Level 1: Spacetime — \(0\text{s}\)

• Fundamental quanta

  • Size: \(10^{-35} \text{m}\)

  • Initial state: Big Bang monistic substance

  • Production process: Gravity decoupling

  • Meta-stable equilibrium: GUT Quantum fields

  • Stable future: Vacuum with macro-fluctuations

• The Universe

  • Temperature: \(10^{32} \text{K}\)

  • Initial state: Big Bang monistic substance

  • Production process: Compactification

  • Meta-stable equilibrium: Curved 4D universe

  • Stable future: Frozen / Ripped universe

Level 1 describes structures that appeared at the same time as our universe. We’ve talked a lot about other worlds and the possibility of the multiverse in previous sections. To be clear, when talking about "the universe" here, I refer to the entity that originated in the Big Bang (which we know occurred thanks to the Cosmic Microwave Background surrounding us). It might well be part of a larger structure (such as a world undergoing eternal inflation, with Big Bangs simmering here and there), but I’ll stick to the conventional universe.

It is tempting to picture the Big Bang as the explosive growth of a tiny "egg" at an unimaginable pace. But remember that the universe might well be infinite. In that case, it hasn’t been growing in size following the Big Bang. Expansion just means that the density decreases everywhere: an infinite fabric is getting stretched more and more — but it was already infinite, and stays infinite. That is why there is no size information under \(1\Omega\) but rather a temperature metric.

To study what happened at or right after the Big Bang requires what Gamow called "\(G\,\hbar c\) physics".

The physics cube

\(G\), \(\hbar\) and \(c\) are respectively the gravitational constant, the speed of light and the Planck constant.

In classical mechanics, one can ignore gravitation (\(G=0\)), light is assumed to move infinitely fast (\(c=\infty\)) and energy transfers are taken to be completely continuous (no quanta: \(\hbar=0\)). For each constant, we have the choice to "switch it on" or not, resulting in the "physics cube".

• Newton’s Universal Gravitation (UG) is introduced.

• Einstein’s special relativity (SR) reconciles classical physics with a finite \(c\).

• Planck, Einstein and Bohr consolidate early quantum mechanics (QM).

• Einstein’s general relativity (GR) elegantly expands SR to explain gravity.

• Dirac’s quantum field theory (QFT) incorporates QM with SR.

• Non-relativistic quantum gravity (QG) has never been seriously attempted.

• The holy Grail of physics, supposedly the key to a "theory of everything" (TOE).

The current dichotomy between the Standard Model of quantum physics and General Relativity is tenable because each theory has its own domain. At macroscopic scales, quantum effects are negligible, so \(h=0\) (the semi-classical limit) is a good approximation. At microscopic scales (molecular or below), quantum effects become significant but it is gravitation that is unimportant: the mass of the systems under consideration is too small, or to put it another way, spacetime curvature looks locally flat over the tiny patch that we’re observing.

However, those remarks are true only in the context of "normal physics", that is at the levels of energy and at the scales that we can find in outer space, on earth, or in a star’s core. In extreme conditions, both gravitational and quantum effects become essential, hence the quest for quantum gravity as the only type of model that can teach us about the early universe.

Yet quantum gravity is not just as a needlessly precise version of our current physics, nor one that is only valid in specific environments such as black holes: if we choose to focus on a very small patch of spacetime, then by virtue of Heisenberg’s uncertainty principle, quantum fluctuations will be enormous enough to bring us back to energy levels comparable to the very early universe.

Planck’s natural units

In fact, \(G\, \hbar c\) also yields the system of Planck units which define the scales at which quantum gravity effects dominate.

\[\begin{aligned} &\text{Planck length} && l_P= \sqrt{\frac{\hbar G}{c^3}} && 1.6 \times 10^{-35} \text{m}\\ &\text{Planck time} && t_P= \sqrt{\frac{\hbar G}{c^5}} && 5.4 \times 10^{-44} \text{kg}\\ &\text{Planck mass} && l_P= \sqrt{\frac{\hbar c}{G}} && 2.2 \times 10^{-8} \text{kg}\\ &\text{Planck energy} && l_P= \sqrt{\frac{\hbar c^5}{G}} && 1.2 \times 10^{19} \text{GeV}\\ &\text{Planck temperature} && l_P= \sqrt{\frac{\hbar c^5}{Gk_B^2}} && 1.4 \times 10^{32} \text{K}\\ \end{aligned}\]

These are indeed the characteristic values of physical metrics right after the Big Bang. And as noted above, since Heisenberg’s relations teach us that \(\Delta E \Delta t \geq \hbar\), then zooming on the Planck scales should introduce quantum fluctuations with amplitudes comparable to the Planck energy, and virtual particles as heavy as the Planck mass popping up...

The Planck epoch: \(0\) to \(10^{-43} \text{s}\)

The period betwen the Big Bang and one Planck time is known as the Planck epoch. Energy levels throughout the period are unknown (it’s not even clear if that concept would make sense), but at the end of that period, temperature is still at an unfathomable \(10^{32} \text{K}\).

Entropy is also very low (estimated around \(10^{88} k_B\)), which is quite reassuring since it gives the universe room to grow before becoming fully homogeneous (and therefore time for complexity to develop). At the same time, something might seem counter-intuitive. Didn’t we say that highly homogeneous states had high entropy, and isn’t the early universe expected to be largely homogeneous? In fact, the universe did not have any structure or asymmetries indeed, but the combination of an extremely strong interaction and of immense quantum (therefore, random) fluctuations actually yields a state of very low entropy (relative to the energy).

If I had to summarily depict the universe in the Planck epoch, I would say: one object, one interaction. Indeed, the kind of distinctions that seem fundamental in physics: between spacetime and its content, between interactions and the particles they act upon, between radiation and matter, make no sense at the Planck energy level. There’s only one object, the universe, and no clear way to distinguish anything more fundamental.

Unified forces

The root cause is the unification of all forces at such energies. To fully unpack what that means, let’s refresh our memories a tad...

There are four fundamental forces in the world today (interactions) that explain all physical phenomena.

1. The strong nuclear force binds quarks together into protons, neutrons and other hadrons. These structures neutralize the strong force but the residual is strong enough to induce a powerful attraction that "glues" protons and neutrons together, ensuring that atomic nuclei can exist. It doesn’t behave like forces we are more used to, such as electromagnetism or gravitation: between quarks the strong force actually grows with distance, ensuring that apart from super-high energy environments, quarks have no way of being free.

2. The electromagnetic force is more familiar: it is the source of all the electric, magnetic, and optical phenomena (including, more generally, all EM waves). The attraction or repulsion between electrically charged bodies quickly decreases with distance.

3. The weak nuclear force is mostly evident at our scale in the form of beta radioactive decay. Unlike the other three, it is characterized by a short range: long-distance interactions are always possible but vanishingly unlikely beyond microscopic scale.

4. The gravitational force is a well-known attractive force between all entities that have mass or energy (which covers every physical object). It has an infinite range. General relativity interprets it as a manifestation of spacetime curvature.

None of the objects we’ve just mentioned (protons, neutrons, electrons...) exist at the Planck epoch. But we want to give a sense of what the forces do, and how they could be unified during the Planck epoch... First, what does that even concretely mean? Unification has always been a powerful driving force in the history of science, taking seemingly disparate theories and finding they’re entirely reducible to a more fundamental one. That leads to ontological parsimony, epistemological efficiency, and most of all to great beauty. Physics offers textbook examples of unification:

• Universal gravitation: Johannes Kepler’s laws of planetary movement and Galileo Galilei’s mathematical representation of gravity were both landmark achievements of the seventeenth century, but must have seemed to contemporaries as very disjointed in their scope. What could be in common between an apple falling to the ground and a planet orbiting the Sun? Isaac Newton’s concept of gravitation elegantly explains both by integrating a new gravitational force with his laws of dynamics. The apple that I throw and which falls on the ground is as much in orbit as the moon, only given its low speed, it is an unstable orbit that ends in a crash. And the moon, in a stable orbit, is constantly "falling on the earth", and missing.

• Electromagnetism: Electricity and magnetism are amongst the oldest physical phenomena to have been discussed by "natural philosophers". Similarities had always been observed between the two, despite one major difference. Electrically charged objects can exist independently from each other, exerting an attractive or repulsive force (unlike gravitation, electrical interactions act on charges that can be positive or negative). But magnetic objects found in nature are always dipoles, endowed with a North and a South pole (like the needle of a compass, which obviously inspired this terminology). Coulomb’s law and later Gauss’s formulation aimed to describe these two forces in isolation.
  But important connections were evidenced between them. Ampère, Biot and Savart’s work formalized the earlier observation that an electrical current generates a magnetic force; in fact a circular current would create a field that approximates an actual magnet, suggesting that magnets’ properties have their source in micro-currents. Lastly, Michael Faraday discovered induction of an electrical current by a varying magnetic field.
  James Clerk Maxwell, in a work of physical and mathematical acumen that can only be compared to Newton and Einstein’s achievements, re-framed all these laws as part of a simple theorization of electromagnetism as a whole. The objects under consideration are not charged bodies; that Newtonian style of physics was replaced by a focus on fields: Maxwell’s theory is a dynamics of the electromagnetic field. From these equations in a vacuum, Maxwell landed on a wave equation, whose solutions has a propagation speed remarkably close to the speed of light. Maxwell had proved that light was in fact not an irreducible entity, but an electromagnetic wave, governed by the laws of electromagnetism. All the conclusions of optical theories could then be folded into Maxwell’s.

• Statistical mechanics: The development of thermodynamics during the nineteenth century had turned the old calorics (science of heat) into a rigorous theory of gases, where pressure, temperature, entropy played the same fundamental role as mass or velocity in classical mechanics. The newly minted concept of energy played the role of a bridge between these domains. Boltzmann’s incredible insight was not only to espouse the right side in the controversy over atomism (thereby positing that a gas is made up of molecules in an excited state), but also to run away with the idea, developing the appropriate statistical model, and showing that all thermodynamic quantities were just statistical moments (average, volatility...) of the molecules’ mechanical properties (mass, momentum...). Boltzmann’s work was the first step in unifying thermodynamics with classical mechanics.

Based on this triumphant history of unification in physics, it may come as no surprise that similar successes bloomed in the twentieth century. But the unification of fundamental interactions that quantum physics has achieved is different: it is both conceptual and historical.

Newton, Maxwell and Boltzmann proved a connection between phenomena that were thought to be grounded in different forces, but actually never were. The acceleration of a falling apple relates to the orbit of the moon in a way that’s always been the same, since there is no qualitative difference, only circumstantial ones (the earth is big and I cannot throw the apple fast enough). They are two facets of the same interaction.

The modern unification of the fundamental forces does not attempt to correct the record on our understanding of the present: we are right in considering those four interactions as distinct since they do have very different strengths. The idea is that:

1. Their relative strength varies with energy levels: they are closer to each other at higher energy.

2. Since the energy density of the universe has decreased over time, so have the fundamental forces diverged from each other.

3. Past the energy level where two or more forces have reached an equal strength, that relationship remains stable: the forces can be considered to be unified.

All of this ought to be described by a single theory, which would describe both the high-energy, unified state of affairs, and the low-energy environment we’re experiencing today.

This is quite remarkable. Unification is not just a metascientific process indicating progress in our understanding of the world. It is both that and a time machine showing us what actually was. We’re not just unifying theories, we also describing the very physical, very concrete fact that if we turn the temperature high enough, these forces will be unified again!

Coming back to our main throughline: what marks the end of the vanishingly short Planck epoch is that gravity separates from the electronuclear force (the unified composite of the strong, electromagnetic and weak interactions). The intensity of gravitational interactions, which was equivalent to that of electronuclear interactions, starts to undergo a relative decline.

What does it mean for an interaction to be more or less intense than another? Let’s put it simply: if you drop an iron nail on the floor, a small magnet should be enough to have it "fly" towards you. The attractive force from that small magnet has handily beaten the attraction from the whole planet Earth... Today, electromagnetism is about \(10^{37}\) stronger than gravity. The only reason that gravity seems so mighty, creating galaxies, solar systems, keeping our feet stuck to our planet, is that most matter in the universe is electrically neutral, being either a plasma, or made of atoms. When a stronger force gets neutralized that way, and only then, a weaker one can dominate.

The strong nuclear force is even more intense, 100 times more than electromagnetism. The mere existence of atoms is striking evidence of that: how else could positive protons be so tightly bound within a nucleus otherwise? Electromagnetic forces are desperately trying to separate them, but they stubbornly stay together, so much so that it takes a tremendous energy to break nuclei... And we’re only talking about the very frail residual of the strong nuclear force: the one that acts between nucleons, and not between their quark components! The weak nuclear force is a specific case: despite its name, it is not the weakest. It’s a million times weaker than electromagnetism, but still \(10^{31}\) stronger than gravitation. However, its concrete effects are alleviated by its tiny range.

The idea that interactions’ intensity has varied over time may seem strange. After all, aren’t we touching to the most fundamental aspects of the physical world? That stems in fact from the deep nature of quantum field theories. They don’t represent forces the way Newton did, as two particles acting on each other, or like Maxwell did, as field acting on a charge. In the quantum world, what we at our scale perceive as an attractive or a repulsive force is the result of exchanging a virtual boson (a force-mediating particle). For instance two electrons exchange a photon (the mediator for the electromagnetic interaction). But that’s not all: one of the main teachings of quantum field theory is that the vacuum, far from being empty, is constantly disturbed by virtual particle/antiparticle pairs popping up. The more energy around, the more such events. The quantum field of a charge particle carries energy, and that particle is then "dressed" by a could of virtual pairs. An electron is never just an electron then: it is constantly surrounded by myriads of electron-positron pairs.

All those virtual particles intervene in the calculations of concrete, measurable values, and ultimately of the coupling constant (the measure of interaction strength) that we observe today. But now, it should be clearer how that is not really a constant: at very high energy levels, the production of virtual particles is bound to change according to quantum-relativistic effect; therefore, the interaction’s coupling "constant" is equally bound to vary.

Quantum gravity and spacetime compactification

As we turn back time from today to the most distant past (or equivalently, "turn up the heat" from a cool summer night to the fiery Big Bang furnace), we should then see the weak interaction become unified with electromagnetism, then the resulting electroweak force wed the strong force to make the electronuclear interaction, which would then become one with gravitation.

Conceptually, it is now understood. Mathematically, it also means that we should have a unique formalism that could encompass general relativity and quantum field theories at lower energies. We don’t have that yet, although the work done to outline the superstring landscape is giving us precious hints, which we’ll go on to describe soon. But most of all, we should understand concretely why the Planck epoch reality was so different and monistic.

In classical physics, things are pretty clear: space and time are the background onto which physics happen, not an actor of physics itself. In modern physics, things are a bit more complicated: spacetime is definitely an actor in general relativity, which is at its core a law of dynamics for spacetime itself. But quantum field theories ignore that, using the flat spacetime of special relativity as a fixed background for their physics. Attemps at doing quantum physics in curved spacetime keep that wall up: you can try to solve equations at the quantum scale inside a star, or near a black hole, under severely stressed conditions of spacetime curvature, but that still stays fixed at over the timeframes under consideration.

At the Planck scale, this distinction has no meaning. If you insisted to talk about spacetime and particles (despite my warnings!), you could say that particles’ interactions warp spacetime, while spacetime distorsions themselves are quantum fields interacting with the others and between themselves... But that is encouraging confusion. Mathematically there was indeed a topological entity commonly called "spacetime" in the literature, but which doesn’t resemble ours, neither in its dimensionality (11 dimensions is a fairly consensual, though still speculative value) nor its properties. It cannot be the "neutral background" where quantum physics takes place that we commonly picture.

However, the key event at the end of the Planck epoch, besides the divergence of gravity (and of course it is not unrelated) is the spontaneous compactification of the extra dimensions. A quick historical note may be warranted, lest all of this appears as a fanciful deus ex machina.

As I’ve stated a couple times I think, the theoretical combination of general relativity is a massive challenge. We know that now, but it didn’t have to be a priori. Maxwell’s electromagnetic theory was quantized in the form of QED (Quantum Electrodynamics); it was a long and complex endeavor for sure, but in the end a success, and QED has made some of the most accurate predictions in science to have been experimentally confirmed. Why not try and do the same?

General relativity is entirely governed by Einstein’s field equation: \[R_{\mu \nu} - \frac{1}{2} R = T_{\mu\nu}\] The left-hand side term depends curvature of spacetime, and the right-hand side is the stress-energy tensor. Together with the usual spacetime symmetries (the Poincaré group), from which we deduce all the conservation laws (of energy, momentum, etc.), that’s the whole theory: Curvature = Energy.

In a modern field-theoretic context, we prefer to start from the lagrangian density, from which the field equations can be derived via the Euler-Lagrange equations anyway. In general relativity that is simply: \[\mathcal{L}_{GR} = \sqrt{-g}R\]

What QFT then goes on to quantize a lagrangian by turning each variable into a quantum field. Then, in principle, the resulting term could be added to any other lagrangian, say the one for the free electron, processed through the machinery, to get some verifiable prediction at the end. For QED that is far from being an easy process. The calculations involve lots of seemingly infinite term, which end up cancelling each other — technically, we say the QED is renormalizable. However a basic attempt to quantize general relativity doesn’t work that way. All the "equipment" is there: where QED identified Maxwell’s electromagnetic potential with a mediating particle: the photon, quantum gravity has the graviton. But in general relativity, the force field is a \((0,2)\) tensor field while in electromagnetism, it is a vector (or \((0,1)\) tensor) field. That makes this basic quantization non-renormalizable.

Once that was ascertained, many different candidates for quantum gravity were proposed. One interesting observation was that gravity arose naturally in the context of locally supersymmetric theories. Supersymmetry is an extension of the familiar symmetries of quantum field theories. It builds a bridge between bosons (force-mediating particles) and fermions (particles subject to force). It was discovered that imposing supersymmetry as a local gauge symmetry of the lagrangian, a procedure that was introduced to solve a range of independent issues, made gravity a necessary addition. That model, called supergravity, still wasn’t renormalizable, however work continued on various models including higher dimensions (\(D>4\)).

It might seem fanciful to do so. After all, the world as we observe it is definitely four-dimensional (and more precisely, (3,1)-D: three spatial dimensions, one time dimension). That said, the idea of compact, and therefore invisible dimensions had been around for a while. Theodor Kaluza, only a few years after Einstein publicized general relativity, developed his own version of a unified field theory, tentatively unifying gravity and electromagnetism. The theory was a 5D version of general relativity but with a catch: the extra spatial dimension didn’t have the same geometry: it was curled up and microscopic, so that a circular condition could be imposed, with a very small radius (around what we now call the Planck length). This both explains why that dimension is undetectable to us and negligible in terms of our regular experience, but it also is the cause, mathematically, for the appearance of Maxwell-style electromagnetic terms in the lagrangian! Kaluza’s theory was not verified, buy the compactification idea stayed.

The largest possible "realistic" supergravity theory (that is, with only one time dimension and one graviton) has 11 dimensions. It is to this day a promising avenue for quantum gravity in the context of superstring theory. In short, string theory is an approach to field theory that considers quantum fields to be describing tiny strings, open or closed instead of points. The extra dimension involved is not visible in our 4D world, so compact dimensions are a normal feature of string theory. What was a fringe projects suddenly became mainstream in the 1980s when supersymmetry applied to strings ("superstring") was shown to be anomaly-free and perturbatively finite for certain dimensions and certain gauge symmetries, including some that could include all the Standard Model symmetries.

The 1990s brought a second "superstring revolution". By that time, several superstring theories existed that were thought of as rivals. In comes "M-theory", whose main architect was Edward Witten. I should be clear that M-theory is not a full-blown theory in the same sense as Einstein’s, or QED. It has no fundamental equations, and makes no testable predictions. All I can say is that it is still, thirty years on, a promising field of research, that it has proved to be a fertile ground for mathematical discoveries, and that whatever happens, it does give us insights about the ultimate reality.

M-theory is a hypothetical 11-dimensional theory of branes. Out of the five rival superstring theories, it admits two as 10D-compactified approximations, and connects the other three via dualities. Even better, its "low-energy" limit is 11D supergravity! Branes are simply generalizations of strings to N dimensions: for a 2D brane, imagine a sheet. A strong argument in favor of superstring theories in general, and hopefully, transitively, of M-theory, is the compactification solution it’s developed. Extra spatial dimensions don’t end up curled over a tiny cylinder like in the Kaluza model, but on a much more complicated Calabi-Yau or \(G_2\) manifold. But that compactification not only yields the correct result, compactifying 7 dimensions, leaving 4 large ones as required: it also should determine the nature of the residual 4D physics... which should encompass everything we know about general relativity and the standard model.

We have been delving into speculative territories, but this is how I would sum up the various steps during the Planck epoch, should this speculation, or more likely some heavily amended version, become reality:

• At the beginning, the universe is a single object, a single substance, taking the form of an eleven-dimensional brane field.

• Around the Planck scale, the universe’s energy density is "low enough" that 11D supergravity is a good approximation, with individual branes appearing as distinct entities over an 11D spacetime.

• Above the Planck scale, characteristic lengths become superior to the string coupling constant and spontaneous compactification kicks in: 7 dimensions now appear as compact, 4 as "large". Brane- or string-related effects become negligible and everything behaves as per an 11D, (7-4)-compactified supergravity theory that includes a supersymmetric grand-unified QFT.

The Grand Unification epoch: \(10^{-43}\) to \(10^{-35} \text{s}\)

Thus begins the Grand Unification epoch. That phrase has historical roots: when physicists worked on the unification of the three microscopic forces, they called the fruit of their labor "Grand Unified Theories" (GUT), but that unification is not more "grand" as the one including gravitation.

The Planck epoch was the wont of highly speculative physics, full of promising ideas but far from any hope of verifiable predictions or even theoretical consistency. With the GU epoch, we are on firmer ground. There’s no verified theory as of yet, but scores of viable attempts, based on the tried and true methods that made the standard model so successful.

We need to go on a detour — I won’t promise it will be quick, but hopefully it may be enlightening and lift some of the veil that obscures the profund beauty behind the complexity of the mathematical apparatus.

Symmetry as a generating principle

First things first: I said that post-compactification, branes and strings would recede into the background, their body "curled up" into the extra dimensions, leaving only the traditional quantum fields of QFT to be seen. We could say that these fields represent "elementary particles", although it would be more correct to say that the particles are excited eigenstates of the fields. What are those particles? To raise ourselves up to the particle content of the likely GUTs, we need to start from the particles that we do know to exist: those of the Standard Model.

Actually, let’s fast forward a bit more, to the early 1930s, when quantum field theory was nascent. Things seemed simple at the time: the microscopic universe was thought to be composed of atoms and light. Atoms are made of electrons (negatively charged) around a nucleus, and that nucleus is composed of nucleons, some positively charged (protons), some neutral (neutrons). Light, it had been known since Einstein’s discovery of the photo-electric effect, is not only an electromagnetic wave, it can be represented as a quantum particle: the photon. Even the more exotic radiation like X-rays or gamma rays have been shown to be photons outside of the visible light spectrum.

