The Dirac Equation

Contents

Construction of the equation
Lorentz covariance and the spinor
Spin as a consequence, not an assumption
The hydrogen atom solved exactly
Negative energies, holes and the positron
Phenomena of the one-particle theory
The systematic nonrelativistic expansion
The massless limit and the boundary of the theory

The Klein–Gordon equation of The Klein–Gordon Equation fails as a one-particle theory because it is second order in time. Dirac asked in 1928 what a first-order relativistic wave equation would have to look like, and found that the coefficients multiplying the derivatives cannot be numbers: they must be matrices satisfying an anticommutation relation, and the smallest such matrices are $4\times4$ [Dirac:1928] [Dirac:1928b]. Everything else follows from that one algebraic demand, and what follows is extraordinary. The wavefunction acquires four components; two of them describe an internal angular momentum of magnitude $\hbar/2$ with a gyromagnetic ratio $g=2$, so that electron spin — inferred from spectra by Uhlenbeck and Goudsmit [Uhlenbeck:1925] and grafted onto the Schrödinger equation by hand by Pauli [Pauli:1927b] — ceases to be an assumption and becomes a theorem. The hydrogen spectrum comes out exactly, fine structure included [Darwin:1928] [Gordon:1928a], where the spinless equation had failed. And the other two components, carrying negative energy, forced Dirac to predict a particle of the electron's mass and opposite charge [Dirac:1930a] [Dirac:1931] — found four years later in a cloud chamber (Experiment: The Positron).

This chapter states the equation and its consequences as relativistic quantum mechanics, that is, for a single particle in an external field, which is the domain in which the results above are correct. It also marks the boundary of that domain honestly: hole theory needs an infinite filled sea and does not survive (Canonical Quantization of Fields); the Klein paradox [Klein:1929b] and the Zitterbewegung [Schroedinger:1930] are one-particle artefacts of a theory that cannot conserve particle number; and the exact Coulomb spectrum is falsified at the ninth significant figure by the Lamb shift [Lamb:1947] and the anomalous moment [Kusch:1948], both of which require the quantized field of Part XI — Quantum Field Theory and the Standard Model. The group-theoretic reading — spin $\tfrac{1}{2}$ as the $(\tfrac{1}{2},0)\oplus(0,\tfrac{1}{2})$ representation of the Lorentz group — is developed in Particles as Poincaré Representations. Standard treatments are [Bjorken:1964] [Dirac:1958] [Weinberg:1995] [Sakurai:2017].

Derivation pending.

The Dirac Equation: all derivations of this chapter are pending.

Construction of the equation

Dirac's requirement: first order in time

[Reserved: the argument of Dirac's 1928 paper [Dirac:1928] — a probability density free of time derivatives requires an equation first order in $\pp_{t}$, and Lorentz covariance then requires it to be first order in $\pp_{i}$ as well; the ansatz $\ii\hbar\pp_{t}\psi=\left(c\,\vect{\alpha}\cdot\hat{\vect{p}} +\beta mc^{2}\right)\psi$, and the demand that each component satisfy the Klein–Gordon equation of The Klein–Gordon Equation, which fixes $\alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=2\delta_{ij}$, $\alpha_{i}\beta+\beta\alpha_{i}=0$ and $\beta^{2}=\identity$; why no numbers satisfy these relations and matrices of dimension below four do not either.]

The Clifford algebra of the gamma matrices

[Reserved: the covariant form $\left(\ii\hbar\gamma^{\mu}\pp_{\mu}-mc\right)\psi=0$ and the defining relation $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu} =2\eta^{\mu\nu}\identity$, an instance of the algebra introduced by Clifford [Clifford:1878] and developed as an algebraic structure in Algebraic Structures; the trace identities, the completeness of the sixteen products $\set{\identity,\gamma^{\mu},\sigma^{\mu\nu}, \gamma^{\mu}\gamma_{5},\gamma_{5}}$, and $\gamma_{5}$ as the chirality element; the dimension count that forces $4\times4$ in $3+1$ dimensions.]

Representations and the choice of basis

[Reserved: Pauli's theorem, that any two irreducible sets of gamma matrices are related by a similarity transformation unique up to a factor, so nothing physical depends on the choice; the Dirac (standard) basis in which $\beta$ is diagonal, adapted to the nonrelativistic limit of Section 93.7; the Weyl (chiral) basis in which $\gamma_{5}$ is diagonal, adapted to the massless limit of Section 93.8; the Majorana basis in which all gammas are purely imaginary and the equation admits real solutions [Majorana:1937].]

