Quantum Dirac equation

Short description: Relativistic quantum mechanical wave equation

Dirac equation is the relativistic wave equation for spin-12 particles. Introduced by Paul Dirac in 1928, it provided the first quantum-mechanical description fully consistent with special relativity and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.[1][2] It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.[3]

The equation acts on a four-component spinor field, or Dirac spinor, rather than on a single complex wavefunction. In this way it naturally incorporates spin, positive- and negative-energy solutions, and the correct relativistic dispersion relation.[4]

## Contents

* 1 Mathematical formulation
* 2 Why the Dirac equation was needed
* 3 Spinors, spin, and antimatter
* 4 Relation to other equations
* 5 Conserved current and symmetry
* 6 Lagrangian form
* 7 Physical significance
* 8 See also
* 9 Table of contents (185 articles)
* 10 Index
* 11 Full contents
* 11.1 References

## 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][5]

## 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.[6]

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][7]

## 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.[5][8]

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.[3][9]

## 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.[5]
* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.[4]
* In the massless case the equation reduces to the Weyl equation, relevant for chiral fermions.[10]

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.[4][11]

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.[12]

## 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.[5][4]

## 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.[13][14]

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.[11][12]

## See also

# Table of contents (185 articles)

# Index

Core theory

1\. Foundations

2\. Conceptual and interpretations

3\. Mathematical structure and systems

4\. Atomic and spectroscopy

5\. Wavefunctions and modes

6\. Quantum dynamics and evolution

7\. Measurement and information

8\. Quantum information and computing

Applications and extensions

9\. Quantum optics and experiments

10\. Open quantum systems

11\. Quantum field theory

12\. Statistical mechanics and kinetic theory

13\. Condensed matter and solid-state physics

14\. Plasma and fusion physics

15\. Timeline

16\. Advanced and frontier topics

Book II

* Matter by scale

Book III

* Methods and tools

Book IV

* Data Analysis Techniques

# Full contents

1\. Foundations (11) ↑ Back to index

1\. Physics:Quantum basics

2\. Physics:Quantum Postulates

3\. Physics:Quantum Hilbert space

4\. Physics:Quantum Observables and operators

5\. Physics:Quantum mechanics

6\. Physics:Quantum mechanics measurements

7\. Physics:Quantum state

8\. Physics:Quantum system

9\. Physics:Quantum superposition

10\. Physics:Quantum probability

11\. Physics:Quantum Mathematical Foundations of Quantum Theory

2\. Conceptual and interpretations (14) ↑ Back to index

12\. Physics:Quantum Interpretations of quantum mechanics

13\. Physics:Quantum Wave–particle duality

14\. Physics:Quantum Complementarity principle

15\. Physics:Quantum Uncertainty principle

16\. Physics:Quantum Measurement problem

17\. Physics:Quantum Bell's theorem

18\. Physics:Quantum Hidden variable theory

19\. Physics:Quantum nonlocality

20\. Physics:Quantum contextuality

21\. Physics:Quantum Darwinism

22\. Physics:Quantum A Spooky Action at a Distance

23\. Physics:Quantum A Walk Through the Universe

24\. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

25\. Physics:Quantum measurement problem

3\. Mathematical structure and systems (13) ↑ Back to index

26\. Physics:Quantum Density matrix

27\. Physics:Quantum Exactly solvable quantum systems

28\. Physics:Quantum Formulas Collection

29\. Physics:Quantum A Matter Of Size

30\. Physics:Quantum Symmetry in quantum mechanics

31\. Physics:Quantum Angular momentum operator

32\. Physics:Quantum Runge–Lenz vector

33\. Physics:Quantum Approximation Methods

34\. Physics:Quantum Matter Elements and Particles

35\. Physics:Quantum Dirac equation

36\. Physics:Quantum Klein–Gordon equation

37\. Physics:Quantum pendulum

38\. Physics:Quantum configuration space

4\. Atomic and spectroscopy (14) ↑ Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

39\. Physics:Quantum Atomic structure and spectroscopy

40\. Physics:Quantum Hydrogen atom

41\. Physics:Quantum number

42\. Physics:Quantum Multi-electron atoms

43\. Physics:Quantum Fine structure

44\. Physics:Quantum Hyperfine structure

45\. Physics:Quantum Isotopic shift

46\. Physics:Quantum defect

47\. Physics:Quantum Zeeman effect

48\. Physics:Quantum Stark effect

49\. Physics:Quantum Spectral lines and series

50\. Physics:Quantum Selection rules

51\. Physics:Quantum Fermi's golden rule

52\. Physics:Quantum beats

5\. Wavefunctions and modes (9) ↑ Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