Might these four building blocks give birth to all the matter in our universe? The Presocratic dream of pinning down the "four elements" had never seemed so close. However the picture had to get more complex before it got simpler again (and even more beautiful).

• Antimatter: One prediction of Dirac’s work on relativistic quantum mechanics (the ground for quantum field theory) was that each particle had an antiparticle, with exactly opposite charges (but same mass). Anti-electrons (called positrons), antiprotons and antineutrons then join the party (the photon is its own anti-particle), as do antiparticles for all fields we’ll mention going forward.

• The neutrino: Studying experimental results about beta decay, Enrico Fermi hypothesized that a new particle was needed to preserve the conservation laws from relativity. It had to be electrically neutral and nearly massless: the neutrino was born. Despite behaving a lot like a photon (moving at light speed, electrically neutral), the neutrino had a spin 1/2, like the electron, proton and neutron (photon has spin 1). Spin-1/2 particles would be called fermions going forward.

• Quarks: The theory of the strong nuclear force was developed by Hideki Yukawa, who made the successful prediction that it had a mediator: pion particles that are constantly exchanged between nucleons in their nucleus straitjacket. Troublingly however, a whole bestiary of particles were found while trying to verify this theory: some were force-mediating particles like the pions (they were called mesons), others were matter particles (spin 1/2 fermions) like the nucleons; they were called baryons. Together they form the group of hadrons. Intense work to organize that zoo led to the realization by Gell-Mann and Zweig that all hadrons known at the time could be describe as composite of even smaller particles, which they called quarks. Three of them were theorized at the time: up, down and strange. The up and down quarks make up the nucleons from our daily experience.

• Generations: that’s still not all. Not only were new particles discovered all the time, strange "echos" of existing ones were too: particles that shared all the same characteristics and quantum numbers, except for mass: they were heavier. The first one was the muon, a heavier electron analog observed in the 1930s. Later it was found the muon had its very own neutrino. With that in mind, it was easy to observe that the strange quark shared all the characteristics of the down quark, only it was heavier. With the discovery of the charm quark, a heavier copy of the up quark, it was clear this was not a coincidence, and the concept of second generation was coined. To the original group of [up & down quarks, electron, neutrino] would then correspond [charm & strange quarks, muon, muon-neutrino]. In following decades, a complete third generation was found too: [top & bottom quarks, tau, tau-neutrino].

This covers all the known matter content of the universe. But to get a more complete picture, we need to understand the forces, and their mediating particles, better. That is the great achievement of the standard model.

I cannot give a very detailed exposé of the standard model here. If I could briefly communicate one idea, though, it would be this one: symmetry is the source of dynamics.

That is by far the most striking feature of modern physics. It is not intrinsically connected with QFT or the standard model at all, but only with those models has it openly reached its current prominence. It had been a long time in the making though.

First, "rational mechanics" in the eighteenth century had to re-frame Newton’s mechanics under the guise of a variational method. Maupertuis established the principle of least action: that the physical evolution of a system must minimize the quantity of action spent (a quantity linked to the energy of the system). This changed the perspective of mechanics from trying to solve equations of dynamics, to trying to optimize for the "best" motion possible.

Action can be defined as the integral, over a path of the lagrangian, a functional of all the parameters defining a motion (for instance, in classical mechanics, position and velocity).

From that definition of the lagrangian \[\mathcal{S} = \int_t \mathcal{L}(t,\mathbf{x},\dot{\mathbf{x}}) dt\] and the principle of least action \[\delta S =0\] we can derive the Euler-Lagrange equations: \[\frac{\partial \mathcal{L}}{\partial \mathbf{x}} - \frac{d}{dt} (\frac{\partial \mathcal{L}}{\partial \dot{\mathbf{x}}})=0\]

With this approach, we can start from the data of a lagrangian, and deduce the dynamics of the theory from it. It is only a mathematical re-phrasing so far, but of the kind that will open wide horizons.

For instance, it enabled the most fundamental incursion of symmetry, with Noether’s theorem. We all have an idea of what a symmetry is in daily life. What does it mean for a physical theory to have a symmetry.

We can define it as being invariant under a transformation. There are very simple examples: classical physics is invariant under spatial translations. It doesn’t matter where I set the origin of space \(\mathbf{0}\): it can be at the center of the earth, at my feet, in outer space, the dynamics of the system will always be the same. In other words, there is no privileged point in space. The same goes for time: whether \(t=0\) is today, 0 A.D. or 15 billion years ago does not change physical phenomena.

Since we’ve just argued that a theory should be described by its lagrangian, to each symmetry should correspond a transformation that leaves the lagrangian unchanged.

To take a simple example, the free Lagrangian of Newtonian mechanics is: \[\mathcal{L} = \tfrac{1}{2}m\dot{\mathbf{x}}^2\] It doesn’t depend on \(\mathbf{x}\) nor \(t\), so spatial and temporal translation invariances are automatically verified.

What Noether’s theorem says, in plain English, is that every global symmetry of the lagrangian (that is a transformation that doesn’t depend on spacetime coordinates and applies everywhere the same) determines a conserved quantity.

The spatial translation invariance from above yields the conservation of momentum. The invariance by time translation leads directly to the conservation of energy. Noether has showed that the conservation of these quantities is not a posit but a natural consequence of the theory’s symmetries!

The preoccupation with symmetry had then opened the way for the cross-pollination of physics with algebra. Indeed, the kind of global symmetries we’ve mentioned are not isolated: they belong to a structure. For instance, the combination of two translations is still a translation. The kind of structure where transformations can be composed while preserving the symmetry is called a group.

The set of symmetries of classical physics was represented by the Galilean group and encompassed space and time translations, rotations around an axis, and a "boost" (change of Galilean referential). The latter invariance is just another way of stating Galileo’s principle of relativity! In special relativity, mechanics could be re-phrased in the 4D Minkovski space, and the group of interest was the Poincaré group.

The next sea change, as so many times before, was courtesy of Maxwell’s electromagnetic theory. Let’s remind ourselves of Maxwell’s equations: \[\begin{aligned} \nabla \boldsymbol{\cdot} \mathbf{E} & = \rho \\ \nabla \boldsymbol{\cdot}\mathbf{B}& = 0 \\ \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{E} & = -\partial_t \mathbf{B} \\ \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{B} & = \mathbf{j}+\partial_t\mathbf{E} \end{aligned}\]

\(\mathbf{E}\) and \(\mathbf{B}\) are the electric and magnetic fields, and \(\rho\) and \(\mathbf{j}\) are the charge density and the current vector. The corresponding Lagrangian is

\[\mathcal{L}_{EM} = \underbrace{\tfrac{1}{2}(\mathbf{B}^2-\mathbf{E}^2)}_\text{Field energy} - \underbrace{\rho\phi + \mathbf{j}\cdot \mathbf{A}}_\text{Matter-field interaction}\]

\(\phi\) and \(\mathbf{A}\) are respectively the electrical potential and the magnetic (vector) potential. They are defined directly from the fields as:

\[\begin{aligned} \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{A} &=\mathbf{B} \\ -\boldsymbol{\nabla} \phi &= \mathbf{E}+\partial_t \mathbf{A} \end{aligned}\]

Their existence is guaranteed by Maxwell’s equations together with differential calculus (divergence of a curl is \(0\); curl of a gradient is \(\mathbf{0}\)). However they are not unique.

Clearly for a global transformation that just adds a constant term: \[(\phi,\mathbf{A}) \rightarrow (\phi + \phi_0, \; \mathbf{A}+\mathbf{A}_0)\] the electromagnetic field, which is a derivative of the potentials, is unaffected, and therefore the whole dynamics of the system as well.

To this global symmetry corresponds this conservation law:

\[\partial_t \rho + \boldsymbol{\nabla} \cdot \mathbf{j}=0\]

That is the conservation of electric charge! Noether’s theorem works perfectly to derive this basic principle.

Nothing so far is too different from previous examples: a global symmetry yields a conservation law. But a remarkable fact about Maxwell’s theory is that it’s stable under local transformations. For any function \(f(\mathbf(x),t)\) that is twice derivable (i.e. sufficiently "smooth"), Maxwell’s equation is invariant under the following transformation: \[(\phi,\mathbf{A}) \rightarrow (\phi - \partial_t f, \; \mathbf{A}+\boldsymbol{\nabla} f)\]

This type of local symmetry is called gauge invariance (because originally the choice of potential among all the possible ones helped "gauge" the system).

This looks like it could be mere chance: we just "happened" to find a gauge invariance in Maxwell’s theory, which he had not especially noticed himself — great! But there might be something more. Digging in, we can easily prove that Maxwell’s theory is the simplest field theory verifying this gauge invariance.

It’s starting to look as if the symmetry, far from being an incidental feature of the theory, is instead the very root of the theory.

Quantum Electrodynamics: the original quantum field theory

To go to the next step, we need to switch to quantum field theory. Namely Quantum Electrodynamics (QED), the QFT version of Maxwell’s electromagnetism.

QFT is based on the relativistic version of quantum mechanics whose chief architect was Paul Dirac.

The original version of quantum mechanics was not relativistic, as we can see looking at Schrödinger’s equation: \[-\frac{1}{2m}\Delta \psi = i \partial_t \psi\] On the left-hand side we have a laplacian operator, so a second-derivative of spatial parameters, and on the right-hand side a first-derivative of time. That the four dimensions are treated differently is a good indicator that the theory is not relativistic (that is, not invariant under Poincaré transformations.

Dirac used a beautiful "factorization" of the Klein-Gordon equation (an early relavistic quantum field equation), to derive Dirac’s equation, which turned out to perfectly describe the behavior of an electron:

\[i\gamma^\mu \partial_\mu \psi = m \psi\]

In relatvistic notation, the \(\mu\) index runs over the four dimensions of space and time, and the conventional raising/lowering rules apply. The \(\gamma^\mu\) matrices are defined by the following equalities: \[\{ \gamma^\mu,\gamma^\nu \} \hat{=} \gamma^\mu \gamma^\nu+\gamma^\nu\gamma^\mu=- 2\eta^{\mu\nu} I\]

\(\eta\) is the metric of the Minkowski space, so just \(\text{diag}(-1,1,1,1)\). This definition of the gamma matrices actually yields all sorts of algebraic insights, which we can’t delve into here. Suffice to say they define a Clifford algebra, a notion we’ll come back on later.

The smallest matrices that can fulfil these constraints are \(4\times 4\). So the Dirac quantum field is not just a complex function of space and time, like the Schrödinger field, it is a vector-like object with four components, called a spinor. However by the way it transforms and behaves, a spinor is not a relativistic four-vector (those are visible thanks to their \(\mu\) index, conventionally).

The lagrangian that yields Dirac’s equation as its Euler-Lagrange equation is:

\[\mathcal{L}_{Dirac}=\bar{\psi} (i\gamma^\mu\partial_\mu - m) \psi\]

To get the QED lagrangian, we only need to take the Dirac lagrangian applied to a free electron, and add the Maxwell lagrangian we’ve seen before. There’s jsut some "translation" to be done since we’re using relativistic 4D notations now:

• The 4-current \(j_\mu\) is the 4-vector \((\rho, \mathbf{j})\).

• The 4-potential \(A_\mu\) is the 4-vector \((\phi, \mathbf{A})\).

We also define a kinetic field tensor defined as: \[F_{\mu\nu} \hat{=} \partial_{[\mu}A_{\nu]} \hat{=} \partial_\mu A_\nu - \partial_\nu A_\mu\] Taking first derivatives is analogous to the way we go from potential to field in classical electromagnetism, and indeed this anti-symmetric \((0,2)\) tensor, with its 6 degrees of freedom, contains the same information as the two 3-vectors \(\mathbf{E}\) and \(\mathbf{B}\).

We can obtain the QED lagrangian by adding to Dirac’s free electron lagrangian the terms from Maxwell’s classical lagrangian.

\[\mathcal{L}_{QED}=\underbrace{\bar{\psi} (i\gamma^\mu\partial_\mu - m) \psi}_\text{Electron (Dirac free field)} -\underbrace{\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu}}_\text{Photon (Maxwell free field)} - \underbrace{e j^\mu A_\mu}_\text{Electron-photon interaction}\]

It’s part of the great beauty and plasticity of the lagrangian framework that we can just do that: a simple addition. Taming the complexity of the calculations that ensue is another story, but lagrangian manipulation is an intuitive and powerful framework.

Now let’s come back to our gauge invariance. We could re-trace our steps and apply the same global gauge invariance as in the classical version: \(A_\mu \rightarrow A_\mu + A^0_\mu\) (i.e. adding a constant 4-vector to the potential). The result of applying Noether’s theorem is unsurprisingly that \(j^\mu\) is conserved (\(\partial_\mu j^\mu\)=0).

But there is another global gauge invariance: rotations in the complex plane:

\[\psi \rightarrow e^{i \theta_0} \psi\]

The conserved quantity according to Noether’s theorem is:

\[\bar{\psi} \gamma^\mu \psi\]

We will identify this to the electric current \(\tfrac{j^\mu}{e}\), allowing to re-write the lagrangian:

\[\mathcal{L}_{QED}=\underbrace{\bar{\psi} (i\gamma^\mu\partial_\mu - m) \psi}_\text{Electron (Dirac free field)} -\underbrace{\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu}}_\text{Photon (Maxwell free field)} - \underbrace{e \bar{\psi} \gamma^\mu \psi A_\mu}_\text{Electron-photon interaction}\]

Now, does local gauge invariance work? Let’s now assume that \(\theta\) is not a constant but a function over spacetime points:

\[\psi \rightarrow e^{i \theta(x_\mu)} \psi\]

Unfortunately, we can see that the first term (the free electron) is not invariant under this transformation. That is because spacetime derivative \(\partial_\mu\) does not transform like \(\psi\) (i.e. it’s not just rotated, there is an extra term that appears):

\[\partial_\mu \psi \rightarrow e^{i \theta} (\partial_\mu + i \partial_\mu \theta)\psi\]

But we still have a lever to action: the lagrangian doesn’t have only \(\psi\) as an input, it had \(A_\mu\). If we imposed that vector’s transformation, we could retrieve the full lagrangian’s invariance. This can be done by imposing:

\[A_\mu \rightarrow e^{i \theta} A_\mu + \tfrac{1}{e} \partial_\mu \theta\]

The pesky extra term from the free electron will cancel with the one from the interaction term. We can now use this fact to define the covariant derivative, which transforms in "safe" way:

\[\begin{aligned} D_\mu \psi \; \hat{=} \; & (\partial_\mu - i e A_\mu) \psi \\ \rightarrow & (\partial_\mu + i \partial_\mu \theta) e^{i \theta}\psi - ie(A_\mu + \tfrac{1}{e} \partial_\mu \theta ) e^{i \theta}\psi \\ =& e^{i \theta} D_\mu \psi \end{aligned}\]

We can even compute the commutator of between covariant derivatives (the commutator between regular derivatives if \(0\)):

\[[D_\mu \psi,D_\nu \psi] = ieF_{\mu \nu}\psi\]

All of this allows us to rewrite the lagrangian entirely in terms of this covariant derivative.

\[\mathcal{L}_{QED}=\underbrace{\bar{\psi} (i\gamma^\mu D_\mu - m) \psi}_\text{Covariant Dirac field} +\underbrace{\tfrac{1}{4e^2}[D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]}_\text{Covariant gauge field}\]

Using the mathematical techniques developed for Quantum Field Theory, one can describes the physics of electricity, magnetism and light with an astounding accuracy, that was entirely unthinkable a few decades earlier. QED turned out to be the stepping stone onto which the whole Standard Model was built.

Even more, it can be the best and simplest example of how symmetry is the new ur-principle, the true queen of fundamental physics. Let’s recap quickly the steps we went through.

• We found the QFT form of the global invariance evident in classical electrodynamics, and quickly found that it still held.

• Since local gauge invariance was also a feature of Maxwell’s theory, we made sure that both \(\psi\) (the electron) and \(A_\mu\) (the force) had the appropriate ways to transform under a gauge symmetry.

• We found that the lagrangian could be expressed in terms of a QED-covariant derivative: all we had to do was take the free Dirac lagrangian, and replace ordinary derivatives with the QED-covariant ones (to be precise, the ordinary ones are Lorentz-invariant: compatible with special relativity, while the new ones are both Lorentz- and QED-gauge invariant: compatible with both SR and Maxwell’s theory).

The gauge symmetry that we’ve found is not any symmetry, it corresponds to the group \(U(1)\) (the unitary group of order 1).

\[U(1)= \{ e^{i\theta}, \theta \in \mathbb{R} \}\]

It is a Lie group. As a reminder, a group is a set characterized by its elements (here, complex numbers) and an operation (here, multiplication). To be a group it must verify these properties:

• It is not empty.

• It is stable (if \(a\) and \(b\) are group members, so is \(a \cdot b\)

• It has a neutral element \(e\) (\(e\cdot a =a\)).

• Every element has an inverse (\(a \cdot a^{-1} = a^{-1} \cdot a = e\)).

• The operation is associative (\(a\cdot (b \cdot c) = (a \cdot b ) \cdot c\))

A Lie group is a group that also happens to be a differentiable manifold. In simpler terms, it means that :

• A Lie group is "smooth" and continuous ("between" two elements, we can find other elements.

• Locally (if we "zoom in"), it looks linear, like a Euclidean space.

Think about another group: the group of rotations by a right angle in a plane. It’s a group with just four elements: rotations by \(90^{\circ}\), \(180^{\circ}\),\(270^{\circ}\),and no rotation at all. It’s clearly a legitimate group, but not continuous. There is a "gap" between the elements.

\(U(1)\), the group of all rotations, verifies the characteristics of a Lie group: between a rotations by angles \(\alpha\) and \(\beta\) there’s an infinity of intermediate angles. And locally all rotations can be linearized since \(e^{i\theta} ~_0 1+i \theta\).

From the dynamics of the Dirac and Maxwell equations, we’ve evidenced the \(U(1)\) symmetry. That’s the traditional approach. The modern approach flips the table and asks: what if we started from the symmetries, and deduced the dynamics?

The steps would be:

• Find the simple quantum-mechanical lagrangian that preserves \(SL(3,1)\) (the Poincaré-Lorentz group of relativistic spacetime transformations): that’s the Diract lagrangian.

• Find the simplest way to make it locally \(U(1)\)-invariant: thereby emerges the 4-vector potential and the covariant derivative.

• What about the kinetic gauge field tensor term? Without it, the lagrangian is already invariant enough for our stated goals, so why do we need it?
  Without this term, the Euler-Lagrange equation applied to \(A_\mu\) is entirely trivial \(j^\mu=0\). It describes a world with a potential but no charges.
  Since we want to interpret \(A\mu\) as an electromagnetic potential, and since we know there are electric charges, it cannot be. The next simplest term involving \(A_\mu\) giving a non-trivial dynamics is the one that we’ve encountered, \(F^{\mu \nu}F_{\mu \nu}\), or \([D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]\).

By imposing these constraints (relativistic invariance, \(U(1)\) invariance, existence of charges), we have mathematically developed the simplest possible lagrangian. Of course, we’ve used with savvy our knowledge of the physical world to guide our steps, but this was still an abstract progression.

The audacious leap of faith of the standard model is that we can continue this process, and by imposing more complex symmetries, keep answering more and more of our long-held questions.

Quantum Chromodynamics: the platonic ideal of Yang-Mills gauge theories

The standard model is composed of two theories: quantum chromodynamics (the theory of the strong nuclear force) and the electroweak model (the unified theory of the electromagnetic and weak nuclear forces).

Remember that quarks were discovered when the numbers of particles similar to the good ol’ protons and neutrons had ballooned beyond all reason. It seemed like more hadrons were discovered with each week that passed. Surely, God, logos or the Universe could not allow that complexity! That’s when Murray Gell-Mann and others developed the “eightfold way”, a system of classification of these pesky particles that was the seed of the idea there might be a sub-structure there. When the existence of quarks was hypothesised, nothing was known about the interaction that bound them together without hadrons. Since quarks in baryons went by groups of three, a new quantum charge was imagined, called color. A proton for instance, would assemble three quarks of different colours: red, blue and yellow. A pion (made of a quark and an antiquark) would assemble opposite charges, for instance a blue quark and an “anti-blue” anti-quark.

There’s no space here to get into the exact details why, but careful consideration of the various hadrons showed that quarks must be governed by \(SU(3)\) symmetry. That is a Lie group that generalizes the simple \(U(1)\) that we already know. Its fundamental representation (one of the ways to mathematically embody the underlying symmetry) is as the set of complex \(3\times 3\) matrices that are unitary (that’s the “U”) and of positive determinant (that’s the “S” for “special).

\[SU(3) = \{ U \in GL(3,\mathbb{C}) \; | \; U^\dagger U = I \text{ and } \text{det}(U)=1 \}\]

So, we have matrices, but what do they act on? In electromagnetism, we had \(U(1)\), whose members were scalars (complex numbers). The \(U(1)\) transformation just multiplied the Dirac spinors (representing the electron) by a complex function of spacetime. Now the \(SU(3)\) transformation multiplies something by a matrix of complex numbers. We know that multiplying a matrix of size 3 by a vector with 3 components yields another vector with 3 components. It looks like the fundamental object of this new theory then ought to be a triplet. This is consistent with the fact we’ve just posited there might be three different color charges. A triplet would then contain blue, red and yellow quarks:

\[\mathbf{q} = \begin{pmatrix} q_b \\ q_r \\ q_y \end{pmatrix}\]

Note that a triplet contain the same type of quark: it can be a triplet of three up quarks, of three charm quarks, of three strange anti-quarks, but there can be no mixing. Let’s remember that each component like \(q_b\) is a full-blown Dirac spinor, mathematically the same type of object as an electron \(\psi\).

We can use this triplet language to write the interaction-free QCD lagrangian:

\[\mathcal{L}_{QCD}^{free}=\bar{\mathbf{q}} (i\gamma^\mu \partial_\mu - m) \mathbf{q}\]

It’s the same as the Dirac free lagrangian but with triplets of spinors replacing the single spinors. In fact, using \(c\) as an index running over colours:

\[\bar{\mathbf{q}}\mathbf{q} = \sum_c{\bar{q_c} q_c }= \bar{q_b} q_b+ \bar{q_r} q_r+ \bar{q_y} q_y\]

So plugging this into the lagrangian, we get:

\[\mathcal{L}_{QCD}^{free}=\bar{q_b} (i\gamma^\mu \partial_\mu - m) q_b + \bar{q_r} (i\gamma^\mu \partial_\mu - m) q_r + \bar{q_y} (i\gamma^\mu \partial_\mu - m) q_y\]

This simply means that the free QCD lagrangian describes three regular particles, independent from each other since the interaction terms are not in there yet. The triplet formalism works as we expect so far.

