The Dirac equation is a fundamental, relativistic wave equation for spin-½ particles such as electrons and quarks. Proposed by Paul Dirac in 1928, it unites quantum mechanics with special relativity in a single formalism. In covariant form (setting $\hbar =c=1$ for simplicity) the free Dirac equation is. $(i\gamma ^{\mu }\partial _{\mu }-m)\,\psi =0,$ where $\gamma ^{\mu }$ are the gamma matrices, $\psi$ is a four-component spinor field, and $m$ is the particle’s mass. This equation naturally incorporates the concept of spin, correctly describes the fine details of the hydrogen atom’s spectrum, and even predicted antimatter. The four-component object $\psi$ (often called a Dirac or bispinor) encodes two degrees of freedom for a spin-½ particle and two for its antiparticle.

Historical Context and Evolution

Dirac Equation
Type	Relativistic wave equation
Key terms	gamma matrices; spinors; probability current
Related	antiparticles; plane-wave solutions; hydrogen fine structure
Domain	Relativistic quantum mechanics; Quantum field theory; Mathematical physics
Examples	covariant form (iγ^μ∂_μ−m)ψ=0; representations (Dirac, Weyl); conserved current
Wikidata	Q272621

Before Dirac’s work, quantum mechanics lacked a fully consistent relativistic equation for electrons. The straightforward Klein–Gordon equation (second order in time) applied to spin-0 particles and gave problems for probability interpretation. Dirac sought a first-order, Lorentz-invariant equation so that the probability density would remain positive-definite and satisfy a continuity equation. He hypothesized linear momentum and energy operators involving matrices $\alpha _{i}$ and $\beta$ , discovering that the wavefunction must have four components. This led to the identifications $\beta =\gamma ^{0}$ and $\alpha _{i}=\gamma ^{0}\gamma ^{i}$ , where $\gamma ^{\mu }$ satisfy certain anticommutation relations. In 1928 Dirac published this equation, which not only reproduced known nonrelativistic results in the low-speed limit but also predicted new phenomena. The equation accounted exactly for the hydrogen atom’s fine structure (splitting of energy levels) and implied the existence of positive- and negative-energy solutions. Interpreting the negative-energy states led Dirac to propose the existence of antiparticles: the hole theory, ultimately explaining the positron discovered by Carl Anderson in 1932. Dirac’s insight that four-component spinors can describe relativistic electrons was revolutionary, and the equation became a cornerstone for later developments, including quantum electrodynamics and the Standard Model of particle physics.

Formulation of the Dirac Equation

In covariant notation, the Dirac equation is. $(i\gamma ^{\mu }\partial _{\mu }-m)\psi =0,$ with $\partial _{\mu }=(\partial _{t},-\nabla )$ and $\gamma ^{\mu }$ the gamma matrices. Here we use the “mostly-minus” metric $\eta ^{\mu \nu }=\mathrm {diag} (1,-1,-1,-1)$ . The gamma matrices are $4\times 4$ complex matrices satisfying the Clifford algebra. $\{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\,\eta ^{\mu \nu }\,I_{4\times 4}.$ This anticommutation relation encodes the Minkowski metric and ensures Lorentz invariance. There are many equivalent choices (representations) of $\gamma ^{\mu }$ , but all will satisfy these relations. Often $\gamma ^{0}$ is chosen Hermitian and $\gamma ^{i}$ anti-Hermitian. In practical work one sometimes uses Dirac’s original form of the equation implied by a Hamiltonian. $i\hbar {\frac {\partial \psi }{\partial t}}={\Bigl (}c\,{\boldsymbol {\alpha }}\cdot \mathbf {p} +\beta mc^{2}{\Bigr )}\psi ,$ where ${\boldsymbol {\alpha }}$ and $\beta$ are $4\times 4$ matrices related to the $\gamma ^{\mu }$ . In units with $c=1$ , this is equivalent to the covariant form above. Each component of the spinor $\psi$ satisfies the Klein–Gordon equation $(\Box +m^{2})\psi =0$ , so the energy–momentum relation $E^{2}=\mathbf {p} ^{2}+m^{2}$ holds for each.

