Editing Relativistic wave equations (section)

== Linear equations ==

{{further|Linear differential equation}}

The following equations have solutions which satisfy the [[superposition principle]], that is, the wave functions are [[additive map|additive]].

Throughout, the standard conventions of [[tensor index notation]] and [[Feynman slash notation]] are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wave functions are denoted ''{{math|ψ}}'', and {{math|∂<sub>''μ''</sub>}} are the components of the [[four-gradient]] operator.

In [[matrix (mathematics)|matrix]] equations, the [[Pauli matrices]] are denoted by {{math|''σ<sup>μ</sup>''}} in which {{math|1=''μ'' = 0, 1, 2, 3}}, where {{math|''σ''<sup>0</sup>}} is the {{math|2 × 2}} [[identity matrix]]:
<math display="block">\sigma^0 = \begin{pmatrix} 1&0 \\ 0&1 \\ \end{pmatrix} </math>
and the other matrices have their usual representations. The expression
<math display="block">\sigma^\mu \partial_\mu \equiv \sigma^0 \partial_0 + \sigma^1 \partial_1 + \sigma^2 \partial_2 + \sigma^3 \partial_3 </math>
is a {{math|2 × 2}} matrix [[Operator (mathematics)|operator]] which acts on 2-component spinor fields.

The [[gamma matrices]] are denoted by {{math|''γ''<sup>''μ''</sup>}}, in which again {{math|''μ'' {{=}} 0, 1, 2, 3}}, and there are a number of representations to select from. The matrix {{math|''γ''<sup>0</sup>}} is ''not'' necessarily the {{math|4 × 4}} identity matrix. The expression
<math display="block">i\hbar \gamma^\mu \partial_\mu + mc \equiv i\hbar(\gamma^0 \partial_0 + \gamma^1 \partial_1 + \gamma^2 \partial_2 + \gamma^3 \partial_3) + mc \begin{pmatrix}1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{pmatrix} </math>
is a {{math|4 × 4}} [[matrix (mathematics)|matrix]] [[Operator (mathematics)|operator]] which acts on 4-component [[spinor field]]s.

Note that terms such as "{{math|''mc''}}" [[scalar multiplication|scalar multiply]] an [[identity matrix]] of the relevant [[Dimension (vector space)|dimension]], the common sizes are {{math|2 × 2}} or {{math|4 × 4}}, and are ''conventionally'' not written for simplicity.

{| class="wikitable"
|-
! scope="col" width="100px" | Particle [[spin quantum number]] ''s''
! scope="col" width="200px" | Name
! scope="col" width="300px" | Equation
! scope="col" width="200px" | Typical particles the equation describes
|-valign="top" 
| 0
| [[Klein–Gordon equation]]
| <math>(\hbar \partial_{\mu} + imc)(\hbar \partial^{\mu} -imc)\psi = 0</math>
| Massless or massive spin-0 particle (such as [[Higgs boson]]s).
|-valign="top" 
|scope="row" rowspan="5"| 1/2
| [[Weyl equation]] 
| <math> \sigma^\mu\partial_\mu \psi=0</math>
| Massless spin-1/2 particles.
|-valign="top" 
| [[Dirac equation]]
| <math>\left( i \hbar \partial\!\!\!/ - m c \right) \psi = 0 </math>
| Massive spin-1/2 particles (such as electrons).
|-valign="top" 
| [[Two-body Dirac equations]]
| <math>[(\gamma_1)_\mu (p_1-\tilde{A}_1)^\mu+m_1 + \tilde{S}_1]\Psi=0,</math>{{br}}
<math>[(\gamma_2)_\mu (p_2-\tilde{A}_2)^\mu+m_2 + \tilde{S}_2]\Psi=0.</math>
| Massive spin-1/2 particles (such as electrons).
|-valign="top" 
|[[Majorana equation]]
| <math> i \hbar \partial\!\!\!/ \psi - m c \psi_c = 0</math>
| Massive [[Majorana particle]]s.
|-valign="top" 
|[[Breit equation]]
|<math> i\hbar\frac{\partial \Psi}{\partial t} = \left(\sum_{i}\hat{H}_{D}(i) + \sum_{i>j}\frac{1}{r_{ij}} - \sum_{i>j}\hat{B}_{ij} \right) \Psi </math>
| Two massive spin-1/2 particles (such as [[electron]]s) interacting electromagnetically to first order in perturbation theory.
|-valign="top" 
|scope="row" rowspan="2"| 1
| [[Maxwell's equations]] (in [[Quantum electrodynamics#Equations of motion|QED]] using the [[Lorenz gauge]])
|<math>\partial_\mu\partial^\mu A^\nu = e \overline{\psi} \gamma^\nu \psi </math>
| [[Photon]]s, massless spin-1 particles.
|-valign="top" 
|[[Proca equation]]
|<math>\partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+\left(\frac{mc}{\hbar}\right)^2 A^\nu=0</math>
| Massive spin-1 particle (such as [[W and Z bosons]]).
|-valign="top" 
|3/2
|[[Rarita–Schwinger equation]]
|<math> \epsilon^{\mu \nu \rho \sigma} \gamma^5 \gamma_\nu \partial_\rho \psi_\sigma + m\psi^\mu = 0</math>
| Massive spin-3/2 particles.
|-valign="top" 
|scope="row" rowspan="2"|''s''
|[[Bargmann–Wigner equations]]
|<math>\begin{align}
(-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2s}} &= 0 \\
(-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2s}} &= 0 \\
&\;\; \vdots \\
(-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_{2s} \alpha'_{2s}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2s}} &= 0
\end{align}</math>{{br}}
where {{math|''ψ''}} is a rank-2''s'' 4-component [[spinor]].
|Free particles of arbitrary spin (bosons and fermions).<ref name="E.A. Jeffery 1978"/><ref>
{{cite news
 |author      = R.Clarkson, D.G.C. McKeon
 |year        = 2003
 |title       = Quantum Field Theory
 |pages       = 61–69
 |url         = http://www.apmaths.uwo.ca/people/QFTNotesAll.pdf
 |url-status  = dead
 |archive-url = https://web.archive.org/web/20090530181029/http://www.apmaths.uwo.ca/people/QFTNotesAll.pdf
 |archive-date = 2009-05-30
}}</ref>
|-
|[[Joos–Weinberg equation]]
| <math> [(i\hbar )^{2s}\gamma ^{\mu _{1}\mu _{2}\cdots \mu _{2s}}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{2s}}+(mc)^{2s}]\psi =0</math>
|Free particles of arbitrary spin (bosons and fermions).
|-
|}

=== Linear gauge fields ===

The [[Duffin–Kemmer–Petiau algebra#Duffin–Kemmer–Petiau equation|Duffin–Kemmer–Petiau equation]] is an alternative equation for spin-0 and spin-1 particles:
<math display="block">(i \hbar \beta^{a} \partial_a - m c) \psi = 0</math>