Everything so far used the Newtonian energy $p^{2}/2m$. Near the speed of light the energy of a free particle is $E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}$: a rest energy $mc^{2}$, the Newtonian kinetic energy, and corrections that grow with $p/mc$. For the electron in hydrogen, moving at about $\alpha c$, where $\alpha\approx1/137$ is the fine-structure constant, the first correction is about one part in seventy-five thousand of its kinetic energy — the fine structure. But relativity changes more than numbers. The relation is quadratic in $E$, so every equation built on it has solutions of negative energy as well as positive. And pinning a particle down to within its Compton wavelength $\hbar/mc$ gives it momenta of order $mc$, energy enough to make a new particle and its antiparticle: a theory of one particle at a time cannot hold at those scales. This chapter follows both threads, to the Dirac equation and its antiparticles, and the next turns them into fields.

A Newtonian map of the Earth drawn on a flat sheet is fine for a town and fails for a continent. Newtonian quantum mechanics is the town map: exact enough for atoms, and wrong at speeds near light.

The energy of a free particle against its momentum, in units where $m=c=1$: $\sqrt{p^{2}c^{2}+m^{2}c^{4}}$, and the Newtonian , dashed, with the rest energy added for comparison. They agree while ; beyond it the relativistic energy grows only in proportion to .

A free particle of mass $m$ and momentum $\mathbf{p}$ has the energy

• The first correction to the kinetic energy is $-p^{4}/8m^{3}c^{2}$, a fraction of about $v^{2}/4c^{2}$ of it: in hydrogen.

Factor the rest energy out of the square root and expand it by the binomial series in the small quantity $p^{2}/m^{2}c^{2}$. The first term is the rest energy, the second the Newtonian kinetic energy, and the third the first relativistic correction, whose size relative to the kinetic energy is a quarter of $(v/c)^{2}$.

The energy–momentum relation of special relativity (stated).

Take out $mc^{2}$.

The binomial series.

Multiply back by $mc^{2}$.

For hydrogen, $v\approx\alpha c$ gives $\alpha^{2}/4\approx1.3\times10^{-5}$.

Squeeze a particle into a region of size $\hbar/mc$ and its momentum spread reaches $mc$: enough energy to make pairs of particles.

The relation $E^{2}=p^{2}c^{2}+m^{2}c^{4}$ is quadratic in , so a wave equation built on it has solutions of negative energy as well.

The electron in hydrogen moves at about $\alpha c$, so relativity changes its energies by about $\alpha^{2}$, a part in twenty thousand.

The kinetic energy is $E-mc^{2}$. At $p=mc$ the relativistic energy is $\sqrt{2}mc^{2}$, so the kinetic energy is .

Great work! You now know how to:

• expand the relativistic energy and find its first correction
• explain where negative energies come from
• say why one particle at a time fails below the Compton wavelength