---
title: Antiparticles and Hole Theory
module: Relativistic Wave Equations
moduleNumber: 5
lessonNumber: 3
order: 503
summary: >
  The negative-energy solutions of the Dirac equation refuse to go away, so they
  must mean something. Dirac read them as a filled sea of occupied negative-energy
  states whose holes are positive-energy antiparticles, predicting the positron
  before its discovery. The picture works for fermions but not bosons, and the
  Feynman-Stückelberg interpretation replaces it: an antiparticle is a
  negative-energy solution propagating backward in time, equivalent to a
  positive-energy antiparticle going forward. Crossing symmetry ties incoming
  particles to outgoing antiparticles in a single amplitude.
topics: [Relativistic Wave Equations]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 7, §7.3 (hole theory) and Ch. 2, §2.1 (antiparticles, the Feynman-Stückelberg picture)"
  - book: Thomson
    ref: "§4.7 (antiparticles and the negative-energy solutions)"
  - book: Tipler & Llewellyn
    ref: "§12-1 (the Dirac sea and the positron)"
---

The Dirac equation solved the Klein-Gordon density problem, but it did not remove
the negative-energy solutions — the spinors $v(p)$ from the previous lesson are as
much a part of the complete solution set as the positive-energy $u(p)$. An electron
sitting in a positive-energy state could in principle emit a photon and drop into a
negative-energy state, and then keep dropping, since the negative branch is
unbounded below. Ordinary matter would be catastrophically unstable. That this does
not happen means the negative-energy solutions are telling us something physical
rather than signalling a broken theory. Two readings extract the physics. Dirac's
hole theory came first and predicted the positron; the Feynman-Stückelberg
interpretation is cleaner, works for bosons too, and is the one the Feynman
calculus uses. This lesson develops both and ends with crossing symmetry, the
practical payoff.

Natural units $\hbar = c = 1$ throughout.

## The negative-energy solutions will not go away

Recall the four plane-wave solutions of the Dirac equation for each momentum: two
positive-energy spinors $u^{(1,2)}(p)\,e^{-iEt+\dots}$ with $E = +E_p$, and two
"negative-energy" spinors $v^{(1,2)}(p)\,e^{+iEt+\dots}$ with $E = -E_p$, where
$E_p = \sqrt{\vec p^{\,2} + m^2}$. Completeness forbids discarding either pair: an
arbitrary spinor field cannot be expanded in the positive-energy solutions alone,
so the theory is only consistent if the negative-energy states are kept and
interpreted. The problem is sharpest as a stability question. Nothing in the free
equation prevents an electron from cascading down through the negative-energy
continuum, releasing an unbounded amount of energy. Since atoms are stable, the
negative-energy states must somehow be inaccessible.

## Dirac's hole theory

Dirac's 1930 resolution uses the exclusion principle. Electrons are fermions, so no
two can occupy the same state. Suppose the vacuum is not empty but is the state in
which **every** negative-energy level is already filled — an infinite "sea" of
occupied negative-energy electrons. Then a positive-energy electron cannot fall
into the sea: every target state is taken. Stability is restored by declaring the
filled sea to be the vacuum, its infinite negative charge and energy unobservable
because they are the baseline against which everything else is measured.

The picture makes a prediction. Give one sea electron enough energy — at least
$2m$, since it must jump from $-E_p$ to $+E_p$ — and promote it to a positive-energy
state. Two things result: a real positive-energy electron, and a **hole** in the
sea. The hole is the absence of a negative-energy, negative-charge electron, so
relative to the filled vacuum it behaves as a particle of **positive** energy and
**positive** charge, with the same mass as the electron. That particle is the
**positron**. Hole theory thus predicts a positive electron and predicts that
supplying energy $\ge 2m$ to the vacuum creates an electron–positron pair — pair
production — while an electron meeting a hole falls in and both vanish, releasing
$\ge 2m$ of energy as photons — annihilation. Anderson's 1932 observation of the
positron in cosmic rays confirmed the prediction, one of the great successes of
theoretical physics.[^dirac-hole]

