Fermions: Meat In Our Particle Soup

Part Of: Demystifying Physics sequence
Prerequisite Post: An Introduction To Energy
Content Summary: 2100 words, 21 min reading time.

Prerequisite Mindware

Today, we’re going to go spelunking into the fabric of the cosmos! But first, some tools to make this a safe journey.

Energy Quanta

As we saw in An Introduction to Energy,

Energy is the hypothesis of a hidden commonality behind every single physical process. There are many forms of energy: kinetic, electric, chemical, gravitational, magnetic, radiant. But these forms are expressions of a single underlying phenomena.


Consider the analogy between { electrons spinning around protons } and { planets spinning around stars }. In the case of planets, the dominant force is gravitational. In the case of the atom, the dominant force is electromagnetic.

But the analogy strength of the above is weak. In contrast to gravitational acceleration, an accelerating electric charge emits electromagnetic waves. Thus, we would expect an orbiting charge to steadily lose energy and spiral into the nucleus, colliding with it in a fraction of a second. Why have atoms not gone extinct?

To solve this problem, physicists began to believe that in some situations, energy cannot be lost. Indeed, they abandoned the intuitive idea that energy is continuous. On this new theory, at the atomic level energy must exist in certain levels, and never in between. Further, at one particular energy level, something we will call the ground state, an electron may never lose energy.



Let’s talk about antiparticles. It’s time to throw out your “science fiction” interpretive lens: antiparticles are very real, and well-understood. In fact, they are exactly the same as normal particles, except charge is reversed. So, for example, an antielectron has the same mass and spin as an electron, but instead carries a positive charge.

Why does the universe contain more particles than antiparticles? Good question. 😛

Meet The Fermions

Nature Up Close

Consider this thing. What would you name it?

Atomic Structure

One name I wouldn’t select is “indivisible”. But that’s what the “atom” means (from the Greek “ἄτομος”). Could you have predicted the existence of this misnomer?

As I have discussed before, human vision can capture only a small subset of physical reality. Measurement technology is a suite of approaches that exploit translation proxies, the ability to translate extrasensory phenomena into a format amenable to perception. Our eyes cannot perceive atoms, but the scanning tunneling microscope translates atomic structures to scales our nervous systems are equipped to handle.

Let viable translation distance represent the difference in scale between human perceptual foci and the translation proxy target. Since translation proxies are facilitated through measurement technology, which is in turn driven by scientific advances, it follows that we ought to expect viable translation distance to increase over time.

We now possess a straightforward explanation of our misnomer. When “atom” was coined, its referent was the product of that time’s maximum viable translation distance. But technology has since moved on, and we have discovered even smaller elements. Let’s now turn to the state of the art.

Beyond The Atom

Reconsider our diagram of the atom. Do you remember the names of its constituents? That’s right: protons, neutrons, and electrons. Protons and neutrons “stick together” in the nucleus, electrons “circle around”.

Our building blocks of the universe so far: { protons, neutrons, electrons }. By combining these ingredients in all possible ways, we can reconstruct the periodic table – all of chemistry. Our building blocks are – and must be – backwards compatible. But are these particles true “indivisibles”? Can we go smaller?

Consider the behavior of the electrons orbiting the nucleus. After fixing one theoretical problem (c.f., Energy Levels section above), we now can explain why electrons orbit the nucleus: electromagnetic attraction (“opposites attract”). But here is a problem: we have no such explanation for the nucleus. If “like charges repel”, then the nucleus must be something like holding the same poles of a magnet close together: you can do it, but it takes a lot of force. What could possibly be keeping the protons in the nucleus together?

Precisely this question motivated a subsequent discovery: electrons may well be indivisible, but protons and neutrons are not. Protons and neutrons are instead composite particles made out of quarks. Quarks like to glue themselves together by a new force, known as the strong force. This new type of force not only explains why we don’t see quarks by themselves, it also explains the persistence of the nucleus.

The following diagram (source) explains how quarks comprise protons and neutrons:


Okay, so our new set of building blocks are: { electron, up, down }. With a little help from some new mathematics – quantum chromodynamics – we can again reconstitute chemistry. biology, and beyond.

Please notice how some of our building blocks are more similar than others: the up and down particle comprise particles with charge divisible by three, the electron particle carries an integer charge. Let us group like particles together.

  • Call up and down particles part of the quark family.
  • Call electrons part of the lepton family.


So far in this article, we’ve gestured towards gravitation and electromagnetism. We’ve also introduced the strong force. Now is the time to discuss Nature’s last muscle group, the weak force.

A simple way to bind the weak force to your experience: consider what you know about radioactive material. The types of atoms that are generated in, to pick one source, nuclear power do not behave like other atoms. They emit radiation, they decay. Ever heard of the “half-life” of a material? That term defines how long is takes for half of an unstable radioactive material to decay into a more stable form. For example, { magnesium-23 → sodium-23 + antielectron }.