The bispinor $\psi (\mathbf {x} ,t)$ has four complex components. These can be thought of as two two-component Weyl spinors (left-handed and right-handed parts). Under rotations and boosts (Lorentz transformations), $\psi$ transforms in a spinor representation. Notably, a $360^{\circ }$ rotation changes the sign of a single Weyl spinor; only after a $720^{\circ }$ rotation does it return to itself. This property reflects the half-integer spin of the electron.

Gamma Matrices and Spinors

The gamma matrices are a key element of the Dirac formalism. One common choice (the Dirac basis) takes. $\gamma ^{0}=<<<M0>>>,\qquad \gamma ^{i}=<<<M1>>>,$ where $\sigma ^{i}$ are the Pauli matrices and $I_{2}$ is the $2\times 2$ identity. In this basis, $\alpha _{i}=<<<M2>>>$ and $\beta =\gamma ^{0}$ . Another popular choice is the chiral or Weyl basis, defined by. $\gamma ^{0}=<<<M3>>>,\qquad \gamma ^{i}=<<<M4>>>,$ with $\gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}=<<<M5>>>$ . In the chiral basis, the upper and lower two components of $\psi$ correspond to left-handed and right-handed Weyl spinors. A third choice is the Majorana basis, wherein all $\gamma ^{\mu }$ are purely imaginary matrices. This lets one impose a Majorana condition $\psi =\psi ^{C}$ (the spinor equals its own charge-conjugate), which is useful for describing Majorana fermions (particles that are their own antiparticles).

No matter the representation, all $\gamma ^{\mu }$ satisfy the same algebraic relations, so the physics is independent of basis. The spinor $\psi$ itself is often called a Dirac spinor. Its Lorentz transformation properties can be written as $\psi (x)\to S(\Lambda )\psi (\Lambda ^{-1}x)$ under a Lorentz transform $\Lambda$ , with $S(\Lambda )$ a corresponding $4\times 4$ matrix in the spinor representation. Tensor quantities like ${\bar {\psi }}\psi$ (a scalar) or ${\bar {\psi }}\gamma ^{\mu }\psi$ (a four-vector) can be constructed using the Dirac adjoint ${\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}$ .

Probability Current and Continuity

A major virtue of the Dirac equation is that it admits a conserved positive-definite probability density. Taking the Dirac equation and its Hermitian conjugate, one can derive the continuity equation. ${\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0,$ where the probability density and current are. $\rho =\psi ^{\dagger }\psi \;,\qquad \mathbf {j} =c\,\psi ^{\dagger }{\boldsymbol {\alpha }}\,\psi =\psi ^{\dagger }{\boldsymbol {\gamma }}\psi \;.$ (Equivalently, in covariant form one writes the four-current $j^{\mu }={\bar {\psi }}\,\gamma ^{\mu }\psi$ , whose time component is $\rho$ .) Here ${\boldsymbol {\alpha }}=\gamma ^{0}{\boldsymbol {\gamma }}$ and ${\boldsymbol {\gamma }}=(\gamma ^{1},\gamma ^{2},\gamma ^{3})$ . Importantly, $\rho =\psi ^{\dagger }\psi$ is always non-negative (it is the sum of absolute squares of the spinor components) and integrates to 1 for a properly normalized state. This overcomes a key difficulty of the Klein–Gordon equation, whose naive probability density can be negative. The Dirac current $j^{\mu }$ is constructed to transform as a Lorentz 4-vector, ensuring relativistic consistency of the conservation law.

