at least!) formula, the Lagrangian, shown below left.

(this and other formulae you see in this article are screenshots taken from the relevant Wikipedia articles)

This is just to show you what it looks like in its simplest form. It is described in more detail in both the Wikipedia QCD article here as wells as in physicist Frank Wilczek's article called QCD Made Simple, which is a great reference for QCD beginners and I highly recommend it. The Lagrangian contains a symbol G (shows up twice in the Lagrangian). This represents the gluon field strength tensor. This is a second order tensor field that describes the gluon interaction between quarks and it is analogous to the electromagnetic field strength tensor  F in the QED Lagrangian shown below (you can see a structural similarity between the two formulas),

This field tensor describes the electromagnetic field of a system. Both of these field strength tensors are four-dimensional mathematical objects (they are formulated in special relativity) that can describe a field (the gluon field in QCD or the electromagnetic field in QED) of any physical system with great precision.

Quarks/Gluons Versus Electrons/Photons

In a sense, QCD is an expanded version of the earlier QED theory. In QED there is just one kind of charge - the electrical charge of electrons; in QCD there are three kinds of colour charge in addition to electrical charge. Evidence for colour charge comes from electron-positron annihilation experiments; the design is well described at Hyperphysics. These colour charges are called red, green and blue but they have nothing to do with physical colours. They are analogous to electrical charge in several ways. Like electrical charge, they are conserved. In QED, the massless spin-1 photon responds to electrical charge. In QCD, massless spin-1 gluons respond to colour charge. Whereas the electron, a spin 1/2 fermion, carries electrical charge, the quark, also a spin 1/2 fermion, carries colour charge (we will explore additional comparisons between gluons and photons later on in this article).

In QED, the quark's analogue is the electron. Quark dynamics, however, are a bit more complicated than electron dynamics. Quarks come in six possible flavours - up, down, strange, charm, top and bottom. These six flavours are grouped into three generations.

First generation quarks are up and down quarks. These have the lowest mass and they are the only stable quarks, the only flavours of quark found in atomic matter in other words. All flavours of quarks are perfectly symmetrical with respect to colour charge. This means you can have any flavour of quark with any one of the three colour charges. This symmetry is described by the Lie group SU(3) in the Lagangian and it is part of the gauge group that the Standard Model is built upon. The electron in a similar fashion is one of six possible lepton flavours - electron, electron neutrino, muon, muon neutrino, tau and tau neutrino. These six flavours also happen to be organized into three generations. The lowest mass electron and all the neutrino flavours are stable but only the electron is a part of atomic matter.

Adding bosons, particles of force, to our list of quarks and leptons we have the following table (below left) useful for reference (this includes only the stable quarks and leptons as well as their antiparticles):

Unlike an electron's single e- charge, a quark can carry a single unit of any of three possible colour charges IN ADDITION TO an electrical charge. More specifically, quarks carry fractional electrical charge. Up, charm and top quarks have a +2/3 electrical charge, whereas down, strange and bottom quarks have a -1/3 electrical charge. This is why a proton (UUD) has a +1 electrical charge and a neutron (UDD) has no charge.

QCD Coupling Constant Versus QED Coupling Constant

The strong force, as mentioned, is much stronger than any other fundamental force. It is only effective up to a distance of about 10-15 m, but it is 137 times stronger than electromagnetism, the next strongest force. Gluons respond to the colour charge of quarks much more vigorously than photons respond to electrical charge. The QCD coupling constant in other words, is much stronger than the QED coupling constant. The Lagrangian of either theory has two general parts - an interaction part and a kinetic part. The coupling constant determines the strength of the interaction part. As we will see later on, knowing the relative strength of different coupling constants gives you a good idea of how the dynamics of a system will play out in reality (a hint: quarks have both electric and colour charge, but they have very different coupling constants - how do these forces play out inside a proton or neutron?)