The conserved positive-definite current

[Reserved: the adjoint $\bar{\psi}=\psi^{\dagger}\gamma^{0}$, the current $j^{\mu}=c\,\bar{\psi}\gamma^{\mu}\psi$, its conservation $\pp_{\mu}j^{\mu}=0$, and the density $j^{0}/c=\psi^{\dagger}\psi\geq0$ — the property the Klein–Gordon current lacked (The Klein–Gordon Equation) and the one Dirac was hunting [Dirac:1928]; the honest qualification that positivity of the density does not rescue the one-particle interpretation, because the energy spectrum is still unbounded below (Section 93.5).]

Lorentz covariance and the spinor

The spinor transformation law

[Reserved: covariance requires $\psi'(x')=S(\Lambda)\psi(x)$ with $S^{-1}\gamma^{\mu}S=\Lambda^{\mu}{}_{\nu}\gamma^{\nu}$; the explicit generators $\sigma^{\mu\nu}=\tfrac{\ii}{2} \comm{\gamma^{\mu}}{\gamma^{\nu}}$ and the finite transformation $S=\exp\left(-\tfrac{\ii}{4}\omega_{\mu\nu}\sigma^{\mu\nu}\right)$; that $S$ is unitary for rotations and Hermitian for boosts, so the representation is not unitary — the group being non-compact (Particles as Poincaré Representations); the $4\pi$ periodicity of a spinor under rotation and its experimental confirmation by neutron interferometry [Werner:1975], which is why the covering group $\SL(2,\C)$ and not the Lorentz group itself acts.]

Phenomenon 93.1 (A spinor changes sign under a full rotation).

Rotating a spin-$\tfrac{1}{2}$ particle through $2\pi$ multiplies its state by $-1$; only a rotation through $4\pi$ returns it to itself. The sign is not a convention that could be redefined away: in a neutron interferometer, one arm of which passes through a magnetic field that precesses the spin, the interference intensity varies with the precession angle at a period of $4\pi$ and not $2\pi$, so the amplitude demonstrably reverses at a single full turn [Werner:1975].

Derivation. In the spinor representation of Section 93.2.1 the generator of rotations about the $z$-axis is $\tfrac{1}{2}\Sigma_{3}$, with $\Sigma_{k}=\diag\left(\sigma_{k},\sigma_{k}\right)$ built from the Pauli matrices, so a rotation through an angle $\theta$ acts on the spinor as

\begin{equation}\tag{93.1} S(\theta)=\exp\left(-\frac{\ii}{2}\theta\Sigma_{3}\right) =\identity\cos\frac{\theta}{2} -\ii\Sigma_{3}\sin\frac{\theta}{2}\ec \end{equation}

the exponential series collapsing because $\Sigma_{3}^{2}=\identity$. At $\theta=2\pi$ this is $-\identity$ and at $\theta=4\pi$ it is $+\identity$. The half-angle is the whole content: $\SO(3)$ is not simply connected, its closed loops falling into two classes, and the spinor representation is a representation not of $\SO(3)$ but of its double cover $\SU(2)$, in which the lift of a single full turn is an open path ending at $-\identity$. An overall sign is unobservable on a state taken alone, which is why the effect can be seen only by interfering a rotated amplitude against an unrotated one.

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Bilinear covariants

[Reserved: the sixteen bilinears $\bar{\psi}\Gamma\psi$ and their transformation types — scalar, vector, antisymmetric tensor, axial vector and pseudoscalar; their role as the only Lorentz-covariant objects a spinor field can build, hence as the complete list of possible interaction terms, which is what makes the $V-A$ structure of the weak interaction a choice among five rather than a free invention (Weak Interactions and Experiment: Parity Violation); the Fierz rearrangement identities relating different pairings [Fierz:1937].]

Plane-wave solutions and spin projectors

[Reserved: the four independent plane waves at each momentum, two with $p^{0}>0$ and two with $p^{0}<0$; the spinors $u(p,s)$ and $v(p,s)$, their normalization and orthogonality, the energy projectors $\left(\pm\gamma^{\mu}p_{\mu}+mc\right)/2mc$ and the spin projector from the Pauli–Lubanski vector of Particles as Poincaré Representations; the completeness relations that make every later trace calculation mechanical.]

