Antiparticles and the negative-energy sea

At the close of last section, the Dirac equation had won the probability and lost the spectrum: every momentum carries a negative-energy solution $E=-\sqrt{p^2c^2+m^2c^4}$, with levels running from $-mc^2$ all the way down to negative infinity. This section tells the story of the boldest patch in the history of physics — the patch itself was later dismantled, but what it predicted stayed, and collected a Nobel Prize.

How concrete the disaster is

First compute the consequences of a “bottomless spectrum”, to appreciate how lethal the problem is.

In a quantum system, excited states decay to lower levels and emit photons — hydrogen’s $2p\to1s$ takes only about 1.6 ns. Now the Dirac equation says: there are levels below $1s$, below $-mc^2$, and no lowest rung. Then nothing stops the cascade:

$$e^-(1s)\;\to\;e^-(E<-mc^2)+\gamma\;\to\;e^-(E\ll-mc^2)+\gamma+\gamma\;\to\;\cdots\tag{12.4.1}$$

Each rung down releases another photon; the energy falls ever more negative, the photons multiply without end. Estimates put the rate of this “downward radiation” at nanosecond scale — no stable atom should exist anywhere in the universe, and you and I should at this moment be a puff of gamma rays. This is worse than KG’s negative probability: negative probability merely defied interpretation, while a bottomless spectrum collides head-on with the observed fact that things exist.

Can we just delete the negative-energy solutions? No — same reason as last section: incompleteness. Worse, in an external field the transition matrix elements between positive- and negative-energy states are non-zero — the dynamics itself will ferry particles into the states you deleted.

Dirac turns the tables: fill the vacuum

In 1930 Dirac proposed a scheme that still reads as breathtakingly audacious.

The pair-creation threshold: why 1.022 MeVbasic~6 minexpand

The minimum energy. Lifting a sea electron from the highest negative level ($-mc^2$, at $p=0$) to the lowest positive level ($+mc^2$) requires at least

$$E_{\text{min}}=mc^2-(-mc^2)=2mc^2 =2\times0.511\ \text{MeV}=1.022\ \text{MeV}\tag{12.4.4}$$

This is the energy threshold for a gamma photon to create an electron-positron pair, matched by experiment to the last decimal. (Strictly, a single free photon creating a pair also needs a nucleus nearby to carry off the recoil momentum, or energy and momentum conservation cannot both be satisfied — the threshold value itself is unchanged.)

The ledger in reverse. A positron and electron at rest annihilate into two photons, each carrying away

$$E_\gamma=mc^2=511\ \text{keV}\tag{12.4.5}$$

The two photons fly out back to back (momentum conservation). This 511 keV line is the positron’s “fingerprint” in astronomy and medicine: a hospital PET scan (positron emission tomography) injects the patient with a positron-emitting tracer, the detector ring catches each back-to-back pair of 511 keV photons, and the annihilation point is reconstructed along the line between them — that is how the metabolic map of a living body is drawn. Antimatter is no science-fiction prop; it clocks in at the hospital every day.

From paper to cloud chamber

The birth of this prediction was not dignified. Dirac at first did not dare invent a new particle and proposed that the hole “is the proton” — after all, the proton was then the only known positive particle. The idea was swiftly executed by mathematics: Weyl proved the hole’s mass must exactly equal the electron’s, and the proton is 1836 times heavier; besides, “the electron in a hydrogen atom falling into its own proton-hole” would mean hydrogen atoms annihilate spontaneously. In 1931 Dirac accepted his fate and wrote it down in black and white: there should exist a new particle “as yet unobserved by experimental physics, with the same mass as the electron and opposite charge”.

