Appearance
7.10 — Quantum Fields, Antimatter and Why Particles Are Not Things
The Schrödinger equation of Chapter 7.3 describes a fixed number of particles. One electron, or two, forever.
Reality does not work that way. An atom emits a photon that did not exist a moment earlier. A neutron decays into a proton, an electron and an antineutrino — three particles where there was one. Collide two protons at the LHC and hundreds of particles come out.
And the equation is not relativistic. It uses E = p^2/2m, which Chapter 6.4 replaced.
Fixing either problem forces the other, and fixing both produces quantum field theory — the framework that describes every particle and every force except gravity, and which predicted antimatter before anyone had seen it.
Why relativity forces particle creation
The argument is short and it is the reason field theory is not optional.
Chapter 6.4 established E = mc^2. Energy can become mass.
Chapter 7.5 established \Delta E\,\Delta t \geq \hbar/2. Energy is uncertain over short times.
Put them together. Over a time \Delta t, the energy is uncertain by \hbar/2\Delta t. If that uncertainty exceeds 2mc^2 — enough to make a particle and its antiparticle — then particle number is not a well-defined quantity over that timescale.
The length scale where this happens comes from setting the localisation energy equal to the pair-creation threshold. Confining a particle to a region \lambda gives it momentum \hbar/\lambda and energy \hbar c/\lambda. Setting that equal to mc^2:
\lambda_C = \frac{\hbar}{mc}
The Compton wavelength, which appeared in Chapter 7.1's scattering formula. For the electron it is 3.86\times10^{-13} m.
So trying to localise an electron more precisely than its Compton wavelength delivers enough energy to create electron–positron pairs. You cannot pin down "the electron" because you cannot guarantee there is still just one.
A single-particle relativistic quantum theory is therefore impossible. The number of particles is not conserved, so the theory must be able to describe any number — which is what a field does.
The Dirac equation
Paul Dirac set out in 1928 to find a relativistic wave equation for the electron.
The obvious attempt. Take E^2 = p^2c^2+m^2c^4 and substitute operators:
-\hbar^2\frac{\partial^2\psi}{\partial t^2} = -\hbar^2c^2\nabla^2\psi+m^2c^4\psi
This is the Klein–Gordon equation, and it has two problems. It is second order in time, which spoils the probability interpretation and produces negative probability densities. And it turns out to describe spin-0 particles, not electrons.
Dirac's requirement: an equation first order in time, like Schrödinger's, and therefore first order in space too, for relativistic symmetry.
i\hbar\frac{\partial\psi}{\partial t} = \left(c\vec{\alpha}\cdot\vec{p}+\beta mc^2\right)\psi
For this to square to the Klein–Gordon equation, the coefficients must satisfy:
\alpha_i\alpha_j+\alpha_j\alpha_i = 2\delta_{ij}, \qquad \alpha_i\beta+\beta\alpha_i = 0, \qquad \beta^2 = 1
No ordinary numbers can do this, because ordinary numbers commute and these must anticommute. Dirac found that 4\times4 matrices work, and nothing smaller does.
Two consequences came out unbidden, and both were spectacular.
Consequence 1: spin. Because \psi must have four components to be acted on by 4\times4 matrices, the electron automatically has an internal two-valued degree of freedom, with the right magnetic moment g = 2. Spin was not assumed. It fell out of demanding a first-order relativistic equation. Chapter 7.7 introduced spin as a mysterious extra property; here it is explained.
Consequence 2: negative energies. Four components means four solutions, and two of them have E < 0. From E^2 = p^2c^2+m^2c^4, the negative square root is mathematically legitimate, and Dirac could not discard it — the four solutions form a complete set and throwing two away breaks the mathematics.
This looked fatal. An electron could cascade down through the negative energy states forever, radiating unlimited energy. All matter should collapse in about 10^{-10} s.
Antimatter
Dirac's first answer, in 1930, was the "sea". All negative energy states are already occupied, everywhere, and the Pauli exclusion principle (Chapter 7.7) forbids a normal electron from dropping into them.