Spin as a consequence, not an assumption

This section is the chapter's principal result: an equation written to be first order in time turns out to describe a particle with intrinsic angular momentum $\hbar/2$ and the right magnetic moment, neither of which was put in.

Orbital angular momentum is not conserved

[Reserved: the computation $\comm{\hat{\vect{L}}}{\hat{H}}\neq0$ for the Dirac Hamiltonian, and the identification of the operator $\hat{\vect{S}}=\tfrac{\hbar}{2}\vect{\Sigma}$ whose sum with $\hat{\vect{L}}$ is conserved [Dirac:1928]; the eigenvalues $\pm\hbar/2$, matching the two beams of the Stern–Gerlach experiment (Experiment: Stern–Gerlach) and the algebra of Angular Momentum and Spin; the contrast with the spinless equation, where $\hat{\vect{L}}$ alone is conserved.]

Phenomenon 93.2 (Electron spin).

A beam of neutral silver atoms, whose single valence electron is in an $S$ state carrying no orbital angular momentum, splits into exactly two components on traversing an inhomogeneous magnetic field — not into the odd number of components that any integer angular momentum would give, and not into the continuum a classical moment would give [Stern:1921] [Gerlach:1922a] [Gerlach:1922b]. The electron therefore carries an intrinsic angular momentum with just two projections, $\pm\hbar/2$, on any axis; the measurement is Experiment: Stern–Gerlach. In any central field it is the total $\hat{\vect{J}}=\hat{\vect{L}}+\hat{\vect{S}}$, and not $\hat{\vect{L}}$, that is found to be conserved.

Derivation. Both halves are theorems about the Hamiltonian $\hat{H}=c\,\vect{\alpha}\cdot\hat{\vect{p}}+\beta m_{e}c^{2}$ of Section 93.1.1. Using $\comm{\hat{L}_{i}}{\hat{p}_{j}}=\ii\hbar\epsilon_{ijk}\hat{p}_{k}$, and noting that $\beta$ and $\vect{\alpha}$ are constant matrices that commute with $\hat{\vect{L}}$,

\[ \comm{\hat{L}_{i}}{\hat{H}} =c\,\alpha_{j}\comm{\hat{L}_{i}}{\hat{p}_{j}} =\ii\hbar c\,\epsilon_{ijk}\alpha_{j}\hat{p}_{k} =\ii\hbar c \left(\vect{\alpha}\times\hat{\vect{p}}\right)_{i}\neq0\ec \]

so orbital angular momentum is not conserved even for a free particle — an absurdity unless something else carries the balance. Set $\hat{\vect{S}}=\tfrac{\hbar}{2}\vect{\Sigma}$ with $\Sigma_{k}=\diag\left(\sigma_{k},\sigma_{k}\right)$. Since $\comm{\Sigma_{i}}{\alpha_{j}}=2\ii\epsilon_{ijk}\alpha_{k}$ and $\comm{\Sigma_{i}}{\beta}=0$,

\[ \comm{\hat{S}_{i}}{\hat{H}} =\frac{\hbar c}{2}\comm{\Sigma_{i}}{\alpha_{j}}\hat{p}_{j} =\ii\hbar c\,\epsilon_{ijk}\alpha_{k}\hat{p}_{j} =-\ii\hbar c \left(\vect{\alpha}\times\hat{\vect{p}}\right)_{i}\ec \]

which cancels the previous line exactly, so $\comm{\hat{\vect{L}}+\hat{\vect{S}}}{\hat{H}}=0$. The operator $\hat{S}_{z}$ has eigenvalues $\pm\hbar/2$ and no others, because $\Sigma_{3}^{2}=\identity$. Nothing about spin was assumed anywhere: it was forced by the demand that the equation be first order in time, which is what made the coefficients matrices in the first place, and the two Stern–Gerlach beams are the observed count of its eigenvalues.

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The gyromagnetic ratio $g=2$

[Reserved: minimal coupling $\pp_{\mu}\mapsto\pp_{\mu}+\tfrac{\ii e}{\hbar}A_{\mu}$, squaring the Dirac operator, and the appearance of the term $-\tfrac{e\hbar}{2mc}\,\sigma^{\mu\nu}F_{\mu\nu}$ that carries a magnetic moment $g=2$ in units of the Bohr magneton [Dirac:1928] — twice the orbital value, exactly the factor Uhlenbeck and Goudsmit had to postulate [Uhlenbeck:1925] and Pauli had to insert [Pauli:1927b]; the measured departure $g-2\approx2.3\times 10^{-3}$ [Kusch:1948] and its computation from field theory [Schwinger:1948], deferred to Experiment: The Electron Anomalous Magnetic Moment.]

