Dirac Equation · aiwiki.page

The Dirac equation is a relativistic wave equation for particles with spin ½, formulated by Paul Dirac in 1928. It combines quantum mechanics with special relativity and was originally developed to describe the electron. Its mathematical structure accounts for electron spin and magnetic properties and led to the prediction of the positron. In modern quantum field theory, it describes the dynamics of a spinor field whose quantized excitations include particles and antiparticles. (nobelprize.org)

Historical development

The nonrelativistic Schrödinger equation did not provide a complete relativistic theory of the electron. Dirac sought an equation that was first order in both time and spatial derivatives while respecting the relativistic relation between energy, momentum, and mass. His solution required matrix-valued coefficients and a multicomponent wave function rather than a single scalar function. Electron spin and its associated magnetic moment then appeared as consequences of the theory. (nobelprize.org)

The equation also admitted negative-energy solutions. Dirac developed a “hole theory” interpretation that eventually identified an electron counterpart with the same mass and opposite charge. Carl Anderson discovered the positron experimentally in 1932. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger for their contributions to atomic theory. The later field-theoretic interpretation replaced the need to regard the vacuum as a literal sea of occupied negative-energy electron states. (nobelprize.org)

Hamiltonian form and linearization

For a free particle, the equation can be written

i\hbar\frac{\partial\psi}{\partial t} = \left(c\,\boldsymbol{\alpha}\cdot\hat{\mathbf p} +\beta mc^2\right)\psi, \qquad \hat{\mathbf p}=-i\hbar\nabla.

Here $c$ is the speed of light, $\hbar$ is the reduced Planck constant, and $\psi$ is a four-component wave function. The matrices $\alpha_1,\alpha_2,\alpha_3,\beta$ act on its components. The operator on the right is the free-particle Hamiltonian. (ocw.mit.edu)

Dirac’s construction starts from

E^2=c^2\mathbf p^2+m^2c^4.

Requiring the square of the proposed Hamiltonian to reproduce this relation gives

\{\alpha_i,\alpha_j\}=2\delta_{ij}I, \qquad \{\alpha_i,\beta\}=0, \qquad \beta^2=I,

where $\{A,B\}=AB+BA$ is the anticommutator. Ordinary numerical coefficients cannot satisfy these conditions; matrices can. In three spatial dimensions, the smallest complex matrix realization has size $4\times4$ , accounting for the four-component form of the equation. (nobelprize.org)

One conventional representation is

\alpha_i= \begin{pmatrix} 0&\sigma_i\\ \sigma_i&0 \end{pmatrix}, \qquad \beta= \begin{pmatrix} I_2&0\\ 0&-I_2 \end{pmatrix},

where $\sigma_i$ are the Pauli matrices. Different representations change the component notation, not the physical predictions. (ocw.mit.edu)

Covariant form and spinors

The relativistic structure becomes explicit in the covariant form

\left(i\hbar c\,\gamma^\mu\partial_\mu-mc^2\right)\psi=0.

Using $x^\mu=(ct,\mathbf x)$ and the metric convention $\eta^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ , the gamma matrices satisfy

\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I_4.

Repeated indices are summed. The relations define a Clifford algebra, and the Hamiltonian notation is recovered through $\gamma^0=\beta$ and $\gamma^i=\beta\alpha_i$ . In units with $\hbar=c=1$ , the equation becomes $(i\gamma^\mu\partial_\mu-m)\psi=0$ . (damtp.cam.ac.uk)

The field $\psi$ is a spinor, not an ordinary four-vector. Its transformation under a Lorentz transformation acts both on the spacetime argument and on its spinor components. This transformation law makes the equation Lorentz covariant and gives its solutions the transformation properties of spin-½ states. (damtp.cam.ac.uk)

Applying the complementary operator $i\hbar c\,\gamma^\mu\partial_\mu+mc^2$ shows that every component of a free Dirac solution also satisfies the Klein–Gordon equation:

\left(\Box+\frac{m^2c^2}{\hbar^2}\right)\psi=0, \qquad \Box=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2.

The converse does not hold: the first-order Dirac equation imposes additional relations among the components. (damtp.cam.ac.uk)

Conserved current and physical interpretation

Define the Dirac adjoint by

\bar\psi=\psi^\dagger\gamma^0.

The equation and its adjoint imply a conserved current,

j^\mu=c\,\bar\psi\gamma^\mu\psi, \qquad \partial_\mu j^\mu=0.

In a single-particle interpretation, this gives

\rho=\psi^\dagger\psi, \qquad \mathbf j=c\,\psi^\dagger\boldsymbol{\alpha}\psi, \qquad \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j=0.

