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Dirac equation: relativistic wave equation for spin-1/2 particles

A relativistic quantum wave equation introduced by Paul Dirac in 1928 that describes spin-1/2 fermions, predicts antimatter, and underpins modern quantum field theory, condensed-matter models, and differential geometry.

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Overview

The Dirac equation is a fundamental relativistic wave equation formulated within quantum mechanics to describe particles with intrinsic spin one-half. Proposed by Paul Dirac in 1928, it reconciles quantum theory with special relativity and provides a unified description of many properties of fermions, including electrons and quarks. One of its most striking predictions was the existence of antimatter, later confirmed by experiment.

Mathematical structure and properties

Unlike the nonrelativistic Schrödinger equation, the Dirac theory is a first-order differential equation in both space and time and is constructed to be invariant under Lorentz transformations. Its unknown is a multi-component spinor that encodes particle and intrinsic spin degrees of freedom. The equation employs a set of matrices (commonly called gamma matrices) so that its square reproduces the relativistic energy–momentum relation. The differential object associated with it—the Dirac operator—also appears in modern mathematics and plays a central role in differential geometry and index theory.

Key consequences

  • Negative-energy solutions and their reinterpretation led to the prediction and discovery of the positron and the concept of antiparticles with the same mass and opposite charges.
  • The equation naturally gives spin-½ and explains the magnetic behavior of particles, including phenomena observed in the Stern–Gerlach experiment.
  • Taking the low-velocity limit yields the Pauli equation, connecting relativistic and nonrelativistic spin descriptions.

History and experimental confirmation

Dirac’s theoretical work predicted antiparticles before direct detection. Within a few years the predicted positron was identified in an experimental cloud-chamber observation, confirming a major conceptual advance. The Dirac formalism also motivated the development of relativistic quantum field theory, where particle creation and annihilation processes are treated consistently.

Applications and modern relevance

Beyond particle physics, the Dirac equation and its massless variants appear in condensed-matter systems. Electrons in materials such as graphene behave like two-dimensional massless Dirac fermions, producing high mobility and unusual transport properties. In theoretical physics the Dirac framework underpins quantum electrodynamics and relativistic treatments in atomic and molecular calculations.

Distinguishing features and remarks

Important distinguishing aspects are its relativistic covariance, first-order character, and spinor structure. Mathematically, the Dirac operator links analysis, geometry, and topology; physically, it predicts spin, magnetic moments, and antiparticles. Its broad impact spans foundational theory, experiment, and diverse applications in modern physics.

Dirac equation of an uncharged particle

The Dirac equation is a system of four coupled partial differential equations for the four component functions of the Dirac spinor ψ \psi (x). The variable xhere stands for x=(x^{0},x^{1},x^{2},x^{3})\,,where the upper subscript 0 (x^{1},x^{2},x^{3})denotes the time t=x^{0}and the subscripts 1 to 3 denote the location coordinates

In natural units of measure with c=1=\hbar , the Dirac equation for an uncharged particle of mass m

\left[{\mathrm i}\sum _{{\mu =0}}^{3}\gamma ^{{\mu }}{\frac {\partial }{\partial x^{{\mu }}}}-m\right]\,\psi (x)=0\,.

The expression in square brackets is the standard form of a Dirac operator.

The constant gamma or Dirac matrices γ \gamma ^{0},\gamma ^{1},\gamma ^{2} and γ \gamma ^{3}act in the space of the four components of the spinor and couple them together. The products of two gamma matrices have the following properties:

\gamma ^{{\mu }}\,\gamma ^{{\nu }}=-\gamma ^{{\nu }}\,\gamma ^{{\mu }}\quad (\,\mu ,\nu \in \{0,1,2,3\}\,,\mu \neq \nu ),

\gamma ^{{0}}\gamma ^{{0}}=-\gamma ^{{1}}\gamma ^{{1}}=-\gamma ^{{2}}\gamma ^{{2}}=-\gamma ^{{3}}\gamma ^{{3}}=1

