Optional — Relativity

Toward relativistic quantum mechanics

Hospitals routinely image the body with positrons, the electron's antiparticle: when one meets an electron, the two usually vanish into a pair of gamma rays of 511 keV each. Antimatter had been predicted before it was found, by an equation written in 1928 to make the electron obey both quantum mechanics and relativity (though the man who found it, in 1932, was not looking for it). What was the equation, and why did it demand a new particle?

Schrödinger's equation was built on the Newtonian relation $E = p^2/2m$. It is accurate for slow electrons, which covers most of atomic physics, but it cannot be right in general, since it does not respect relativity. This essay follows Paul Dirac's attempt to fix that. It explains the construction and its consequences, and leaves the full mathematics to a fold-out: why matrices appear, why the electron then has spin and a magnetic moment of exactly the right size, and why the equation insisted on solutions with negative energy.

The problem: a square root

The wavefunction essay reached Schrödinger's equation by letting derivatives stand in for momentum and energy, $\hat p = -i\hbar\,\partial/\partial x$ and $\hat E = i\hbar\,\partial/\partial t$, and applying them to $E = p^2/2m$. The relativistic relation gives the energy as

$$E = \sqrt{p^2c^2 + m^2c^4}.$$

Replacing $p$ by a derivative inside a square root gives something awkward. Expanded as a series, the square root contains derivatives of every order, so the equation at one point would involve the wavefunction far away, and it treats time and space quite differently, which sits badly with relativity. Dirac wanted an ordinary differential equation, relating the wavefunction to its derivatives at the same point. The obvious escape is to use the squared relation, $E^2 = p^2c^2 + m^2c^4$, with two time derivatives on the left. That was tried in 1926 and gives the Klein–Gordon equation. It is a perfectly good relativistic equation, and it is the correct one for particles with no spin. But it has second derivatives in time, unlike Schrödinger's equation, and as a description of a single electron its candidate for the probability density can be negative. Dirac wanted an equation first order in time, like Schrödinger's, with a probability density that is never negative. Since relativity treats space and time alike, that meant first order in the space derivatives too.

Dirac's step: take the square root with matrices

Dirac looked for an energy that is linear in the momentum components but whose square is still $p^2c^2 + m^2c^4$:

$$E = c\,(\alpha_x p_x + \alpha_y p_y + \alpha_z p_z) + \beta\,mc^2 .$$

Squaring the right side produces terms like $\alpha_x^2 p_x^2$, which must add up to $p^2c^2$, and cross terms like $(\alpha_x\alpha_y + \alpha_y\alpha_x)p_xp_y$, which must vanish. So the four coefficients must satisfy

$$\alpha_x^2 = \alpha_y^2 = \alpha_z^2 = \beta^2 = 1,$$

and every pair must anticommute: $\alpha_x\alpha_y = -\alpha_y\alpha_x$, and so on. No ordinary numbers do that. Matrices can. The Pauli matrices are exactly three such objects, each squaring to 1 and each pair anticommuting, but Dirac needs four, and there is no fourth 2×2 matrix that anticommutes with all three. Matrices of odd size cannot do it at all, as the example below shows, so the smallest that work are 4×4, and the wavefunction they act on has four components.

Worked example: why two dimensions are not enough

Three that work. Multiply out $\sigma_x\sigma_y$ and $\sigma_y\sigma_x$:

$$\begin{aligned}\sigma_x\sigma_y &= \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&-i\\i&0\end{pmatrix}\\[6pt] &= \begin{pmatrix}i&0\\0&-i\end{pmatrix},\end{aligned}$$ $$\sigma_y\sigma_x = \begin{pmatrix}-i&0\\0&i\end{pmatrix}.$$

They are negatives of each other, so $\sigma_x\sigma_y + \sigma_y\sigma_x = 0$. The same holds for the other two pairs, and each $\sigma_i^2 = 1$.

No room for a fourth. Any 2×2 matrix can be written as $M = a\,1 + b_x\sigma_x + b_y\sigma_y + b_z\sigma_z$ for some numbers $a, b_x, b_y, b_z$. Using the three relations just checked, $\sigma_xM + M\sigma_x = 2a\,\sigma_x + 2b_x\,1$. For $M$ to anticommute with $\sigma_x$ this must be zero, so $a = 0$ and $b_x = 0$. Repeating with $\sigma_y$ and $\sigma_z$ forces $b_y = b_z = 0$. The only such $M$ is zero, which cannot square to 1.