Before going further we need to move from Lie groups to Lie algebras. The fact that Lie groups are locally Euclidean (linear and “flat”) is not an afterthought, but an absolutely crucial fact. It means that if we focus only on infinitesimal transformations, the Lie group elements are correctly approximated by the elements of an algebra (essentially a vector space with some multiplication structure). The linearity of the vector space will make calculations infinitely easier! . The Lie algebra that corresponds to \(SU(3)\) is usually written as \(\mathfrak{su}(3)\). There are lots of beautiful and clean mathematical ways to develop the theory of Lie algebras, but we’ll stick to a high-level “physicist” introduction. When Lie groups are represented by sets of matrices, as in the fundamental representation, we can prove that all of their elements \(U\) are the exponential of another matrix \(M\). That’s how we define the Lie algebra!

\[\mathfrak{su}(3) = \{ M \text{ such as } U \in SU(3) \text{ and } U = \exp{M} \}\]

Algebras are vector spaces, and this one has to be of finite dimensions (since it is included in the vector spaces of all size-3 matrices, which has 9 dimensions), Therefore it is entirely defined by a basis of generators which we will call \(( \lambda_a )\).

\[\mathfrak{su}(3)= \{ \theta^a \lambda_a , \text{ for } (\theta_a) \in \mathbb{R}^d \}\]

But that’s not all. Products of \(SU(3)\) matrices also belong to \(SU(3)\) (by group stability). If we switch to “algebra view” it means that if \(M\) and \(N\) are part of \(\mathfrak{su}(3)\), then \(\exp(M)\) and \(\exp(N)\) are part of \(SU(3)\) and there is some matrix \(P\) of \(\mathfrak{su}(3)\) such that \(\exp(M)\exp(N) = \exp(P)\).

We so happen to have a formula telling us exactly what a product of matrix exponentials is: \[\exp (M) \exp (N) =\exp ( M + N + [M,N] + \dots )\]

The notation we have used here is the matrix commutator: \([A,B] = AB-BA\). All the following terms also involve commutators. The vector space \(\mathfrak{su}(3)\) is stable by addition so \(M+N\) is part of it. So to ensure that \(M + N + [M,N] + \dots\) is part of \(\mathfrak{su}(3)\), it should be enough to have it that \([A,B]\) is. Since \(\mathfrak{su}(3)\) is a vector space fully generated by its generator basis \(( \lambda_a )\), then that basis must be stable against the action of the commutator:

\[[\lambda_a, \lambda_b] = i f^{abc} \lambda_c\]

The \(f^{abc}\) are called the structure constants. They entirely define the structure of the Lie algebra.

We are now equipped with enough knowledge about Lie groups and algebras to implement the programme we set out to develop: from symmetry shall emerge the dynamics, in the form of a full QCD lagrangian. First we can easily verify that the free lagrangian is invariant under global \(SU(3)\) transformations:

\[\mathbf{q} \rightarrow U_0 \mathbf{q}\]

\(U_0\) is a constant complex matrix from \(SU(3)\). The global invariance is proved by noting that:

\[\mathbf{q} \rightarrow \overline { U_0 \mathbf{q} } = \bar{\mathbf{q}} U_0^\dagger\]

So, by the definition of unitary matrices, terms like \(\bar{\mathbf{q}}\mathbf{q}\) are automatically invariant. And since the matrix is constant, the spatial derivative can just be absorbed (that is: \(\bar{\mathbf{q}}\gamma^\mu \partial_\mu \mathbf{q}\) is also invariant).

However the free lagrangian is not invariant under a local \(SU(3)\) transformation, that is one where the matrix can be different for different spacetime co-ordinates. We know we can write that transformation as:

\[\mathbf{q} \rightarrow U \mathbf{q} = \exp \left( i \frac{\lambda_a}{2} \theta^a \right) \mathbf{q}\]

The \(\theta^a\) are now functions over spacetime. We also know that we can linearise, since we are only looking at infinitesimal transformations.

\[\mathbf{q} \rightarrow \mathbf{q} + i \frac{1}{2} \left( \lambda_a \mathbf{q} \right) \delta\theta^a\]

The transformation of the free lagrangian yields an extra term:

\[\mathcal{L}_{QCD}^{free} \rightarrow \mathcal{L}_{QCD}^{free} - \frac{1}{2} \lambda_a \bar{\mathbf{q}} \gamma^\mu ( \partial_\mu \delta\theta^a ) \mathbf{q}\]

Just like we did for \(U(1)\) we can build a covariant derivative of this form:

\[D_\mu \hat{=} \partial_\mu + i g \frac{\lambda_a}{2} G^a_\mu\]

The covariance is ensured by imposing this transformation to the \(G^a_\mu\):

\[G^a_\mu \rightarrow G^a_\mu + \frac{1}{g} \partial_\mu \delta \theta^a - f^{abc} \delta \theta_c G^b_\mu\]

With our brand new QCD-covariant derivative in tow, we can define the full lagrangian exactly in the way we did for QED, except that It’s now acting on quark triplets instead of electron singlets.

\[\mathcal{L}_{QCD}=\underbrace{\bar{ \mathbf{q}} (i\gamma^\mu D_\mu - m) \mathbf{q}}_\text{Covariant Dirac field} +\underbrace{\tfrac{1}{4g^2}[D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]}_\text{Covariant gauge field}\]

Based on the covariant derivative definition, we can make the last term explicit in terms of the \(G^a_\mu\) and define a field tensor of order 2, like we did for electromagnetism.

\[\tfrac{-i}{g}[D_\mu,D_\nu] \; \hat{=} \; G_{\mu \nu} \; \hat{=} \; \lambda_a G^a_{\mu \nu}\]

\[G^a_{\mu \nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f^a_{bc} G^b_\mu G^c_\mu\]

We get the following:

\[\mathcal{L}_{QCD}=\underbrace{\bar{ \mathbf{q}} (i\gamma^\mu D_\mu - m) \mathbf{q}}_\text{QCD field} -\underbrace{\tfrac{1}{4}G^{\mu \nu}_a G^a_{\mu \nu}}_\text{Kinetic Gluon field}\]

The last term is very similar to the photon term in QED, except that it’s summing over the index \(a\). Therefore, there are as many force-mediating fields now as there are generators of the Lie algebra. In the case of \(SU(3)\) that number is \(8\). We have then shown that there are eight bosons (force-mediating particles derived from a 4-vector potential and represented by a field tensor) in QCD. They are called gluons.

We can develop the lagrangian even more and make interesting observations.

\[\begin{aligned} \mathcal{L}_{QCD} = & \underbrace{\bar{ \mathbf{q}} (i\gamma^\mu \partial_\mu - m) \mathbf{q}}_\text{Free quark} -\underbrace{\tfrac{1}{4} \partial^{[/mu} G^{\nu]}_a \partial_{[/mu} G_{\nu]}^a }_\text{Free Gluon} \\ &+ \underbrace{ g G^\mu_a \bar{\mathbf{q}} \gamma_\mu \frac{\lambda ^a}{2} \mathbf{q} }_{\text{Quark-Gluon interaction}} \\ & - \underbrace{ \frac{g}{2} f^{a}_{bc} \partial^{[\mu} G^{\nu]}_a G_\mu^b G_\nu ^c}_{\text{ Cubic gluon interaction}} - \underbrace{ \frac{g^2}{4} f^{abc}f_{ade} G_\mu^b G_\nu ^c G_\mu ^d G_\nu ^e }_{\text{ Quartic gluon interaction}} \end{aligned}\]

The QCD field term decomposes into the free quark terms (the free lagrangian) and a gluons-quarks interaction (similar to the electron-photon interaction in QED).The kinetic gluon field also can be decomposed, first to a term that looks exactly like the QED field tensor (except that it’s not gauge-invariant alone) and that we’ll identify with free gluon propagation. The other terms are shaped in a way that when applying Feynman rules, we find that they involve interactions between three gluons, and between four gluons!

This is different from QED: photons could not interact with each other since they have no electric charge. Gluons have no electric charge but they do have a color charge: that is because they can transform elements of the quark triplets: a blue quark can become a red quark by emitting a gluon for instance. While an electron emitting a photon has an effect on its dynamics, but doesn’t change its electric charge. Gluons being color-charged is the reason the cubic and quartic terms are present.

These results are spectacular: by imposing a local \(SU(3)\) gauge symmetry to the Dirac Lagrangian, we get a full-blown theory of strong interactions. All the steps we’ve gone through could have been followed the same for the general case of \(SU(N)\) though. The general case is called the Yang-Mills non-abelian gauge theory ("non-abelian" just means that the elements of the group don’t commute, unlike \(U(1)\) for instance).

The simple Yang-Mills theory for \(SU(N)\) then ultimately gives us:

• An \(N\)-uplet of fermions (the particles that are subject to forces, with half-integer spin). Fermions can have mass.

• \(N^2-1\) generators that are identified as spin-1 bosons ("force-carrying" particles). These bosons are massless and charge-carrying (i.e. they can mediate the transformation of a fermion particle into another).

Electroweak unification

As I said, the standard model is made up of two theories. Why not three, since there are three fundamental interactions at the quantum level (we have seen that gravitation is a whole other story)? We have QCD now (strong interaction), QED (electromagnetism), why not add some theory of weak interactions?

The primary reason is that even today, with their vastly different strength levels, electromagnetism and weak interactions seem thoroughly entangled. Remember that the phenomenon that started investigations in the weak force was beta decay of a neutron into a proton (with the emission of an electron and an anti-neutrino).

\[n \rightarrow p + e^- + \bar{\nu}\]

We now know that since \(n=udd\) and \(p=uud\), what is actually happening is the transformation of a down quark into an up quark.

\[d \rightarrow u + e^- + \bar{\nu}\]

Furthermore, we can interpret antiparticles are particles that experience time in reverse. So the antineutrino being "emitted" can be seen as a neutrino "arriving" to be transformed into an electron.

All in all, the weak force seems to be transforming particles into new particles of different electric charges. That’s new! The strong force could turn a blue \(u\) quark into a red \(u\) quark, but color is the strong charge. All other conserved charges were left alone. In a way, the strong force is minding its own business.

In a beta decay, the overall electric charge is conserved, of course. But the standard model does not handle the mixing of leptons (electrons, neutrinos) with quarks. So we’re really looking at two reactions happening "side-by-side": up-down quarks, and electron-neutrino.

Both of them involve a "charged current", and therefore, an electrically charged boson. Gluons were charged, but only with color, the conserved quantity corresponding to "their" symmetry. It seems that to describe electrically charged weak bosons, we need to incorporate electromagnetism as part of the theory!

We could be quite confident. After all we’ve just learned how to develop an \(SU(N)\) gauge theory before. We know that particles that can transform into each other should be placed in doublets. Perhaps those might do:

\[\begin{pmatrix} e \\ \nu \end{pmatrix} , \begin{pmatrix} d \\ u \end{pmatrix}\]

Alas, things are not that simple. As it turns out we need to consider if those particles are left-handed or right-handed. This can seem like a very eccentric statement, so let me explain.

Every quantum particle has a "spin", which can be presented as a rotation degree of freedom (let’s stick to that rough characterization, lest we’re gone on an even longer tangent). The spin can "point" in the same direction as a particle’s momentum: then that particle’s is right-handed. If it points the other way, the particle is left-handed.

Of course, we are in the quantum world, so everything is about fields in superposed states, rather than well-defined particles. But we can use Dirac’s theory to define operators that "isolate" the left-handed and right-handed parts of the fields:

\[\frac{1}{2}(1-\gamma^5)\psi = L\psi = \psi_L \quad \frac{1}{2}(1+\gamma^5) \psi = R\psi = \psi_R\]

\(\gamma^5\) is deduced from the other Dirac matrices (\(\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3\)). With that we can prove that these operators actually do behave like projections:

\[\begin{aligned} L+R &= 1 \\ LR=RL&=0 \\ L^2&=L \\ R^2&=R \end{aligned}\]

From all this we can derive these useful equalities: \[\begin{aligned} \bar{\psi}\psi &= \bar{\psi}_R\psi_L+\bar{\psi}_L\psi_R \\ \bar{\psi}\gamma^\mu \psi &= \bar{\psi}_R \gamma_\mu \psi_R+\bar{\psi}_L \gamma_\mu \psi_L \end{aligned}\]

The first equation denotes a mass term: only massive particles have that one featured in the lagrangian. It shows that the left- and right-handed components are mixed together. The second equation is a gauge current: it shows that conversely, those components are not mixed! When two charged particles interact, the interaction of their left-handed components, and that of their right-handed components are perfectly superposed: there’s no mixing.

So far, so good, but what does this handedness business have to do with the weak force. Well, we know it has a lot to do since to a crucial experiment by Chien Shiung Wu. She proved that weak interaction violate parity, in other words, that it doesn’t apply the same to left- and right-handed particles.

To understand how surprising that is, consider that left-handed and right-handed particles are literally mirror reflections of each other. "Common sense", as well as all our physical observations to that point suggest that it’s an absurd thought. Take any physical phenomenon, and flip everything left to right (momenta, forces, positions...), and everything will still follow the law of physics. If I film Rafael Nadal (a lefty) and flip the video so that the court appears inverted, that is still a physical situation. It doesn’t correspond to our world, because in the "flipped world", Nadal is now a righty, he’s serving on the "wrong" side when the game starts, etc. But no law of physics appears to be violated.

Well, Chien Shung Wu, by observing the decay of Cobalt atoms, showed that beta decay could not be flipped. That is remarkable. If you show a photograph of someone I’ve never met, and then its mirror version, both likely look fine and "physical". But in fact, if I had godlike powers allowing me to "see" the many beta decays happening on those photos (say, of Carbon-14), then I could be able to tell for sure which photo is the original and which one was flipped!

More precisely, what has been shown is that weak interactions only concern the left-handed component of particles. The right way to write our beta decay reaction is then:

\[d_L \rightarrow u_L + e_L + \bar{\nu}_L\]

Therefore, our suggestion to use the doublets \((e,\nu)\) and \((d,u)\) does not work. If it did, the right-handed components of different particles could interact. The real \(N\)-uples are then doublets for the left-handed fermion components, and singlets for the right-handed ones:

\[\begin{pmatrix} e_L \\ \nu_L \end{pmatrix} , \begin{pmatrix} d_L \\ u_L \end{pmatrix} , (e_R),(d_R), (u_R)\]

In what follows, to simplify this brief overview, we will consider only leptons, and define:

\[\mathbf{L} = \begin{pmatrix} \nu_L \\ e_L \end{pmatrix} \quad , \quad \mathbf{R} = (e_R ) \quad , \quad \Psi = \begin{pmatrix} \nu \\ e \end{pmatrix}\]

You may notice that there is no right-handed neutrino. That is the hypothesis the standard model makes: the only neutrinos we’ve observed and identified as such were interacting through the weak force (therefore they were left-handed). Since neutrinos were thought to be massless, and therefore constantly travel at \(c\) (the speed of light), they couldn’t change their helicity and would stay left-handed forever (making it an intrinsic property, unlike for massive fermions).

This suggests that the right symmetry group is not of the form \(SU(N)\). Instead we will use a semi-simple Lie group: \(SU(2) \times U(1)\).

\(SU(2)\) will act on the doublets (left-handed particles) and \(U(1)\) on both the doublets and the singlets.

Just like \(SU(3)\) had \((3^2-1)=8\) generators, \(SU(2)\) has \((2^2-1=3)\) generators, 3 matrices of order 2 called the Pauli matrices.

\[\tau^1=\frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \tau^2=\frac{1}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad \tau_3=\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]

The commutation relations are: \[[\tau_1, \tau_2] = i \tau_3 \quad [\tau_2, \tau_3] = i \tau_1 \quad [\tau_3, \tau_1] = i \tau_2\]

The \(U(1)\) generator \(\tilde{Y}\) is required by the properties of the algebra to commute with the \(SU(2)\) generators: \([\tilde{Y},\tau^i]=0\). The only solution is for that generator to be a multiple of the identity matrix: \(\tilde{Y} = Y \text{Id}\).

The factor \(Y\) is arbitrary. Furthermore, \(U(1)\) is acting on both doublet and singlet representations, and in principle it doesn’t need to be the same factor (\(Y_L\neq Y_R\)). Conventionally, we choose \(Y_L=-1\). This is called the weak hypercharge of the doublet.

Now, \(U(1)\) seems very familiar from our QED exploration. Is this the same symmetry that governs electromagnetism? We can see it’s not the case; indeed:

\[\tilde{Y}_L \mathbf{L} = Y_L \mathbf{L} = - \begin{pmatrix} \nu_L \\ e_L \end{pmatrix}\]

This means that the neutrino and electron have the same hypercharge. But we know it’s not the case for the electric charge. so \(U(1)_Y\) is not the symmetry of electromagnetism.

The electric charge operator is supposed to have eigenvalues that correspond to the components’ vrious electric charges. In this case:

\[Q\begin{pmatrix} \nu_L \\ e_L \end{pmatrix} = \begin{pmatrix} 0 \\ -e_L \end{pmatrix}\]

Which suggests:

\[Q= \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix}\]

Looking at our algebra’s generators, we find that:

\[Q = \tau^3 + \frac{\tilde{Y}_L}{2}\]

This matrix equation is also valid in eigenvalue form. Let’s call \(T_3\) the conserved charge from \(\tau^3\), and we get:

\[q = T_3 + \frac{Y_L}{2}\]

For singlets, there is no action of the Pauli matrices and therefore no \(T_3\), so we can conclude \(Y_R= 2q = -2\) (for the electron singlet).

We continue doing what we did for the non-abelian and abelian cases: for \(SU(2)\) we know there will be 3 force bosons (for the 3 generators), the \(W^i_\mu\), and for \(U(1)\) one boson, \(B_\mu\). Then the covariant derivatives applying to left-handed and right-handed components are.

\[\begin{aligned} (D_L)_\mu &= \partial_\mu + i g \tau_i W^i_\mu + i \frac{g'}{2}Y_L B_\mu = \partial_\mu + i g \tau_i W^i_\mu - i \frac{g'}{2} B_\mu \\ (D_R)_\mu &= \partial_\mu + i \frac{g'}{2} Y_R B_\mu = \partial_\mu - i g' B_\mu \end{aligned}\]

The free electroweak lagrangian (without interaction) will look like the previous one we’ve worked on.

We also define:

\[\tilde{m}= \begin{pmatrix} m_e & 0\\ 0 & m_\nu \end{pmatrix}\]

It will be useful since the two particles involved don’t have the same mass (unlike quarks in a \(SU(3)\) triplet). The free Dirac lagrangian is then:

\[\begin{aligned} \mathcal{L}_{EW} &= \bar{\Psi}(i\gamma^\mu \partial_\mu - \tilde{m}) \Psi\\ &= \overline{e} ( i\gamma^\mu \partial_\mu - m_e) e + \bar{\nu} ( i\gamma^\mu \partial_\mu - m_\nu) \nu \end{aligned}\]

So far, so good: the two particles are unmixed, as we expect in the absence of interactions. However this lagrangian is not invariant under global \(SU(2) \times U(1)\) transformations. That is because the mass term mixes the left- and right-handed components, as we’ve seen. That’s a roadblock right from the start! The only way to make it invariant is to impose \(m_e = m_\nu = 0\). But we know that electrons have mass (and we think neutrinos have)... All we can do is accept it for now. It will be resolved in spectacular fashion later.

The full EW lagrangian (still massless) is then given by:

\[\mathcal{L}_{EW}=\underbrace{ \bar{ \mathbf{L}} i\gamma^\mu (D_L)_\mu \mathbf{L} + \mathbf{R} i\gamma^\mu (D_R)_\mu \mathbf{R} }_\text{Fermion fields} -\underbrace{\tfrac{1}{4} ( W^{\mu \nu}_i W^i_{\mu \nu} + B^{\mu \nu}B_{\mu \nu})}_\text{Kinetic Boson fields}\]

Let’s focus on the fermion fields:

\[\mathcal{L}_{EW}^{fermions}=\underbrace{ \bar{ \Psi} i\gamma^\mu (D_L)_\mu \Psi }_\text{Free fermions} + \underbrace{ g W^\mu_i \bar{\mathbf{L}} \gamma_\mu \tau ^i \mathbf{L} - \frac{g'}{2} B^\mu \bar{\mathbf{L}} \gamma_\mu Y \mathbf{L} - g' B^\mu \bar{\mathbf{R}} \gamma_\mu Y \mathbf{R} }_{\text{Fermions-Bosons interactions}}\]

Let’s consider the interaction terms, starting with the one involving non-diagonal matrices:

\[g\bar{\mathbf{L}} \gamma_\mu (\tau^1 W^\mu_1 + \tau^2 W^\mu_2 ) \mathbf{L} = \frac{g}{2}\bar{\mathbf{L}} \gamma_\mu \begin{pmatrix} 0 & W_\mu^1 -iW_\mu^2 \\ W_\mu^1 + iW_\mu^2 & 0\end{pmatrix} \mathbf{L}\]

It makes sense to rewrite the bosons as \(W_\mu^\pm \hat{=} \frac{1}{\sqrt{2}} W_mu^1 \mp i W^\mu_2\). Then the term above becomes:

\[\begin{aligned} \mathcal{L}_{EW}^{\pm int}& = \frac{g}{2\sqrt{2}} ( \overline{\nu_L} \gamma^\mu e_L W_\mu^+ + \overline{e_L} \gamma^\mu \nu_L W_\mu^- ) \end{aligned}\]

As expected, we have weak charged currents transforming the doublet particles into each other.

Let’s look at the neutral term now:

\[\begin{aligned} \mathcal{L}_{EW}^{0 \, int}& = g \bar{\mathbf{L}} \gamma_\mu \tau^3 W^\mu_3 \mathbf{L} - \frac{g'}{2} B^\mu \bar{\mathbf{L}} \gamma_\mu \mathbf{L} - g' B^\mu \bar{\mathbf{R}} \gamma_\mu \mathbf{R} \\ &= \frac{g}{2} ( \overline{\nu_L} \gamma^\mu \nu_L - \overline{e_L} \gamma^\mu e_L) W_\mu^3 - \frac{g'}{2} ( \overline{\nu_L} \gamma^\mu \nu_L + \overline{e_L} \gamma^\mu e_L + 2\overline{e_R} \gamma^\mu e_R) B_\mu \\ &= g j_3^\mu W^3_\mu + g' j^\mu_Y B_\mu \end{aligned}\]

\(j_3^\mu\) and \(j^\mu_Y\) are the currents corresponding to the two neutral generators, but none of them corresponds to the electromagnetic current, since they involved neutrinos. The EM current looks like this: \[j_{EM}^\mu = \bar{e} \gamma^\mu e = \overline{e_L} \gamma^\mu e_L+ \overline{e_R} \gamma^\mu e_R\]

We can note that the following holds: \[j_{EM}^\mu = - ( j_{3}^\mu+ \frac{j_{Y}^\mu}{2})\]

What if we could re-combine the \(W^3\) and \(B\) bosons to retrieve something like a photon? The idea is to rotate the bosons vector by an angle \(\theta_W\) to get a new coupling involving \(A\) and \(Z\) bosons. We’ll be hopeful, of course, that \(A\) can be identified with the photon.

\[\begin{pmatrix} A_\mu \\ Z_\mu \end{pmatrix} = \begin{pmatrix} \cos{\theta_W} & \sin{\theta_W}\\ -\sin{\theta_W} & \cos{\theta_W} \end{pmatrix} \begin{pmatrix} B_\mu \\ W^3_\mu \end{pmatrix}\]

We can plug this into our lagrangian:

\[\begin{aligned} \mathcal{L}_{EW}^{0 \, int}& = \begin{pmatrix} \frac{g'}{2} j_Y^\mu & g j_3^\mu \end{pmatrix}\begin{pmatrix} B_\mu \\ W^3_\mu \end{pmatrix} \\ &= \begin{pmatrix} \frac{g'}{2} j_Y^\mu & g j_3^\mu \end{pmatrix} \begin{pmatrix} \cos{\theta_W} & -\sin{\theta_W}\\ \sin{\theta_W} & \cos{\theta_W} \end{pmatrix} \begin{pmatrix} A_\mu \\ Z_\mu \end{pmatrix} \\ \end{aligned}\]

This gives us the answer we wanted: to have an electromagnetic-like term, we need to impose:

\[\tan{\theta_W}=\frac{g'}{g} \quad e=g'\cos{\theta_W}=g\sin{\theta_W}\]

And finally:

\[\begin{aligned} \mathcal{L}_{EW}^{0 \, int}& = e j_{EM}^\mu A_\mu + \frac{e}{\sin{\theta_W}\cos{\theta_W}} ( \frac{j_3^\mu}{2} - \sin^2{\theta_W} j_{EM}^\mu ) Z_\mu\\ \end{aligned}\]

The \(Z\) boson then mediates a weak interaction that acts on all particles: left-handed and right-handed electrons, and the neutrino.