Conservation of energy dictates that such decay reactions must preserve energy. However, when you actually calculate the energetic content of decay process given above, you find a mismatch. And so, scientists were confronted with the following dilemma: either reject conservation of energy, or posit the existence of an unknown particle to “balances the books”. Which would you chose?

The scientific community began to speculated that a fourth type of fermion existed, even with an absence of physical evidence. And they found it 26 years later, in 1956.

Why did it take relatively longer to discover this fourth particle? Well, these hypothesized neutrinos do not carry an electric charge or a color charge. As such, they only interact with other particles via the weak force (which has a very short range) and the atomic force (which is 10^36 times less powerful than electromagnetic force). Due to these factors, neutrinos such as those generated by the Sun pass through the Earth undetected. In fact, in the time it takes you to read this sentence, hundreds of billions of neutrinos have passed through every cubic centimeter of your body without incident. Such weak interactivity explains the measurement technology lag.

Are you sufficiently creeped out by how many particles pass through you undetected? 🙂 If not, consider neutrino detectors. Because of their weak interactivity, our neutrino detectors must be large, and buried deep inside the earth (to shield from “noise” – more common particle interactions). Here we see a typical detector, with scientists inspecting their instruments in the center, for contrast:


The Legos Of Nature

Here, then, is our picture of reality:

Fermions- One Generation

Notice that all fermions have spin ½; we’ll return to this fact later.

A Generational Divide

Conservation of energy is a thing, but conservation of particles is not. Just as particles spontaneously “jump” energy levels, sometimes particles morph into different types of particles, in a way akin to chemical reactions. What would happen if we were to pump a very large amount of energy into the system, say by striking an up quark with a high-energy photon? Must the output energy be expressed as hundreds of up quarks? Or does nature have a way to “more efficiently” spend its energy budget?

It turns out that you can: there exist particles identical to these four fermions with one exception: they are more massive. And we can pull this magic trick once more, and find fermions even heavier than these fermions. To date, physicists have discovered three generations of fermions:

Fermions- Three Generations


The latter generation took lots of time to “fill in” because you only see them in high-energy situations. Physicists had to close the translation distance gap, by building bigger and bigger particle accelerators. The fermion with the highest mass – the Top quark – was only discovered in 1995. Will there be a fourth generation, will we discover some upper bound on fermion generations?

Good question.

Even though we know of three generations, in practice only the first generation “matters much”. Why? Because the higher-energy particles that comprise the second and third generations tend to be unstable: give them time (fractions of a second, usually), and they will spontaneously decay – via the weak force – back into first generation forms. This is the only reason why we don’t find atomic nuclei orbited by tau particles.

Towards A Mathematical Lens

General & Individual Descriptors

The first phase of my lens-dependent theorybuilding triad I call conceptiation: the art of carving concepts out of a rich dataset. Such carving must be heavily dependent on descriptive dimensions: quantifiable ways that an entity may differ from one another.

For perceptual intake, the number of irreducible dimensions may be very large. However, for particles, this set is surprisingly small. There is something distressingly accurate in the phrase “all particles are the same”.

Each type of fermion is associated with one unique value for the following properties (particle-generic properties):

  • mass (m)
  • electric charge (e)
  • spin (s)

Fermions may differ according to their quantum numbers (particle-specific properties). For an electron, these numbers are:

  • principal. This corresponds to the energy level of the electron (c.f., energy level discussion)
  • azimuthal. This corresponds to the orbital version of angular momentum (e.g., the Earth rotating around the Sun). These numbers correspond to the orbitals of quantum chemistry (0, 1, 2, 3, 4, …) ⇔ (s, p, d, f, g, …); which helps explain the orbital organization of the periodic table.
  • magnetic. This corresponds to the orientation of the orbital.
  • spin projection. This corresponds to the “spin” version of angular momentum (e.g., the Earth rotating around its axis). Not to be confused with spin, this value can vary across electrons.

Quantum numbers are not independent; their ranges hinge on one another in the following way:

Quantum Numbers

Statistical Basis

With our fourth building block in place, we are in a position to answer the question: what does the particulate basis of matter have in common?

All elementary particles of matter we have seen have spin ½. By the Spin-statistics Theorem, we must associate all such particles with Fermi-Dirac statistics. Let us name all particles under this statistics – all particles we have seen so far – “fermions”. It turns out that this statistical approach generates a very interesting property known as the Pauli Exclusion Principle. The Pauli Exclusion Principle states, roughly, that two particles cannot share the same quantum state.

Let’s take an example: consider a hydrogen atom with two electrons. Give this atom enough time, and both electrons will be on its ground state, n=1. What happens if the hydrogen picks up an extra electron, in some chemical process? Can this third electron also enter the ground state?