Why not 3×3. Since $\beta^2 = 1$, the eigenvalues of $\beta$ are $+1$ and $-1$. Take a vector $v$ with $\beta v = v$. Because $\alpha_x$ anticommutes with $\beta$, $\beta(\alpha_xv) = -\alpha_x\beta v = -(\alpha_xv)$: the matrix $\alpha_x$ turns every $+1$ eigenvector into a $-1$ eigenvector, and the same argument run backwards turns every $-1$ eigenvector into a $+1$ one. Since $\alpha_x^2 = 1$, nothing is lost on the way, so there are as many of one kind as of the other. The size of the matrices is therefore even. Two is ruled out, so the smallest possible is four.

Four that work. Build 4×4 matrices from 2×2 blocks: $\beta$ has the identity in the upper block and minus the identity in the lower, and each $\alpha_i$ has $\sigma_i$ in the two off-diagonal blocks. Multiplying these out, which the script for this essay does, confirms that all four square to 1 and every pair anticommutes. Dirac's equation is then

$$i\hbar\,\frac{\partial\psi}{\partial t} = \bigl(c\,\vec\alpha\cdot\hat{\vec p} + \beta\,mc^2\bigr)\psi ,$$

with $\psi$ a column of four complex functions.

What the four components mean

Spin. Part of the extra room goes to spin. The Pauli matrices sit inside Dirac's matrices, and the equation turns out to conserve not the orbital angular momentum alone but the orbital angular momentum plus an extra piece, $\tfrac12\hbar\,\vec\sigma$: spin ½, arriving without being put in by hand. The scope of this is worth stating carefully. Relativity alone does not demand spin: the Klein–Gordon equation is relativistic and describes spinless particles. An equation first order in time does not demand it either. What produced it was Dirac's particular construction: an equation linear in the energy and in all three components of momentum, whose coefficients have to anticommute.

The magnetic moment. Put the electron in a magnetic field and take the slow-electron limit. The Dirac equation reduces to Schrödinger's equation plus a spin term, with the spin's magnetic moment twice as large, for its angular momentum, as an orbiting charge would have. That factor, called $g$, is 2, which is exactly what spectroscopy had required. Measured, $g = 2.0023$: the tiny excess is explained by quantum electrodynamics, the later theory of electrons interacting with the electromagnetic field.

Fine structure. Solved for hydrogen, the equation reproduces Bohr's levels plus small corrections of relative size $\alpha^2$, where $\alpha = e^2/4\pi\varepsilon_0\hbar c = 1/137.036$ is the fine-structure constant. (The Bohr essay's ground-state speed, $2.19\times10^{6}$ m/s, is $c/137$: that ratio is $\alpha$, and relativistic corrections go as its square, $5\times10^{-5}$.) The corrections split the $n = 2$ level into two sublevels about 11 GHz apart, one with total angular momentum (orbital plus spin) $j = \tfrac12$ and one with $j = \tfrac32$. Spectroscopists call this the $2P_{3/2}$–$2P_{1/2}$ splitting. It produces the close pairs of lines the hydrogen essay mentioned.

The energies allowed by $E^2 = p^2c^2 + m^2c^4$ for an electron: a positive branch starting at $+mc^2 = 0.511$ MeV and a negative branch, its mirror image, starting at $-0.511$ MeV, with a gap of 1.022 MeV between. Classical physics could ignore the lower branch; Dirac's equation could not. Dashed: Newton's kinetic energy added to $mc^2$, which matches the upper branch only at low momentum.

The negative energies, and antimatter

The rest of the extra room goes to negative energy. At each momentum the equation has four independent solutions: two spin states with positive energy, and two with negative energy, the lower branch in the figure. The division is between solutions, not between rows of the column: for an electron at rest the positive-energy solutions use only the upper two components, but for a moving electron they involve all four. In classical physics a particle starting with positive energy never jumps to the lower branch, since energy changes continuously. In quantum mechanics it can jump, giving the difference to a photon, and an electron could then fall forever to lower and lower energies. Atoms would not be stable.