In particle physics, where high-energy colliders are used, you may hear the term QCD scale to describe mass-energy. It is around 200 MeV. Above this energy density, quarks and gluons are thought to exist as separate components (free particles) in quark-gluon plasma (QGP). Below this energy, they are confined inside hadrons. This is the energy at which QCD coupling gives way. The temperature at which quark gluon plasma is thought to exist, 2 x 1012 K, corresponds to the kinetic energy of about 200 MeV per particle. Colliding two large nuclei together (at CERN SPS (http://home.web.cern.ch/about/accelerators/super-proton-synchrotron) (lead nuclei) and BNL RHIC (gold nuclei) colliders) can achieve this energy, and there is evidence that droplets of QGP have been created in such "fireball" collision events.

A neutron star is much cooler than 1012 K, but its core is so dense that quark matter may exist there. At around 400 MeV of chemical potential energy, hadrons are squeezed together so tightly that they should melt into a quark liquid (rather than the plasma state described above).

The strength of coupling changes depending on the energy of the system. This is where QED and QCD show remarkable and interesting differences. In QED, according to perturbative theory, the beta function is positive. Perturbation is a set of mathematical approximation schemes that allow you to study a very complicated system based on your knowledge of a simpler one. A subfield of QCD is perturbative QCD. These approximation techniques can be used in high-energy cases where the strong force coupling constant is small. It is limited in scope but absolutely essential to make testable predictions of QCD possible because under most circumstances QCD is almost impossible to solve mathematically.

A QED positive beta function tells us that as energy increases, the coupling becomes stronger - photons respond much more strongly to electrical charge at high energy. Quark coupling is different in two important ways. First, the coupling strength DECREASES logrhythmically as energy increases. This is a phenomenon known as asymptotic freedom, which means coupling increases with decreasing energy. In atoms at everyday temperatures, quarks and gluons are very tightly bound inside the nucleus but at very high temperatures (high energy) quarks and gluons are only very loosely bound and transition into a quark-gluon plasma instead.

The second difference is something called confinement. In QED, the attractive force between (virtual) photons decreases with distance. Think of pulling two play magnets apart. It gets very easy as soon as you have a few millimetres of distance between them. In QCD the attractive force between gluons does not decrease with distance. This is also very different from the electric field of charges such as electrons in the atom. The electric field extends into space and weakens with distance. This means that electrons furthest away from the nucleus experience less attraction than those closer to the nucleus. These outermost electrons are freer to interact electrically with other electrons to create chemical bonds and they can leave the atom altogether creating an ion. Quarks are tightly confined inside the proton or neutron.

The Strong Coupling Constant Means Jets In Collider Experiments

When a quark-antiquark pair is produced in a high-energy situation and it then separates, the gluon field is believed to form a narrow tube of colour field between them, much like an elastic band. The strong force remains constant so eventually, as the tube narrows, it becomes thermodynamically favourable for the tube to snap and when it does a brand new quark/antiquark pair appears in its place. This means that you can never pull a quark off of a proton or neutron and study it as a free particle. One quark just leaves and a new one pops up instantly in its place because the strong force has enough energy to make a replacement.

When quarks do become separated from a nucleon or when new quarks are formed, as happens in high-energy particle colliders, physicists see jets, narrow cones in other words, of mesons (a quark/antiquark pair) and baryons (particles containing three quarks) rather than free new quarks. No free quarks have been found in any collider.

Below is a Fermilab image, showing the production and decay of a top quark and antiquark (CDF top quark event), after a proton and antiproton are smashed together. The top quark and anti-top quark live too briefly to be seen, but they show themselves by their decays into jets of other particles (green and yellow lines) and two weak force W boson particles. One W boson decays to a positron (light magenta block, lower left) and an unseen neutrino (red arrow, bottom); the other W boson turns into two mesons (part of the spray, right).

The easy part of this kind of experimentation is that mass-energy going in always equals mass-energy coming out. Here, we have two proton masses going in at extremely high velocity. Remember that the proton is about 100 times more massive than its constituent quarks - gluons are massless - because the strong force contributes significant mass-energy. What comes out is the most massive particle/antiparticle pair known; the pair is about 70,000 time more massive (about 173 GeV x 2) than either up or down quarks (1.7 to 5.8 MeV each x 6). But we don't see the top quark because it has a life span of about 5 x 10-25 seconds. Interestingly this is not enough time for the top quark to decay into hadrons as other quarks do. Instead, it decays through the weak interaction producing W bosons and other particles. The complex-looking image above is the closest physicists have come to a "portrait" of this super-massive particle (has about the same mass as an atom of tungsten!). The top quark was predicted to exist in 1973 but was not discovered until 1995 with the CDF experiment at Fermilab.