Phenomenon 93.3 (The electron magnetic moment).

The magnetic moment associated with the electron's spin is twice as large, per unit of angular momentum, as the moment associated with its orbital motion:

\begin{equation}\tag{93.2} \vect{\mu}=-g\,\frac{\mu_{\mathrm{B}}}{\hbar}\hat{\vect{S}}\ec \qquad \mu_{\mathrm{B}}=\frac{e\hbar}{2m_{e}}\ec \qquad g\simeq2\ec \end{equation}

where an orbital current of the same angular momentum would give $g=1$. This factor is what the Zeeman splittings of the alkali doublets demand, and it had to be inserted by hand into the prerelativistic theory. Its measured magnitude is $2.00231930436$ [Mohr:2025] — equal to 2 to eleven significant figures, the residual departure of about one part in $10^{3}$ being the anomaly of Experiment: The Electron Anomalous Magnetic Moment.

Derivation. Couple the electron minimally to an external field, writing $\hat{\vect{\pi}}=\hat{\vect{p}}-q\vect{A}$ with $q=-e$, and use $\left(\vect{\sigma}\cdot\vect{a}\right) \left(\vect{\sigma}\cdot\vect{b}\right) =\vect{a}\cdot\vect{b} +\ii\vect{\sigma}\cdot\left(\vect{a}\times\vect{b}\right)$ with $\vect{a}=\vect{b}=\hat{\vect{\pi}}$. The cross term does not vanish, because the components of $\hat{\vect{\pi}}$ fail to commute:

\[ \comm{\hat{\pi}_{j}}{\hat{\pi}_{k}} =-q\left(\comm{\hat{p}_{j}}{A_{k}} +\comm{A_{j}}{\hat{p}_{k}}\right) =\ii\hbar q\left(\pp_{j}A_{k}-\pp_{k}A_{j}\right) =\ii\hbar q\,\epsilon_{jkl}B_{l}\ec \]

so that $\left(\hat{\vect{\pi}}\times\hat{\vect{\pi}}\right)_{i} =\tfrac{1}{2}\epsilon_{ijk}\comm{\hat{\pi}_{j}}{\hat{\pi}_{k}} =\ii\hbar q B_{i}$ and hence

\begin{equation}\tag{93.3} \left(\vect{\sigma}\cdot\hat{\vect{\pi}}\right)^{2} =\hat{\vect{\pi}}^{2}-q\hbar\,\vect{\sigma}\cdot\vect{B}\ep \end{equation}

Eliminating the lower components of the spinor at leading order in $1/m_{e}c^{2}$ (Section 93.3.3) leaves the upper two-component amplitude obeying $\ii\hbar\pp_{t}\chi=\left[\left(\vect{\sigma}\cdot \hat{\vect{\pi}}\right)^{2}/2m_{e}+q\phi\right]\chi$, and inserting Equation (93.3) produces the interaction $-\left(q\hbar/2m_{e}\right)\vect{\sigma}\cdot\vect{B} =-\vect{\mu}\cdot\vect{B}$ with

\[ \vect{\mu}=\frac{q\hbar}{2m_{e}}\vect{\sigma} =-\frac{e}{m_{e}}\hat{\vect{S}} =-2\,\frac{\mu_{\mathrm{B}}}{\hbar}\hat{\vect{S}}\ec \]

using $q=-e$ and $\hat{\vect{S}}=\tfrac{\hbar}{2}\vect{\sigma}$: this is Equation (93.2) with $g=2$. The same substitution applied to the orbital term gives $-\left(q/2m_{e}\right) \hat{\vect{L}}\cdot\vect{B}$, that is $g=1$ for orbital motion. The factor of two is produced entirely by the algebra of $\left(\vect{\sigma}\cdot\vect{a}\right)^{2}$; it is not fitted, and its exactness at this order is what makes the measured departure from 2 informative rather than merely a correction.

