Dirac Equation

Prerequisites: Klein-Gordon Equation, Spinors. Full path (23 pages)

Best explanations

Concrete David Tong's Quantum Field Theory notes, chapter 4

Intuitive
Concrete
Abstract
Why is it interesting?
Definitions
History

$(i\gamma_\mu \partial^\mu - m ) \Psi =0$

Intuitive

The Dirac equation describes how the state of a relativistic (= fast moving) quantum system with half-integer spin changes in time.

The analogous equation for systems without spin is the Klein-Gordon equation.

If the system only moves slowly, the Dirac equation becomes the Pauli equation.

Concrete

A nice discussion can be found in chapter 4 of Klauber's Student Friendly QFT book
For an elementary derivation and a discussion of what the solution of the Dirac equation mean, see Physics from Symmetry by Schwichtenberg
Alternative, two possible derivations can be found at page 100ff in Relativistic Quantum Mechanics by Paul Strange
A great discussion can be found in chapter 10 of the quantum field theory book by Matthew Schwartz.
Another great discussion, especially of question how many degrees of freedom a Dirac spinor has can be found in http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf

Gamma Gymnastics:

There are many important rules for the $\gamma$ matrices that appear in the Dirac equation. These rules are important for many practical calculations.

For a nice description, see section 7.4.3 "Diracology" in the book The Conceptual Framework of Quantum Field Theory by Duncan

Graphical Summary

The diagram below shows the Dirac equation and its Lagrangian in various forms. For a more detailed explanation see Fun with Symmetry.

No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Free Spinor Fields and the Dirac Equation"

Abstract

We use the metric signature $(+,-,-,-)$ and note $\gamma_\mu \partial^\mu = \gamma^\mu \partial_\mu$ . The gamma matrices are a representation of the Clifford algebra of Minkowski space,

$\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}\,\mathbb{1}.$

The smallest irreducible representation of this algebra over $\mathbb{C}$ is four-dimensional, so $\Psi$ takes values in $\mathbb{C}^4$ . The matrices $S^{\mu\nu} = \frac{i}{4}[\gamma^\mu,\gamma^\nu]$ satisfy the commutation relations of the Lorentz algebra and exponentiate to a representation $\Lambda_{1/2}$ of $\mathrm{Spin}^+(1,3) \cong SL(2,\mathbb{C})$ , the double cover of the proper orthochronous Lorentz group. This representation is reducible: it is the direct sum $(\tfrac12,0)\oplus(0,\tfrac12)$ of a left-handed and a right-handed Weyl representation, which the projectors $\frac12(1\pm\gamma^5)$ separate. The Dirac equation is covariant because

$\Lambda_{1/2}^{-1}\,\gamma^\mu\,\Lambda_{1/2} = \Lambda^\mu{}_\nu\,\gamma^\nu ,$

so that $\Psi'(x) = \Lambda_{1/2}\Psi(\Lambda^{-1}x)$ solves the equation whenever $\Psi$ does. The mass term couples the two Weyl components, which is why a massive spin $1/2$ field needs both chiralities (or a Majorana condition relating them).

The first-order operator $i\gamma^\mu\partial_\mu$ is a square root of the wave operator: $(i\gamma^\mu\partial_\mu)^2 = -\partial_\mu\partial^\mu$ , hence

$(i\gamma^\mu\partial_\mu + m)(i\gamma^\mu\partial_\mu - m) = -(\partial_\mu\partial^\mu + m^2),$

and every component of a solution of the Dirac equation solves the Klein-Gordon equation. The equation follows from the Lagrangian $\mathscr{L} = \bar\Psi(i\gamma^\mu\partial_\mu - m)\Psi$ with $\bar\Psi = \Psi^\dagger\gamma^0$ .

Geometrically, on a spacetime manifold $M$ with a spin structure, $\Psi$ is a section of the spinor bundle $S \to M$ , a vector bundle associated to the principal $\mathrm{Spin}$ bundle (see fiber bundles). The Levi-Civita connection lifts to a spin connection $\nabla$ on $S$ , and with a vierbein $e_a{}^\mu$ the Dirac operator is $D = i\gamma^a e_a{}^\mu \nabla_\mu$ . Coupling to a gauge field replaces $S$ by $S\otimes E$ for a vector bundle $E$ with connection and $\nabla$ by the tensor product connection; this is minimal coupling. On a compact even-dimensional Riemannian spin manifold, the Dirac operator $D = \sum_i e_i\cdot\nabla_{e_i}$ (with an orthonormal frame $e_i$ and the Clifford convention $v\cdot v = -|v|^2$ ) satisfies the Lichnerowicz formula $D^2 = \nabla^*\nabla + \frac{R}{4}$ , where $R$ is the scalar curvature. The chiral part of $D$ is a Fredholm operator whose index is computed by the Atiyah-Singer index theorem as the $\hat A$ -genus of $M$ (twisted by the Chern character of $E$ when a gauge field is present).

Why is it interesting?

The Dirac equation is the correct equation of motion that describes free spin $1/2$ particles.

In fact, Dirac's equation for the electron must be rated, alongside the Maxwell and Einstein equations, as one of the Great Field Equations of physics.page 289 in "The Emperors new Mind" by Penrose

Definitions

$\partial _{\mu}$ denotes the partial derivative and $\gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} = \gamma_0 \partial^0 + \gamma_1 \partial^1 +\gamma_2 \partial^2 +\gamma_3 \partial^3$ ,
$m$ denotes the mass of the particle,
$\Psi$ is either the wave function of the spin $1/2$ particle if we use the Dirac equation in a particle theory, or describes the spin $1/2$ field if we work in a field theory. In any case, $\Psi$ is not a vector but a spinor.
$\gamma_\mu$ are the Dirac gamma matrices.

History

“A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It should be added, however, that it was Dirac who found most of the additional insights.” Weisskopf on Dirac

Niels Bohr: “What are you working on Mr. Dirac?” Paul Dirac: “I’m trying to take the square root of something”

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