We should note that at this stage the switch from \((W^3,B)\) to \((Z,A)\) is just a mathematical "trick". The underlying physics have not changed at all.

Our electroweak theory now describes the following:

• Massless fermions

• 2 massless charged bosons (\(W^\pm_\mu)\)

• 2 massless neutral bosons (\(Z\),\(A\))

This is still not consistent with reality. Obviously, electrons do have mass. And we also know that the weak bosons have mass, because that’s the only way to explain their very short range. If they were massless like the photon, then the interaction would have infinite range and, among other effects, neutron decay would be quasi-instantaneous, making the world a very different place (with no possible physical structure).

Above all else, we know that the bosons’ mass terms would cause the lagrangian to not be invariant anymore under the symmetry group. That means that in the current universe, the symmetry is broken.

The Higgs mechanism, crowning achievement of the Standard Model

This obviously glaring issue was solved by the application of the now-famous Higgs mechanism to the spontaneous symmetry breaking (SSB) of the electroweak theory. This is not the place to come back on the origins of this discovery, so at the risk of making it a deus ex machina, I will simply present the procedure, which I think is still important given its elegance.

We start by introducing a scalar boson doublet, with one positively charged particle, and a neutral one (a scalar boson means it’s a spin-0 particle which is just a complex function of spacetime, and follows the Klein-Gordon equation, unlike the fermions \(\psi\), which are spinors, or the vector bosons like \(A\_mu\) which are 4-vectors).

\[\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\]

We associate the following potential with it:

\[V (\Phi^\dagger \Phi)= \mu \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2\]

It might seem arbitrary, and to an extent it is — this potential was stumbled upon while studying ferromagnetism and then ported to quantum field theory. But it also happens to be the only acceptable potential for a scalar field. Indeed, we know that we need it to be a function of the norm \(\Phi^\dagger \Phi\) (to be gauge invariant). We also know that if the term with highest power is \((\Phi^\dagger \Phi)^n\), \(n\) has to be an even number; otherwise the potential is not physical because it doesn’t have a minimum (and then no stable state). Lastly, we can show that \(n>3\) doesn’t work because it makes the lagrangian non-renormalizable (in other terms: it is impossible to compute any physical quantity).

This is all well and good, but how can this additional field help us explain the broken symmetry?

To understand this, we need to talk briefly about the vacuum expectation value (VEV). QFT is a perturbative theory. It means that all the calculations we make are from "small" fields. This doesn’t mean that the theory is only approximately true. But the (relatively) simple picture we’ve developed of fermions interacting with force-mediating bosons is an approximation that is incredibly accurate for "small" perturbations of the vacuum. Note that "small" here is very relative. Even the energies from the early universe (say, at \(10^{-30}\text{s}\)) would still qualify as small perturbations. Closer to the Big Bang is when the perturbative approach starts to break down, in a way we’ve described previously.

So, the perturbations are small, but compared to what? It seems natural to say they are close to zero. However, that’s not a given because in a QFT, the quantum vacuum can have a non-zero background energy level! And we will see that such a non-zero VEV is precisely what interacts with particles to give them mass.

This explains why we need this scalar field: it is possible to prove that in a QFT with the basic relativistic symmetry, only such terms can potentially generate a VEV other than zero. Simply speaking the VEV is the minimum value of the potential \(V\) (as always in physics, a system tends to "fall" into the potential well).

As it turns out, the value of that minimum depends on the sign of \(\mu\). For \(\mu\) positive, it is simply zero. So there’s no change compared to before.

\[\mu>0 \rightarrow \Phi^\dagger \Phi \rangle_0 = 0 \rightarrow \langle \Phi \rangle_0 = 0\]

But for negative values of \(\mu\), the minimum of the norm is not nil: it takes a value we will call \(v^2\).

\[\mu<0 \rightarrow \langle \Phi^\dagger \Phi \rangle_0 = -\frac{\mu}{2\lambda} \hat{=} v^2\]

This could be verified for any doublet such that \(\phi^+{*}\phi^+ + \phi{^0*}\phi^0=v^2\). Thanks to the \(SU(2)\) symmetry we can pick any value that’s convenient (whatever the exact VEV we can rotate the whole theory to that value).

The key requirement of the symmetry breaking operation is that we preserve the \(U(1)_{EM}\) symmetry, which we know is still preserved. This happens if the doublet is invariant under \(U(1)_{EM}\) transformations. Since those can be written as \(U=\exp{i\theta Q}\), this is equivalent for this doublet to:

\[Q \langle \Phi \rangle_0 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \langle \Phi \rangle_0 = 0\]

We can then choose:

\[\langle \Phi \rangle_0 = \begin{pmatrix} 0 \\ v \end{pmatrix}\]

Remember that QFT is analysing the Lagrangian perturbatively around the VEV. So now instead of looking at \(\Phi\) we can look at the perturbation around \(v\). This perturbation will be called \(H\), the Higgs boson. (Since none of this can break the electromagnetic symmetry, \(H\) needs to be electrically neutral, hence the field is real).

\[\Phi = \begin{pmatrix} 0\\ v + H \end{pmatrix}\]

Now all we need to do is examine our scalar doublet’s lagrangian. This is a regular Klein-Gordon lagrangian, but since we want to enforce \(SU(2)\) gauge invariance, we use the relevant covariant derivative (with \(Y_\Phi=1):\)

\[\begin{aligned} \mathcal{L}_\Phi &= D_\mu \Phi^\dagger D^\mu \Phi - V(\Phi^\dagger \Phi)\\ \end{aligned}\]

Before symmetry breaking, there’s not much to say about it. It’s entirely invariant under \(SU(2)\times U(1)_{Y}\), so it doesn’t bring anything new. If we developed each term, we would find \(\phi^0\) and \(\phi^+\) interacting with each others as well as with the weak bosons.

Things change when we plug in the new value around the post-SSB VEV.

\[\begin{aligned} \mathcal{L}_\Phi &= D_\mu (v+H)^\dagger D^\mu (v+H) - V((v+H)^2) = \dots =\\ =& \underbrace{ \partial^\mu \partial_\mu H}_{\text{Higgs (scalar) boson}} + \underbrace{\frac{g^2}{2} v^2 (W^+_\mu W^{-\mu} + \frac{1}{2\cos^2{\theta_W}}Z_\mu Z^\mu)}_{\text{Weak bosons' mass}} \\ &+ \underbrace{\frac{g^2}{2} (2vH+H^2) (W^+_\mu W^{-\mu} + \frac{1}{2\cos^2{\theta_W}}Z_\mu Z^\mu)}_{\text{Higgs-weak interactions}} \\ & + \underbrace{2\mu H^2}_{\text{Higgs boson mass}} + \underbrace{ \mu^2(\frac{2}{v}H^3 + \frac{1}{2v^2}H^4-\frac{1}{4} ) }_{\text{Cubic and quartic Higgs terms}} \end{aligned}\]

After some calculations, we find mass terms for the weak bosons! Even better, the calculations confirm that the actual fields getting masses are \(W^+, W^-, Z\). That is not a huge surprise, since \(\Phi\) was by construction invariant under \(U(1)_{EM}\), therefore even after symmetry breaking, the interaction with the \(A\mu\) would just cancel out.

Beyond that, we also get the lagrangian for a scalar boson, the Higgs boson, as well as its self-interaction and interactions with the weak bosons.

It might be enlightening to think in terms of degrees of freedom: the original Higgs field was complex scalar doublets: it had 4 real degrees of freedom. The single real scalar \(H\) only has one. But each of the three massive bosons has one extra degree of freedom than its massless predecessor. It’s like the bosons have "absorbed" part of the Higgs field, but we can see that mathematically we are still consistent.

The first role of the Higgs field is then fulfilled. But that’s not all... It will perform a second role! For a scalar field, we can also introduce an interaction term with the fermions, which takes this form:

\[-G_{H} ( \bar{\mathbf{R}} \Phi^\dagger \mathbf{L} +\bar{\mathbf{L}} \Phi \mathbf{R} )\]

Now, remember why the mass term for the fermion wasn’t gauge invariant. We have shown that it would look like this:

\[-m_e \bar{e}e = -m_e (\overline{e_R}e_L + \overline{e_L}e_R )\]

It cannot be \(SU(2) \times U(1)\)-invariant, because it mixes together a singlet and one component only from a doublet. The interaction term we’ve shown above is invariant though, since all the factors involved are invariant in their own right.

After developing we end up with a mass term! There is also a fermion-Higgs interaction which is called a Yukawa term.

\[-G_{H} ( \bar{\mathbf{R}} \Phi^\dagger \mathbf{L} +\bar{\mathbf{L}} \Phi \mathbf{R} ) = \underbrace{- G_H v \bar{e}e}_{\text{Electron mass}} - \underbrace{G_H \bar{e}e H}_{\text{Yukawa interaction}}\]

The theory doesn’t specify fermion masses since the coupling \(G_H\) is arbitrary and we express it in terms of the empirically observed masses. But the Higgs SSB has played its second role in providing gauge-invariant mass terms for the fermions!

To finish about the Higgs field, we need to talk about the process itself. We’ve showed that symmetry was broken \(\mu\) was negative, but we never explain why its sign would change. As it turns out, all the "constants" that appear in QFT, including \(e\), \(g_S\), \(g_W\) change with the energy level, as we’ve explained while discussiong the evolution of forces’ relative strengths. \(\mu\) is no exception, and as the universe cools down, its value will also diminish, down to the crucial zero point when the symmetry breaks...

So, here we are. Symmetry implies dynamics. We have:

• started from Dirac’s QFT lagrangian, and the technical apparatus around it,

• imposed the right symmetry group (for our world): \(SU(3)_C\times SU(2)_L \times U(1)_Y\),

• used a Higgs mechanism to break the symmetry, leaving us with \(SU(3)_C\times U(1)_{EM}\).

Wit that, we get the Standard Model: the crowning achievement, together with general relativity, of modern physics. It is incredibly exhaustive and displays staggering accuracy at the highest energies we can yet produce.

And it still all comes from that simple lagrangian :

\[\mathcal{L}_{SM}=\sum_G{ ( \underbrace{\bar{\Psi_G} (i\gamma^\mu D_\mu - m) \Psi_G}_\text{Covariant Dirac fields} +\underbrace{\tfrac{1}{4g_G^2}[D^{\mu},D^{\nu}] [D_{\mu},D_{\nu}]}_\text{Covariant gauge fields} ) }+ \underbrace{ \mathcal{L}_\Phi}_{\text{Higgs lagrangian}}\]

In the above, we sum over \(G\), representing the three simple components of the Lie group, with \(\Psi_G\) representing the fermion content as multiplets. It is time to present the field content of the standard model in a more algebraic manner.

Each multiplet of fields corresponds to a representation of the symmetry group. We’ve been talking about \(SU(N)\) as a group of \(N\times N\) matrices, but that’s not the case. What defines the group is the algebraic relationship between its elements. The group of size-\(N\) matrices happens to be the definition that we’ve used, so we call it the fundamental representation. Since \(N\times N\) matrices act on \(N\)-uplets, we say those belong to the fundamental representation.

In \(SU(3)_C\times SU(2)_L \times U(1)_Y\), each multiplet is defined by three data points: the representations that it belongs to relative to each of those groups. Conventionally, this is written as \((R_{SU(3)},R_{SU(2)})_Y\), where \(Y\) is the hypercharge, and defines the representation in \(U(1)\).

We end up with this content:

This is a total of 29 particles. We should note that antiparticles also exist (only for the fermions) and that the corresponding representations are written as \(\bar{\mathbf{3}}\), \(\bar{\mathbf{6}}\) etc., unless the representations are their own conjugate.

A few points deserve an explanation:

• Fermions are in the fundamental representation, which for \(SU(N)\) has \(N\) dimensions. Right-handed quarks are \(SU(3)\) triplets but \(SU(2)\) singlets so their representation is \((\mathbf{3},\mathbf{1})\) (the \(Y\) is deduced from our empirical observation about electric charges). Conversely, left-handed leptons form a \(SU(2)\) doublet but a \(SU(3)\) singlet, hence \((\mathbf{1},\mathbf{2})\). Right-handed electrons are singlets under all symmetries. And left-handed quarks are both triplets and doublets, so they live in \((\mathbf{3},\mathbf{2})\).
  The \(i\) indices are here to remind us that there are three generations of fermions (so \(e^3\) would be the \(\tau\) lepton, \(d^2\) the strange quark, etc.).

• Gauge bosons are in the adjoint representation (a group defined by the generators of the Lie algebra, so it has the same number of dimensions as there are generations. For \(SU(N)\) that is \(N^2-1\).

• The scalar Higgs field was defined as a \(SU(2)\) doublet.

Beyond the Standard Model: the minimal \(SU(5)\) GUT

The standard model is incredibly exhaustive and accurate, to the point that it’s been a challenge finding empirical facts that would directly contradict it. Yet, for the past 50 years, the quest for Grand Unification has been the overarching goal of fundamental physics.

Despite the incredible success of the standard model, there are reasons to think it does not offer a complete description of reality. Those are the most compelling arguments:

• Charge quantization: How come that protons and electrons have exactly opposite electric charges? From the standard model perspective, it is just a concidence that the electron’s charge is \(-1\) and the quarks’ are \(+2/3\) and \(-1/3\). Yet we know that it’s not an approximation: this has been measured to an extremely high accuracy. I may add it’s rather consequential, since there could be no matter at all without this "coincidence": atoms couldn’t exist if a set of quarks couldn’t exactly counterbalance the proton’s electric charge.

• Free parameters: There are 18 free parameters in the standard model (interactions’ mixing parameters, Yukawa couplings (which determine fermion masses), Higgs potential parameters...).

• Generations: Three full generations of fermions have been observed, and further data seems to prove there cannot be fourth one. Yet the presence of three identical families (except for their mass) is completely sui generis in the Standard Model.

• The hierarchy problem: Why is the weak interaction so much stronger than gravity? One way to look at this is to consider that gravitation’s strength is characterized by the Planck mass \(m_p\) while for the weak force the relevant quantity is the mass of its bosons \(m_W\) or \(m_Z\), which depend directly on the Higgs boson’s mass \(m_H\). In fact, the value of \(m_H\) is not a prediction of the standard model, it is an observation we’ve made. The issue is that in QFT, the renormalization procedure that helps us go from the "bare masses" of the lagrangian to the "physical mass" that we observe, relies on the cancellation of certain terms that require fine-tuning. The form of Higgs renormalization makes the cancellation terms proportional to \(\Lambda / m_H\), where \(\Lambda\) is roughly the ambient energy level. So if we wanted the standard model to be valid up to energies comparable to the Planck mass, then we would need terms \(10^34\) times larger than the Higgs mass to "magically" cancel out and give us the right result. This is mathematically acceptable but appears as a "coincidence" which could only be explained by the anthropic principle (only a universe where that coincidence happens could have us in it, so we must be in that one), or this procedure can be avoided thanks to a higher-level theory.

• Matter-antimatter asymmetry: The imbalance between matter and antimatter in our universe is difficult to square with the symmetries of the standard model (where fermions and anti-fermions only get produced, or annihilate, in pairs)/

• Neutrino mass: Observations now strongly suggest that neutrinos have a small mass, which means that right-handed neutrinos could exist. That is not ruled out but unlike right-handed electrons, right-handed neutrinos would need to be completely separate from the other fermions: not part of any representation of the basic gauge group.

A great way to explain the first two items is through unification of the interactions, which as we’ve seen, also fulfils a long-held aesthetic preference that has also borne fruit in spectacular fashion in physics history.

We know that unification cannot happen in the context of the \(SU(3)\times SU(2) \times U(1)\) group. Since that group is a product of three simple groups, there can be three different coupling constants (interaction strength, which we called \(g_s\), \(g\) and \(g'\) above). We can compute their dependence to temperature based on the standard model. When we increase it (or equivalently, when we "turn back time" closer to the Big Bang), there is no point when the three constants would be equal. So, unification is a miss.

One way to avoid this is to build a theory around one simple Lie group, which by definition has only one coupling constant (say, \(g_{GUT}\)). That symmetry can then be broken into \(SU(3) \times SU(2) \times U(1)\), at which point we retrieve the standard model.

The strategy should then be as follows:

• Find a simple Lie group \(G\) that contains the Standard Model group and has complex representations (lest it couldn’t account for the difference between left-handed and right-handed particles which we observe in the Standard Model).

• Find representations matching the known particle content of the standard model.

• Find a symmetry breaking procedure that leads to the Standard Model group.

• Check that the predictions are not inconsistent with our observations (in particular, extra particles should all be heavy enough to explain why they were not observed before).

Luckily, the set of possible groups is very small. First let’s remember that the classification of all simple Lie groups is itself parsimonious.

• The special unitary groups: \(A_n = SU(n+1)\)

• The "odd-dimensional" special orthogonal groups: \(B_n = SO(2n+1)\)

• The symplectic groups: \(C_n = Sp(n)\)

• The "even-dimensional" special orthogonal groups: \(D_n = SO(2n)\)

• \(E_6\),\(E_7\),\(E_8\),\(F_4\),\(G_2\)

That list covers all the possible simple Lie groups. This can be proved very elegantly using Dynkin diagrams, which we can only show here for illustration.

First of all, we should note that the rank of the standard model group is \(4\). The rank can be defined as the largest number of generators that commute with each other. We know that \(SU(3)\) has rank \(2\), \(SU(2)\) has rank \(1\), and \(U(1)\) only has one generator. By construction of the group product, generators of distinct simple groups commute, hence the rank of \(SU(3)\times SU(2) \times U(1)\) ought to be \(2+1+1=4\).

The nomenclature we’ve presented above uses the rank as an index. So the simple groups of rank 4 are \(A_4\),\(B_4\),\(C_4\),\(D_4\) and \(F_4\). But Lie group theory (an aesthetically delighting discipline) tells us exactly which groups fulfill our basic criteria (to embed the Standard Model, and to admit complex representations). They are: the \((A_n)_{n \geq 4}\), the \((D_{2n+1})_{n\geq 2}\), and \(E_6\).

For rank \(4\), we know that the only eligible group is \(A_4=SU(5)\). At rank 5, there are \(A_5=SU(6)\) or \(D_5=SO(10)\). At rank 6 the eligible groups are \(A_6=SU(7)\) and the exceptional group \(E_6\).

SU(5) has a particular status, being the minimal supergroup of the Standard Model: the smallest supergroup that contains it. Building a gauge theory, thanks to all the rules inherited from group theory, is a lot like Meccano at this stage! So let’s get on with it...

In theories we’ve considered so far, fermions have been placed in the fundamental representation and its anti-fundamental counterpart. For \(SU(N)\) groups, these were alway \(N\)-dimensional. No difference here, the fundamental representation is indeed \(\mathbf{5}\), and the anti-fundamental one is \(\bar{\mathbf{5}}\).

What are the generators of \(SU(5)\). Remember that they form a basis of the Lie algebra \(\mathfrak{su}(N)\), which is the space of \(N\times N\) traceless Hermitean matrices. We already know that the basis for \(\mathfrak{su}(3)\) is the family of 8 Gell-Mann matrices \((\lambda^i)\), and for \(\mathfrak{su}(2)\) is the family of 3 Pauli matrices \((\tau^i)\).

These are already traceless Hermitean matrices in their own space, so it’s easy to see that those matrices also form a free family of \(\mathfrak{su}(5)\):

\[\begin{pmatrix} \lambda^i & 0 \\ 0 & 0 \end{pmatrix}_{i=1\dots8} \quad , \quad \begin{pmatrix} 0 & 0 \\ 0 & \tau^i \end{pmatrix}_{i=1\dots3}\]

This means that if we divide the quintuplet representation into a triplet (on top) and a doublet (on the bottom), those will be respectively \(SU(3)\)- and \(SU(2)\)-invariant. It makes sense, based on the way we’ve modelized fermions to fill out the quintuplet with a right-handed quark triplet and a left-handed lepton doublet.

If we continue to explore the generators of \(SU(5)\), we can show there is a diagonal matrix among them: \(\text{diag}(2,2,2,-3,-3)\). From the independent perspectives of a triplet and of a doublet, this is acting as a simple scalar (a multiple of the identity). We can then identify it with \(\tilde{Y}\), the operator corresponding to the weak hypercharge \(Y\).

The difference is that in the standard model, the generator of \(U(1)_Y\) does not need to be a traceless matrix, and indeed it is not. Here, the matrix is a generator of \(SU(5)\) and thus had to be traceless, so it could not be a multiple of the identity. The fact is traceless ensures that the sum of hypercharges within the quintuplet. Since the electric charge matrix \(Q\) will ultimately result from the same kind of symmetry breaking as within the Standard Model and will be some linear combination of \(\tau^3\) (also traceless) and \(\tilde{Y}\). Therefore \(Q\) will be traceless and the sum of electric charges in the quintuplet will be nil.

This means that \(3q_Q + q_e=0\) (because quarks of each color have the same electric charge, and we know the neutrino’s charge is nil). Furthermore, the quark has positive hypercharge and the leptons negative hypercharge. Based on our standard model study, this means the quark triplet is actually an antiquark triplet (since only antifermions carry positive hypercharge). The charged lepton is then an electron (not an antiparticle). We end up with \(q_e=-1\) and then \(q_Q=+1/3\), which we then identify with an anti-down quark.

But most importantly, this relationship is imposed by the theory! Right off the bat, \(SU(5)\) solves the puzzle of charge quantization.