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The Pauli equation as the nonrelativistic limit

[Reserved: elimination of the small components at leading order, yielding the Pauli equation $\ii\hbar\pp_{t}\chi=\left[\tfrac{1}{2m} \left(\hat{\vect{p}}-\tfrac{e}{c}\vect{A}\right)^{2} -\tfrac{e\hbar}{2mc}\vect{\sigma}\cdot\vect{B}+e\phi\right]\chi$ [Pauli:1927b], previously an empirical two-component equation and now a derived limit; the identification of the Pauli matrices with $\vect{\Sigma}$ restricted to the upper components; the systematic continuation of this expansion in Section 93.7.1.]

The hydrogen atom solved exactly

The Dirac–Coulomb spectrum

[Reserved: separation of the equation in a Coulomb field and the closed-form spectrum obtained independently by Darwin [Darwin:1928] and by Gordon [Gordon:1928a], depending on $n$ and on the total angular momentum $j$ only; its expansion in powers of $Z\alpha$, reproducing the Bohr levels of Atomic Models and Spectra at leading order and Sommerfeld's fine-structure formula [Sommerfeld:1916] at the next — obtained by Sommerfeld from a relativistic orbit calculation with no spin at all, one of the most instructive coincidences in physics, which the chapter should state and explain rather than repeat; the breakdown at $Z\alpha>1$ and what it signals.]

Phenomenon 93.4 (Fine structure).

The spectral lines of atomic hydrogen are not single. Each level of principal quantum number $n$ is resolved into components separated by intervals of relative order $\alpha^{2}\approx5.3\times 10^{-5}$, and the pattern is governed by the total angular momentum $j$ alone: two states sharing $n$ and $j$ but differing in orbital $\ell$ — $2s_{1/2}$ and $2p_{1/2}$, for instance — are observed to coincide. The splittings were fitted, in advance of any notion of spin, by Sommerfeld's relativistic orbit formula [Sommerfeld:1916], and the Dirac equation reproduces the same expression in closed form,

\begin{equation}\tag{93.4} E_{nj}=m_{e}c^{2}\left[1+\frac{\left(Z\alpha\right)^{2}} {\left(n-\left(j+\tfrac{1}{2}\right) +\sqrt{\left(j+\tfrac{1}{2}\right)^{2} -\left(Z\alpha\right)^{2}}\right)^{2}}\right]^{-1/2}\ec \end{equation}

now as a consequence of the spin it does not assume.

Derivation pending.

Separation of the Dirac equation in a Coulomb potential and solution of the resulting radial system in closed form, together with its expansion in powers of the nuclear charge times the fine-structure constant, which must reproduce the Bohr levels at leading order and the fine-structure intervals at the next.

Fine structure: spin–orbit, Darwin and mass terms

[Reserved: the three order-$\alpha^{4}$ terms separated by the Foldy–Wouthuysen expansion — the relativistic kinetic correction, the spin–orbit coupling with the factor $\tfrac{1}{2}$ that Thomas supplied kinematically [Thomas:1926], and the Darwin term [Darwin:1928] that acts only on $s$ states and reflects the smearing of the electron over a Compton wavelength; the resulting degeneracy of states with equal $j$, e.g. $2s_{1/2}$ and $2p_{1/2}$, which is a prediction of the exact spectrum and is the one the next subsection reports as false; comparison with the perturbation-theory route of Approximation Methods.]

Where the exact spectrum fails: the Lamb shift

[Reserved: Lamb and Retherford's microwave measurement of a splitting of about $1058\,\mathrm{MHz}$ between $2s_{1/2}$ and $2p_{1/2}$ [Lamb:1947], states the Dirac equation makes exactly degenerate; the shift as the first quantitative evidence that the electron interacts with the quantized electromagnetic field, and therefore as the experiment that ended relativistic quantum mechanics as a fundamental theory; its calculation belongs to Quantum Electrodynamics and Renormalization and the modern spectroscopy to Experiment: Precision Spectroscopy and Atomic Clocks. This subsection must state the falsification plainly: the chapter's best result is wrong at the level current experiments resolve easily.]

Phenomenon 93.5 (The Dirac–Coulomb degeneracy is broken).

The states $2s_{1/2}$ and $2p_{1/2}$ of hydrogen, which Equation (93.4) makes exactly degenerate because they share both $n$ and $j$, are observed to be split, the $s$ state lying higher [Lamb:1947] [Bezginov:2019]. The interval is driven directly as a microwave resonance between the two levels, so it is measured rather than inferred from a difference of optical frequencies, and it exceeds every uncertainty of that measurement by orders of magnitude. The disagreement with Phenomenon 93.4 is therefore not marginal: the exact spectrum of this chapter is false. The measurements are Experiment: Precision Spectroscopy and Atomic Clocks.

