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{{Short description|Relativistic quantum mechanical wave equation}}
{{Quantum book backlink|Mathematical structure and systems}}

'''Dirac equation''' is the relativistic wave equation for spin-<math>\tfrac{1}{2}</math> particles. Introduced by [[Biography:Paul Dirac|Paul Dirac]] in 1928, it provided the first quantum-mechanical description fully consistent with [[Physics:Special relativity|special relativity]] and successfully accounted for electron spin and the fine structure of the hydrogen spectrum.<ref name="Dirac1928">{{cite journal |last=Dirac |first=P. A. M. |title=The Quantum Theory of the Electron |journal=Proceedings of the Royal Society A |volume=117 |issue=778 |pages=610–624 |year=1928 |doi=10.1098/rspa.1928.0023 |bibcode=1928RSPSA.117..610D |jstor=94981 |doi-access=free}}</ref><ref name="Atkins1974">{{cite book |last=Atkins |first=P. W. |title=Quanta: A Handbook of Concepts |publisher=Oxford University Press |year=1974 |page=52 |isbn=978-0-19-855493-6}}</ref> It also implied the existence of antimatter, later confirmed experimentally through the discovery of the positron.<ref name="Anderson1933">{{cite journal |last=Anderson |first=Carl D. |title=The Positive Electron |journal=Physical Review |volume=43 |issue=6 |page=491 |year=1933 |doi=10.1103/PhysRev.43.491 |bibcode=1933PhRv...43..491A |doi-access=free}}</ref>

The equation acts on a four-component spinor field, or [[Physics:Dirac spinor|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.<ref name="Thaller1992">{{cite book |last=Thaller |first=B. |title=The Dirac Equation |series=Texts and Monographs in Physics |publisher=Springer |year=1992}}</ref>

<div style="float:right; border:1px solid #ccc; padding:4px; background:#f9f9f9; margin:0 0 1em 1em; width:470px;">
[[File:Quantum_Dirac_equation.jpg|450px|Conceptual visualization of the Dirac equation, showing the union of quantum mechanics, special relativity, spinor structure, and antimatter.]]
</div>

== Mathematical formulation ==

In covariant form, the Dirac equation is

{{Equation box 1
|title='''Dirac equation'''
|indent=:
|equation = <math>(i\hbar \gamma^\mu \partial_\mu - mc)\psi(x)=0</math>
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

and in natural units <math>\hbar=c=1</math>,

{{Equation box 1
|title='''Dirac equation (natural units)'''
|indent=:
|equation = <math>(i\gamma^\mu \partial_\mu - m)\psi(x)=0</math>
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

Here <math>\psi</math> is a four-component spinor and the gamma matrices satisfy the anticommutation relation

<math display="block">\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I_4.</math>

This algebraic structure ensures Lorentz covariance and makes the equation first order in both space and time derivatives.<ref name="Dirac1928" /><ref name="BjorkenDrell1964">{{cite book |last1=Bjorken |first1=J. D. |last2=Drell |first2=S. D. |title=Relativistic Quantum Mechanics |publisher=McGraw-Hill |year=1964}}</ref>

== Why the Dirac equation was needed ==

The nonrelativistic [[Physics:Schrödinger equation|Schrödinger equation]] works well at low velocities, but it does not incorporate special relativity. A naive relativistic replacement leads to the [[Physics:Klein–Gordon equation|Klein–Gordon equation]], which is second order in time and does not naturally describe spin-<math>\tfrac{1}{2}</math> electrons.<ref name="RaeNapolitano2015">{{cite book |last1=Rae |first1=Alastair I. M. |last2=Napolitano |first2=Jim |title=Quantum Mechanics |edition=6th |publisher=Routledge |year=2015 |isbn=978-1482299182}}</ref>

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.<ref name="Dirac1928" /><ref name="Shankar1994">{{cite book |last=Shankar |first=R. |title=Principles of Quantum Mechanics |edition=2nd |publisher=Plenum |year=1994}}</ref>

== 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 <math>v/c</math>.<ref name="BjorkenDrell1964" /><ref name="Schiff1968">{{cite book |last=Schiff |first=L. I. |title=Quantum Mechanics |edition=3rd |publisher=McGraw-Hill |year=1968}}</ref>