$$
% caption: Dirac's sea. Every negative-energy level below minus m is filled
% (shaded), so the exclusion principle blocks a positive-energy electron from
% falling in. Positive-energy levels above plus m are empty. The gap of width 2m
% between the two continua is the minimum energy to create a pair.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black!70] (0,-2.6) -- (0,2.9) node[above, font=\scriptsize] {energy};
  % positive continuum (empty) : draw empty level lines
  \foreach \y in {1.1,1.5,1.9,2.3}
    \draw[black] (0.5,\y) -- (3.5,\y);
  \node[black, right, font=\scriptsize] at (3.6,1.7) {empty levels};
  % negative continuum (filled) shaded band
  \fill[acc!14] (0.5,-2.5) rectangle (3.5,-1.1);
  \foreach \y in {-1.1,-1.5,-1.9,-2.3}
    \draw[acc, thick] (0.5,\y) -- (3.5,\y);
  \node[acc, right, font=\scriptsize] at (3.6,-1.7) {f\/illed sea};
  % gap markers at +-m
  \draw[black] (0.4,1.1) -- (0.6,1.1);
  \draw[black] (0.4,-1.1) -- (0.6,-1.1);
  \node[black, left, font=\scriptsize] at (0.35,1.1) {plus m};
  \node[black, left, font=\scriptsize] at (0.35,-1.1) {minus m};
  \draw[black, <->] (2.0,-1.1) -- (2.0,1.1);
  \node[black, right, font=\scriptsize] at (2.05,0) {gap 2m};
\end{tikzpicture}
$$

$$
% caption: Pair production in hole theory. A photon of energy at least 2m promotes
% one electron out of the filled negative-energy sea into a positive-energy state.
% The promoted electron is a real particle; the vacancy it leaves is a hole that
% behaves as a positron.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black!70] (0,-2.4) -- (0,2.7) node[above, font=\scriptsize] {energy};
  % filled sea with one vacancy
  \fill[black!8] (0.6,-2.3) rectangle (3.4,-1.0);
  \foreach \y in {-1.0,-1.4,-1.8,-2.2}
    \draw[thick] (0.6,\y) -- (3.4,\y);
  % the hole (open circle) at a sea level
  \draw[black, very thick, fill=white] (2.0,-1.4) circle (3.2pt);
  \node[black, right, font=\scriptsize] at (2.2,-1.4) {hole = positron};
  % promoted electron above
  \fill[acc] (2.0,1.4) circle (3.2pt);
  \node[acc, right, font=\scriptsize] at (2.2,1.4) {electron};
  % promotion arrow
  \draw[->, black, very thick] (1.2,-1.3) -- (1.2,1.3);
  \node[black, left, font=\scriptsize] at (1.15,0) {photon};
\end{tikzpicture}
$$

## Where hole theory breaks down

Hole theory is a triumph of intuition, but it is not the final word, and its
limits are worth stating plainly. The whole construction rests on the exclusion
principle: the sea is stable only because fermions cannot double-occupy a level.
For **bosons** — a spin-$0$ Klein-Gordon particle such as the pion, or the $W$ and
$Z$ — there is no exclusion principle, no filled sea is possible, and yet those
particles have antiparticles just as electrons do. So a mechanism that relies on
Fermi statistics cannot be the general explanation of antimatter. A second
discomfort is the infinite unobservable charge and energy of the sea, carried along
as excess baggage. A third is that the picture is inescapably many-body — the
"vacuum" is an infinite collection of particles — which sits awkwardly with the
one-particle wave equation it was meant to interpret. These are not fatal for the
electron, but they signal that a better reading exists.