We have successfully identified the fundamental representation with known fermion content: \[\begin{aligned} \bar{\mathbf{5}} &\rightarrow (\bar{\mathbf{3}},\mathbf{1})_{2/3} \oplus (\mathbf{1},\mathbf{2})_1\\ & = \overline{d_R} \oplus L_L = \begin{pmatrix} \overline{d_R^b} & \overline{d_R^r} & \overline{d_R^y} & \nu_L & e_L \end{pmatrix} \end{aligned}\]

However, we’re still missing lots of our fermions. To find them we can go to the next smallest representation, the ten-dimensional \(\mathbf{10}\), which we can evidence by taking the product of two fundamental representations.

\[\mathbf{5} \otimes \mathbf{5} = \mathbf{10} \oplus \mathbf{15}\]

\(\mathbf{10}\) is the anti-symmetric part of that product. It can be represented as an anti-symmetric \(5\times 5\) matrix, which indeed has 10 degrees of freedom. One way to think about this product is that each component of \(\mathbf{10}\) is the "sum" of two distinct components of \(\mathbf{5}\) (or rather, the particle whose conserved charges are the sum of the two components’ respective charges).

\[\begin{aligned} \mathbf{10} &\rightarrow (\bar{\mathbf{3}},\mathbf{1})_{4/3} \oplus (\mathbf{3},\mathbf{2})_{1/3} \oplus (\bar{\mathbf{1}},\mathbf{1})_{2} \\ & = \overline{u_R} \oplus Q_L \oplus \overline{e_R}\\ & = \begin{pmatrix} 0 & \overline{u_R^b} & -\overline{u_R^y} & u_L^r & d_L^r\\ - \overline{u_R^b} & 0 & \overline{u_R^r} & u_L^y & d_L^y\\ \overline{u_R^y} & -\overline{u_R^r} & 0 & u_L^b & d_L^b\\ -u_L^r & -u_L^y & -u_L^b & 0 & \overline{e_R}\\ -d_L^r & -d_L^y & -d_L^b & -\overline{e_R} & 0\\ \end{pmatrix} \end{aligned}\]

Between \(\mathbf{5}\), \(\bar{\mathbf{5}}\),\(\mathbf{10}\) and \(\bar{\mathbf{10}}\), we’ve retrieved the 15 fermions and 15 anti-fermion of the Standard Model!

What about the bosons? As usual, they occupy the adjoint representation, which is 24-dimensional (\(5^2-1\)).

\[\begin{aligned} \mathbf{5} \otimes \bar{\mathbf{5}} &= \mathbf{24} \\ & \rightarrow (\mathbf{8},\mathbf{1})_0 \oplus (\mathbf{3},\mathbf{1})_0 \oplus (\mathbf{1},\mathbf{1})_0 \oplus (\mathbf{3},\mathbf{2})_{-10/3} \oplus (\bar{\mathbf{3}},\mathbf{2})_{10/3} \end{aligned}\]

This matches or analysis of the generators. In this sum we find:

• 8 gluons \(G\)

• 3 weak bosons \(W^\pm\) and \(W^3\)

• 1 hypercharge boson \(B\)

• 12 \(SU(5)\)-specific bosons called \(X\) and \(Y\)

The \(X\) and \(Y\) bosons have charges correspond to generators that govern interactions between quarks and leptons. That’s why they’re sometimes called leptoquarks. They carry weak, electric and color charges.

Among the reactions allowed by the presence of the \(X\) and \(Y\), some involve the decay of these boson in two fermions or two anti-fermions. Therefore, in the context of a very fast cooling of the universe, we are much closer to explaining the matter-antimatter asymmetry!

One important task remains: we have a wonderful symmetry, but as for all unbroken local gauge symmetries, it only allows massless fermions and bosons.

As for the Standard Model, we’ll use the wonderful Higgs mechanism, which has abundantly proven its worth. At this stage it is very important to highlight the difference between GUT symmetry breaking and the electroweak symmetry breaking.

The electroweak Higgs process breaks the symmetry this way: \[SU(3) \times SU(2) \times U(1)_Y \rightarrow SU(3) \times U(1)_{EM}\]

On both sides we have a product of simple Lie groups, each one can have its own coupling constant (interaction strength). The symmetry breaking has caused the coupling of the Higgs field with other particles which gave mass to the \(W^\pm\), the \(Z\), the fermions and the Higgs boson. But it has not caused the divergence of the \(SU(2)\) and \(U(1)_Y\) coupling constants: they were already distinct. In that sense, it is an exaggeration when we speak of the "electroweak" force. And the symmetry breaking process is much better described as a phase transition than as a "force de-unification".

The \(SU(2)\) and \(U(1)_Y\) coupling constants vary with the energy level, and they do meet at some point, but that is a different one, at a much higher energy than the Higgs boson mass (which is where the electroweak phase trsnsition happens). The goal of the GUT builders is to have the three couplings meeting at the same energy level, so there’s need for only one GUT breaking. And the running couplings’ calculation, after incorporating all the \(SU(5)\)-specific terms, shows them almost converging!

The \(SU(5)\) model then uses two Higgs field, one in the adjoint representation (made of 24 real scalar fields) breaks \(SU(5)\) directly into the SM group. It gives mass to the 12 \(X\) and \(Y\) bosons, and to 12 Higgs bosons. Another Higgs field in the fundamental representation is a complex quintuplet, with the lower doublet being the usual SM Higg field, and with at the top a color-charged Higgs triplet, resulting into 5 massive Higgs boson (why this doesn’t give mass to the gluon is a problem called triplet-doublet splitting that introduces significant fine-tuning in the theory).

Let’s look at our GUT checklist. \(SU(5)\) has successfully explained charge quantization, and majorly improved the situation regarding matter/antimatter asymmetry, and coupling unification. It has also established relationships between parameters, however it also introduces new free ones...

Despite this overall success, \(SU(5)\) GUT as such is not enough: it is contradicted by experience, the most striking one being the half-life of the proton, which has to be much higher than what it predicts (based on our observations). It also makes a number of wrong predictions, for instance about the ratios between fermion masses.

From \(SO(10)\) to \(E_8\)

\(SU(5)\) was our only option at rank \(4\). But not to worry, we have plenty of higher-rank ones. At rank \(5\), it is \(SO(10)\) that seems to offer the best prospects at this stage.

First off, the fundamental representation of that group is sixteen-dimensional: \(\mathbf{16}\). We can soon verify that the whole SM fermion content can fit in that representation (all fifteen of them) plus a particle we can identify with the righ-handed neutrino! This is considered more elegant since all the matter and antimatter content can fit in the fundamental and anti-fundamental representations.

The adjoint representation is 45-dimensional, which means that we should find the 24 bosons from \(SU(5)\), plus 21 new ones.

\(SU(5)\) is a subgroup of \(SO(10)\), and sure enough, the following symmetry breaking route is available:

\[\begin{aligned} SO(10) & \rightarrow SU(5) \times U(1) \\ &\rightarrow SU(3)_C \times SU(2)_L \times U(1)_Y \end{aligned}\]

This means there will be two, instead of one, GUT breaking scales. At least this route allows us to retrieve the familiar \(SU(5)\) multiplets, accompanied by the new fermion (the RH neutrino) and the new bosons.

\[\mathbf{16} \rightarrow \mathbf{10} \oplus \bar{\mathbf{5}} \oplus \mathbf{1}\]

\[\mathbf{45} \rightarrow \mathbf{24} \oplus \bar{\mathbf{10}} \oplus \mathbf{10} \oplus \mathbf{1}\]

However there is another route:

\[\begin{aligned} SO(10) & \rightarrow SU(4)_C \times SU(2)_L \times SU(2)_R \\ &\rightarrow SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L} \\ &\rightarrow SU(3)_C \times SU(2)_L \times U(1)_R \times U(1)_{B-L} \\ &\rightarrow SU(3)_C \times SU(2)_L \times U(1)_{Y} \\ \end{aligned}\]

This is obviously a more complex scheme that involves four symmetry breaking scales. \(SU(4)_C \times SU(2)_L \times SU(2)_R\) corresponds to the Pati-Salam theory, which pre-dates even the \(SU(5)\) Georgi-Glashow model. It posits a four-color symmetry, where all leptons would have the same color, called "liliac". The two \(SU(2)\) groups correspond to a left-right symmetry, which imposes the existence of the right-handed neutrino. The breakdown of the \(SU(4)\) color group yields the QCD group and a \(B-L\) symmetry that preserves the Baryon-minus-Lepton number. The last step to reach the Standard Model gives the neutrino its mass and mixes the two \(U(1)\) symmetries into the familiar hypercharge one.

The \(SO(10)\) representations’ branching rules when it breaks into the Pati-Salam group are as follows:

\[\mathbf{16} \rightarrow (\mathbf{4},\mathbf{2},\mathbf{1}) \oplus (\bar{\mathbf{4}},\mathbf{1},\mathbf{2})\]

\[\mathbf{45} \rightarrow (\mathbf{15},\mathbf{21},\mathbf{1}) \oplus (\mathbf{6},\mathbf{2},\mathbf{2}) \oplus (\mathbf{1},\mathbf{3},\mathbf{1}) \oplus (\mathbf{1},\mathbf{1},\mathbf{3})\]

The Higgs field content and mechanism depends of course on the route chosen. It typically involves scalars in at least the \(\mathbf{16}\) and \(\mathbf{45}\) representations.

\(SO(10)\) theories are then richer in terms of the phenomenology they may explain, but introduces even more new heavy particles. Unlike \(SU(5)\), \(SO(10)\) has not been ruled out by observations, but none of these extra particles have been observed at experimental energy levels.

The next step is usually the exceptional \(E_6\) group. It contains \(SO(10)\) and can break its way:

\[\begin{aligned} E_6 \rightarrow SO(10)\times U(1) \end{aligned}\]

The fundamental representation has 27 components. We then end up with many additional fermions to which we need to give high mass to explain why they don’t pop up now. The adjoint representation has 78 bosons.

\[\mathbf{27} = \mathbf{16} \oplus \bar{\mathbf{10}} \oplus \mathbf{1}\]

\[\mathbf{78} = \mathbf{45 } \oplus\mathbf{16} \oplus \bar{\mathbf{16}} \mathbf{1}\]

\(E_6\) is interesting because it has as a maximal subgroup \(SU(3)^3\), called the trinification group. Although that is by no means fleshed out, it is thought that group can implement flavorgenesis, and at last explain the existence of three fermion generations as the result of a specific symmetry. It can then break into the same left-right group we’ve seen above as an outcome of Pati-Salam symmetry breaking.

\[\begin{aligned} E_6 & \rightarrow SU(3)_C \times SU(3)_L \times SU(3)_R &\rightarrow SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L} \\ \end{aligned}\]

\[\mathbf{27} = (\mathbf{3},\mathbf{3},\mathbf{1}) \oplus (\bar{\mathbf{3}},\bar{\mathbf{3}},\mathbf{1}) \oplus (\mathbf{1},\bar{\mathbf{3}},\mathbf{3})\]

\[\mathbf{78} = (\mathbf{8},\mathbf{1},\mathbf{1}) \oplus (\mathbf{1},\mathbf{8},\mathbf{1}) \oplus (\mathbf{1},\mathbf{1},\mathbf{8}) \oplus (\mathbf{3},\bar{\mathbf{3}},\bar{\mathbf{3}}) \oplus (\bar{\mathbf{3}},\mathbf{3}),\mathbf{3})\]

The Higgs field content to achieve these complex breaking chains from bigger and bigger groups gets overabundant, with a 351-dimensional representation involved in most models.

All in all \(E_6\) almost seems oversized, despite the potential added value from trinification. Unlike \(SO(10)\), it entails the addition of new, unobserved fermions. The main attractive factor may then be aesthetic. \(E_6\), being an exceptional group, appears like the end of the road:

\[SU(3) \times SU(2) \times U(1) \supset SU(5) \supset SO(10) \supset E_6\]

No Lie group other than \(E_7\) and \(E_8\) contain \(E_6\) as a subgroup, and we know their lack of complex representations make them unlikely candidates to be the GUT group. So picking \(E_6\) offers the advantage of singularity: it is not just "some group".

It is worth noting, however that the picture changes once superstrings come into play. We have seen that superstring theory, at a "low" energy , can be approximated by a Lie-gauge-group invariant QFT, the likes of which we’ve been considering here ("low energy" for superstrings means the high-end of the range for every GUT theory).

And it so happens that one of the only gauge groups compatible with the superstring framework is \(E_8\times E_8\), a rank-16 gauge group acting over strings from a 10D spacetime. Now, of course, we’ve seen that \(E_8\), lacking real representations, is ineligible for GUTs... but that’s only true in 4D spacetimes! As it turns out, the compactification to 4D and the use of "stringy" procedures would allow to retrieve, as an approximation, the gauge group \(E_6 \times SU(3)\) for our GUT. Alternately, \(E_8\) can break into \(SO(10) \times SU(4)\).

This connection to superstrings makes the study of \(E_6\) and \(SO(10)\) GUTs particularly attractive. \(E_8\) is also interesting to physicists since it is a unique case where the fundamental and adjoint representations have the same dimensionality (in this case, 248). Since they typically house the fermion and boson contents respectively, this makes any \(E_8\) theory a natural for supersymmetry!

From a purely aesthetic perspective, \(E_8\) is really the end of the line, since it is not a subgroup of any simple Lie group.

"Magical" supersymmetry

To close this exploration of the Grand Unification epoch, we need to evoke supersymmetry. We’ve already introduced this important concept in our discussion of supergravity and superstrings. The reason there is such high conviction that our reality is somehow “super” in that way, despite the lack of convincing observations, is that supersymmetric theories find themselves at the nexus of our two main concerns right now. They could be the key to the biggest issues faced by “regular” GUTs. They also offer, as we’ve seen, a natural way to have gravity emerge in a quantum field or quantum string framework.

It is not pure chance that supersymmetry finds itself in this enviable position. Indeed it acts very much as a bridge between internal and spacetime symmetries. Let’s review briefly the different type of continuous transformations pertaining to these symmetries:

• Spacetime transformations are part of the Poincaré group \(ISO(3,1)\) (the group of isometries in Minkovski space), which is an Abelian Lie group, and itself is assembled from two types of transformations:

  • Lorentz group transformations preserve the Minkowski metric \(\eta\), so they verify \(\Lambda^T \eta \Lambda = \eta\). They belong to the group \(SO(3,1)\), which is isomorphic to \(SL(2,\mathbb{C})\), which itself can be seen as \(SU(2) \otimes SU(2)\) (that is why the Dirac spinor is made of two Pauli spinors). Concretely speaking, the group has 6 generators: three of them generate the spatial rotations, while the other three are the “boosts”: the switch of Galilean referential along a specific axis (which can be seen as rotation in spacetime). The six generators are represented as the components of an anti-symmetric 4-tensor \(M_{\mu\nu}\).

  • 4 spacetime translations \(P_\mu\).

  We know special relativity to be valid in all circumstances, so all of our theories need to be Poincaré-invariant. Interestingly, all of these transformations were known and accepted as symmetries in the pre-Einstein era. But because the maximum speed was assumed to be infinite, their most general form was not found until the twentieth century.

• Internal transformations do not change the spacetime properties of the objects they transform at all. The internal symmetry group \(G\) is the Lie group corresponding to the Standard Model or to any other GUT we are studying. We can call the generators of the corresponding Lie algebra \(g_l\).

The overall symmetry group is the direct product of the Poincaré group and the internal symmetry group. This means that two generators from different groups always commute: there can be no simple groups that would incorporate all these types of symmetry together. That has been shown to be rigorously true, under a number of desirable circumstances (like causality, locality, finiteness…) as the Coleman-Mandula theorem.

Therefore the group of continuous symmetries of the Lagrangian ought to look something like \[ISO(3,1) \otimes G\] And the following commutation relations must be true: \[[P_\mu,g_l] =0 \quad [M_{\mu\nu},g_l]=0\]

However, there is one of condition of Cole-Mandula that can be weakened without jeopardizing the underlying physics. It is the assumption that all symmetry groups must be Lie groups (and thus that all their generators define a Lie algebra). It is actually simpler to focus on the algebra picture. What defines a Lie algebra is the existence of a "bracket" operator \([\cdot,\cdot]\). When we worked with matrix representations we just accepted that \([A,B]=AB-BA\), for brevity sake, but that is only true (by definition) in those matrix representations. Rigorously speaking the Lie bracket of a Lie algebras \(L\) verifies the following properties:

\[\begin{aligned} &[l_1,l_2] \in L &\quad (\text{stability})\\ &[l_1, (l_2+l_3)] = [l_1,l_2] + [l_1,l_3] & \quad ( \text{linearity})\\ &[l_2,l_1] = -[l_1,l_2] &\quad (\text{anti-symmetry})\\ &[l_1,[l_2,l_3]]+[l_2,[l_3,l_1]]+[l_3,[l_1,l_2]]=0 & \quad (\text{Jacobi identity}) \end{aligned}\]

There is a way to extend any Lie algebra through grading. In particular, \(\mathbb{Z}_2\)-graded Lie algebras are called superalgebras. Let’s call \(S\) the superalgebra or \(L\). Then there is some vector space \(V\) (which is not a Lie algebra) such as: \[S=L \oplus V\]

What we will do is define a linear superbracket \([\cdot,\cdot\}\) over \(S\), with the following properties.

\[\begin{aligned} [l,l'\} &\in L & [l,v\} &\in V & [v,v'\} &\in L\\ [l,l'\} &= - [l',l\} & [l,v\} &= -[v,l\} & [v,v'\} &= [v',v\} \end{aligned}\]

In addition, it verifies some generalized Jacobi identities which we don’t reproduce here. The main point is that the restriction of the superbracket over \(L\times L\) is simply the Lie bracket. \(L\) is stable under the superbracket, however \(V\) is not, so \(V\) cannot be a Lie algebra.

Besides, the antisymmetry/symmetry laws show that over \(L\times L\) as well as \(L \times V\), the superbracket behaves like a commutator, but over \(V \times V\) it behaves like an anticommutator (and indeed in matrix representations, it can be realized as \(\{A,B\}=AB+BA\)). We will colloquially write:

\[\begin{aligned} [l,l'\} &=[l,l'] \\ [l,v\} &=[l,v] \\ [v,v'\} &={v,v'} \end{aligned}\]

We can apply that procedure to a specific Lie algebra, like the Poincaré algebra, and obtain its super algebra, often called the SuperPoincaré algebra.

As we’ve said before, Coleman and Mandula have shown that the group of continuous symmetries governing our universe must be of the form \(\text{Poincaré} \otimes G\). There is no larger Lie algebra than Poincaré that we could use instead (it needs to be larger so that it includes the Poincaré transformations, since we know special relativity to be true).

However Coleman-Mandula has nothing to say about spaces that are not Lie algebras. We could show that the SuperPoincaré algebra could a larger entity that we could use.

The Haag–Łopuszański–Sohnius theorem does just that, and goes even further: it shows that using the "super", graded version of the Poincaré algebra, is the one and only way to extend the symmetry group. That’s a beautiful mathematical result that could be promising from a physical perspective.

So what do the commutation relations that characterize this SuperPoincaré algebra look like? There are different options already: we can show that generators of the "\(V\)" space (the new ones) come in pairs. The simplest case: 2 generators, is called \(N=1\). Since we’re looking at the classic Poincaré algebra in (3,1) spacetime, the simplest SuperPoincaré is labelled \(D=4,N=1\), even though other versions are used in more complex theories, including superstrings of course.

We’ll call these new generators \(Q_\alpha\) and \(\overline{Q}_{\dot{\alpha}}\). The underlying mathematics leads to the following superbracket relations:

\[\begin{aligned} [M_{\mu\nu},M_{\rho\sigma}]&= i\eta_{[\mu[\sigma}M_{\nu]\rho]} \quad & (1)& \quad & [M_{\mu\nu},Q_\alpha] & = i(\sigma_{\mu\nu})_\alpha^\beta Q_\beta &\quad(7)\\ [P_\mu,P_\nu]&=0 \quad &(2) & \quad & [P_\mu, Q_\alpha]&=0 &\quad(8)\\ [M_{\mu\nu},P_\rho]&= i \eta_{\rho[\nu}P_{\mu]} \quad &(3) & \quad & [Q_\alpha,g_l]&=-Q_\alpha &\quad(9)\\ [P_\mu,g_l]&=0 \quad &(4) & \quad & \{ Q_\alpha , Q_\beta\} &=0 &\quad(10)\\ [M_{\mu\nu},g_l]&=0 \quad &(5) & \quad & \{ Q_\alpha , \overline{Q}_{\dot{\beta}} \} &= 2\sigma^\mu_{\alpha \dot{\beta}}P_\mu &\quad(11)\\ [g_l,g_m]&= f^n_{lm} g_n\quad &(6) & \\ \end{aligned}\]

• (1) and (2) define respectively the Lorentz (rotations and boosts) and the Euclidean (translations) groups. (3) completes the overall Poincaré algebra.

• (4) and (5) is a consequence of the Coleman-Mandula theorem. Poincaré generators and internal symmetry group generators commute.

• (6) is the generic definition of \(G\)’s structure constants.

• (7) shows that the \(Q_\alpha\) generators transform like spinors under the action of the Poincaré group. That is why they are called fermionic generators. It is not altogether surprising: these generators are characterized by anticommutation laws, just like the Dirac matrices, and elements from higher-dimensional spinor representations.
  If we remember that \(M_{12}=J_3\), we also get the momentous consequence that \([J_3,Q_1]=+\tfrac{1}{2}Q_1\) and \([J_3,Q_2]=-\tfrac{1}{2}Q_2\): the supersymmetry generators can change a particle’s sping by a half integer, meaning it turns a boson into a fermion and vice versa!

• (8) implies that, like standard model generators, supersymmetry transformations preserves mass.

• (9) shows that unlike the Poincaré generators, SUSY generators don’t commute with the internal symmetries of the theories. In practice this results in another \(U(N)\) symmetry called R-symmetry.

• (10) and (11) are the anticommutation relations governing the fermionic sectors of the SuperPoincaré algebra. At \(N=1\) they imply that each particle has a single superpartner.

A more abstract, but perhaps simpler way to look at the SuperPoincaré algebra is to introduce the concept of superspace. It starts from the idea that since the generators \(P_\mu\) correspond to translation along spacetime coordinates, the generators \(Q_\alpha\) and \(\overline{Q}_{\dot{\alpha}}\) could similarly correspond to translations along some new coordinates \(\theta_\alpha\) and \(\overline{\theta}_{\dot{\alpha}}\). Of course, because of the anticommutation relations, these are not "normal" real coordinates; instead they are Grassmann numbers. Ultimately, it appears that the SuperPoincaré algebra is just the analog of the Poincaré algebra over the SuperMinkowski space, with coordinates \((x_\mu,\theta_\alpha,\overline{\theta}_{\dot{\alpha}})\).

Let’s go back to the most concrete feature of supsersymmetry so far: it introduces a group of transformations that can turn a fermion (a matter particle with half-integer spin) into a boson (a force-mediating particle with integer spin), and vice versa. It is definitely a spacetime symmetry, since changing the spin of a particle does change its spacetime properties. At the same time, it is closer to the way we think of internal symmetries: we consider those as able to “change” a particle into another, for instance a neutron into a proton, or a blue quark into a red quark. Supersymmetry seems to generalize this by allowing fermion-boson transformations, and thus to constitute one more step in the unification process. But it does that by elegantly integrating the symmetries of special relativity with those of quantum physics (pursuing what Dirac had started by formalizing spinors).