53\. Physics:Quantum Wavefunction

54\. Physics:Quantum Superposition principle

55\. Physics:Quantum Eigenstates and eigenvalues

56\. Physics:Quantum Boundary conditions and quantization

57\. Physics:Quantum Standing waves and modes

58\. Physics:Quantum Normal modes and field quantization

59\. Physics:Number of independent spatial modes in a spherical volume

60\. Physics:Quantum Density of states

61\. Physics:Quantum carpet

6\. Quantum dynamics and evolution (17) ↑ Back to index

62\. Physics:Quantum Time evolution

63\. Physics:Quantum Schrödinger equation

64\. Physics:Quantum Time-dependent Schrödinger equation

65\. Physics:Quantum Stationary states

66\. Physics:Quantum Perturbation theory

67\. Physics:Quantum Time-dependent perturbation theory

68\. Physics:Quantum Adiabatic theorem

69\. Physics:Quantum Scattering theory

70\. Physics:Quantum S-matrix

71\. Physics:Quantum tunnelling

72\. Physics:Quantum speed limit

73\. Physics:Quantum revival

74\. Physics:Quantum reflection

75\. Physics:Quantum oscillations

76\. Physics:Quantum jump

77\. Physics:Quantum boomerang effect

78\. Physics:Quantum chaos

7\. Measurement and information (9) ↑ Back to index

79\. Physics:Quantum Measurement theory

80\. Physics:Quantum Measurement operators

81\. Physics:Quantum Projective measurement

82\. Physics:Quantum POVM

83\. Physics:Quantum Weak measurement

84\. Physics:Quantum Measurement collapse

85\. Physics:Quantum entanglement

86\. Physics:Quantum Zeno effect

87\. Physics:Quantum limit

8\. Quantum information and computing (10) ↑ Back to index

88\. Physics:Quantum information theory

89\. Physics:Quantum Qubit

90\. Physics:Quantum Entanglement

91\. Physics:Quantum Gates and circuits

92\. Physics:Quantum Computing Algorithms in the NISQ Era

93\. Physics:Quantum Noisy Qubits

94\. Physics:Quantum random access code

95\. Physics:Quantum pseudo-telepathy

96\. Physics:Quantum network

97\. Physics:Quantum money

9\. Quantum optics and experiments (5) ↑ Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

98\. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

99\. Physics:Quantum optics beam splitter experiments

100\. Physics:Quantum Ultra fast lasers

101\. Physics:Quantum Experimental quantum physics

102\. Physics:Quantum optics

103\. Template:Quantum optics operators

10\. Open quantum systems (9) ↑ Back to index

104\. Physics:Quantum Open systems

105\. Physics:Quantum Master equation

106\. Physics:Quantum Lindblad equation

107\. Physics:Quantum Decoherence

108\. Physics:Quantum dissipation

109\. Physics:Quantum Markov semigroup

110\. Physics:Quantum Markovian dynamics

111\. Physics:Quantum Non-Markovian dynamics

112\. Physics:Quantum Trajectories

11\. Quantum field theory (20) ↑ Back to index

Structural dependency map of quantum field theory.

113\. Physics:Quantum field theory (QFT) basics

114\. Physics:Quantum field theory (QFT) core

115\. Physics:Quantum Fields and Particles

116\. Physics:Quantum Second quantization

117\. Physics:Quantum Harmonic Oscillator field modes

118\. Physics:Quantum Creation and annihilation operators

119\. Physics:Quantum vacuum fluctuations

120\. Physics:Quantum Propagators in quantum field theory

121\. Physics:Quantum Feynman diagrams

122\. Physics:Quantum Path integral formulation

123\. Physics:Quantum Renormalization in field theory

124\. Physics:Quantum Renormalization group

125\. Physics:Quantum Field Theory Gauge symmetry

126\. Physics:Quantum Non-Abelian gauge theory

127\. Physics:Quantum Electrodynamics (QED)

128\. Physics:Quantum chromodynamics (QCD)