Wilson Loop: Why Quark/antiquark Mesons Can Exist

You might wonder why mesons (quark/antiquark pairs) don't always instantaneously annihilate when they are formed (note: they are also unstable and quickly decay anyway). The answer has to do with something called a Wilson loop. This is the path in spacetime traced by such a pair created at one point and annihilated at another point. In a confining theory such as QCD, the loop action is proportional to its area, so the area is proportional to the separation of the quark and antiquark. This loop action therefore suppresses free quarks. In mesons, however, which are quark/antiquark pairs, the Wilson loop contains another loop in the opposite direction, leaving only a small area between the loops, meaning that meson particles are allowed. The Wilson loop creates an excitation of the colour quantum field that is localized on the loop. In QED, Faraday's fluxtubes are analogous excitations of the quantum electromagnetic field.

Deconfinement: A Phase Transition of QCD

Quark-gluon plasma (QGP) is currently only theoretical although there is some indirect evidence it has formed in extremely high-energy collider experiments. For example, RHIC (Relativistic Heavy Ion Collider) released a press release last year (2013) that indicates they may have created tiny droplets of the plasma by smashing gold nuclei together at almost the speed of light. This plasma is also called the deconfining phase of quarks. It consists of asymptotically (weaker coupling as energy increases) free quarks and gluons. Plasma is a state of matter in which charges are screened by a sea of other mobile charges. In the plasmas we encounter on Earth (neon signs are an example), in the Sun and in outer space, it is the electrical charges of ions and free electrons that are screened. In QGP, the colour charges of quarks and gluons are screened.

Lattice QCD Theory Handles Deconfinement

This colour charge is too great to deal with using perturbative computations so it must be dealt with using a different branch of QCD called lattice gauge theory instead. The phase transition from confined quarks and gluons inside hadrons to deconfined quark/gluon plasma is best modelled using lattice QCD, and a number of those models are being tested at both the RHIC and the Large Hadron Collider, two colliders that can collide heavy nuclei together and reach the incredibly high energy density required to deconfine quarks.

One recent formulation of lattice QCD can accurately describe both the masses and the symmetries of the quarks, and using this data, it can predict the critical temperature where the phase transition occurs and it predicts that it is a smooth process rather than an abrupt one. To do this, lattice QCD puts quarks and gluons on a four-dimensional spacetime lattice. Then it turns the problem of summing over all possible quantum configurations of quarks and gluons into a very high dimensional integral that is estimated by using statistical sampling. This method greatly simplifies the daunting number of calculations that would be required (by supercomputers). It describes a smooth transition beginning to take place at around 155 MeV of energy. It can also be used to describe the equation of state of the plasma, and this in turn allows physicists to explore the physical characteristics of it.

Lattice QCD also offers insight into the nature of hadrons. For example, QCD computations show that a meson is not just a paired up quark and antiquark. "Fluxtubes" of gluon fields are also significant components of the meson structure. The image below is from a QCD lattice computation of meson's colour fields (the directions of the flux tubes shown as black lines). It is a computation of a hybrid Wilson loop.

The word fluxtube is borrowed from the analogous magnetic fluxtube of QED, a cylinder of space containing a magnetic field. Sunspots on the Sun are associated with large magnetic fluxtubes, for example, and they form a part of the structure of the Sun's atmosphere.

Chiral Symmetry Breaking: A Challenge For Lattice QCD Theory

There is one drawback to lattice QCD however. It does not account for the chiral symmetry of quarks. Chiral symmetry in QCD is analogous to magnetization in QED. Above a critical temperature, a magnetic material such as iron has no magnetic moment and it is rotationally symmetric. It is demagnetized in other words. Below the critical temperature, however, it spontaneously remagnetizes, and this phase transition breaks its SO(3) rotational symmetry (this is the theory behind why a heated magnet loses its magnetization and then regains it once it cools, although its new magnetic orientation may now be completely different).