The Minimal Supersymmetric Standard Model (MSSM) is then a theory whose symmetry group is \(\text{SuperPoincaré} \otimes G_{SM}\) instead of \(\text{Poincaré} \otimes G_{SM}\). The theory predicts that each particle predicted by the Standard Model msut have a superpartner:

• The superpartners of fermions (spin 1/2) are spin 0 bosons (scalar bosons) called sfermions (including selectrons, sneutrinos, squarks, etc.).

• The superpartners of the gauge bosons (spin 1) are spin 1/2 fermions and are called gauginos (including photinos, winos, bino, gluinos).

• The superpartner of the Higgs bosons (spin 0) are spin 1/2 fermions, the Higgsinos (unlike the SM which can make do with a single Higgs, the MSSM requires a Higgs doublet to generate fermion masses).

The theory predicts that both superpartners have the exact same mass. Obviously we know that is not the case since we’ve never observed any superpartner in the observable mass range. We also know that no observed fermion-boson couple can be a superpartner doublet, since they would need to share all quantum numbers (except spin), which is never the case. This means that supersymmetry is broken, so the MSSM must include a symmetry breaking process as well.

Since supersymmetric theories aim to improve upon the Standard Model it also makes sense to apply that concept to Grand Unified theories. We can work with \(\text{SuperPoincaré} \otimes G_{GUT}\) where the GUT group could be \(SU(5)\), \(SO(10)\), \(E_6\), or any of the other groups we’ve encountered. All the extra bosons or fermions predicted by those theories would then all have their own superpartners.

By the way, since the Poincaré algebra has been rendered "super", could \(G_{GUT}\) also become a Lie superalgebra? That is indeed allowed by the Haag–Łopuszański–Sohnius theorem. However, it seems to be decisively proved that the resulting theories are not renormalizable: none of our calculations could converge and yield actual physical quantities.

Now to conclude, the perceiving reader might wonder if we haven’t gone on a flight of fancy. Supersymmetry is mathematically pleasing, but it introduces numerous extra particles and phenomena. Can it give us anything in exchange? The answer is a resounding "yes!", and then some.

• The hierarchy problem is neatly solved by supersymmetry. As we’ve seen, that problem relates to the un-naturalness of perturbative calculations relative to the Higgs boson. Pesky quadratic terms would force us into picking highliy un-natural combinations of free parameters for the quadratic terms to cancel and leave us only with finite results. However the quadratic terms pertaining to each fermion (resp. boson) would be exactly opposite for a fermion (resp. boson) with the exact same quantum numbers, save for spin. That is the exact definition of a superpartner! So, in a supersymmetric theory, the cancellation of the quadratic terms happens in the most natural way. This was actually the initial motivation to explore theories with a fermion/boson symmetry.

• Gauge unification is also magically facilitated by supersymmetry. We’ve explained that in GUTs such as \(SU(5)\), the coupling constants of interactions would meet at a single point, however that was a shortcut: although the situation is much improved in GUTs versus the Standard Model, the couplings only approximately meet. Thus, large, quite unnatural corrections would be needed to enforce exact gauge unification. However, supersymmetry seems to always result in much more precise unification, requiring small corrections only even in the MSSM (that is, without Grand Unification). This is even truer for supersymmetric GUTs.

• We have seen that the prediction of \(SU(5)\) theory for the proton decay were not compatible with experimental observations. One way to make that decay more unlikely, and therefore to push up the GUT scale, is to use a larger GUT group like \(SO(10)\). Another one is to use supersymmetry: the GUT breaking scale in supersymmetric \(SU(5)\) is already much higher.

• Dark matter is more of a cosmological problem than one for quantum physics. Yet it is a fact that most of the universe’s matter (non-luminous, non-baryonic matter which we know is there from studying the large-scale properties of the observable universe) does not appear to be made of our commonplace particles. Light, semi-stable superpartners seem to be good candidates to play that role.

• Supergravity is by far the most tantalizing prospect that is encouraging research into supersymmetry. Remember that: \(\{ Q_\alpha , \overline{Q}_{\dot{\beta}} \} = 2\sigma^\mu_{\alpha \dot{\beta}}P_\mu\). Everything we’ve done so far is talk about global supersymmetry. What happens if we try to develop a local gauge theory that incorporates supersymmetry? This means that we will make the theory invariant with regards to supersymmetry transformations that are functions of spacetime parameters \(\epsilon(x_\mu)\). Because of the anticommutator bracket above, this means that the theory is also invariant under translations that depend on some \(\epsilon'(x_\mu)\). That is the definition of a theory that invariant under general coordinate transformations: in other words: general relativity! We can show that conversely, if we start from a global supersymmetry and impose invariance under general coordinate transformation, we get local supersymmetry as a consequence! \[\text{Global SUSY} + \text{Gravity} \leftrightarrow \text{Local SUSY}\]

Now we can at last seize back the chronological thread that we had introduced. At the end of the Planck epoch, around \(10^{-43}\text{s}\), the Grand unification era starts, with a compactified spacetime over which the perturbative approach works: in other words, identifiable quantum fields are recognizable, and this brand new spacetime is rife with GUT denizens: leptoquark bosons, all the additional content that higher-ranked GUT groups may yield, various Higgs scalars, and all of the superpartners that accompany these exotic creatures.

The stable, (very) far future

From the post-Big Bang compactification, a meta-stable state has emerged: the expanding (3+1)D curved spacetime that we now live in.

Over a long period of time, all the matter content of the universe will have decayed, leaving only the ultimate stable particles (potentially only photons). Even with no matter, gravitational structures could still persist, chiefly black holes (spacetime singularities).

It is well accepted, though, that black holes evaporate, emitting Hawking radiation. Hawking radiation is mostly made of photons, but the smaller the black hole, the likelier it is to take the form of particle-antiparticle pairs. Over time, those will either meet or decay further, reducing the universe’s contents to photons, plus a tiny amount of stable fermionic matter, reflecting the matter-antimatter asymmetry.

Post-black hole evaporation, we then get a uniformly curved universe with uniform energy density, and no complexity whatsoever! There is no telling at which point exactly this would happen, but this will likely be north of \(10^{100}\) years (1 googol years): a small black hole with a mass like our Sun’s takes \(10^{64}\) years to evaporate, and it is believed that the future collapse of galaxy clusters will give rise to black holes \(10^{14}\) as heavy as the Sun.

At the end of a black hole’s lifecycle, the temperature is such that it would actually appear to be luminous. The Universe may then get a last luminous hurrah... But unlike the light that we see from the stars, this will not be the harbinger of glorious supernovae, creators of complexity, but a dying light.

Once the universe is entirely uniform, two futures are possible, depending on the dark energy content. In one scenario, the "Big Freeze", expansion continues forever but at an ever-slower pace. The temperature converges to zero, and the Universe ends up as a static, forever-frozen fabric. However an alternate scenario can be envisioned: the "Big Rip". If the expansion rate is instead increasing, the curve of expansion will one day become exponential. Every distance will thus tend towards infinity, and the observable universe from any spacetime point will have a radius equal to at most the Planck length.

Absent an accepted theory to unify quantum physics and general relativity, it is difficult to envision how these cosmological scenarios will impact the behavior of quantum fields. What we know however is that at extremely low energies, quantum effects can manifest at larger and larger scale. A Big Freeze universe might then witness macroscopic quantum fluctuations on larger and larger scales, until they become Universe-wide, coming full circle, since this was also the case right after the Big Bang. The familiar picture of fields and particles would then become again meaning less. It may turn out that the Big Freeze might look more like the paradoxical image of an infinite ocean of boiling water at the absolute zero...

Level 2: Forces — \(10^{-35}\text{s}\)

• Elementary particles

  • Size: \(10^{-18} \text{m}\)

  • Initial state: GUT quantum fields

  • Production process: Symmetry breaking

  • Meta-stable equilibrium: First-generation particles

  • Stable future: Stable particles (photon, neutrino, LSP)

• The 4D Universe

  • Temperature: \(10^{29} \text{K}\)

  • Initial state: Curved 4D universe

  • Production process: Cosmic inflation

  • Meta-stable equilibrium: Locally flat 4D universe (radiation-dominated)

  • Stable future: Flat 4D universe

This level will witness the emergence of the fundamental forces. At Level 1, we’ve dabbed quite a bit into theoretical physics, to attempt a coherent presentation. But the period when Level 1 structures emerged, up until \(10^{-35}\text{s}\) was really about the emergence of the primordial distinction between the spacetime background (a many dimensional object that gets compactified into our familiar 4D spacetime) and its physical content (which we can “unpack” as a number of quantum fields).

The universe is still an extremely dense, unfathomably hot, curved spacetime, inhabited by the complete bestiary that was inherited from the "true" Grand Unified Theory (whichever one it is): quarks, leptons, gluons, \(B\), \(W\), \(X\), \(Y\) bosons, sfermions and gauginos, higgs and higgsinos, and likely so many more... All are interacting in a massive tohu-bohu that makes the expression "primordial soup" feel appropriate.

The Standard Model epoch: \(10^{-35}\) to \(10^{-12} \text{s}\)

By the \(10^{-35}\text{s}\) mark, the expansion of the universe has cooled it to a "measly" \(10^{29}\text{K}\). It is in that neighborhood that the likely GUT scale is located, which means that symmetry is broken past that threshold, either in one go or in several successive steps, as we’ve seen. The result, down the line, is the separation of the strong force coupling from the electroweak coupling, inaugurating the Standard Model epoch, when an unbroken \(SU(3)\times SU(2) \times U(1)_Y\) still reigned over quantum interactions. \(X\) and \(Y\) leptoquarks, alongside the whole exotic GUT zoo, acquire mass and disappear from the picture, save from more and more infrequent short-distance interactions. The strong interaction displays the familiar features from today’s world, but the electroweak bosons are still all massless.

Around that time, cosmic inflation starts. It is a period of extremely fast expansion of space, which is dilated by a factor of at least \(10^{26}\) within a period of around \(10^{-32}\text{s}\). If we were to dilate a human cell (\(~1\mu \text{m}\)) by that much, we would end up with an object 100 quadrillion kilometers wide! Yet after inflation, the patch of spacetime that is now the whole observable universe woudl have been less that 3 meters long. As a result of inflation, the universe rapidly cools down to \(10^{22} \text{K}\).

Inflation is still a hypothesis, since its cause cannot be explained in terms of known quantum fields nor in the context of general relativity. Yet it is widely accepted as the solution to the overbearing horizon and flatness puzzles. The universe appears to us to be nearly flat, homogeneous and isotropic on a large scale. Without inflation, this would be incompatible with the laws of physics, but the faster-than-light metric expansion from the inflationary epoch would explain how a minuscule causal patch could still end up much larger than our observable universe, resulting in the flatness and homogeneity we’re now perceiving. The coincidental timing around the GUT breaking scale suggests a possible link between the inflation-causing inflaton field and the symmetry breaking, although that is still highly speculative.

Cosmic inflation stopped abruptly. Although the universe was now cooler by many orders of magnitude, the temperature was still too high for any equilibrium to emerge. The next disturbing even was the breaking of supersymmetry. Very little is known about the one, be it about the exact process or the energy scale at which it happened; we only know, almost certainly, it occured before the breakdown of the electroweak symmetry. Post-supersymmetry breaking, all superpartners become massive. As cooling proceeds, they quickly decay, typically into a lighter superpartner and some ordinary particles. At the end of that process, there can be either no superpartner at all or, in R-symmetry-preserving theories, only the LSP (lightest superpartner). For instance, in the MSSM, the LSP is the neutralino, a gaugino fermion that’s a composite of the bino, neutral wino and higgsinos. It has become a sort of beacon for cosmologists in their desperate attempt to decipher the riddle of "dark matter".

There is a glaring mismatch between the observed gravitational dynamic of the universe and the ordinary matter that we can see (galaxies, nebulae, clouds,...). That visible matter only accounts for 15% of the mass that would explain large-scale dynamics! Obviously, "visible" matter either emits, reflects, or somehow interacts with electromagnetic waves, or some other sort of radiation. That means that the missing matter doesn’t; it is therefore called "dark matter". Neutrinos may seem like good candidates: they don’t interact with anything else, and are now thought to have mass. However their tiny mass, and what we know about their total population form other observations, doesn’t gel well with what we would expect of dark matter. Neutralinos, whose all quantum charges are nil, and have significant mass (100 billion times heavier than the neutrino) seem like good candidates! More generally, supersymmetric LSPs are considered very promising.

The quark epoch: \(10^{-12}\text{s}\) to \(10\mu\text{s}\)

At approximately \(10^{-12} \text{s}\), the temperature has cooled down to \(10^{15} \text{K}\) and we reach the electroweak breaking scale. We reach solid ground at last with this amply confirmed and well-known Higgs mechanism.

All fermions become massive, the electroweak bosons recombine into \(W^\pm\) and \(Z_0\), which become massive, and the photon \(\gamma\), which is massless. The unbroken segment of the internal symmetry group reduces to \(SU(3) \times U(1)_{EM}\). The familiar picture of particle physics emerges: matter, represented by fermions, interacting by exchanging bosons which mediate the fundamental forces.

The era right after electroweak symmetry breaking is called the "quark epoch" because it is still hot enough for quarks to be free, unfettered still from the confining forces that would bind them together into hadrons. As a result, since the strong interaction is already much more intense than the others, and color-charged quarks and gluons are still freely roaming the universe, the strong force entirely dominates. The result is a quark-gluon plasma that is radiation-dominant (that is, most of the universe’s energy is from the massless gluons rather than the massive quarks). Gluons are energetic enough to generate quark-antiquark pairs, which in turn disintegrate into gluons. That plasma can end up locally in a thermal equilibrium.

As cooling continues, the heavier fermions from the second and third generations become more and more rare. The ordinary matter content of the universe is then mostly limited to the classic "roster" of particles that make up us humans: up and down quarks, electrons and neutrinos (with some muons and strange quarks the lightest from the second generation joining the party at times).

The stable future

15 billion years later, the universe’s ordinary matter is still made up of the same basic components: up and down quarks, electrons and neutrinos. Indeed, on such a scale they are almost perfectly stable. The half life of quarks has a lower bound of \(10^{34}\) years (ten decillion!). In a cool universe, quarks are bound into nucleons. Since neutrons decay into protons on much shorter scales, quark decay will largely happen in the form of proton decay, for which in most GUTs the principal reaction is \(p \rightarrow e^+ + \pi^0 \rightarrow e^+ + 2 \gamma\).

Crucially, we also know that the universe is electrically neutral (since the electromagnetic force has infinite range, like gravitation, but is much more intense, gravitational effects couldn’t dominate the macroscopic scales as they do, if it wasn’t the case). This means that the positrons will in turn disintegrate with electrons. The ultimate matter content of the universe will then be limited to neutrinos and potentially an LSP (like the neutralino). There will be no mediator bosons since those fermions are chargeless. As today, there will be one freely travelling boson: the photon.

Level 3: Matter — \(10^{-5}\text{s}\)

• Atoms

  • Size: \(10^{-10} \text{m}\)

  • Initial state: First-generation particles

  • Production process: Primordial nucleosynthesis, atomic combination

  • Meta-stable equilibrium: Light atoms and molecules (first period)

  • Stable future: Dihydrogen and Helium.

• The transparent universe

  • Temperature: \(10^{12} \text{K}\)

  • Initial state: Locally flat 4D universe (radiation-dominated)

  • Production process: Photon decoupling

  • Meta-stable equilibrium: Matter-dominated, transparent universe

  • Stable future: Dark energy-dominated universe

The hadron epoch : \(10\mu\text{s}\) to \(0.5 \text{ms}\)

The quark epoch ends when the universe’s temperature crosses \(10^{12}\text{K}\) and is not high enough to sustain unconfined quarks. The intensity of the strong interaction is such that its attractiveness grows with distance, which leads to an incredibly powerful pull on every color-charged particle. Quarks can be confined with baryons (3 quarks of distinct colors) or mesons (one quark and one antiquark, with opposite colors), which are color-neutral particles. Those particles are collectively called hadrons, hence this epoch’s name.

The temperature at the start of the hadron epoch is \(175\text{MeV}\), which means that only up, down and strange quarks were in thermal equilibrium with their antiparticles in the plasma. During the phase transition, the quarks and antiquarks quickly assemble into baryons and mesons, but the temperature is already low enough that hadron-antihadron pairs are not spontaneously produced. The story of the hadron epoch is then one of annihilation. Due to matter-antimatter asymmetry, there will be residual hadrons (or ordinary matter would not exist and the universe would have jumped right to the photon-and-neutrino-only era).

The many heavy hadrons or exotic hadrons (tetraquarks, pentaquarks, hexaquarks, glueballs...) that have been either observed or merely predicted, have then never existed in a meta-stable state, although they are constantly being created to this day, either in extreme environments (supernovae, neutron star collapse...), or more prevalently, when high-energy cosmic rays interact with ordinary matter.

From the up, down and strange quarks that were around, the possible baryons are nucleons or lambdas (including strange quarks among their components). The possible mesons are pions (no strangeness) or kaons (strangeness involved).

By the time the energy is down to \(30\text{MeV}\) all the anti-baryons have annihilated, leaving only the residual output of the baryogenesis in the form of nucleons and lambdas (with density \(10^{10}\) lower than before). The mesons were already dominating the mass content of ordinary matter, but at that point they represent almost all of it.

The meson epoch : \(0.5 \text{ms}\) to \(5 \text{ms}\)

That marks the start of the meson epoch, when initially, a mix of pions and kaons dominate matter. However, as pointed out above, energy is too low to support the production of meson-antimeson pairs. The annihilation process is thus on-going, as well as the decay process of the remaining mesons.

By \(20\text{MeV}\), both the remaining Kaons and the lambda baryons (the "strange" content) are at equal density

The electron epoch : \(10 \text{ms}\) to \(10 \text{s}\)

At that energy level, pions then dominate the hadronic mass in the universe, but their density, although still \(10^{10}\) higher than the nucleons’, has decreased enough that it becomes inferior to that of "matter leptons". With that term we are excluding neutrinos, whose low mass makes them ultra-relativistic particles that we classify as radiation. Let’s not forget the hierarchy at the universe’s level: dark matter represents most of its, follow by photon and neutrino radiation, with ordinary matter only representing a few percentage points. The successive "domination" by quarks, hadrons, mesons and now leptons only refer to an advantage within ordinary matter.

Further, we know that by that point, the muon and tau leptons have almost entirely decayed. The only matter particle that keeps being produced in particle-antiparticle pairs is the electron. Despite its lower mass (2000 times lower than a nucleon’s, 200 times lower than a pion’s), it can dominate by virtue of that continued production.

Pion decay continues, with nucleon-pion equality around \(6\text{MeV}\), and the pions’ extinction at \(3\text{MeV}\). The only remaining hadrons are the nucleons, but they’re now entirely drowned in a sea of electrons and positrons where electromagnetic interactions dominate.

The density is still high enough that the short-distance nature of the weak interaction (since electroweak symmetry breaking) is not really consequential yet, and neutrinos still interact freely with all those particles. The reaction that we know as neutron decay is not only hyper-frequent, it is also reversible, with protons and neutrons thus being in perfect parity (because they are constantly turning into each other).

That starts to change as cooling marches on, and the proton-to-neutron ratio keeps increasing, until \(1\text{s}\) (\(1\text{MeV})\), when neutrino entirely decouple from matter and the ratio freezes around 5 protons to 1 neutron. Neutrinos then adopt the behavior we know today, as a form of radiation that’s travelling freely accross the universe, with only the occasional interaction when it gets close enough to another particle.

The nucleon epoch : \(10 \text{s}\) to \(2 \text{min}\)

At the end of the electron epoch, pairs stop being produced and electron-positron density decreases quickly as they annihilate. Since nucleons are heavier, they end up dominating the mass of ordinary matter (as they do today). The neutron-to-proton ratio has been frozen out at 0.21 and due to electromagnetic neutrality there is one electron for each proton. Neutron, protons and electrons now cohabit in a structure-less plasma where they’re constantly interacting electromagnetically via photons.

The nucleosynthesis epoch : \(2\) to \(20 \text{min}\)

2 minutes after the Big Bang, the primordial nucleosynthesis starts. You may have a notion that the elements we know today were created through a nucleosynthesis process happening in stars’ core. That is indeed the case: this primordial nucleosynthesis only yielded Helium nuclei.

When it began, the environment was cool enough that stable structures could form under the "residual nuclear force". That term designates the attraction between nucleons: although they are individually color-neutral, the sheer intensity of the strong interaction is enough to generate attraction between them (similar to the way that atoms, electrically neutral objects, can still form molecules due to residual electrostatic forces).

Protons and neutrons were already strongly attracted enough to form deuterium, the simplest possible nucleus, but temperature was so high that it immediately decomposed right back. Once that bottleneck is passed, two deuterium nuclei are available to react and yield a Helium-3 (two protons, one neutron) and a free neutron. Then Helium-3 can react with yet another deuterium to create Helium-4 (plus a free proton).

Deuterium and Helium-3 are unstable,: left to their own devices, they decay quickly. However at this energy level, Helium-4 is extremely stable. It doesn’t undergo fission (reverting to lighter elements), and it rarely undergoes fusion (yielding heavier elements). Since the universe is cooling at the same time that this is happening, nucleosynthesis ends before all the hydrogen has been transformed.

The expected composition of the universe’s hadronic mass after that phase is 75% hydrogen, 25% Helium-4 and only traces amount of anything else (0.01% in deuterium and Helium-3, and less than a millionth % in Lithium nuclei). Indeed, that is consistent with the evolution of the neutron-to-proton ratio. Remember that after neutrino decoupling, this had been frozen at 0.21. But free neutrons natural decay into protons via the weak interaction. Accounting for the energy levels and the progress of nucleosynthesis, we expect the final neutron-to-proton ratio to be very close to 1:7. And that is exactly what a 75/25 mass split between hydrogen and helium translates into! (Since helium is 4 times as heavy as hydrogen, this split means there are 12 free protons for each Helium nucleus (2 protons and 2 neutrons); then there are 2 neutrons for every 14 protons).

As a matter of fact, this is exactly the ratio we observe to this day! The helium produced via stellar nucleosynthesis is only a small amount most of it originates in the primordial nucleosynthesis. This is then a resounding success for the whole theory...

The plasma epoch : \(20 \text{min}\) to \(47\text{k}\) years

Nucleosynthesis then leaves a universe composed of free protons, Helium nuclei, and electrons, still interacting with photons in a hot electromagnetic plasma. We could say that matter is in an ionic state, since electrons are moving around freely, as the temperature, at \(10^7 \text{K}\) or so, is still too high to allow electrically neutral objects like atoms to emerge. Even though we are focusing a lot on the evolution of ordinary matter, it is always worth repeating that most of the universe’s energy is stored in dark matter’s mass, and that the second store is in radiation, with ordinary matter’s contribution a distant third.