Thus they form a Clifford or Dirac algebra. If the Dirac operator

{\displaystyle \mathrm {i} \sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}-m}

applied to both sides of the Dirac equation, the four differential equations decouple and the Klein-Gordon equation is obtained for each component of ψ : \psi

{\Bigl [}{\frac {\partial ^{2}}{(\partial {x^{{0}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{1}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{2}}})^{2}}}-{\frac {\partial ^{2}}{(\partial {x^{{3}}})^{2}}}+m^{2}{\Bigr ]}\,\psi (x)=0

Thus, applying a Dirac operator twice leads to the Klein-Gordon equation, which is why the Dirac equation is also considered the "root" of the Klein-Gordon equation. For a particle in a momentum eigenstate, the Klein-Gordon equation yields (in the order of its terms) -E^{2}+{\vec {p}}^{{\,2}}+m^{2}=0, that is, the relativistic energy-momentum relation of a particle of mass m\,.

Each irreducible representation of the Dirac algebra consists of 4\times 4matrices. In the standard or Dirac representation they have the following form (vanishing matrix elements with value zero are not written here):

{\displaystyle {\begin{array}{c c}\gamma ^{0}={\begin{pmatrix}1&&&\\&1&&\\&&-1&\\&&&-1\end{pmatrix}}\,,&\gamma ^{1}={\begin{pmatrix}&&&1\\&&1&\\&-1&&\\-1&&&\end{pmatrix}}\,,\\\,&\,\\\gamma ^{2}={\begin{pmatrix}&&&-\mathrm {i} \\&&\mathrm {i} &\\&\mathrm {i} &&\\-\mathrm {i} &&&\end{pmatrix}}\,,&\gamma ^{3}={\begin{pmatrix}&&1&\\&&&-1\\-1&&&\\&1&&\end{pmatrix}}\,.\end{array}}}

Thus, the first two components of γ \gamma _{0} form the two-component unit matrix, and the last two components form its negative. Similarly, the two upper components of the second, third, and fourth γ \gamma matrices, respectively, yield the three 2×2 Pauli matrices σ {\displaystyle \sigma _{j}}and the two last components of γ \gamma _{j}their negatives. The latter go to zero in the nonrelativistic limit case like v/c. Thus this representation, the standard one, is particularly suitable for the treatment of slowly moving electrons. In the mathematically and physically equivalent Weyl representation, the spinor transformation behavior for Lorentz transformations is particularly simple, and in the likewise equivalent Majorana representation, the Dirac equation is a real system of equations. Further representations are obtained by equivalence transformations.

The four gamma matrices can be written symbolically as the contravariant 4-vector

{\displaystyle \gamma ^{\mu }={\begin{pmatrix}\gamma ^{0}\\\gamma ^{1}\\\gamma ^{2}\\\gamma ^{3}\end{pmatrix}}}

summarize. Then the first term of the Dirac equation has the form of a scalar product of the vectors γ\gamma ^{{\mu }}and ∂{\frac {\partial }{\partial x^{{\mu }}}} . However, this is not invariant under Lorentz transformation, because γ\gamma ^{{\mu }}remains constant. The Lorentz invariance of Dirac theory arises only because the Dirac operator \psi acts on a spinor ψ whose four components are suitably co-transformed. In the final result a solution ψ \psi (x)the Dirac equation is transformed by Lorentz transformation into a solution of the correspondingly transformed Dirac equation.

Momentum space and slash notation

In addition to the form just described in spatial space, the Dirac equation can also be written down in momentum space. It then reads

{\Bigl [}\gamma ^{\mu }p_{\mu }-m{\Bigr ]}\,\psi (p)=0\,,

where for abbreviation the Einstein summation convention was used (which states that summation is done over equal indices). In the even further simplified Feynman-Slash notation, the scalar product with the gamma matrices is expressed by a slash symbol. In the space of places it results in

{\displaystyle {\Bigl [}i\partial \!\!\!/\ -m{\Bigr ]}\,\psi (x)=0\,,}

and in momentum space

{\displaystyle {\Bigl [}p\!\!\!/\ -m{\Bigr ]}\,\psi (p)=0\,.}

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