## The Feynman-Stückelberg interpretation

Stückelberg and Feynman supplied it. Return to a single negative-energy solution
and look at its time dependence,

$$
\psi \sim e^{-iEt}
\quad\text{with}\quad
E = -E_p < 0,
\qquad\text{so}\qquad
\psi \sim e^{+iE_p t}.
$$

A negative-energy state evolving _forward_ in time carries the phase $e^{+iE_p t}$.
But that coincides with the phase of a _positive_-energy state evolving _backward_ in
time, $e^{-iE_p(-t)}$. The Feynman-Stückelberg interpretation takes this
mathematical identity as physics: **a negative-energy particle moving backward in
time is equivalent to a positive-energy antiparticle moving forward in time.** The
electric charge flips too — running the film backward reverses the direction of
charge flow — so a negative-energy electron running backward is a positive-energy
positron running forward. No sea, no exclusion principle, no infinities. The
interpretation applies to bosons and fermions alike, supplying the
generality hole theory lacked.

This is why antiparticle lines in Feynman diagrams are drawn with their arrows
pointing **against** the time direction: the arrow tracks the flow of (negative)
charge / fermion number, not the flow of time, and for an antiparticle the two
oppose. A process in which an electron is absorbed and a positron emitted is the
_same_ line — a single electron line — that happens to reverse its sense in time at
a vertex.

$$
% caption: The Feynman-Stuckelberg picture. Time runs upward. A positive-energy
% electron enters, and at the vertex the line reverses its time sense: read
% downward it is a negative-energy electron going backward in time, read upward it
% is a positive-energy positron going forward. The arrow follows charge, not time,
% so the antiparticle arrow points down.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % time axis
  \draw[->, black] (-3.2,-2.2) -- (-3.2,2.4) node[above, font=\scriptsize] {time};
  % vertex
  \fill[black] (0,0) circle (2pt);
  % electron incoming from lower left, arrow up toward vertex
  \draw[acc, very thick, ->] (-1.8,-1.8) -- (-0.9,-0.9);
  \draw[acc, very thick] (-0.9,-0.9) -- (0,0);
  \node[acc, left, font=\scriptsize] at (-1.8,-1.8) {electron};
  % antiparticle line leaving vertex to lower right, arrow pointing DOWN (against time)
  \draw[acc, very thick, ->] (0,0) -- (0.9,-0.9);
  \draw[acc, very thick] (0.9,-0.9) -- (1.8,-1.8);
  \node[acc, right, font=\scriptsize] at (1.8,-1.8) {positron};
  % emitted photon up (wavy simplified as dashed)
  \draw[black, very thick, dashed] (0,0) -- (1.4,1.6);
  \node[black, right, font=\scriptsize] at (1.4,1.6) {photon};
  \node[black, above, font=\scriptsize] at (-1.0,0.3) {one line, arrow follows charge};
\end{tikzpicture}
$$

## Antiparticle spinors

The two readings agree on the spinor bookkeeping. The antiparticle is described by
the $v(p)$ spinors, which solve $(\gamma^\mu p_\mu + m)\,v(p) = 0$ with $p$ now the
**physical** (positive-energy) four-momentum of the antiparticle. Writing the
negative-energy solution with the substitution $E \to -E$, $\vec p \to -\vec p$
converts the awkward $v(p)\,e^{+ip\cdot x}$ into a positive-energy antiparticle
wave $v(p)\,e^{-i p\cdot x}$ with the sign flips of the Feynman-Stückelberg rule
built in. The two $v$ spinors carry the antiparticle's two spin states, so the
counting of the previous lesson is complete and physical:

- $u^{(1)}(p), u^{(2)}(p)$ — particle, spin up and spin down, energy $+E_p$;
- $v^{(1)}(p), v^{(2)}(p)$ — antiparticle, spin up and spin down, energy $+E_p$.

Four solutions, all of positive physical energy, two particle and two antiparticle.
The negative-energy branch of the Dirac equation is not discarded and not swept
under a sea — it _is_ the antiparticle, expressed in the variables an experimenter
measures.