Since dark matter doesn’t interact with anything and thus has no bearing on the evolution of the universe at the scale under consideration, it is common to say the universe is then radiation-dominated.

The atomic epoch : \(47\text{k}\) to \(370\text{k}\) years

The ion epoch lasted much longer than its predecessors, and it is fair to say that beyond continuous cooling, not much happened during the next millenia. But that was always bound to change. It may seem that the forces on the battlefield were now fixed. Unlike in previous epoch, particles were not changing into each other much, nor were they combining, decomposing, or annihilating a lot. Yet the fact that the universe kept expanding was far from neutral.

It is a well-known fact that energy density doesn’t react in the same way to space expansion for all energy sources. For a scaling factor \(a\), the energy density of matter (both ordinary and dark) evolves as \(a^{-3}\): nothing is more natural since expansion increases all distances by a factor \(a\), and thus all volumes by \(a^3\). But for radiation, that density follows an \(a^{-4}\) law. Lastly for what is sometimes called "dark energy" but can be seen as a "cosmological constant" an unchanging curvature term in the equations governing the universe’s large-scale dynamics , the energy density embedded in spacetime geometry is entirely independent of \(a\).

The first consequence is that the energy density from matter is bound to overtake that of radiation. That is precisely what happens, around 47,000 years after the Big Bang. What practical consequences could this possibly have? They are extremely momentous, since matter domination ushers in the era of large-scale structures. More exactly, it makes that era possible: it will take many million years for meaningful structures to emerge. But it is the end of radiation prevalence that makes it all possible.

We’ve seen somewhere above that cosmic inflation was a neat way to explain how structures (concentration of mass in specific sectors of the universe) could arise from a purely homogeneous and isotropic universe. Inflation was so fast (literally, more than lightning fast), that ordinary quantum fluctuations were frozen and instantly blown up many orders of magnitude in size. After the mighty stretch that inflation inflicted upon spacetime, those relics looked like minuscule variations in curvature, completely negligible factors vs the chaotic plasma where more intense quantum fields deployed their exuberance. But dark matter was not subject to any interactions with other sorts of matter or radiation. Since only gravity could affect it, it started to slowly migrate towards specific regions which will be later known as filaments.

At the time immediately following matter domination though, hadronic matter was still much too agitated in its electromagnetic plasma to feel such gravitational effects.

The premices of change were not long to come though. As temperature crossed \(10,000\text{K}\), combination started. To be more precise: the electromagnetic combination of electrons with nuclei. Up to that point, the natural attraction between the electron’s negative and the nuclei’s positive charges was more than counter-balanced by the prevalent agitation in the giant pinball game that was the primordial plasma. If such combinations formed, they were immediately taken down by a super-energetic photon. Most of the time they could not form at all since particles were too "busy" scattering the photons constantly colliding with them.

With cooling, it became possible for a fraction of the electrons to assemble with nuclei for longer and longer periods, resulting a in a growing ratio of matter in ionic or atomic state. It may seem like the simplest possible compound is the hydrogen atom (1 proton and one electron), but in fact, the hydrogen ionization temperature is much lower than for helium.

The first object to be steadily produced to the point it becomes of some abundance is then the helium cation, around the 120,000 years mark: \[\text{He}^{2+} + \text{e}^- \longrightarrow \text{He}^+\]

Later on, the cation is able to consistently absorb another electron and form the first atom: helium.

\[\text{He}^+ + \text{e}^- \longrightarrow \text{He}\]

At lower temperature, helium can form the first covalent bond with a proton to generate a helium hydride molecular ion.

\[\text{He} + \text{p} \longrightarrow \text{HeH}^+\]

It’s only around 370,000 years after the Big Bang that the hydrogen ionization energy is crossed, an the remaining protons form hydrogen atoms.

\[\text{p} + \text{e}^- \longrightarrow \text{H}\]

With more cooling, hydrogen atoms tend to form bonds with the still-free protons to create dihydrogen cations. That object also appears when all the helium hydride reacts with the newly formed hydrogen. Lastly, an electron is absorbed to form dihydrogen, the first neutral molecule in the universe.

\[\text{H} + \text{p} \longrightarrow \text{H\_2\^+}\] \[\text{HeH\^+} + \text{H} \longrightarrow \text{H\_2\^+} + \text{He}\] \[\text{H\_2\^+} + \text{e\^-} \longrightarrow \text{H\_2}\]

The golden epoch : \(370\text{k}\) to \(2\text{M}\) years

The last sequence from the point where hydrogen atoms appear is very fast. That combination process yields a very large energy in the form of energetic photons, first when the atoms assemble in an excited state, then (almost immediately) when they emit an extra photon during their transition to the bound state.

Now that ordinary matter is entirely atomic or molecular (made largely of dihydrogen and helium), photon scattering is not a significant event anymore, thermically speaking, and these photons are free to travel the universe untethered, like neutrinos have been doing for some time already. That is photon decoupling. That specific batch of photons is known as the Cosmic Microwave Background.

The CMB was originally the same frequency as orangey light, hence the name of this epoch, but with the universe’s expansion it naturally redshifted. At the end of this epoch, roughly two million years after the Big Bang, the universe was filled with a hydrogen/helium gas, some free infrared photons (not interacting with matter in a significant way) as well as neutrinos and dark matter. To a human eye, it would then appear entirely dark.

That is the situation we know today: outer space is generally black-looking, but of course it is now sprinkled with the light from trillions of stars. How those came to be is a topic for our next level.

The stable future

The ultimate fate of structures at this particular level is an interesting case. Remember that this section is often a thought experiment rather than an actual prediction of the future. The question we’re asking is: if we’re only considering the forces under which action the objects from Level 3 emerged, at the range of energy under consideration, and all else being equal, what would happen in the far future?

This is often a way of asking what the most stable state of organization at that level is, since in nature, meta-stable structures always decay into more stable ones, and some that may seem stable even over a billion years, may prove unstable in the longer run.

Atoms and molecules are extremely robust structures. Yet, it is clear that in the real world, they won’t be eternal. A Big Rip might happen; in that case the universe would expand so much that the size of the observable universe from a proton’s perspective would be shorter than any atom’s radius, which would automatically entail all of the atoms’ breakdown. But that’s Level 1’s far future, it involves forces, timescales and phenomena that go beyond Level 3. We also know that protons are likely to decay, in which case hydrogen atoms could very briefly turn into positronium (a positron-electron atom), but ultimately would still annihilate into photons. But that’s Level 2.

Hydrogen and Helium atoms, although they are extremely stable nuclei, could still go through cold fusion, since the bound states of heavier nuclei have lower energies. But we haven’t encountered heavier nuclei yet: that’s Level 4.

At Level 3, it then seems that the electrically neutral, stable structures we’ve encountered: the dihydrogen \(\text{H\_2}\) molecule and the Helium atom are, for all accounts and purposes, pretty much eternal, if temperatures stay below their ionization level. If not for universe-bending relativistic effects, for GUT-only quark decay, or for nuclear cold fusion, they would just persist forever, motionless, all the way down to absolute zero.

Dynamochora — the Physical domain

Terminology

"Physics" is derived from the Greek physis, meaning "nature", and that root is therefore better to denote the natural sciences in general rather than this more limited discipline. The universality it suggests is at odds with my clarification goals in this work.

The diversity of covered phenomena makes the identification of a single, central object quite elusive. A robust proposal is the concept of force, or that of power. Although strictly speaking, the concept of force is more present in Newtonian mechanics, and that of power in thermodynamics, analogs exist in every single theory, and the flow of energy or momentum they involve is central to the Lagrangian approach that henceforth constitutes the lingua franca of physical formalism. Hence the choice to use as a root for this domain the Greek dyna- or dynamo-

Dynatoms (physical objects)

It may look like a peculiar goal to ringfence the objects of the "physical domain". Have we not spoken for many pages of the "physical realm", one of three realms alongside the mental and the abstract? Indeed, it is true that physics, as it came to be known, was always distinguished in the minds of practitioners by its universality and generality. An object can be inanimate and outside of biology’s purview. A substance can be chemically inert. But nothing in the physical realm can be "physically inert". A quark, a rock, a human being, the earth or the whole universe: none of them can escape the laws of physics.

Is that all? Not quite. We have also explained above that each domain and each discipline would be associated to stages of the Scala Naturae. Physics (the dynamochora)) takes as its object the first four levels: fundamental quanta, elementary particles, atoms, elements and (small) molecules on the micro side, and the physical universe, together with its internal high-scale structures, on the macro side.

That is familiar terrain for modern physics. After all, for the past century, its most spectacular progress has been in understanding reality at extremely small or extremely large scales. So it’s pretty intuitive to map the physical "stages" onto modern theories: molecules on molecular physics, elements on nuclear physics, atoms on atomic physics, particles on the Standard Model of particle physics, fundamental quanta on possible theories "beyond the Standard model", and the universe on relativistic cosmology.

Yet, as we’ve just said, physics is not a specialized science of very small, very large or very fast things. The key is that we are considering ordinary objects rather than physical objects per se. For instance this gives us the ability to consider a human being as a living being, as a chemical compound or as a physical object. In the latter case, we would only consider it to have physical properties like mass, temperature, electrical charge, etc.

In that sense, we could say that, although a whole human being is not to be analyzed by microphysics, to consider it as only a physical object and to "do physics" about it only requires us to mobilize properties that emerge at or before the molecule stage, and nothing above. Let’s take a specific physical theory as example. If I need to use Galileo’s law of gravity to predict my free fall while bungee jumping, all that matters about "me" as a physical object is my mass, its distribution, and my shape. A mannequin with similar characteristics would not be treated differently by the theory. That is understandable, since atoms and molecules do have a mass, a position, a momentum: at common speed and energy levels, these properties are fully emerged at the atomic stage. For the purpose of physical theories, I can therefore acceptably be considered as "a bunch of atoms". If I were instead considered as "a group of living cells", we would already be talking of biology, not physics.

That principle is implicit in classical physics, although it is never laid out very rigorously. Any student just beginning to study Newton’s dynamics will be told about the movements of "material points". Mathematically, that is simply a quintuplet \((x,y,z,t,m)\) that denotes the presence of a mass at a given position and a given time. Of course, there is no such thing as a material point with zero volume and infinite density. But all students understand that the point represents something like a billiard ball, or in other contexts, a planet. A modern student would also know that the point might well be an atom. After all, an atom might be the closest thing to a material point in classical physics, since it is about the smallest object for which a classical approximation stands. To model a "point-like" planet Earth in elementary studies of orbital mechanics, we can then mentally picture a very heavy atom, right at the centre of the Earth.

That is a remarkable fact that cannot be taken for granted. The Earth, or a billiard ball, have plenty of internal structural properties beyond their mass and position. In another universe, the approximation that is classical physics may well rely on the elasticity, viscosity, refraction index or chemical solubility of objects to compute their trajectory. But it is not the case in ours. The form of classical equations give rises to Gauss’s theorem, which allows us to be completely blind to a body’s volume. As long as we "look" from outside its boundaries and it has spherical symmetry, all that counts is its mass and the position of its center. Arise, "material point"!

Hopefully it is now clearer why the physical domain can be considered to cover ordinary objects up to atoms and molecules: this lets us include in that domain studies of not only their transparent physical analogues (a single atom, a free elementary particle...), but also collections, as long as their behavior is considered from the perspective of those same physical properties attributable to the single objects. That is enough to build a concept of the "material point" or "rigid body" of classical mechanics, the "charged particle" of electrostatics or the "charged current" of electrokinetics and magnetostatics. Other objects of classical physics like the light or the gravitational field are now demonstrably grounded directly in "elementary" objects (respectively, the photon and spacetime), while thermodynamic systems’ proper characteristics emerge from the collective mechanical behavior of atoms or simpler particles.

Dynamoplases (physical events)

We can apply the same rationale to get a sense of physical events: those events that transform the physical properties of a physical body.

For general mechanics they would include motion or propagation (change of position), collision or scattering (change of momentum), annihilation (that one is specific to relativistic quantum mechanics). In relation to gravity we can evoke free fall, orbital capture, or gravitational collapse.In electromagnetic theories one could talk of electrical charging or of induction. Optical events include reflection, refraction, interference, diffraction, polarization. Thermodynamic events include heat transfer, phase transitions, compression or expansion. Atomic physics, which is concerned with electronic layers, oversees excitation and ionization, while nuclear physics, which only concerns atoms’ nuclei, consider radioactive decay, nuclear fission and nuclear fission.

Dynamonoia (physical concepts)

We could well have perused physical theories, past and present, and come up with scores and scores of additional physical objects and events. We could also explore the vast expanses of that field of research and reap thousands of physical concepts. But if we have to pick one, and only one, "big idea" to represent this domain as its dynamonoia, it should be action.

Let us not get stopped in the minutiae of this choice. The goal here is not to sanctify a given physical quantity among others, whose exact formulation differs from theory to theory, nor to elevate an principle ("least action" or "stationary action"), as incredibly fruitful as it is.

Action is a nodal point within a network of important ideas that form the background of any form of physical knowledge, in particular variation (action is grounded in some for of parametrization; there can be physics without time but not without a concept of change), symmetry (the symmetries of the action determine those of the theory), geometry (even as our experience of space has become more and more remote from the actual foundations of physics, geometric principles have risen anew in generalized context; to minimize the action is often expressed as "drawing a straight line" in some exotic phase space).

Dynamosophy (physical philosophy)

The applied philosophy of the physical domain should not be equated with philosophy of physics, which is largely a philosophy of science. Of course, the relevant objects, events and concepts have largely been conjured by scientists using scientific methods, which makes the distinction porous. But we must carefully follow our distinctions.

Dynamosophy would encompass the ontology, metaphysics, semantics and value theory of the dynamochora. The metaphysics of time, space, of quantum systems, and a plethora of other objects is a common theme. The epistemics of physical phenomena (what can be known about them and how to justify that knowledge) certainly lies at the border of philosophy of science. The aesthetics of physical objects or phenomena is a sub-field of the broader aesthetics of nature (which is distinct from the aesthetics of physical theories, since those are not physical objects).

Dynamology (physical science)

Our focus on specific ordinary objects up to molecules as well as collections thereof restrics the scope of dynalogy, as compared to the common usage of "physics", which has grown further and further to become a nearly all-encompassing physical science, on par with the old "natural philosophy".

In describing that scope, we need to be careful about two category mistakes. The first is the frequent identification of a sub-discipline with a theory, e.g. to say that quantum field theory is to elementary particles what solid mechanics is to large solids. Solid mechanics is a generic term that has been applied to various theories over time, while quantum field theory is a theory, that is unfathomably good at describing elementary particles, but might still be replaced. The second mistake is the lack of procedural clarity in how to delineate the scope: is physics a science of certain things? or a science that uses certain methods? Why is the change of internal structure of a material through its magnetization usually considered to be physics, while the change of its internal structure through ionization is normally part of chemistry? Why is the decay of a uranium nucleus into a thorium nucleus part of physics, while the combustion of glucose into carbon dioxyde and water is chemistry? Negligent approaches to that question lead to fuzzy boundaries with chemistry; that may not be a huge issue for the teacher or learner, but is one to the classificatory-minded. Our distinction between dynalogy and hylelogy aims to remedy these ailments.

Sub-disciplines

Level-specific sub-disciplines can be directly described as the study of specific entities in nature.

• Particle physics has been thought of as the study of fundamental particles. Of course, we don’t know what is really fundamental. Around the time that term was first used, it was applied to nucleons (protons and neutrons), electrons and photons. Conventionally, particle physics now apply to those objects and their analogs at the same level of complexity, as well as anything "smaller". In terms of settled physics today, leptons, neutrinos, EW bosons and Higgs fields are described in the Standard Model of Electroweak interactions, while quark (and their hadrons aggregates) and gluons are tackled by Quantum Chromodynamics (QCD)

• Nuclear physics, at the next level of organization, is concerned with the formation and breakdown of atomic nuclei, through the aggregation of hadrons bound by the residual strong force. Predecessors to the Standard Model like the Yukawa theory of nuclear force, or the Fermi theory of radioactive decay, can still be used as approximations and feed into modern-day nuclear models.

• Atomic physics uses either non-relativistic quantum mechanics combined with classical electromagnetism, or the relativistic quantum field theory of electromagnetism (QED), to study the internal structure of atoms and their interaction with radiation like light.

• Molecular physics obviously has large overlaps with chemical physics and physical chemistry, but also includes specific studies like molecular dynamics.

• Universal physics (often called cosmology, which I avoid here to be consistent with my rebranding of astronomy with that term) is the part of physics that directly studies the universe or its large-scale components. It relies mostly today on General Relativity and cosmological extensions like the Lambda-CDM model.

Level-agnostic sub-disciplines study physical objects of arbitrary sizes, from the perspective of properties that are fully emerged at molecular, atomic or lower levels. Unsurprisingly, we can break them down into one very general practice (mechanics), and two more that examine the effects of the fundamental interactions that manifest at macroscopic scale. In other sections we have described the four fundamental interactions, but also noted the nuclear forces’ extremely short range at our energy level (the strong force is already neutralized by the existence of protons and neutrons; the weak force is limited in range by the mass of its mediator bosons: even macroscopic effects like radioactivity, once they’ve been identified as distinct from those of electromagnetic radiations, have only been described from the fundamental level as the decay of subatomic particles).

• Mechanics is the general science of motion and force as it relates to matter. Those concepts used to be split between kinematics and statics before being unified into the Newtonian dynamics. The adjunction of additional quantities like energy and work, of new techniques grouped under the category of analytical mechanics (lagrangians, hamiltonians...), and the transition from point-like particles to spatially extended, rigid solids led to the durable synthesis of classical mechanics. The principles of special and general relativity, or equivalently, new descriptions of spacetime, led to its replacement by relativistic mechanics at high energy levels. Meanwhile, an even fuller break occurred with the development of quantum mechanics. Since both quantum and relativistic principles become important at high energy, their combination (involving only special relativity) was explored in detail and developed into quantum field theory. It is crucial to acknowledge two things:

  1. Relativistic and quantum mechanics have specific domains of application but their domain of validity is much wider. Current quantum mechanics might not be valid at levels of energy above the ones we’ve been able to observe in particle accelerators, but it is at all other levels. A single electron or photon might be archetypes of "quantum objects", but a rock, a star or a mouse can also be described as wavefunctions. They can equally be described as relativistic objects in 4D space, of course. But obviously, most practical applications will solely use classical mechanics to represent them. This doesn’t make classical physics "wrong", or "just an approximation": it can contribute physical facts, about physical properties, of physical objects. There will always be a small error, within the implicit uncertainty associated with Newtonian dynamics under regular conditions. It would be true of any theory.

  2. There is also a tendency to set in stone the relation between different versions of a theory and their most common mathematical expressions. Quantum theory nowadays describes complex operators and wavefunctions in Hilbert space. Newtonian dynamics describes n-uplet points in a vector space called phase space (the position and momentum of massive points). It is sometimes said that because quantum theory is more fundamental and its validity domain more extensive, the world is really described by complex vectors in Hilbert space, which supposedly also explains many a "strange" feature of the quantum world. That is a fallacy: the capacity of mathematical theories to be re-formulated in new instances calling entirely different images to the human mind is boundless. In fact, there is a formulation of classical mechanics due to Koopman and von Neumann that is entirely based on square-integrable, complex wavefunctions in Hilbert space, and uses a host of mathematical techniques developed for quantum mechanics. Like in classical mechanics though , there is no uncertainty about measurements and interferences do not occur in a double-slit experiments. That is not due to the "nature" that our mentality attributes to its objects, but instead to intrinsic mathematical simplifications. In a world where Newton would have somehow couched his own theory in that language, there would be no inclination to explain the lesser validity range of classical mechanic through the "dull" nature of its mathematical framework. We would simply be presented with two very similar theories mathematically speaking, with one being made "more right" than the others by the unction of empirical experience only.

• Gravity is concerned with the movement of massive bodies subjected to the Earth’s gravity (in Galileo’s original formulation), to a gravitational force (in Newton’s theory), or evolving in a curved spacetime (in Einstein’s general relativity theory). In classical physics, gravity is conceptually distinct from general dynamics (which connect forces with acceleration, without ever specifying what forces ought to look like). However the practical connection was impossible to avoid due to the complete identification of inertial mass with gravitational mass: most macroscopic objects are electrically and magnetically neutral, but none (except, in the classical worldview, for light) are "gravitationally neutral". With general relativity, the identification is justified, and what had been perceived as a "gravitational interaction" is explained by spacetime geometry, thus entirely subsuming the macroscopic "theory of gravity" into a general mechanics.

• Electromagnetism & Optics designate a group of sub-disciplines that study electrically or magnetically charged bodies, electric currents, and light. Respectively, those sub-disciplines have been called electrostatics, magnetostatics, electrokinetics and optics. Electro- and magnetostatics were early on seen as somewhat subsumed into mechanics since charged bodies were still endowed with mechanical properties. Electrostatics and electrokinetics merged into electrodynamics when the interaction between charges and currents was better understood. Together with magnetostatics, it was then unified into electromagnetic theory following the discovery of magnetic induction. The perfect, classical expression of electromagnetism was in Maxwell’s equations which explained all these phenomena via the dynamics of an electromagnetic vector field. Meanwhile, light, the only visible type of radiation, was described in successive paradigms as a geometric "line", a beam of corpuscles, and a wave. In Maxwell’s theory, the free electromagnetic field, an alternating wave, was soon identified with light, thus subsuming optics into an electromagnetic theory of light and helping to explain newly discovered radiations like X-Ray, infrared or ultraviolet, in one fell swoop. Like classical mechanics, that theory is still in use for a wide range of energy levels, and the go-to modelization for practical purposes. Yet, the most precise description of these phenomena by far is provided by Quantum Electrodynamics (QED) (a post-symmetry-breakup product of the Standard Model), which describes the electromagnetic field as a photon wavefunction, and the effect of electromagnetic interactions as caused by photon exchange between charged particles (arguably, the theory of electromagnetism at that degree of precision is now entirely part of mechanics, since that interaction can be represented as the relativistic quantum-mechanical behavior of elementary particles).

Statistical sub-disciplines explain the behavior of systems with a very large number of components. All other sub-disciplines are typically expressed in terms of a single object, or a few objects interacting, but can be expanded to cover "many-body" problems. Statistical physics, however, consider ensembles with arbitrary large number of components. Given that the dimension of the phase space under consideration can then quickly tend towards infinity, the same techniques cannot be applied, and emergent concepts appear, such as partition functions, state function (like entropy, free energy...), etc.

All of the above theories can be expanded into a statistical version. The original kinetic theory of gases, due to Maxwell, Boltzmann and Gibbs, has been tremendously successful at deriving pre-existing results in thermodynamics, therefore achieving its rigorous reduction (in certain cases) to mechanics. It also provided the statistical blueprint for futures theories.

• Classical statistical mechanics start from the initial kinetic theory, which studies a low-density gas with a large number of identical, but indistinguishable molecules, that only interact with the system’s borders or with each other via binary, elastic shocks. From that simple physical concept sprang the Maxwell-Boltzmann distribution, which describes the probability distribution of micro-states in phase space. That approach can be used for many other case studies in classical mechanics, such as denser gases, inelastic shocks, or interacting particles (gases in a gravitational field, or a classical plasma of charged particles).