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.<ref name="Anderson1933" /><ref name="Penrose2004">{{cite book |last=Penrose |first=Roger |title=The Road to Reality |publisher=Jonathan Cape |year=2004 |isbn=0-224-04447-8 |page=625}}</ref>

== Relation to other equations ==

The Dirac equation contains several important limiting cases and connections:

* In the low-energy limit it reduces to the [[Physics:Pauli equation|Pauli equation]], and then further to the Schrödinger equation when spin effects are neglected.<ref name="BjorkenDrell1964" />
* Applying another Dirac operator shows that each spinor component satisfies the relativistic Klein–Gordon equation.<ref name="Thaller1992" />
* In the massless case the equation reduces to the [[Physics:Weyl equation|Weyl equation]], relevant for chiral fermions.<ref name="Ohlsson2011">{{cite book |last=Ohlsson |first=Tommy |title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory |publisher=Cambridge University Press |year=2011 |page=86 |isbn=978-1-139-50432-4}}</ref>

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

<math display="block">J^\mu=\bar{\psi}\gamma^\mu\psi,</math>

where the Dirac adjoint is defined by

<math display="block">\bar{\psi}=\psi^\dagger\gamma^0.</math>

The conservation law

<math display="block">\partial_\mu J^\mu = 0</math>

follows directly from the Dirac equation and reflects a global <math>U(1)</math> symmetry of the theory.<ref name="Thaller1992" /><ref name="Griffiths2008">{{cite book |last=Griffiths |first=D. J. |title=Introduction to Elementary Particles |edition=2nd |publisher=Wiley-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>

This symmetry becomes especially important in field theory, where replacing <math>\partial_\mu</math> by a covariant derivative <math>D_\mu</math> produces the coupling to the electromagnetic field and leads directly to [[Physics:Quantum electrodynamics|quantum electrodynamics]].<ref name="HalzenMartin1984">{{cite book |last1=Halzen |first1=Francis |last2=Martin |first2=Alan |title=Quarks & Leptons: An Introductory Course in Modern Particle Physics |publisher=John Wiley & Sons |year=1984 |isbn=9780471887416}}</ref>

== Lagrangian form ==

The Dirac equation can be derived from the Lagrangian density

<math display="block">\mathcal{L}=i\hbar c\,\bar{\psi}\gamma^\mu\partial_\mu\psi-mc^2\bar{\psi}\psi.</math>

In natural units, the corresponding action is

{{Equation box 1
|title='''Dirac action'''
|indent=:
|equation = <math>S=\int d^4x\,\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi</math>
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

This formulation makes the symmetry structure of the theory transparent and is the natural starting point for relativistic quantum field theory.<ref name="BjorkenDrell1964" /><ref name="Thaller1992" />

== 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.<ref name="HeyWalters2009">{{cite book |last1=Hey |first1=T. |last2=Walters |first2=P. |title=The New Quantum Universe |publisher=Cambridge University Press |year=2009 |page=228 |isbn=978-0-521-56457-1}}</ref><ref name="PhysicsWorld2000">{{cite web |last=Zichichi |first=Antonino |title=Dirac, Einstein and physics |website=Physics World |date=2000-03-02 |url=https://physicsworld.com/a/dirac-einstein-and-physics/ |access-date=2023-10-22}}</ref>

In modern physics it is interpreted not merely as a single-particle wave equation, but as the field equation for spin-<math>\tfrac{1}{2}</math> fermion fields such as electrons and quarks. It therefore stands at the foundation of both [[Physics:Quantum electrodynamics|QED]] and the broader framework of the Standard Model.<ref name="Griffiths2008" /><ref name="HalzenMartin1984" />

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

{{Author|Harold Foppele}}

[[Category:Quantum mechanics]]
[[Category:Relativistic quantum mechanics]]
[[Category:Quantum field theory]]
[[Category:Spinors]]
[[Category:Partial differential equations]]

{{Sourceattribution|Physics:Quantum Dirac equation|1}}

Physics:Quantum Dirac equation - Revision history

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