The density $\rho$ is nonnegative, an important advantage over the analogous conserved density obtained by treating a scalar Klein–Gordon field as a single-particle wave function. In field theory, the corresponding current is interpreted as a conserved charge current rather than a universal particle-position probability current. (damtp.cam.ac.uk)

For each free momentum, the energy branches are

E_\pm=\pm\sqrt{c^2\mathbf p^2+m^2c^4},

with two independent spin states on each branch. After quantization, the field expansion contains particle annihilation operators and antiparticle creation operators. Both types of physical excitation have positive energy; antiparticles are not physical particles carrying negative energy. (damtp.cam.ac.uk)

Consistent relativistic quantization uses anticommutation relations and produces fermions obeying Fermi–Dirac statistics. This connection belongs to the spin–statistics theorem, rather than following from the unquantized differential equation alone. (damtp.cam.ac.uk)

Electromagnetic coupling and the nonrelativistic limit

For a particle of electric charge $q$ in prescribed electromagnetic potentials, minimal coupling gives the SI-unit form

i\hbar\frac{\partial\psi}{\partial t} = \left[ c\boldsymbol{\alpha}\cdot \left(-i\hbar\nabla-q\mathbf A\right) +\beta mc^2+q\phi \right]\psi.

Here $\phi$ is the scalar potential and $\mathbf A$ the vector potential. This interaction is compatible with the local phase symmetry underlying gauge theory. Quantizing the electromagnetic field as well leads to quantum electrodynamics (QED). (arxiv.org)

For a massive particle at energies small compared with $mc^2$ , after removing the rest-energy phase, the positive-energy sector reduces to the Pauli equation. Its leading Hamiltonian is

H_{\mathrm P} = \frac{(\mathbf p-q\mathbf A)^2}{2m} +q\phi -\frac{q\hbar}{2m}\boldsymbol{\sigma}\cdot\mathbf B,

where $\mathbf B$ is the magnetic field. The magnetic term predicts a spin magnetic moment with $g=2$ . Higher-order terms include the relativistic kinetic-energy correction, spin–orbit interaction, and Darwin term. Systematic separation of particle and antiparticle sectors can be performed through a Foldy–Wouthuysen transformation. (ocw.mit.edu)

Applications and limits

In a Coulomb potential, the equation accounts for the principal relativistic contributions to the fine structure of hydrogen and hydrogen-like atomic spectra. However, the basic one-electron equation does not include radiative corrections such as the Lamb shift or the electron’s anomalous magnetic moment. These require QED beyond the minimally coupled Dirac equation. Nuclear motion, nuclear structure, and many-electron interactions introduce further corrections. (damtp.cam.ac.uk)

A fixed-particle-number interpretation is useful when particle creation can be neglected. It is not a complete description when pair production or annihilation becomes important: those processes require field theory, even though the Dirac equation remains part of the field dynamics. (damtp.cam.ac.uk)

The equation also has effective analogues in condensed-matter physics. Low-energy electronic excitations in graphene obey a two-dimensional, approximately massless Dirac equation. Its characteristic velocity is a material-dependent Fermi velocity rather than $c$ , and the relevant spinor components can encode sublattice degrees of freedom rather than the electron’s physical spin. This analogy concerns the excitations’ dynamics, not a disappearance of the electron’s fundamental rest mass. (arxiv.org)

Related formulations and extensions

A four-component Dirac spinor can be decomposed into left- and right-chiral two-component fields. A Dirac mass couples these components; when the mass vanishes, the equations separate into Weyl equations. A Majorana field instead obeys a reality condition identifying the field with its charge conjugate, so its particle is its own antiparticle. These constructions share spinor mathematics but differ in their independent degrees of freedom and possible charges. (damtp.cam.ac.uk)

On curved spacetime, the equation is generalized using local orthonormal frames and a spin connection. The latter enters a covariant derivative appropriate to spinors, allowing their dynamics to be formulated consistently with general relativity. (damtp.cam.ac.uk)

References

Paul A.M. Dirac – Factsnobelprize.org
The Nobel Prizes in Physics 1901-2000nobelprize.org
Paul A.M. Dirac – Biographicalnobelprize.org
The Quantised Worldeducationalgames.nobelprize.org
Paul A. M. Dirac – Nobel Lecturenobelprize.org
Quantum Physics III Chapter 2: Hydrogen Fine Structureocw.mit.edu
Exact correspondence between classical and Dirac-Pauli spinors in the weak-field limit of static and homogeneous electromagnetic fieldsarxiv.org