• Relativistic versions of statistical mechanics take into account the relevant symmetries, which obviously results in velocity distributions being capped by the speed of light, and other effects.

• Quantum statistical mechanics turn out fundamentally different due to the indistinguishability of the particles. The outcome is also determined by the particles’ spins, with fermions following Fermi-Dirac statistics, while bosons follow Bose-Einstein statistics. Like classical and relativistic distributions, those can be refined for different theoretical backgrounds or more and more general cases.

Dynamotechny (physical technique)

We have seen in previous discussions of the techne level that applied science is closely linked to the concepts of engineering and technology. Dynamurgy, the techne slice of the physical domain, or more colloquially applied physics, is of course the very core of engineering "proper" and the breeding ground for the oldest human inventions.

The progress of humanity’s material civilization is inseparable from the harnessing of energy. Energy is conserved, but humans can act by transforming one form into another, or by altering its expression.

The concept of energy as it is used in applied physics is closer to its understanding in classical physics than to a modern theoretical formulation. It is marked by the classical divisions between mechanics, gravity, electromagnetism, optics and thermodynamics, as well as between kinetic and potential forms of energy.

• Mechanical energy

  • Kinetic mechanical energy represents the energy of a body in motion in a referential: it is the work required to achieve that velocity. For instance, in the simple 1D classical case: \[E_k= \int \mathbf{F} \cdot \mathbf{ds}= \int \frac{d\mathbf{p}}{dt} \cdot \mathbf{ds} =m\int (\mathbf{\dot{v}} \cdot \mathbf{v}) dt = \frac{mv^2}{2}\]

  • Potential mechanical energy represents the energy stored in a certain configuration of a system: changes in potential energy are equal to the work performed by the forces in that mechanical system. In an idealized spring model where the pull and push forces are directly proportional to the distance from equilibrium: \(\mathbf(F)=-k\mathbf(x)\), variations in potential energy are simply expressed as: \[\Delta E_p= \int \mathbf{F} \cdot \mathbf{ds}= - k \int \mathbf{x} \cdot \mathbf{ds} =k\int x \cdot dx = k (\Delta x)^2\] In fact, elasticity emerges from the material characteristics of bodies (and therefore lies outside the strict scope of the physical domain). Since the tendency of molecules in a material to attract each other or push against each other is an electrostatic influence, that form of energy transfer is ultimately electrostatic in nature, but its capacity to convert displacement into momentum at a macroscopic level makes it fall into the mechanical category. By evoking elasticity, we’re anticipating on our next domain, since the porosity between the dynamochora (physics) and hylechora (chemistry + materials) is higher for applied matters.

• Potential gravitational energy Gravitational energy is a form of potential energy that corresponds to the gravitational field surrounding a body. On the surface of a planet, we may use the Galilean approximation of a uniform field, while more generally in classical physics, the Newtonian formulation can be used to derive variations in potential energy.

• Electromagnetic energy

  • Electrical potential energy can manifest in electrostatics as a close analog to the gravitational case, or in electrokinetics as the energy from a circuit’s voltage (\(\frac{1}{2}CV^2\)), which obviously derives from it.

  • Magnetic energy represents in a concrete electric circuit the amount of work done to generate an electric current (\(\frac{1}{2}LI^2\)), and is considered as an electrical analog to kinetic energy.

  • Radiant energy is the energy of the electromagnetic field perturbation in EM waves, which make up all the classically known forms of radiation.

• Thermal energy is proportional to a body’s temperature, corresponding to the intuitive idea that a body one spends energy bring to a higher temperature is absorbing that energy. In the statistical picture, that internal thermodynamic energy is understood as the microscopic kinetic energy from internal agitation. Heat is a form of non-mechanical energy transfer between thermal systems that is now identified with radiant energy (heat at daily-life temperatures is generally invisible infrared radiation, but thermal exchanges can generate EM waves at every frequency).

• Nuclear energy refers to the form of radiative energy, discovered more recently than the ones mentioned above, from radioactive decay. It was soon unveiled that gamma decay could be identified with the generation of a high-energy EM waves, while alpha and beta decays consist in the emission of fast helium nuclei and electrons, respectively.

• Binding energies are forms of potential energies derived from the above that reflect the energy transferred from or to a system when the structure of its microscopic components is altered. For instance, ionization energy is the electrostatic potential energy that corresponds to an electron’s bound to the nucleus. Nuclear binding energy can be released in a nuclear fusion process.

From the perspective of human civilization, there is a number of pre-existing energy stores: radiant (solar), thermal (geo-heat), mechanical (wind, waves, tides...), plus that which can be unbound from the structure of inert materials (fossil fuels, biomass, uranium...) or living beings (human and animal labor). At the other end of the chain, humans strive for radiant energy (light), thermal energy (hot or cold) or mechanical energy (moving things, raising things). One primary goal of applied physics is then to understand and enable the transduction of energy forms into each other. Let’s mention a few examples:

• A turbine converts the thermal energy of steam into mechanical energy.

• A generator converts kinetic energy to electrical energy.

• A dam converts potential mechanical energy to electrical energy.

• An electric motor converts electrical energy to kinetic energy.

• An electric heater converts electrical energy to heat.

• An electric bulb converts electrical energy to light.

• A solar panel converts radiant energy to electrical energy.

• etc.

Obviously, applied physics is not just about the transduction between different energy forms.

From prehistoric tools to modern-times electronic-circuitry, engineering has consisted in the design of systems to harness the flow of energy. We will illustrate that by using the bond graph formalism, an abstract generalization from electric circuitry that can describe universal systems, albeit in an idealized, linearized fashion. Let’s take the example of a mechanical system.

• Power, the rate of energy flowing into elements of the system is the product of two power conjugates: \(\mathcal{P}=\mathbf{F}\cdot\mathbf{v}\).

  • The force \(\mathbf{F}\) is the effort variable, which "pushes" energy through the system.

  • The velocity \(\mathbf{v}\) is the flow variable, which represents the rate at which the energy flows through the system.

• The power conjugates are the time derivatives of momentum (\(\mathbf{F}=\dot{\mathbf{p}}\)) and displacement (\(\mathbf{v}=\dot{\mathbf{x}}\)). The generalized displacement and momentum are the state variables of the system, so if we assume only linear relationships between \(\mathbf{F}\), \(\mathbf{v}\), \(\mathbf{p}\) and \(\mathbf{x}\), a system can be described via a first-degree differential matrix equation.

• In the bond graph formalism, energy flows freely between elements, which are characterized by the number of ports (connections) to the network:

  • 1-port elements:

    • Effort source (Energy-supplying):
      An external input of force, for instance: the gravity field on Earth, or a constant-force spring.

    • Flow source (Energy-supplying):
      An external input of velocity, for instance: a current, a wind, a motor.

    • Resistance (Energy-dissipating):
      Resistance elements impose a constitutive relation between effort and flow (or equivalently, momentum and displacement). In the case of a mechanical system the relation between force and velocity is characteristic of an idealized friction: \(\mathbf{F}=-c\mathbf{v}\).

    • Rigidity (Energy-storing):
      Rigidity elements impose a constitutive relation between effort and displacement. In the case of a mechanical system the relation between force and displacement is characteristic of an idealized spring: \(\mathbf{F}=k\mathbf{x}\). The spring stores potential energy.

    • Inertia (Energy-storing):
      Inertia elements impose a constitutive relation between flow and momentum. In the case of a mechanical system the relation between velocity and momentum is just the integral form of Newton’s law: \(\mathbf{p}=m\mathbf{v}\). The inertia of massive bodies can be seen as a store of kinetic energy.

  • 2-port elements (Energy-preserving): 2-port elements can change the value of effort and flow, but need to preserve power. If one such elements has two ports with state variables \((F_1,v_1)\) and \((F_2,v_2)\), then \(F_1 v_1=F_2 v_2\).

    • Transformers connect the efforts between the two ports, and the flows between the two ports: \(F_1=mF_2\) , \(mv_1=v_2\). A simple example is that of a lever, like a shovel or a wheelbarrow.

    • Gyrators connect the effort and flow of opposite ports: \(F_1=nv_2\) , \(nv_1=F_2\). An example example in the mechanical domain is a gyroscope. Another one in an electromechanic circuit is the ideal electric motor, which connects a torque (rotary mechanical effort) to an electric intensity (electrical flow) and an angular velocity (rotary mechanical flow) to a voltage (electrical effort).

  • N-port elements (Energy-preserving): N-port elements are nodes in the circuit that connect 3 or more elements. To enforce power preservations, two ideal cases emerge:

    • Effort junctions connect branches with equal forces. Power conservation then imposes a relation on velocities.

    • Flow junctions connect branches with equal velocity. Power conservation then imposes a relation on forces.

Let’s recapitulate the equivalence between generalized bond graph nomenclature and that of mechanical and electrical systems. \[\begin{array}{|c c c |c c|c c|} \hline &&\text{Constitutive relation}& \text{Mechanical system}& & \text{Electrical system}& \\ \hline \text{Effort} &e&& \text{Force}& \mathbf{F} &\text{Voltage} & V \\ \hline \text{Flow} & f& &\text{Velocity}&\mathbf{v} & \text{Intensity}& I \\ \hline \text{Momentum}&p &p=\int e \,dt& \text{Momentum}&\mathbf{p} & \text{Flux}& \Phi \\ \hline \text{Displacement}&q &q=\int f \, dt & \text{Displacement}& \mathbf{x} & \text{Charge}&q \\ \hline \text{Rigidity}&K & e=Kq &\text{Spring stiffness}&k &\text{Elastance}& 1/C \\ \hline \text{Resistance} &R &e=Rf & \text{Damper parameter}&c &\text{Resistance}& R \\ \hline \text{Inertia} &I & p=If &\text{Mass}& m &\text{Inductance}& L \\ \hline \text{Power} & \mathcal{P}& \mathcal{P}=ef&\text{Mechanical power}&\mathbf{F}\cdot\mathbf{v} & \text{Electrical Power}&VI \\ \hline \end{array}\]

As will become more and more familiar, as we move from episteme to techne, to delineate sub-disciplines becomes harder and the border with other domains itself becomes porous. The fields of applied physics, engineering science or technology is often divided according to the ultimate application of the knowledge at hand, as in automotive engineering, or power station engineering. That is a good solution for matters of professional training, but a poor one for our purposes, since it is very context-dependent and can hardly be generalized. It could also be divided to closely match the sub-discipline of dynalogy (theoretical physics). For that, I would remark that the level-specific vs level-agnostic distinction is less sensible. Knowledge of atomic physics, nuclear physics or molecular physics is a significant input of modern applied physics for sure. But the nature of the energy being ultimately harnessed to enable the targeted applications is more central. For example, laser science is entirely predicated on atomic physics rather than classical optics, yet its ultimate applications make it more lumpable with the rest of optical and electrical engineering, rather than in a hypothetic "atomic engineering" with mechanical and thermal applications of atomic knowledge.

Our classification will then follow the forms of energy outlined above (a list which itself follows broadly the categories of macroscopic classical physics):

• Mechanical engineering pertains to the transformations of mechanical energy. This includes the "core" of the discipline, the study and design of mechanisms that apply forces and control movements to perform certain actions. by way of these mechanisms, hominids and humans have been able to act on their environment and shape it to their needs.

• Gravitational engineering is not a common category. Yet, it is a legitimate distinction: gravitational potential energy at the surface of the earth can be expressed via a simple approximation (proportional to altitude), and the corresponding energy exchanges are wholly enmeshed within mechanical engineering. In outer space, the control of bodies’ movements in a gravitational field is more complex and energy exchanges with that field become predominant. At this stage of our civilization’s developments, such "engineering" is mostly represented by astronautics, the sub-discipline used to devise the voyage of spacecrafts or probes.

• Electrical engineering is an umbrella term that covers the harnessing of all forms of electromagnetic and radiant energy (electrical engineering, electronic engineering, optical engineering, etc.). That energy domain is not rife with direct applications (those which see humans enjoy that energy as an end upon itself). Simple ones include the generation of radiant energy for humans to enjoy visible light at will (electric bulbs), or of lower-frequency radiation to induce heat transfers (electric heating). That being said, electric circuitry and EM waves emissions are the absolute bedrock of our current civilization as an intermediary between other forms of energy.
  

  • The major asset of electromagnetic and radiant energy is that it is fast. Electromagnetic energy is transmitted through a medium like air, or through conductors like metals, at a speed that is below, but very close to the speed of light. Compare that with mechanical systems that are limited by constraints like friction or inertia.

  • Electromagnetic transmission is also extremely efficient: although it does experience energy loss or signal distortion, those are tiny compared to what mechanical gears or pulleys may experience. Combined with speed, this means that aggregate losses over very long distances are minimal.

  • Electrical systems can operate over a wide range of power levels, while mechanical systems are typically designed to sustain a narrow band only.

  All in all, we can say that electrical systems are extremely scalable: Babbage was able to build a simple mechanical computer. With physical constraints as they are, it is unimaginable for a mechanical engine to replicate the operation of a modern computer, or to transmit information all over the earth as it is now done.

• Thermal engineering deals specifically with the design of heat transfer or temperature control systems. Such systems can be intermediaries in more complex endeavors, or final applications, since human beings naturally seek to control the thermal features of their environment, for their own selves of course, or for food conservation and preparation, for instance.

• Nuclear engineering simply aims to harness nuclear energy. In the civil domain, there is no direct application, since direct exposure to nuclear energy is harmful to human beings; however it can be readily converted into thermal, then electric energy.

Cross-disciplines

Cross-disciplines follow a pattern that will become familiar in the physical sciences. To cross dynalogy with other disciplines from the natural sciences is roughly to consider the "physics of X" where X is the type of objects that’s normally at the forefront of the other discipline. For example biophysics or medical physics consider our body as physical objects: the ductility of our bones, or the characteristics of electrical activity in our brains. They "back up" compared to a discipline (biology) that considers its objects of study as living objects, to one that considers the same as per their physical properties only (which explains why the methods of physics can be used). We’ll call it grounded cross-disciplinarity.

This won’t work with the mathematics, or the human sciences since their objects (minds, social institutions, musical pieces, algebraic structures...) are not instantiations of physical kinds. Quantum physics has very little to say about the Supreme Court or a sonata by Corelli! A dangerous mistake that’s now ingrained in the terminology is to confuse applications of physics with the application of mathematical methods used in physics... "Sociophysics" or the likes will denote the use of methods akin to those of statistical mechanics to explain social behavior. But these are mathematical methods that happen to be famously used in physics, which certainly was an inspiration to those who first applied them to social entities; but they do not stem from physical science itself.

The correct crossing between natural and non-natural sciences is an application to the actual physical objects from the crossed domain: artifacts, edifices, actors. We will call this induced cross-disciplinarity. Importantly, the crossing between two natural sciences can be (in certain cases) grounded, as we’ve seen, but it can also be induced.

• Arithmodynalogy: the application of physical knowledge to the study and design of mathematical artifacts (for instance, the use of mechanical engineering, electrical engineering, electronics, quantum theory to design computer architectures).

• Cosmodynalogy has encompassed gravitational physics since Newton. Today it represents the broader realm of astrophysics, which does not restrain itself to the orbits of astral bodies but explores the physical processes responsible for their whole lifecycle.

• Hyledynalogy: chemical physics is a sort of reductionist enterprise that "zooms in" single instances of chemical reactions to explain their characteristics at the level of molecular physics; physical chemistry is more generally a way to denote the part of chemistry that relates to thermodynamic or optical properties.

• Geodynalogy: geophysics considers planets as complex physical systems with internal structures, mechanical properties as well as thermic or magnetic ones.

• Ecodynalogy: ecophysics apply physics to ecosystems, being at the same time a restriction of geophysics to that smaller scale, and an overlap with biophysics, as living populations mesh with the geological features of the planetary surface.

• Biodynalogy: biophysics has many sub-disciplines itself, echoing the may levels of organization of life, from molecular biophysics (e.g. the physical structure of DNA) to cellular biophysics (the mechanics of division processes) to the biomechanics, thermodynamics or electrical properties of larger entities like bones, organs or whole organisms. It also includes medical physics, which studies the physical properties of living bodies to develop imaging or analytic technology.

It is worth interrupting here and notice that for brevity we’ve skipped over most induced crossings between natural sciences and physics,with the exception of medical physics, a rich field in its own right. The others represent the physical study and engineering of equipment or machines connected to other domain, such as spacecraft or satellites (cosmo-), textile, metalworking or glassmaking equipment (hyle-), mining equipment (geo-), agricultural equipment (bio- and eco-).

• Topodynalogy: geography is the science of location and place from a human perspective. Physical geography is concerned with the physical features of those locations, but to seek a connection with physical sciences is very quickly to cross the line into geology, for which geophysics would be the relevant cross-discipline. The "physics of geography" should rather be concerned with the use of physics towards geospatial aims (maps, localization,...), to develop means of transportation, or to study the physical features of human dwellings (air, heat or water flows in human environments).

• Chremodynalogy covers physical applications to economics Let’s re-iterate, financial or econometric theories that "look like" statistical mechanics are not physics, they are part of Chremoarithmology! Physics can be applied to the economic domain with regards to the physical aspects of warehousing, commerce, and other concrete instances of the economic chain of value.

• Phylodynalogy: the physical imprint of the social domain is not distinctive. Sociology deals largely in institutions; their concrete instantiations like social symbols (decorations, badges, ensign), social buildings, or social activities (firefighting, social care...) are ruled by physics like all things concrete, but are not distinctive enough to delineate a clear cross-discipline.

• Dikaidynalogy: physics may not apply directly to legal theory as an abstract endeavor, but is used for concrete legal investigations and reasoning, forensics being the best-known and most spectacular case. It constitues an application of physics to the legal domain.

• Polidynalogy: a lot of politology could attract the same comment as sociology, in its dealing with physics: being mostly concerned with institutions and fairly benign concrete objects (government buildings, administrative records...), the crossing might not be fruitful. But the political domain also includes a vast military and policing wing, whose physical projection in the world has, sadly, been of serious significance. Applications of physics to weaponry and other military structures and installations is paramount for practitioners.

• Chronodynalogy: physics applied to history include horology (the study of time measurement techniques and apparatus), the science of conservation and restoration of historical artifacts, and the application of physics to various auxiliary disciplines focusing on different types of artifacts (ancient coins, books, sigils, flags, etc.).

• Psychodynalogy: mental states and human minds are generally connected to physics by way of biology. Yet, there is a field of neurophysics considering high-level electromagnetic or quantum parameters as possible (partial) grounds for mental activity, or more simply as its consequences. The psychological domain is also represented in a multitude of artifacts used for furnishing, maintenance, clothing, personal care, which can all be studied as physical objects.

• Glossodynalogy: the study of forms of linguistic expression like writing, can be aided by some physical sub-disciplines, as can the study of recording, printing, or general communication equipment.

• Tropodynalogy: the cultural domain is rich in concrete realizations of its abstract structures, be they in religion, culture, entertainment or recreation, all subject to physical applications. One emblematic case is the physical study of sporting activities.

• Theorodynalogy: scientific investigations often require experimental conditions in a laboratory context, and complex instrumentation to conduct experiments or detect their outcomes. Instrumentation physics or laboratory engineering are natural applications.

• Hermeneodynalogy: physics can be applied to the literary domain only in an ancillary way, for instance as it relates to the science of books as physical objects, or to the acoustics of public reading.

• Kalodynalogy: the intersection with the artistic domain is of much more import. Visual arts are...visual, and the relation to optics has always been extremely deep; painters and sculptors have often be at the forefront to study lightning and color: their perception and effect, but also the physical aspect of their combination. The arts have also led to the development of artifacts to create or present artworks. Physical applications abound when it comes to the study of paint pigments, of photographic and cinematographic equipment, of tools and technical equipment, etc. Performing arts also have a strong imprint in the physical domain, since dancing and other physical performances can be the object of physical studies.

Dynamurgy (physical activities)

Dynamurgy, the set of human activities anchored to the physical domain, apply the knowledge from dynamotechny (the part of engineering anchored to that same domain). Broadly speaking, they represent vast swathes of the following activities, though they do not by any means cover them exhaustively, since they are shared with other poiesis domains.

• Manufacturing, the physical transformation of materials, substances or components into products, to be fed as inputs into further manufacturing processes, or to be used by humans. This extends from simple craftsmanship in prehistoric times to the making of modern microchips

• Construction, the erection of buildings, which could be seen as part of a manufacturing process (making a building and its trimmings), but is generally treated separately due to the ample overlap into the geological domain, as we’ll see.

• Energy production, by which we actually mean the transduction and transmission of natural energy sources to forms and locations required by human life, also has a dynamurgic component (in a gas power plant, the combustion reaction is of a chemical rather than physical nature, but the conversion of heat into electricity and its transmission relates to physical consideration).

Dynamergons (physical artifacts)

The artifacts that are involved in activities of the physical domain can generically be called machines.

In the mechanical realm, machines are composed of structural elements like frame members, bearings, axles, splines, fasteners, seals... which impose constraints on motion, and of mechanisms. When we introduced the bond graph formalism, we explained that power is conserved, but it could dissipate (friction) or be stored (springs). Besides, power conjugates can be made to vary while keeping power constant: by converting velocity into force, a mechanism can generate a mechanical advantage. Complex mechanisms are compounds (kinematic chains) of simpler ones. Efforts to classify and analyze machines have led from the Renaissance era to identifying the six most basic blocks, known as simple machines. All have been known for millenia:

• The wedge, exemplified by an axe, has been around since prehistoric times.

• The inclined plane is probably as ancient, and is thought to have been used to move megaliths or build early structures.

• The wheel and axle mechanism is though to date back as far as 5,000 years ago.

• The lever appeared in the form of a simple balance scale, and would be used later in cranes to move large stone block and build structures like the Giza pyramid.

• The pulley dates back to the 2nd millenium in Mesopotomia.

• The screw was the last simple machine to be invented, in Assyria.

Later on, it was realized that even this short list of 6 machines could be reduced to 2 basic devices: a rotating body (lever, pulley, wheel) and a sliding block on a surface (screw, incline plane, wedge). The difference in mechanisms then really stems from structural elements: the joints (whose four types are: revolute joint, prismatic joint, cam joint and gear joint).

In the electrical realm, the analogous object is a circuit element, which we can even more directly match to the bond graph concepts: the five linear elements are resistors, inductors, conductors, transformers and gyrators.

Dynamodomes (physical edifices)

Buildings where humans engage in manufacturing, in energy production or in construction are variously called factories, plants, mills, yards, workshops, building sites...

Dynamanthropes (physical actors)

The physical facet of human beings is simply represented by their body, the purely physical projection of their being. That body falls, conducts electricity, vibrates with sounds, floats in water, like object that would share its physical properties.

Dynamopraxy (physical experience)

The human experience (segment of the praxis plane) that corresponds the physical domain is the interaction (creation or usage) of artifacts. To interact physically with the world is the wont of dynamanthropes, but the connection with dynamurgy and dynamergons can only be established for man-made physical objects.