Dirac's answer, in 1930, was to suppose that all the negative-energy states are already filled, so that the exclusion principle forbids electrons from falling into them. A gamma ray with enough energy could lift an electron out of this filled "sea", leaving a hole, which would behave as a particle with the opposite charge. At first Dirac took the holes to be protons. When Hermann Weyl and Robert Oppenheimer showed that a hole must have the electron's mass, Dirac in 1931 predicted a new particle, an "anti-electron". In 1932 Carl Anderson found tracks of exactly such particles in cosmic-ray photographs: the positron.

Dirac's picture of a filled sea is no longer how physicists describe this. It does not work for bosons, which obey no exclusion principle, and it was replaced by quantum field theory, in which the electron is a quantum of a field and the negative-energy solutions describe the antiparticle directly, with positive energy. The predictions that survive are the ones that matter: every particle of this kind has an antiparticle of equal mass and opposite charge, and a pair can be created from energy or annihilate into it.

Optional: how the magnetic moment comes out with $g = 2$

Split the four-component $\psi$ into an upper pair $\varphi$ and a lower pair $\chi$. For a slow electron with energy close to $+mc^2$, the equations show $\chi$ is small, about $(\vec\sigma\cdot\vec\pi/2mc)\,\varphi$, where $\vec\pi = \hat{\vec p} + e\vec A$ is the momentum including the magnetic vector potential $\vec A$ for an electron of charge $-e$. Substituting back gives, for $\varphi$ alone,

$$i\hbar\,\frac{\partial\varphi}{\partial t} = \left[\frac{(\vec\sigma\cdot\vec\pi)^2}{2m} + V\right]\varphi$$

measured from $mc^2$. The identity $(\vec\sigma\cdot\vec\pi)^2 = \pi^2 + e\hbar\,\vec\sigma\cdot\vec B$, which follows from the Pauli matrices' multiplication rules, turns the first term into the ordinary kinetic energy plus $(e\hbar/2m)\,\vec\sigma\cdot\vec B$. That is the energy of a magnetic moment of one Bohr magneton, $e\hbar/2m$, carried by a spin of only $\tfrac12\hbar$: twice the ratio for orbital motion, $g = 2$.

Two questions

1. Making a pair. What is the least energy a gamma ray needs to create an electron and a positron? After the positron slows down and annihilates with an electron at rest, what photons come out?

Worked answer

The pair needs at least its rest energy, $2mc^2 = 2\times0.511 = 1.022$ MeV. (A lone photon cannot convert in empty space, since it could not conserve energy and momentum together; the conversion happens near a nucleus, which takes up the momentum, and the threshold is then very slightly above 1.022 MeV.) An electron and positron annihilating at rest have total momentum zero and energy 1.022 MeV. A single photon would carry momentum, so at least two are needed. Exactly two is the commonest outcome (three or more are possible but rarer), and those two are emitted back to back, each with 0.511 MeV: the 511 keV gamma rays of the opening, which PET scanners detect in coincidence.

The slip to avoid is thinking one photon of 1.022 MeV comes out. Momentum conservation, not energy, decides the number of photons.

2. Did relativity predict spin? A popular account says: "Spin is a consequence of relativity; Dirac derived it." How accurate is that?

Worked answer

Partly. Spin ½ and $g = 2$ do come out of Dirac's equation without being put in, which was a genuine success. But the Klein–Gordon equation is just as relativistic and has no spin, so relativity by itself does not force spin. What produced it was Dirac's particular construction, an equation linear in the energy and in all three components of momentum, whose coefficients have to anticommute. Spin had already been described without relativity, by Pauli's two-component theory in the spin essay.

The tempting wrong turn is to credit a result to the most famous ingredient in the derivation. The honest statement is narrower and more interesting: a particular relativistic wave equation for the electron requires spin ½ and predicts its magnetism correctly.

Where this leaves us

Positrons exist because the equation that makes the electron obey both quantum mechanics and relativity has solutions with negative energy, and those solutions describe an antiparticle. Dirac's equation takes the square root of $E^2 = p^2c^2 + m^2c^4$ with 4×4 matrices; spin ½, $g = 2$ and hydrogen's fine structure come out of it, and antimatter came with it, found four years after the equation was written.

This is where the book ends. The negative-energy puzzle is where single-particle quantum mechanics runs out: once particles can be created and destroyed, the right description is quantum field theory, the next step in the story and the natural place to go from here. Everything in the core essays, from a hot poker's glow to a detector's click, remains true within it.

Quantum Mechanics, Explained