A hole in the sea — a missing negative-energy electron — would behave like a particle of positive energy and positive charge.
Dirac initially proposed that the hole was the proton, and Weyl and Oppenheimer quickly pointed out that it must have the same mass as the electron, not 1836 times more. In 1931 Dirac accepted this and predicted a new particle: "a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron."
Carl Anderson found it in 1932, in cosmic ray tracks in a cloud chamber, curving the wrong way in a magnetic field. He named it the positron. Dirac got the Nobel Prize in 1933, Anderson in 1936.
The sea picture is obsolete. It only works for fermions, since it relies on exclusion, and antiparticles exist for bosons too. Modern field theory treats particles and antiparticles symmetrically, with no sea. The prediction survived the death of the reasoning behind it, exactly as Carnot's did in Chapter 3.4.
Every particle has an antiparticle
| Particle | Antiparticle | Note |
|---|---|---|
| Electron e^- | Positron e^+ | Found 1932 |
| Proton | Antiproton | Found 1955 |
| Neutron | Antineutron | Opposite quark content |
| Neutrino \nu | Antineutrino \bar{\nu} | May be its own — Chapter 8.5 |
| Photon \gamma | Itself | Neutral, no distinguishing charge |
| Quark q | Antiquark \bar{q} |
Antiparticles have the same mass and spin, and opposite charge and other quantum numbers.
A few particles are their own antiparticles — the photon, the Z boson, and possibly the neutrino. This requires all their additive quantum numbers to be zero.
Annihilation
e^-+e^+ \to 2\gamma
Why two photons and not one? Momentum conservation. In the centre-of-mass frame the pair has zero total momentum, and a single photon always has momentum E/c \neq 0. Two, going opposite ways.
Each carries m_ec^2 = 0.511 MeV. That specific energy, in a back-to-back pair, is the signature that PET scanners look for (Chapter 6.4).
Making antimatter is expensive. CERN's Antiproton Decelerator produces roughly 10^{7} antiprotons per second, which is 1.7\times10^{-20} kg per year. At the facility's operating cost, antimatter is the most expensive substance ever made by a very wide margin — estimates run to 10^{15} dollars per gram. Total antimatter produced by humanity to date: a few nanograms.
And it has been used to make atoms. CERN's ALPHA experiment has trapped antihydrogen — an antiproton with a positron around it — for over 16 minutes, and measured its spectrum. The 1s–2s transition matches hydrogen's to a few parts in 10^{12}. In 2023 ALPHA-g measured how antihydrogen falls in gravity and found it falls down, at g within experimental error, settling a question people had genuinely wondered about.
Where has all the antimatter gone?
The Big Bang should have made equal amounts of matter and antimatter. They would have annihilated completely, leaving a universe of photons and nothing else.
The observed asymmetry is roughly one part in 10^{9} — for every billion antiquarks in the early universe there were a billion and one quarks, and everything you see is that leftover one.
Sakharov's three conditions, set out in 1967, say what any explanation must include:
- Baryon number violation — some process must change the quark count.
- C and CP violation — the laws must treat matter and antimatter differently.
- Departure from thermal equilibrium — otherwise the reverse process undoes it.
CP violation has been observed, first in neutral kaons by Cronin and Fitch in 1964 and now in B mesons and D mesons. But the amount in the Standard Model is roughly 10^{10} times too small to produce the observed asymmetry.
So this is an unsolved problem, and it is one of the strongest reasons to think the Standard Model is incomplete. Chapter 8.7 covers the candidate explanations, and neutrinos are a leading suspect (Chapter 8.5).
Fields, not particles
The modern picture drops particles as fundamental objects entirely.
There is a field filling all of space for each kind of particle. Particles are localised excitations — quanta — of those fields.
One electron field. One photon field. One up-quark field. One Higgs field. And so on, one per particle species.
What this explains, that a particle picture cannot:
Why all electrons are identical. Every electron in the universe has exactly the same mass, charge and spin, to the limits of measurement. In a particle picture this is an unexplained coincidence. In a field picture it is inevitable — they are all excitations of the same field, so they cannot differ any more than two waves on the same pond can be made of different water.