No, it cannot. Consider the quantum numbers for our first two electrons: { n=1, l=0, m_l=0, m_s=1/2 } and { n=1, l=0, m_l=0, m_s=-1/2 }. Given the range constraints given above, there are no other unique descriptors for an electron with n=1. Since we cannot have two electrons with the same quantum numbers, the third electron must come to rest at the next highest energy level, n=2.

The Pauli Exclusion Principle has several interesting philosophical implications:

  • Philosophically, this means that if two things have the same description, then they cannot be two things. This has interesting parallels to the axiom of choice in ZFC, which accommodates “duplicate” entries in a set by conjuring some arbitrary way to choose between them.
  • Practically, the Pauli Exclusion Principle is the only thing keeping your feet from sinking into the floor right now. If that isn’t a compelling demonstration of why math matters, then I don’t know what is.

Composite Fermions

In this post, we have motivated the fermion particulate class by appealing to discoveries of elementary particles. But then, when we stepped back, we discovered that the most fundamental attribute of this class of particles was its subjugation to Fermi-Dirac statistics.

Can composite particles have spin-½ as well as these elementary particles? Yes. While all fermions considered in this post are elementary particles, that does not preclude composite particles from membership.

What Fermions Mean

In this post, we have done nothing less than describe the basis of matter.

But are fermions the final resolution of nature? Our measurement technology continues to march on. Will our ability to “zoom in” fail to produce newer, deeper levels of reality?

Good questions.


Knowledge: An Empirical Sketch

Table Of Contents

  • Introduction
    • All The World Is Particle Soup
    • Soup Texture
  • Perceptual Tunnels
    • On Resolution
    • Sampling
    • Light Cones, Transduction Artifacts, Translation Proxies
  • The Lens-dependent Theorybuilding Triad
    • Step One: Conceptiation
    • Step Two: Graphicalization
    • Step Three: Annotation
    • Putting It All Together: The Triad
  • Conclusion
    • Going Meta
    • Takeaways


All The World Is Particle Soup

Scientific realism holds that the entities scientists refer to are real things. Electrons are not figments of our imagination, they possess an existence independent of your mind. What does it mean for us to view particle physics with such a lens?

Here’s what it means: every single thing you see, smell, touch… every vacation, every distant star, every family member… it is all made of particles.

This is an account of how the nervous system (a collection of particles) came to understand the universe (a larger collection of particles). How could Particle Soup ever come to understand itself?

Soup Texture

Look at your hand. How many types of particles do you think you are staring at? A particle physicist might answer: nine. You have four first-generation fermions (roughly, particles that comprise matter) and five bosons (roughly, particles to carry force). Sure, you may get lucky and find a couple exotic particles within your hand, but such a nuance would not detract from the morale to the story: in your hand, the domain (number of types) of particles is very small.

Look at your hand. How large a quantity of particles do you think you are staring at? The object of your gaze is a collection of about 700,000,000,000,000,000,000,000,000 (7.0 * 10^26) particles. Make a habit about thinking in this way, and you’ll find a new appreciation for the Matrix Trilogy. 🙂 In your hand, the cardinality (number of tokens) of particles is very large.

These observations generalize. There aren’t many kinds of sand in God’s Sandbox, but there is a lot of it, with different consistencies across space.

Perceptual Tunnels

On Resolution

Consider the following image. What do you see?

Lincoln Resolution

Your eyes filter images at particular frequencies. At this default human frequency, your “primitives” are the pixelated squares. However, imagine being able to perceive this same image at a lower resolution (sound complicated? move your face away from the screen :P). If you do this, the pixels fade, and a face emerges.

Here, we learn that different resolution lens may complement one another, despite their imaging the same underlying reality. In much the same way, we can enrich our cognitive toolkit by examining the same particle soup with different “lens settings”.


By default, the brain does not really collect useful information. It is only by way of sensory transductor cells – specialized cells that translate particle soup into Mentalese – that the brain gains access to some small slice of physical reality. With increasing quantity and type of these sensory organs, the perceptual tunnel burrowed into the soup becomes wide enough to support a lifeform.

Another term for the perceptual tunnel is the umwelt. Different biota experience different umwelts; for example, honeybees are able to perceive the Earth’s magnetic field as directly as we humans perceive the sunrise.

Perceptual tunneling may occur at different resolutions. For example, your proprioceptive cells create signals only on the event of coordinated effort of trillions and trillions of particles (e.g., the wind pushes against your arm). In contrast, your vision cells create signals at very fine resolutions (e.g., if a single photon strikes your photoreceptor, it will fire).

Perceptual Tunneling

Light Cones, Transduction Artifacts, Translation Proxies

Transduction is a physically-embedded computational process. As such, it is subject to several pervasive imperfections. Let me briefly point towards three.

