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Revision as of 11:31, 22 May 2026

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Dirac equation is the relativistic wave equation for spin-12 particles such as the electron. Introduced by Paul Dirac in 1928, it joined quantum mechanics with special relativity in a first-order wave equation and explained electron spin as a natural part of the theory rather than as an added rule.[1]

The equation acts on a four-component Dirac spinor and uses gamma matrices to preserve Lorentz covariance. Its solutions include positive- and negative-energy sectors, leading historically to the prediction of antimatter and the later discovery of the positron.[2] In the Quantum Collection it connects nonrelativistic wave mechanics, spin, the Pauli equation, the Klein-Gordon equation, and the transition toward quantum field theory.

Quantum Dirac equation.

Mathematical formulation

In covariant form, the Dirac equation is

Dirac equation
(iℏγμ∂μ−mc)ψ(x)=0

and in natural units ℏ=c=1,

Dirac equation (natural units)
(iγμ∂μ−m)ψ(x)=0

Here ψ is a four-component spinor and the gamma matrices satisfy the anticommutation relation

{γμ,γν}=2ημνI4.

This algebraic structure ensures Lorentz covariance and makes the equation first order in both space and time derivatives.[1][3]

Why the Dirac equation was needed

The nonrelativistic Schrödinger equation works well at low velocities, but it does not incorporate special relativity. A naive relativistic replacement leads to the Klein–Gordon equation, which is second order in time and does not naturally describe spin-12 electrons.[4]

Dirac’s key insight was to seek an equation linear in both the time and spatial derivatives. This required introducing matrix coefficients acting on a multi-component wavefunction. The resulting formalism explained electron spin from first principles rather than inserting it phenomenologically.[1][5]

Spinors, spin, and antimatter

A Dirac wavefunction has four complex components, often interpreted as encoding two spin states and positive- versus negative-energy sectors. In the nonrelativistic limit, the upper two components reduce to the familiar Pauli spinor, while the lower two become small corrections of order v/c.[3][6]

One of the deepest consequences of the equation is the appearance of negative-energy solutions. Historically this led Dirac to propose hole theory and ultimately to the prediction of antimatter. The later experimental discovery of the positron confirmed this remarkable implication.[2][7]

Relation to other equations

The Dirac equation contains several important limiting cases and connections:

  • In the low-energy limit it reduces to the Pauli equation, and then further to the Schrödinger equation when spin effects are neglected.[3]
  • Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.[8]
  • In the massless case the equation reduces to the Weyl equation, relevant for chiral fermions.[9]

These links make the Dirac equation a central bridge between nonrelativistic quantum mechanics and modern quantum field theory.

Conserved current and symmetry

The Dirac equation admits a conserved current

Jμ=ψ¯γμψ,

where the Dirac adjoint is defined by

ψ¯=ψ†γ0.

The conservation law

∂μJμ=0

follows directly from the Dirac equation and reflects a global U(1) symmetry of the theory.[8][10]

This symmetry becomes especially important in field theory, where replacing ∂μ by a covariant derivative Dμ produces the coupling to the electromagnetic field and leads directly to quantum electrodynamics.[11]

Lagrangian form

The Dirac equation can be derived from the Lagrangian density

ℒ=iℏcψ¯γμ∂μψ−mc2ψ¯ψ.

In natural units, the corresponding action is

Dirac action
S=∫d4xψ¯(iγμ∂μ−m)ψ

This formulation makes the symmetry structure of the theory transparent and is the natural starting point for relativistic quantum field theory.[3][8]

Physical significance

The Dirac equation is one of the great achievements of theoretical physics because it unified quantum mechanics with special relativity, explained intrinsic spin, predicted antimatter, and laid the groundwork for fermionic quantum field theory.[12][13]

In modern physics it is interpreted not merely as a single-particle wave equation, but as the field equation for spin-12 fermion fields such as electrons and quarks. It therefore stands at the foundation of both QED and the broader framework of the Standard Model.[10][11]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 Dirac, P. A. M. (1928). "The Quantum Theory of the Electron". Proceedings of the Royal Society A 117 (778): 610-624. doi:10.1098/rspa.1928.0023. Bibcode: 1928RSPSA.117..610D. 
  2. ↑ 2.0 2.1 Anderson, Carl D. (1933). "The Positive Electron". Physical Review 43 (6): 491. doi:10.1103/PhysRev.43.491. Bibcode: 1933PhRv...43..491A. 
  3. ↑ 3.0 3.1 3.2 3.3 Bjorken, J. D.; Drell, S. D. (1964). Relativistic Quantum Mechanics. McGraw-Hill. 
  4. ↑ Rae, Alastair I. M.; Napolitano, Jim (2015). Quantum Mechanics (6th ed.). Routledge. ISBN 978-1482299182. 
  5. ↑ Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum. 
  6. ↑ Schiff, L. I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. 
  7. ↑ Penrose, Roger (2004). The Road to Reality. Jonathan Cape. p. 625. ISBN 0-224-04447-8. 
  8. ↑ 8.0 8.1 8.2 Cite error: Invalid <ref> tag; no text was provided for refs named Thaller1992
  9. ↑ Ohlsson, Tommy (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. p. 86. ISBN 978-1-139-50432-4. 
  10. ↑ 10.0 10.1 Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2. 
  11. ↑ 11.0 11.1 Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 9780471887416. 
  12. ↑ Hey, T.; Walters, P. (2009). The New Quantum Universe. Cambridge University Press. p. 228. ISBN 978-0-521-56457-1. 
  13. ↑ Zichichi, Antonino (2000-03-02). "Dirac, Einstein and physics". https://physicsworld.com/a/dirac-einstein-and-physics/. 


Author: Harold Foppele

Source attribution: Physics:Quantum Dirac equation

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