Plane‐Wave Solutions

For a free particle (no forces or fields), one can solve the Dirac equation by assuming a plane-wave form $\psi (x)=u(p)e^{-ip\cdot x}$ or $\psi (x)=v(p)e^{+ip\cdot x}$ , where $p^{\mu }=(E,\mathbf {p} )$ is the four-momentum. Substituting into $(\gamma ^{\mu }p_{\mu }-m)\psi =0$ leads to the algebraic condition $(\gamma ^{\mu }p_{\mu }-m)u(p)=0$ for the spinor $u(p)$ and a similar equation $(\gamma ^{\mu }p_{\mu }+m)v(p)=0$ . Consistency of these equations requires the relativistic dispersion $E^{2}=\mathbf {p} ^{2}+m^{2}$ . Thus each of $E=+{\sqrt {\mathbf {p} ^{2}+m^{2}}}$ or $E=-{\sqrt {\mathbf {p} ^{2}+m^{2}}}$ is allowed.

There are four linearly independent spinor solutions in total, corresponding to two spin (or helicity) states for each energy sign. Concretely, one finds two independent 4-component positive-energy solutions $u^{(r)}(\mathbf {p} )\,e^{-iEt+i\mathbf {p} \cdot \mathbf {x} }$ and two negative-energy solutions $v^{(s)}(\mathbf {p} )\,e^{+iEt-i\mathbf {p} \cdot \mathbf {x} }$ , where $r,s=1,2$ label spin-up or spin-down states along a chosen axis. For example, in the rest frame ( $\mathbf {p} =0$ ), the solutions reduce (up to normalization) to basis spinors where either the first two components are nonzero (particle states) or the last two are nonzero (negative-energy states). Boosting to a moving frame mixes the components but the fourfold structure remains.

In the nonrelativistic (low-velocity) limit, the negative-energy solutions are suppressed and the upper two components dominate, reproducing a two-component Pauli spinor with the usual spin-up/down states. The negative-energy plane waves appear to describe particles of “negative energy,” a puzzling feature resolved by the interpretation of antiparticles (see next section).

Antiparticles

The Dirac equation’s allowance of negative-energy solutions presented a conceptual challenge. Dirac’s interpretation was to redefine the vacuum. He proposed that in the true vacuum all negative-energy states are filled (the “Dirac sea” of electrons). A hole – the absence of a negative-energy electron – would then behave like a positive-energy antiparticle with opposite charge. In the case of the electron, such a hole would have the properties of a positively charged electron: namely, the positron. This explanation predicted the existence of antimatter before it was observed. Indeed, the positron (the electron’s antiparticle) was discovered in cosmic ray experiments in 1932 by Carl Anderson, confirming Dirac’s prediction. In modern field theory language, we say that the Dirac field, when quantized, creates both particle and antiparticle excitations. The negative-energy solutions $v^{(s)}(p)$ are reinterpreted as positive-energy antiparticles (for which momentum and current point in the same direction).

Thus one can think of $\psi$ as containing both electron and positron degrees of freedom. Mathematically, one often writes the field operator as a sum of particle and antiparticle parts ( $\psi =u\,a+v\,b^{\dagger }$ , in second quantization). In any case, the Dirac equation naturally incorporates antiparticles as an inescapable consequence of combining quantum mechanics with relativity.

Hydrogen Atom and Fine Structure

One of the greatest early successes of the Dirac equation was its application to the hydrogen atom. By coupling the Dirac field to the electromagnetic field via minimal coupling ( $p_{\mu }\to p_{\mu }-eA_{\mu }/c$ ), one obtains the Dirac equation in a Coulomb potential. Solving this exactly yields energy levels that depend on the principal quantum number $n$ and total angular momentum $j$ , but not separately on orbital ( $l$ ) and spin quantum numbers. The result agrees with the experimentally observed fine structure of hydrogen. In particular, spin–orbit coupling and the so-called Darwin term emerge automatically from the Dirac theory without additional ad-hoc spin assumptions. The Dirac prediction for the energy of a hydrogenic level (for nuclear charge $Ze$ ) can be written in closed form and reproduces the Sommerfeld formula for fine structure.