Derivation pending.

The radiative level shift between the two lowest hydrogen states of equal total angular momentum. Deriving it requires the quantized electromagnetic field, the electron self-energy and the vacuum polarization, and a renormalization prescription, so it lies outside relativistic quantum mechanics altogether and belongs to the quantum-electrodynamics chapter of Part XII.

Negative energies, holes and the positron

Hole theory

[Reserved: the negative-energy solutions of Section 93.2.3 and the catastrophe they imply — an electron radiating indefinitely into ever more negative states; Dirac's proposal that all such states are filled and that the exclusion principle of Identical Particles forbids the transition [Dirac:1930a]; a hole in the sea behaving as a particle of positive energy and opposite charge; the honest catalogue of what is wrong with the picture — an infinite charge density, no analogue for bosons (where The Klein–Gordon Equation needed none [Pauli:1934]), and no Lorentz-invariant meaning for “filled”.]

The prediction of the anti-electron

[Reserved: Dirac's first identification of the hole with the proton [Dirac:1930a] and its refutation — Weyl's symmetry argument and Oppenheimer's observation that the hydrogen atom would then annihilate; the retraction and the correct prediction of an “anti-electron”, a particle of the same mass and opposite charge, in 1931 [Dirac:1931]; the prediction of the antiproton in the same sentence, confirmed only in 1955; the timeline against the observation of Experiment: The Positron, which this chapter should give exactly, because the case is the treatise's cleanest example of a mathematically forced prediction confirmed afterwards.]

Phenomenon 93.6 (Antiparticles).

There exists a particle of the electron's mass and of the opposite charge, the positron; it was first seen in cosmic radiation (Experiment: The Positron), and the same has since proved true of every charged fermion examined. The masses and the charge magnitudes of a particle and of its antiparticle agree to within the precision of the comparisons tabulated in [Navas:2024], and no charged particle without such a partner has ever been found.

Derivation. The pairing is a symmetry of the equation itself, independent of hole theory. Write the minimally coupled equation of Section 93.1.2 for charge $q$,

\[ \left[\gamma^{\mu}\left(\ii\hbar\pp_{\mu}-qA_{\mu}\right) -mc\right]\psi=0\ec \]

and take its complex conjugate, $A_{\mu}$ being real:

\[ \left[\gamma^{\mu *} \left(-\ii\hbar\pp_{\mu}-qA_{\mu}\right)-mc\right]\psi^{*}=0\ep \]

In every representation of the Clifford algebra there is an invertible matrix $C$ with $C\gamma^{\mu *}C^{-1}=-\gamma^{\mu}$; in the Dirac basis one may take $C=\ii\gamma^{2}$, since there $\gamma^{0},\gamma^{1},\gamma^{3}$ are real and $\gamma^{2}$ is purely imaginary. Multiplying the conjugated equation on the left by $C$ and inserting $C^{-1}C$ before $\psi^{*}$ turns it into

\begin{equation}\tag{93.5} \left[\gamma^{\mu}\left(\ii\hbar\pp_{\mu}+qA_{\mu}\right) -mc\right]\psi^{c}=0\ec\qquad \psi^{c}:=C\psi^{*}\ec \end{equation}

which is the original equation with $q$ replaced by $-q$ and $m$ untouched. Solutions therefore come in pairs of equal mass and opposite charge, and since the map is invertible neither member of a pair is the more fundamental. An equation that describes an electron cannot avoid also describing a positron — which is why the prediction was forced rather than proposed, and why its confirmation four years later is the strongest single argument of this chapter. The argument says nothing about why the electron's mass takes the value it does, and nothing about the cosmological asymmetry between matter and antimatter, which Equation (93.5) does not address.

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The Stueckelberg–Feynman reading

[Reserved: the reinterpretation of negative-energy solutions as antiparticles propagating forwards in time, due to Stueckelberg [Stueckelberg:1941] and Feynman [Feynman:1949a]; the propagator prescription and the diagrammatic rules it generates [Feynman:1949b]; why this reading and not hole theory is the one that survives into Canonical Quantization of Fields, and why it applies equally to the spin-0 case where no sea can be built.]