It has been shown experimentally that asymptotically free quarks have chiral symmetry. In terms of theory, left-handed and right-handed quarks (this handedness means spin direction relative to the particle's momentum) are treated exactly the same in the QCD Lagrangian when they are asymptotically free. But when they form hadrons (mesons and the baryons of matter), that symmetry is broken (and it can be restored once again at high energy). QCD theory somehow must undergo spontaneous symmetry breaking, in which a symmetrical state ends up in an asymmetrical state. The Lagrangian obeys chiral symmetry but the lowest energy solutions to it must not exhibit this symmetry (the up and down quarks do not come in left-handed/right-handed pairs - they are all left-handed). This is a big bump in the simple elegance of QCD. Solutions to the theory need to take both processes of chiral symmetry-breaking and confinement into account at low energy and it's not easy because these are two very complex phenomena which the simplest form of QCD just can't handle. However, computers are being used to dig into this complex solving procedure, showing good agreement between the expected calculated masses of hadrons coming out of high-energy collisions and their measured values.

No fermions of matter, particles with mass such as quarks and electrons, exhibit chiral symmetry. Massless bosons, however, do (and at high energy the mathematics of QCD describe a "massless quark" that is chirally symmetrical). For a more in depth discussion of this, chiral perturbation theory is discussed in detail here. The process described by spontaneous chiral symmetry breaking is very significant because it accounts for almost all the mass in the universe (excepting the mass contribution of electrons). This process converts very light mass quarks into baryons (protons and neutrons) which are about 100 times more massive, thanks to the contribution of the mass-energy of the strong force. The lattice formulation we were discussing earlier is not compatible with the theoretical process describing chiral symmetry breaking, prompting a number of other versions of lattice QCD, each of which has advantages and disadvantages.

Gluons Versus Photons: Colour Charge

In keeping with the QED - QCD comparison theme of this article, it is especially interesting to see how the photon of QED compares with the gluon of QCD. The photon comes in one type and it is electrically neutral. In general, photons do not interact with each other (with the exception of recent research indicating photon-photon interaction is possible under specific conditions). Under most circumstances, however, a photon going in one direction will go right through another photon going in the opposite direction. I should make the distinction here that photons do not specifically couple with one another but light does interact through higher order processes (interference, superposition etc.)

Photons don't couple because they carry no charge but gluons do carry colour charge just as quarks do. Gluons, in addition to responding to colour charge, can also change one colour charge into another colour charge, and this means that gluons can and do interact with each other. As an example, a blue quark will change into a red quark when it absorbs a specific gluon, which means that the gluon itself must carry colour charge - one unit of red charge and one unit of antiblue charge in this case. Gluons interestingly carry two colour charges consisting of one colour charge and one anticolour charge. A quark, in contrast, can have just one of three possible colour charges. Antiquarks (antimatter twin particles of quarks) can have one of three possible anticolour charges. Incidentally, both the gluons of QCD and the photons of QED (both are spin-1 bosons) are their own antiparticle.

These three colour charges, this additional quantum number in other words, makes otherwise identical quarks in some hadrons (delta baryons possible without violating the Pauli exclusion principle, which states that two or more fermions cannot occupy the same quantum state at the same time.

All three colours mixed together, or a colour and its anticolour opposite, produce white, or a net charge of zero in other words. This means that free particles (protons, neutrons and mesons) always have zero colour charge.

Overall, there are six possible colour charges and eight possible gluon colours.

Why There Are Eight Gluon Colours

Gluon theory describes an octet of gluon colour fields. Each individual gluon has a combination of two colour charges (one of red, green or blue and one of antired, antigreen or antiblue). These two colour charges exist in a superposition of states, which are mathematically given as Gell-Mann matrices. These are an adjunct representation of the SU(3) component of the gauge group, mentioned earlier (the three colour charges of the quarks is the fundamental representation of the gauge group). The Lie algebra, which represents this group has a dimension of eight. Therefore it is a set of eight linearly independent generators, which can be written down as two rows of four below right (these are mathematically equivalent to the eight Gell-Mann matrices).