Concretely, Dirac’s formula shows that states with the same $n$ and $j$ (but different $l$ ) are exactly degenerate, which was confirmed experimentally (aside from small QED corrections). For example, the $2P_{1/2}$ and $2P_{3/2}$ levels are split by the Dirac theory correctly. The only slight deviation in hydrogen is the Lamb shift, a tiny radiative correction (not predicted by Dirac’s equation itself) arising from quantum electrodynamics. Overall, the Dirac equation was the first to rigorously explain the detailed structure of atomic spectra, and this triumph strongly validated the new theory.

Representations of the Dirac Equation

The Dirac equation’s form depends on the choice of gamma matrices and the basis of spinors, but physical predictions are representation-independent. Common representations include:

Dirac (standard) basis: As given above, with $\gamma ^{0}=\mathrm {diag} (1,1,-1,-1)$ and $\gamma ^{i}$ off-diagonal. In this basis the upper two components of $\psi$ are positive-energy (particle) states, and the lower two correspond to negative-energy solutions in the rest frame. This basis makes the particle–antiparticle symmetry explicit (parity is simple).

Weyl (chiral) basis: Here $\gamma ^{5}=\mathrm {diag} (-1,-1,1,1)$ is block-diagonal, separating left- and right-handed two-component spinors. Massless fermions (when $m=0$ ) decouple into purely two-component Weyl equations, one for each chirality. The Weyl basis is convenient in treatments of massless neutrinos or in chiral gauge theories.

Majorana basis: All $\gamma ^{\mu }$ are chosen purely imaginary. In this basis the condition $\psi =\psi ^{C}$ (charge conjugation) means $\psi$ can be taken real. This is used when dealing with Majorana fermions, which are their own antiparticles (such as Majorana neutrinos, if they exist).

More abstractly, one can classify the Dirac spinor as transforming under the Lorentz group as the sum of two Weyl (spin-½) representations, often denoted $({\tfrac {1}{2}},0)\oplus (0,{\tfrac {1}{2}})$ . The four components correspond to the two spin states of a left-handed Weyl spinor and two spin states of a right-handed one. For any given representation of the gamma matrices, the Dirac equation retains its covariant form $(i\gamma ^{\mu }\partial _{\mu }-m)\psi =0$ .

Associated with these bases is the important matrix $\gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}$ , which anticommutes with $\gamma ^{\mu }$ . Its eigenvalues $\pm 1$ project out the left- and right-chiral components of $\psi$ . Chirality and helicity (spin projection along momentum) coincide for massless particles, leading to conceptually distinct left-handed and right-handed fermions.

Significance and Applications

The Dirac equation revolutionized quantum physics. It was the first relativistic wave equation consistent with both key principles – Lorentz invariance and a positive probability interpretation – for spinning particles. It explained formerly ad hoc concepts (like spin magnetic moment) as natural consequences of relativity. Its success in atomic physics made it indispensable for understanding fine and hyperfine structures in atoms.

In modern physics, the Dirac equation forms the foundation for the description of all charged spin-½ fermions. When quantized (second quantization), it gives quantum field operators that create and annihilate fermions and antifermions. All matter fields in the Standard Model (except possibly neutrinos) are Dirac fields obeying analogous equations. Solutions of the Dirac equation are used in scattering calculations and form a starting point for quantum electrodynamics (QED) and other gauge theories.

Beyond particle physics, Dirac fermions appear in condensed matter contexts: for example, electrons in graphene or topological insulators near the Fermi surface behave according to Dirac-like equations. The concept of spinors and gamma matrices has also found use in fields ranging from curved-spacetime gravity (the Dirac equation has general-relativistic extensions) to mathematics (index theorems, geometry). The deep symmetry properties of the Dirac equation also continue to inspire theoretical advances.