Charge conjugation and the Majorana alternative

[Reserved: the operation $\psi\mapsto\psi^{c}=C\bar{\psi}\transpose$ mapping solutions of charge $e$ to solutions of charge $-e$, and the $CPT$ statement that its existence is not optional (Discrete Symmetries and CPT); Majorana's observation that a neutral spin-$\tfrac{1}{2}$ particle may satisfy $\psi^{c}=\psi$ and so be its own antiparticle [Majorana:1937]; that this is a live experimental question only for neutrinos (Flavour Physics and Neutrinos and The Weyl Equation and Neutrinos), and that neutrinoless double beta decay has not been observed [Navas:2024] — so the alternative is stated as open, not adopted.]

Phenomena of the one-particle theory

Zitterbewegung

[Reserved: Schrödinger's 1930 analysis of free motion [Schroedinger:1930], in which the velocity operator $c\vect{\alpha}$ has eigenvalues $\pm c$ and the position expectation oscillates at angular frequency $2mc^{2}/\hbar$ with amplitude of order the Compton wavelength; the origin of the effect in interference between positive- and negative-energy components, so that it is absent for a state built from one branch alone; the verdict that it is an artefact of the one-particle reading rather than an observable motion of an electron, and the simulations in trapped-ion and cold-atom systems that reproduce the mathematics without vindicating the physics.]

The Klein paradox

[Reserved: Klein's calculation of reflection at a step of height exceeding $2mc^{2}$, where the transmitted current does not vanish and the reflected current exceeds the incident one [Klein:1929b]; the resolution as spontaneous pair creation by a field strong enough to supply $2mc^{2}$ over a Compton wavelength, hence as another statement that particle number is not conserved; the Schwinger critical field $m^{2}c^{3}/e\hbar\approx1.3\times 10^{18}\,\mathrm{V}/\mathrm{m}$ as the scale [Schwinger:1951a], far above anything achieved in the laboratory, so this remains a prediction and must be labelled as one; the spin-0 version noted in The Klein–Gordon Equation.]

Thomas precession and spin transport

[Reserved: the kinematic precession of a spin carried along an accelerated worldline, derived by Thomas [Thomas:1926] from special relativity alone (Relativistic Dynamics) and responsible for the factor $\tfrac{1}{2}$ in the spin–orbit term of Section 93.4.2; the covariant equation of spin transport in an external field of Bargmann, Michel and Telegdi [Bargmann:1959], which is what the storage-ring measurements of Experiment: The Electron Anomalous Magnetic Moment are actually analysed with.]

The systematic nonrelativistic expansion

The Foldy–Wouthuysen transformation

[Reserved: the unitary transformation that decouples upper and lower components order by order in $1/mc^{2}$, constructed by Foldy and Wouthuysen [Foldy:1950]; the transformed Hamiltonian and the mean position operator, which coincides with the Newton–Wigner operator of The Klein–Gordon Equation [Newton:1949] and in terms of which the Zitterbewegung disappears; the free case, where the transformation is exact, against the external-field case, where it is an asymptotic series.]

The corrections order by order

[Reserved: the expansion producing, at order $1/m^{2}c^{2}$, exactly the three fine-structure terms of Section 93.4.2 — kinetic, spin–orbit and Darwin — with their coefficients fixed rather than fitted; the method as the standard route to effective nonrelativistic Hamiltonians for atomic and molecular structure (Atoms and Molecules) and for the relativistic corrections in heavy atoms, where the expansion parameter $Z\alpha$ is no longer small.]

The massless limit and the boundary of the theory

Weyl spinors and chirality

[Reserved: at $m=0$ the equation decouples into two independent two-component equations for the chirality eigenstates, written by Weyl [Weyl:1929] and long rejected because each violates parity; their adoption after parity violation was observed (Experiment: Parity Violation) and the neutrino helicity was measured (Experiment: Neutrino Helicity); the Dirac field as the direct sum of two Weyl fields coupled by the mass term, which is the statement developed in The Weyl Equation and Neutrinos.]

Where relativistic quantum mechanics ends

[Reserved: a closing accounting of the chapter's results by status — the equation, the gamma algebra, the spinor transformation law, $g=2$ at tree level and the Coulomb spectrum to order $\alpha^{4}$ survive intact into field theory; hole theory, the single-particle Hilbert space, the position operator and the exact Coulomb spectrum do not [Pauli:1934] [Feynman:1949a]. The energy scale at which the one-particle reading fails is $2mc^{2}$, and every phenomenon in Section 93.6 is a symptom of that failure. The continuation is Canonical Quantization of Fields.]