129\. Physics:Quantum Electroweak theory

130\. Physics:Quantum Standard Model

131\. Physics:Quantum triviality

132\. Physics:Quantum confinement problem

12\. Statistical mechanics and kinetic theory (9) ↑ Back to index

133\. Physics:Quantum Statistical mechanics

134\. Physics:Quantum Partition function

135\. Physics:Quantum Distribution functions

136\. Physics:Quantum Liouville equation

137\. Physics:Quantum Kinetic theory

138\. Physics:Quantum Boltzmann equation

139\. Physics:Quantum BBGKY hierarchy

140\. Physics:Quantum Relaxation and thermalization

141\. Physics:Quantum Thermodynamics

13\. Condensed matter and solid-state physics (13) ↑ Back to index

142\. Physics:Quantum Band structure

143\. Physics:Quantum Fermi surfaces

144\. Physics:Quantum Semiconductor physics

145\. Physics:Quantum Phonons

146\. Physics:Quantum Electron-phonon interaction

147\. Physics:Quantum Superconductivity

148\. Physics:Quantum Topological phases of matter

149\. Physics:Quantum well

150\. Physics:Quantum spin liquid

151\. Physics:Quantum spin Hall effect

152\. Physics:Quantum phase transition

153\. Physics:Quantum critical point

154\. Physics:Quantum dot

14\. Plasma and fusion physics (8) ↑ Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

155\. Physics:Quantum Fusion reactions and Lawson criterion

156\. Physics:Quantum Plasma (fusion context)

157\. Physics:Quantum Magnetic confinement fusion

158\. Physics:Quantum Inertial confinement fusion

159\. Physics:Quantum Plasma instabilities and turbulence

160\. Physics:Quantum Tokamak core plasma

161\. Physics:Quantum Tokamak edge physics and recycling asymmetries

162\. Physics:Quantum Stellarator

15\. Timeline (8) ↑ Back to index

163\. Physics:Quantum mechanics/Timeline

164\. Physics:Quantum mechanics/Timeline/Pre-quantum era

165\. Physics:Quantum mechanics/Timeline/Old quantum theory

166\. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

167\. Physics:Quantum mechanics/Timeline/Quantum field theory era

168\. Physics:Quantum mechanics/Timeline/Quantum information era

169\. Physics:Quantum mechanics/Timeline/Quantum technology era

170\. Physics:Quantum mechanics/Timeline/Quiz

16\. Advanced and frontier topics (16) ↑ Back to index

171\. Physics:Quantum topology

172\. Physics:Quantum battery

173\. Physics:Quantum Supersymmetry

174\. Physics:Quantum Black hole thermodynamics

175\. Physics:Quantum Holographic principle

176\. Physics:Quantum gravity

177\. Physics:Quantum De Sitter invariant special relativity

178\. Physics:Quantum Doubly special relativity

179\. Physics:Quantum arithmetic geometry

180\. Physics:Quantum unsolved problems

181\. Physics:Quantum Yang-Mills mass gap

182\. Physics:Quantum gravity problem

183\. Physics:Quantum black hole information paradox

184\. Physics:Quantum dark matter problem

185\. Physics:Quantum neutrino mass problem

186\. Physics:Quantum matter-antimatter asymmetry problem

## 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. ↑ Atkins, P. W. (1974). Quanta: A Handbook of Concepts. Oxford University Press. p. 52. ISBN 978-0-19-855493-6.
3. ↑ 3.0 3.1 Anderson, Carl D. (1933). "The Positive Electron". Physical Review 43 (6): 491. doi:10.1103/PhysRev.43.491. Bibcode: 1933PhRv...43..491A.
4. ↑ 4.0 4.1 4.2 4.3 Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer.
5. ↑ 5.0 5.1 5.2 5.3 Bjorken, J. D.; Drell, S. D. (1964). Relativistic Quantum Mechanics. McGraw-Hill.
6. ↑ Rae, Alastair I. M.; Napolitano, Jim (2015). Quantum Mechanics (6th ed.). Routledge. ISBN 978-1482299182.
7. ↑ Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum.
8. ↑ Schiff, L. I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
9. ↑ Penrose, Roger (2004). The Road to Reality. Jonathan Cape. p. 625. ISBN 0-224-04447-8.
10. ↑ 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.
11. ↑ 11.0 11.1 Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.
12. ↑ 12.0 12.1 Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 9780471887416.
13. ↑ Hey, T.; Walters, P. (2009). The New Quantum Universe. Cambridge University Press. p. 228. ISBN 978-0-521-56457-1.
14. ↑ Zichichi, Antonino (2000-03-02). "Dirac, Einstein and physics". https://physicsworld.com/a/dirac-einstein-and-physics/.

Author: Harold Foppele

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