Creation and destruction. Excitations can be added to and removed from a field. An atom emitting a photon is adding a quantum to the electromagnetic field, drawing the energy from the electron field. Nothing is transported; one field's excitation is exchanged for another's.
Antiparticles. The field has both positive and negative frequency parts, giving particles and antiparticles automatically.
The vacuum. The field in its ground state, with the zero-point fluctuations of Chapter 7.9.
Forces are field exchange
In classical physics, a force is a field pushing (Chapter 4.1). In quantum field theory, the interaction is described as an exchange of quanta.
Feynman diagrams are a bookkeeping device, invented by Feynman in the late 1940s, and each one stands for a specific mathematical expression. Sum over all diagrams and you get the amplitude.
They are not pictures of what happens. No electron follows the drawn path, the exchanged quantum is not observable, and the diagram represents one term in a series. Chapter 8.3 teaches how to use them as calculation tools.
Why range depends on mass
The exchanged quantum cannot exist as a real particle — it does not satisfy E^2 = p^2c^2+m^2c^4 — so it must be "borrowed" and returned. Using the energy–time relation as a heuristic, an exchange requiring energy mc^2 can last:
\Delta t \sim \frac{\hbar}{mc^2}
and in that time it travels at most:
R \sim c\Delta t = \frac{\hbar}{mc}
The Compton wavelength again. So:
\boxed{R \sim \frac{\hbar}{mc}}
Massless carrier means infinite range. The photon has m = 0, so electromagnetism reaches forever, and the 1/r^2 law of Chapter 4.1 is a direct consequence of the photon being exactly massless.
Massive carrier means short range. The W boson has m = 80.4 GeV/c²:
R = \frac{\hbar}{mc} = \frac{\hbar c}{mc^2} = \frac{197\ \text{MeV fm}}{80{,}400\ \text{MeV}} = 2.4\times10^{-3}\ \text{fm}
using the useful constant \hbar c = 197 MeV·fm.
Under a hundredth of a proton's radius, which is why the weak force is confined to inside nuclei and why it is called weak — the range is so short that the interaction rate is tiny even though its intrinsic coupling is not much smaller than electromagnetism's.
Yukawa's prediction. In 1935 Hideki Yukawa reversed this argument. The nuclear force has a range of about 1.5 fm, so:
m = \frac{\hbar}{Rc} = \frac{197\ \text{MeV fm}}{1.5\ \text{fm}} \approx 130\ \text{MeV/c}^2
He predicted a particle of about 130 MeV, between the electron's 0.5 and the proton's 938. The pion, mass 139.6 MeV, was found in 1947. Yukawa got the Nobel Prize in 1949. (The muon, found in 1936 at 106 MeV, was mistaken for it for a decade — it turned out to be an entirely unrelated heavy electron, prompting Rabi's remark, "Who ordered that?")
Renormalisation
The first calculations in quantum electrodynamics gave infinity.
The source. Summing over all possible virtual processes includes arbitrarily high momenta, and the integrals diverge. The electron's self-energy — its interaction with its own field — comes out infinite.
Note that classical physics has the same disease. The electrostatic energy of a point charge, ke^2/2r, diverges as r \to 0. The problem is not new.
The resolution, developed by Feynman, Schwinger, Tomonaga and Dyson between 1947 and 1949:
The infinities appear only in a small number of places — in the mass and the charge. And neither of those is ever measured in isolation. What you measure is the electron's mass including its self-interaction, and its charge including vacuum screening.
So absorb the infinities into the definitions. Write the theory in terms of the measured mass and charge rather than the "bare" ones, and every remaining prediction comes out finite — and correct to twelve decimal places.
Was this a swindle? Dirac thought so and said so, repeatedly, until his death. Feynman called it "a dippy process" and "hocus-pocus". The modern view, largely due to Kenneth Wilson in the 1970s, is that it is not a trick at all: a quantum field theory is an effective description valid up to some energy scale, and renormalisation is the systematic procedure for separating what you can predict from what depends on unknown physics at higher energies. Wilson's reformulation won the 1982 Nobel Prize and is now regarded as one of the deepest ideas in theoretical physics.