## Crossing symmetry

The interpretation has an immediate practical consequence for amplitudes. If an
incoming particle and an outgoing antiparticle are the same line traversed in
opposite time senses, then a process with an incoming particle of momentum $p$ and
the process obtained by **crossing** that leg to an outgoing antiparticle of
momentum $-p$ are described by the _same_ amplitude, analytically continued in the
momenta. Schematically,

$$
\mathcal M\bigl(A + B \to C + D\bigr)
\quad\text{and}\quad
\mathcal M\bigl(A + \bar D \to C + \bar B\bigr)
$$

are one function, related by sending a particle's four-momentum to minus the
antiparticle's. This is **crossing symmetry**. It means the amplitudes for a whole
family of related reactions — for example electron–muon scattering, electron–
positron annihilation to a muon pair, and their relatives — need not be computed
independently; one master amplitude covers them all, and the Mandelstam variables
$s, t, u$ introduced in the kinematics module label which physical region each
crossed process lives in. The antiparticle interpretation, born from an apparent
defect of the Dirac equation, thus becomes a labor-saving symmetry at the heart of
the Feynman calculus.

$$
% caption: Crossing. The left diagram has an incoming particle on the lower leg;
% crossing that leg turns it into an outgoing antiparticle (right diagram) with
% four-momentum reversed. Both diagrams are the same amplitude continued to a
% different physical region — the practical face of the antiparticle
% interpretation.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % LEFT diagram
  \begin{scope}[shift={(-3.4,0)}]
    \fill (0,0) circle (1.8pt);
    \draw[acc, very thick, ->] (-1.4,-1.4) -- (-0.7,-0.7);
    \draw[acc, very thick] (-0.7,-0.7) -- (0,0);
    \node[acc, left, font=\scriptsize] at (-1.4,-1.4) {particle in};
    \draw[very thick, ->] (0,0) -- (0.7,0.7);
    \draw[very thick] (0.7,0.7) -- (1.4,1.4);
    \node[right, font=\scriptsize] at (1.4,1.4) {out};
    \node[below, font=\scriptsize] at (0,-1.8) {a reaction};
  \end{scope}
  % arrow between
  \draw[->, black, thick] (-1.3,0) -- (0.1,0);
  \node[black, above, font=\scriptsize] at (-0.6,0.05) {cross};
  % RIGHT diagram
  \begin{scope}[shift={(3.0,0)}]
    \fill (0,0) circle (1.8pt);
    \draw[acc, very thick, ->] (0,0) -- (-0.7,-0.7);
    \draw[acc, very thick] (-0.7,-0.7) -- (-1.4,-1.4);
    \node[acc, left, font=\scriptsize] at (-1.4,-1.4) {antiparticle out};
    \draw[very thick, ->] (0,0) -- (0.7,0.7);
    \draw[very thick] (0.7,0.7) -- (1.4,1.4);
    \node[right, font=\scriptsize] at (1.4,1.4) {out};
    \node[below, font=\scriptsize] at (0,-1.8) {crossed reaction};
  \end{scope}
\end{tikzpicture}
$$

## Summary

The negative-energy solutions of the Dirac equation are not a flaw to be removed
but antimatter waiting to be recognized. Dirac's hole theory fills every
negative-energy level, makes the sea the vacuum, and reads a hole as a
positive-energy positron — a picture that predicted the positron and explains pair
production and annihilation, but that relies on Fermi statistics and so fails for
bosons. The Feynman-Stückelberg interpretation is general: a negative-energy
solution running backward in time _is_ a positive-energy antiparticle running
forward, which is why antiparticle lines carry reversed arrows. In practice the
antiparticle is the $v(p)$ spinor with physical positive energy, giving four
positive-energy solutions per momentum — particle and antiparticle, each with two
spins — and crossing symmetry lets a single amplitude serve a whole family of
reactions. With relativistic wave equations, spin, and antiparticles in place, the
next module turns these ingredients into the Feynman rules of quantum
electrodynamics.

[^dirac-hole]: Griffiths, _Introduction to Elementary Particles_, 2nd ed., §7.3 and §2.1, gives hole theory, its shortcomings, and the Feynman-Stückelberg picture; Thomson, _Modern Particle Physics_, §4.7, treats the negative-energy solutions and the antiparticle spinors; Tipler & Llewellyn, §12-1, recounts the Dirac sea and Anderson's positron. The positron was first reported by C. D. Anderson, _Physical Review_ **43**, 491 (1933); masses and the $2m$ pair-production threshold follow the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).