First, nature precludes the brain from the ability to sample from the entirety of the particle soup. Because your nervous system is embedded within a particular spatial volume, it is subject to one particular light cone. Since particles cannot move faster than the speed of light, you cannot perceive any non-local particles. Speaking more generally: all information outside of your light cone is closed to direct experience.

Second, the nervous system is an imperfect medium. It has difficulty, for example, representing negative numbers (ever try to get a neuron firing -10 times per second?). Another such transduction artifact is our penchant for representing information in a comparative, rather than absolute, format. Think of all those times you have driven on the highway with the radio on: when you turn onto a sidestreet, the music feels louder. This experience has nothing at all to do with an increased sound wave amplitude: it is an artifact of a comparison (music minus background noise). Practically all sensory information is stained by this compressive technique.

Third, perceptual data may not represent the actual slice of the particle soup we want. To take one colorful example, suppose we ask a person whether they perceived a dim flashing light, and they say “yes”. Such self-reporting, of course, represents sensory input (in this case, audio vibrations). But this kind of sensory information is a kind of translation proxy to a different collection of particles we are interested in observing (e.g., the activity of your visual cortex).

This last point underscores an oft-neglected aspect of perception: it is an active process. Our bodies don’t just sample particles, they move particles around. Despite the static nature of our umwelt, our species has managed to learn ever more intricate scientific theories in virtue of sophisticated measurement technology; and measurement devices are nothing more than mechanized translation proxies.

The Lens-dependent Theorybuilding Triad

Step One: Conceptiation

Plato once describes concept acquisition as “carving nature at its joints”. I will call this process (constructing Mentalese from the Soup) theory conceptiation.

TheoryBuilding- Conceptiation

If you meditate on this diagram for a while, you will notice that theory conceptiation is a form of compression. Acccording to Kolmogorov information theory, the efficacy of compression hinges on how many patterns exist within your data. This is why you’ll find leading researchers claiming that:

Compression and Artificial Intelligence are equivalent problems

A caveat: concepts are also not carved solely from perception; as one’s bag of concepts expands, such pre-existent mindware exerts an influence on the further carving up of percepts. This is what the postmoderns attribute to hermeneutics, this is the root of memetic theory, this is what is meant by the nature vs. nurture dialogue.

Step Two: Graphicalization

Once the particle soup is compressed into a set of concepts, relations between these concepts are established. Call this process theory graphicalization.

TheoryBuilding- Graphicalization

If I were ask you to complete the word “s**p”, would you choose “soap” or “soup”?  How would your answer change if we were to have a conversation about food network television?

Even if I never once mention the word “soup”, you become significantly more likely to auto-complete that alternative after our conversation. Such priming is explained through concept graphs: our conversation about the food network activates food-proximate nodes like “soup” much more strongly than graphically distant nodes like “soap”.

Step Three: Annotation

Once the graph structure is known, metagraph information (e.g., “this graph skeleton occurs frequently”) is appended. Such metagraph information is not bound to graphs. Call this process theory annotation.

TheoryBuilding- Annotation

We can express a common complaint about metaphysics thusly: theoretical annotation is invariant to changes in conceptiation & graphicalization results. In my view (as hinted at by my discussion of normative therapy) theoretical annotation is fundamentally an accretive process – it is logically possible to generate an infinite annotative tree; this is not seen in practice because of the computational principle of cognitive speed limit (or, to make a cute analogy, the cognition cone).

Putting It All Together: The Triad

Call the cumulative process of conceptiation, graphicalization, and annotation the lens-dependent theorybuilding triad.

TheoryBuilding- Lens-Dependent Triad


Going Meta

One funny thing about theorybuilding is how amenable it is to recursion. Can we explain this article in terms of Kevin engaging in theorybuilding? Of course! For example, consider the On Resolution section above. Out of all possible adjectives used to describe theorybuilding, I deliberately chose to focus my attention on spatial resolution. What phase of the triad does that sound like to you?  Right: theory conceptiation.


This article does not represent serious research. In fact, its core model – the lens-dependent theorybuilding triad – cites almost no empirical results. It is a toy model designed to get us thinking about how a cognitive process can construct a representation of reality. Here is an executive summary of this toy model:

  1. Perception tunneling is how organisms begin to understand the particle soup of the universe.
    1. Tunneling only occurs by virtue of sensory organs, which transduce some subset of data (sampling) into Mentalese.
    2. Tunneling is a local effect, it discolors its target, and its sometimes merely represents data located elsewhere.
  2. The Lens-Dependent Theorybuilding Triad takes the perception tunnel as input, and builds models of the world. There are three phases:
    1. During conceptiation, perception contents are carved into isolable concepts.
    2. During graphicalization, concept interrelationships are inferred.
    3. During annotation, abstracted properties and metadata are attached to the conceptual graph.