Not all theories are renormalisable, and this is where gravity fails. Attempting to quantise general relativity produces infinities that require infinitely many parameters to absorb, so the theory loses all predictive power. That is the technical statement of why quantum gravity is hard, and Chapter 8.7 covers the attempts.
What quantum field theory delivers
Quantum electrodynamics (QED) — the electron field and the photon field. The most precisely tested theory in existence: the electron's magnetic moment to twelve figures, the Lamb shift, and every calculation involving light and matter.
Quantum chromodynamics (QCD) — quarks and gluons. Explains confinement, why quarks are never seen alone, and why 98 % of your mass is field energy (Chapter 6.4).
Electroweak theory — unifies electromagnetism and the weak force, predicts the W and Z bosons and their masses, and requires the Higgs field.
Together these are the Standard Model, which Part 8 reads completely.
Its record: every prediction it has made has been confirmed, including the W and Z in 1983, the top quark in 1995, and the Higgs in 2012 — the last of these 48 years after it was predicted.
And its known gaps: it does not include gravity, it does not explain dark matter, it does not explain why there is more matter than antimatter, it does not explain the neutrino masses that were found in 1998, and it has around 19 free parameters that must be measured rather than derived.
Where this shows up in your life
PET scans detect the 0.511 MeV annihilation photons.
Every semiconductor is designed with QED-level accuracy in the electron's behaviour.
Cancer radiotherapy uses positron and electron beams whose behaviour is computed relativistically.
Bananas emit positrons. Potassium-40 is a natural radioisotope, about 0.012 % of all potassium, and 0.001 % of its decays are by positron emission. A typical banana emits roughly one positron every 75 minutes, each of which annihilates immediately with a nearby electron. You have handled antimatter.
And your mass is field energy. Over 98 % of the mass of every proton and neutron in your body is the energy of the gluon field, which exists only in the framework this chapter describes.
What Part 7 established
Part 7 started with five experiments classical physics got wrong and ended with fields.
Energy comes in quanta. Planck's h resolved the black-body catastrophe, the photoelectric effect, and the missing heat capacities.
Matter is wavelike. De Broglie's \lambda = h/p was confirmed by accident, and confining a wave makes its energy discrete — which is why atoms have energy levels at all.
The wavefunction is the complete description, evolving by the Schrödinger equation, with |\psi|^2 giving probabilities. Solving it produced the box, the tunnelling barrier that runs the Sun and your SSD, and the harmonic oscillator that underlies everything near equilibrium.
Observables are operators, and pairs that do not commute cannot both be sharp — from which the uncertainty principle follows, and from which the size of every atom follows.
Hydrogen solves exactly, giving three quantum numbers, the orbital shapes, and the 2n^2 that becomes the periodic table.
Spin is a fourth quantum number with no classical analogue, and antisymmetry gives the exclusion principle, which is why matter is solid and why white dwarfs exist.
Entanglement is real and non-local, and Bell's theorem plus fifty years of experiment have ruled out local realism entirely — while forbidding any faster-than-light signal.
Nothing is ever still, and the vacuum has measurable effects.
And particles are excitations of fields, which is forced by combining quantum mechanics with relativity, and which predicted antimatter.
Two threads run out of this Part. The Standard Model of Part 8 is quantum field theory applied to everything that exists. And the failure to reconcile it with general relativity — Part 6's other great success — is the largest open problem in physics.
What the next Part fixes
Everything is now made of fields, and Part 8 asks how many there are. The answer is seventeen, they are arranged in a pattern nobody fully understands, and reading the pattern properly takes a whole Part: every quark, every lepton, every force carrier, with masses spanning eleven orders of magnitude for no known reason. Along the way: how the particle zoo of the 1950s was tamed, what the Higgs field actually does, why a neutrino can pass through a light year of lead, how Super-Kamiokande watches them anyway, and an honest account of what lies beyond — including what string theory is, what it has achieved, and what it has not.