(The square root of 2 and square root of 6 are required for normalization (a corrective adjustment procedure) of the equations)

Notice how this formulation is entirely of mixed colours. If you could directly measure the colour charge of particular gluon, for example red-antiblue plus blue-antired (that's the top left generator above), the gluon would have a 50% chance of being found as red-antiblue and 50% chance of being found as blue-antired. These eight combos are the only possible combinations of states that are totally independent of one another and independent of the one permitted singlet state (see bottom right above). An additional colourless singlet can also be calculated as part of the matrix, shown below,
(there are actually nine matrices in the calculation) but this last one is forbidden in reality because it would mean that this gluon could exist as a free particle (and give rise to the strong force with infinite range as well).

Below right is an animation of the mechanism of the strong force, showing an exchange of gluons and colour charges. (Manishearth;Wikipedia)
Quark Electrical Charges Compared to the Electron's Charge

In contrast to the quark, the electrical charge of an electron comes in just one kind. Positive and negative charges are different aspects of the same fundamental electrical charge - they cancel each other out. The question of why some quarks have a fractional positive charge (up, charm and top quarks have a +2/3 charge) and why others have a fractional negative charge (down, strange and bottom quarks have a -1/3 charge) is a mystery, just as why electrons happen to have a -1 charge is. Assigning positive and negative is arbitrary and interchangeable in the theory and the use of fractions is also in a sense arbitrary because it simply balances the atomic charge - you can say the quarks have charges of  -1 and +2 and the electron has a charge of -3 instead. This does not answer the underlying mysterious question of why the electron carries more charge than either quark group and why the quark electrical charges come in two strengths and opposite signs. The mechanism of how electrical and colour charges arise in particles is unknown.

Strong Force Between Quarks and Between Nucleons

As mentioned, the strong interaction between quarks is so strong that it has enough energy to produce new particles - a new quark is produced in the place of a quark that is "knocked out" of the nucleon during very high-energy collisions. This force is confined to a scale of about 0.8 fm (10-15 metres), which is the radius of a nucleon, but it does not decrease with distance. This is colour confinement and it prevents the emission of the strong force. Instead of the emission of the strong force, one sees jets of massive particles produced in the collider.

The nuclear force binding protons and neutrons together in a nucleus is also called the residual strong force. This force is thought to be the residuum of the strong interaction between the quarks inside. It is more than powerful enough to overcome the electrostatic repulsion that exists between the positively charged protons in the nucleus. It is this force that is also present as nuclear binding energy and this is what is released during nuclear fission reactions, which are used in nuclear reactors and nuclear bombs. This residual version of the strong force operates under different rules that the strong force between the quarks. It is powerfully attractive at distances of about 1 fm but becomes insignificantly small at distances over 2.5 fm. At distances of less than 0.7 fm, it becomes repulsive, thus it maintains a minimum physical nuclear size.

Field Lines Demonstrate The Meaning of Confinement

In analogy to QCD, electric and magnetic field lines can be drawn just as strong force fields can be drawn. Comparing them offers visual clues to the difference between a confined field and an open field. Below we can see how field lines are drawn for electrical fields. Two repulsive positive charges in close proximity are shown left and two attractive opposite charges are shown right. (http://commons.wikimedia.org/wiki/User:Chanchocan;wikipedia)

The magnetic field lines of a magnet are shown below. Notice how they look essentially the same as those between two attractive charges (above right) because the two magnetic poles are also attractive (attractive force is depicted as field lines connecting the opposite charges/pole). Electrical charge and magnetism are two manifestations of one single fundamental force, electromagnetism, and it deals with all the phenomena associated with QED. In all cases, at least some field lines are shown radiating away from the source of the field. Electromagnetism follows the inverse square law, whereby the intensity of the field is inversely proportional to the square of the distance from the source of the field. This is shown as increasing distance between field lines. In theory, then, the field radiates outward to infinity, though it eventually becomes imperceptibly weak. Geek3;Wikipedia

Like electrical field lines, the strong force between quarks can be drawn as field lines depicting the colour field (see below). A key difference between the two is that while electrical field lines can be drawn radiating away as well as arcing between attractive charges or magnetic poles, colour field lines never radiate away and the arcs are narrowed from one charge to another because they are pulling tightly by gluons (this is not really depicted that well). This confinement of the field confines the quarks within the nucleon.

