| Dirac equation | |
|---|---|
| Name | Paul Dirac (originator) |
| Caption | Paul Dirac formulated the equation in 1928 |
| Birth date | 8 August 1902 |
| Death date | 20 October 1984 |
| Nationality | British |
| Fields | Theoretical physics; Quantum mechanics; Quantum field theory |
| Institutions | University of Cambridge; Niels Bohr Institute |
| Notable works | The Principles of Quantum Mechanics; Dirac equation |
Dirac equation
The Dirac equation is a relativistic wave equation formulated by Paul Dirac in 1928 that describes spin‑1/2 particles consistent with both quantum mechanics and special relativity. It unifies the concepts of particle spin and relativistic energy–momentum relations, predicts the existence of antiparticles, and underpins the relativistic treatment of electrons in quantum electrodynamics and modern quantum field theory.
Dirac derived his equation aiming to reconcile the non‑relativistic Schrödinger equation with the relativistic energy relation from Einstein's special relativity and the linearity in time evolution required by quantum theory. Influences include earlier work on relativistic wave equations such as the Klein–Gordon equation and investigations by contemporaries at the University of Cambridge and the Niels Bohr Institute, including correspondence with Werner Heisenberg and Wolfgang Pauli. The equation provided a first‑principles explanation for the intrinsic angular momentum (spin) previously introduced phenomenologically by Pauli and clarified the algebraic role of matrices later recognized as gamma matrices.
The Dirac equation in natural units (ℏ = c = 1) for a free particle is (iγ^μ ∂_μ − m)ψ = 0, where γ^μ are the gamma matrices satisfying the Clifford algebra {γ^μ, γ^ν} = 2η^{μν} and η^{μν} is the Minkowski metric. The field ψ is a four‑component spinor transforming under the spinor representation of the Lorentz group. Dirac introduced operators for energy and momentum consistent with canonical quantization and used matrix algebra to linearize the relativistic dispersion E^2 = p^2 + m^2. The equation admits minimal coupling to electromagnetic potentials via the substitution p_μ → p_μ − eA_μ, leading to the Dirac equation in an external electromagnetic field and forming the basis for quantum electrodynamics.
The Dirac equation predicts intrinsic spin‑1/2 and a magnetic moment for the electron in agreement with measured values up to radiative corrections computed in Richard Feynman's QED framework. A remarkable prediction was negative‑energy solutions, interpreted by Dirac as a filled "sea" of negative states (the Dirac sea), which led to the prediction of the positron; subsequent discovery by Carl Anderson provided experimental confirmation. The equation also encodes relativistic effects such as spin–orbit coupling and the fine structure of hydrogen, and implies phenomena like Zitterbewegung and particle–antiparticle pair creation in strong fields.
Plane‑wave solutions ψ(x) ∝ u(p)e^{-ip·x} and ψ(x) ∝ v(p)e^{ip·x} describe particle and antiparticle states; the spinors u and v are eigenstates of the Dirac Hamiltonian and can be expressed in different bases (Dirac, Weyl/chiral, Majorana). The Weyl representation highlights chiral components and is central to the theory of massless fermions and the electroweak sector of the Standard Model. The Majorana representation makes real spinors manifest and is used in discussions of charge‑conjugation and Majorana fermions. Solutions in external potentials include the hydrogenic bound states solved by applying separation of variables and angular momentum algebra; these yield corrections to spectral lines computed with methods developed by Enrico Fermi and others.
In quantum field theory, the Dirac field is promoted to an operator field and second quantized so that particle number is not fixed and creation/annihilation operators satisfy anticommutation relations. This formalism underlies quantum electrodynamics and the perturbative computation of radiative corrections via Feynman diagrams introduced by Freeman Dyson and Julian Schwinger. Charge conjugation, parity, and time reversal symmetries (C, P, T) are represented on the Dirac field and their combinations (e.g., CPT theorem) are fundamental results in relativistic QFT. The Dirac Lagrangian density L = ψ̄(iγ^μ D_μ − m)ψ couples fermions to gauge fields in Yang–Mills theory and the Standard Model of particle physics.
The Dirac equation is essential in atomic physics for calculating fine structure and Lamb shift corrections (with input from QED), in solid‑state physics for effective descriptions of quasiparticles in materials such as graphene where low‑energy excitations behave as massless Dirac fermions, and in accelerator physics for spin dynamics. Precision tests include measurements of the electron magnetic moment (g‑2) at institutions like Harvard University and CERN, spectroscopy of hydrogen and muonic atoms, and antimatter studies at facilities such as CERN's Antiproton Decelerator. Applications extend to relativistic quantum chemistry methods used in computational packages and to modeling neutrino behavior in the electroweak theory where chiral Dirac or Majorana mass terms are analyzed by collaborations like those at Fermi National Accelerator Laboratory and Institute for Nuclear Research.
Category:Quantum mechanics Category:Quantum field theory Category:Paul Dirac