In 1932, Carl Anderson at Caltech saw a track in a cosmic-ray cloud-chamber photograph: its curvature in the magnetic field showed positive charge, but the ionisation density and radius of curvature showed a mass far below the proton’s — consistent with an electron. To pin down the particle’s direction of flight he placed a 6 mm lead plate across the middle of the chamber: after crossing the plate the particle lost energy and curved more sharply, fixing the direction — and with it the sign of the charge, beyond dispute. The positron had arrived — the first time humanity found a particle in an equation before finding it in nature. Dirac received the Nobel Prize in 1933, Anderson in 1936.

Dismantling the scaffolding: the modern view

The Dirac sea did great service, but it is scaffolding, not the building. Three hard flaws:

1. An infinite background. The filled sea carries infinite negative energy and infinite charge density, waved away with “a uniform background cannot be observed”; gravity does not accept the excuse — energy gravitates however uniform it is.
2. What about bosons? The KG equation has negative-energy solutions too, but spin-0 particles do not obey Pauli exclusion, so “fill the sea” fails in principle. Yet pions have antiparticles all the same — clearly the explanation missed the root.
3. Asymmetric bookkeeping. The electron is a “particle” and the positron a “hole”, yet in experiment the two are perfectly symmetric in status.

∎Key formulas

• The negative-energy catastrophe

  $$E=-\sqrt{p^2c^2+m^2c^4}\;\Rightarrow\;\text{spectrum unbounded below}$$

  Left untreated, atoms cascade downward on nanosecond timescales

• The Dirac sea

  $$\ket{\text{vacuum}}=\prod_{E<0}\ket{\text{occupied}}$$

  Pauli exclusion blocks the downward channel; works only for fermions

• Hole = antiparticle

  $$\text{missing}(-|E|,\,-e)\;\equiv\;\text{surplus}(+|E|,\,+e)$$

  Mass exactly equal to the electron’s — ruling out "the hole is the proton"

• Pair creation and annihilation

  $$E_\gamma\geq2mc^2=1.022\ \text{MeV},\qquad e^+e^-\to2\gamma\,(511\ \text{keV})$$

  511 keV back-to-back photons: the physics behind PET imaging

?Self-check4 questions

1. 1.

   If the negative-energy solutions are left untreated, the direct disaster Dirac theory predicts is:

   • Positive-energy electrons radiate photons and fall rung by rung into the bottomless negative spectrum; atoms collapse on nanosecond timescales
   • The probability density becomes negative
   • Electrons would exceed the speed of light
   • The wavefunction cannot be normalised

   Submit
2. 2.

   In the Dirac-sea scheme, the mechanism preventing electrons from falling into negative levels is:

   • The Pauli exclusion principle: all negative levels are already filled, and fermions cannot double-occupy
   • There are no transition matrix elements between negative and positive levels
   • Energy conservation forbids the transition
   • Coulomb repulsion between the negative-energy electrons

   Submit
3. 3.

   Which statements about the prediction and discovery of the positron are correct? (Select all that apply.)

   Select all that apply

   • Dirac initially proposed the hole was the proton, vetoed by "the hole’s mass must equal the electron’s"
   • Anderson used a lead plate to slow the particle and determine its direction of flight, thereby fixing the sign of its charge
   • The positron was discovered experimentally first and explained by theory afterwards
   • A positron-electron pair at rest annihilates into two back-to-back 511 keV photons — the basis of PET imaging

   Submit
4. 4.

   The energy threshold for a photon to create an electron-positron pair is 2mc². Taking the electron’s mc²=0.511 MeV, find the threshold energy (MeV).

   MeV1% relative tolerance

   Submit

What comes next

Spin emerged from the algebra (12.3), antiparticles from the negative-energy solutions (this section) — and the Dirac equation is not done giving gifts. The last one is also the most precisely verified by experiment: the electron’s magnetic moment. Put the Dirac equation in a magnetic field, take the non-relativistic limit, and $g=2$ drops out automatically; expand one order further and the fine-structure trio hand-stitched in section 6.6 (relativistic correction, spin-orbit coupling, the Darwin term) appears in one stroke, all by itself.