Below, colour fields are shown for a quark (top center), antiquark (top right; the anticolours are shown as cyan, yellow and mauve) and three mesons (bottom). Maschen;Wikipedia
The overall colour charge of each particle shown above is white or neutral. No free particles (mesons, quarks and other exotic baryons) exist with (unbalanced) colour charge.

All of these diagrams are two-dimensional out of necessity, but the mathematics behind them actually describes QED and QCD in four-dimensional spacetime. G (in the diagram above) is the gluon field strength tensor, a second order tensor field (geometric mathematical description) that characterizes the gluon interaction between quarks. In QED, an electromagnetic field tensor (F) is used. Classical electromagnetism is described using two three-dimensional vector fields. In QED electromagnetism is brought into one single description that is described in four-dimensional spacetime.

In QCD, the field tensor additionally describes self-interactions between the gluons as well as asymptotic freedom (you don't see that in the simple diagram above). These additional complications of the QCD field tensor make QCD non-linear compared to QED, which is mathematically linear. Unlike a linear system, a non-linear system does not satisfy the superposition principle, which means the output of the nonlinear system is not directly proportional to the input. This makes nonlinear equations much trickier to deal with and is part of why QCD is so difficult to solve compared to QED problems.

QCD, being a quantum theory that encapsulates special relativity like QED, is described on a four-dimensional spacetime framework. In lattice QCD, that framework describes discrete blocks of spacetime whereas in other QCD theories it is described as continuous.

There is a lot of fancy and elegant theory and math here. Yet, quarks and gluons have never been directly observed in any collider. No (massless) gluons or fractionally (electrically) charged particles ever reveal themselves. In a high-energy collider, for example when two gold nuclei are slammed into each other at near light speed, (coloured) quarks and gluons manifest themselves by fragmenting into more quarks and gluons and these immediately hadronize in colourless free particles such as mesons, protons and neutrons. This process shows up as coplanar jets (bunches of particles all travelling in the same direction) in the collider. Most particles in these collisions appear clustered into one of three jets, called a three-jet event. A three-jet event is the most direct physical evidence there is for gluons. Jets are produced when quarks hadronize, and quarks are always produced in pairs (one quark and one antiquark). This explains two of the jets. Some other particle is needed to explain the third jet, and this is theoretically predicted to be a very energetic gluon radiated by one of the quarks. This gluon is so energetic that it hadronizes as well, just like the quarks do. The theory matches up beautifully with experimental observations in colliders.

Conclusion

While QCD is an especially difficult theory to get a grasp of let alone solve, it is also probably one of the most elegant mathematical formulations in theoretical physics, given its simplicity in terms of describing a multitude of very complex phenomena. It is gratifying that this is one theory that can be experimentally tested (high-energy collision experiments) and there is a great deal of indirect verification of the elusive free quark and the gluon, as well as tantalizing work underway to create and to get a good look at mysterious quark-gluon plasma and quark matter, results which are bound to reveal more secrets about what our universe was like just mico-moments after the Big Bang as well as what's going on deep inside stellar beasts like neutron stars.

It is all too clear to me after researching this material that I've only scratched the thinnest surface film of this remarkable theory and I have neglected most of the theories that branch off of basic QCD. There are several different approach strategies for probing QCD further such as continuing to try to solve it with the aid of computer technology, running ever higher-energy collider experiments that can probe deeper into the strong force, and working up various models for QCD in, for example, fewer dimensions or higher dimensions and/or string theory to see if simpler or easier to solve solutions to the Lagrangian are possible. These are just a few possible directions for research that physicists are undertaking. QCD has lots of space for future theorists and experimentalists to try to tweak apart the tightly held mysteries of the strong force.

Next in Quantum Mechanics, try out Why Do Particles Have Spin?

## 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. JohnCD;Wikipedia
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.

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.