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{{Short description|Wave equations respecting special and general relativity}}
{{Redirect|Relativistic quantum field equations|related concepts|Relativistic quantum mechanics|and|Schrödinger field}}
{{Quantum mechanics|cTopic=Equations}}
{{quantum field theory}}

In [[physics]], specifically [[relativistic quantum mechanics]] (RQM) and its applications to [[particle physics]], '''relativistic wave equations''' predict the behavior of [[particles]] at high [[energy|energies]] and [[velocity|velocities]] comparable to the [[speed of light]]. In the context of [[quantum field theory]] (QFT), the equations determine the dynamics of [[quantum field]]s.
The solutions to the equations, universally denoted as {{math|ψ}} or {{math|Ψ}} ([[Greek language|Greek]] [[Psi (letter)|psi]]), are referred to as "[[wave function]]s" in the context of RQM, and "[[field (physics)|field]]s" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a [[wave equation]] or are generated from a [[Lagrangian density]] and the field-theoretic [[Euler–Lagrange equation]]s (see [[classical field theory]] for background).

In the [[Schrödinger picture]], the wave function or field is the solution to the [[Schrödinger equation]];
<math display="block"> i\hbar\frac{\partial}{\partial t}\psi = \hat{H} \psi</math>
one of the [[Mathematical formulation of quantum mechanics#Pictures of dynamics|postulates of quantum mechanics]]. All relativistic wave equations can be constructed by specifying various forms of the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]] ''Ĥ'' describing the [[Physical system|quantum system]]. Alternatively, [[Richard Feynman|Feynman]]'s [[path integral formulation]] uses a Lagrangian rather than a Hamiltonian operator.

More generally – the modern formalism behind relativistic wave equations is [[Lorentz group]] theory, wherein the spin of the particle has a correspondence with the [[representations of the Lorentz group]].<ref name="T Jaroszewicz, P.S Kurzepa">
{{cite journal
 |author1=T Jaroszewicz |author2=P.S Kurzepa | year = 1992
 | title = Geometry of spacetime propagation of spinning particles 
 | journal = Annals of Physics
 |volume=216 |issue=2 |pages=226–267 | doi=10.1016/0003-4916(92)90176-M
 |bibcode=1992AnPhy.216..226J
}}</ref>

== History ==

=== Early 1920s: Classical and quantum mechanics ===

The failure of [[classical mechanics]] applied to [[molecule|molecular]], [[atom]]ic, and [[Atomic nucleus|nuclear]] systems and smaller induced the need for a new mechanics: ''[[quantum mechanics]]''. The mathematical formulation was led by [[Louis de Broglie|De Broglie]], [[Niels Bohr|Bohr]], [[Erwin Schrödinger|Schrödinger]], [[Wolfgang Pauli|Pauli]], and [[Werner Heisenberg|Heisenberg]], and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The Schrödinger equation and the [[Heisenberg picture]] resemble the classical [[equations of motion]] in the limit of large [[quantum number]]s and as the reduced [[Planck constant]] {{math|''ħ''}}, the quantum of [[action (physics)|action]], tends to zero. This is the [[correspondence principle]]. At this point, [[special relativity]] was not fully combined with quantum mechanics, so the Schrödinger and Heisenberg formulations, as originally proposed, could not be used in situations where the particles travel near the [[speed of light]], or when the number of each type of particle changes (this happens in real [[fundamental interaction|particle interaction]]s; the numerous forms of [[particle decay]]s, [[annihilation]], [[matter creation]], [[pair production]], and so on).

=== Late 1920s: Relativistic quantum mechanics of spin-0 and spin-<sup>1</sup>/<sub>2</sub> particles ===

A description of quantum mechanical systems which could account for ''relativistic'' effects was sought for by many theoretical physicists from the late 1920s to the mid-1940s.<ref name="Esposito">{{cite journal | author = S. Esposito | year = 2011 | title = Searching for an equation: Dirac, Majorana and the others | arxiv = 1110.6878 | doi=10.1016/j.aop.2012.02.016 | volume=327 | journal=Annals of Physics | issue = 6 | pages=1617–1644| bibcode=2012AnPhy.327.1617E | s2cid = 119147261 }}</ref> The first basis for [[relativistic quantum mechanics]], i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the [[Klein–Gordon equation]]:
{{NumBlk||<math display="block">-\hbar^2\frac{\partial^2 \psi}{\partial t^2} +(\hbar c)^2\nabla^2\psi = (mc^2)^2\psi \,,</math>|{{EquationRef|1}}}}
by inserting the [[energy operator]] and [[momentum operator]] into the relativistic [[energy–momentum relation]]:
{{NumBlk||<math display="block">E^2 - (pc)^2 = (mc^2)^2\,,</math>|{{EquationRef|2}}}}

The solutions to ({{EquationNote|1}}) are [[scalar field]]s. The KG equation is undesirable due to its prediction of ''negative'' [[energy|energies]] and [[probability|probabilities]], as a result of the [[quadratic equation|quadratic]] nature of ({{EquationNote|2}}) – inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the [[Schrödinger equation]]) was still of importance. Nevertheless, ({{EquationNote|1}}) is applicable to spin-0 [[boson]]s.<ref>{{cite book|title = Particle Physics |url = https://archive.org/details/particlephysics00mart |url-access = limited | edition = 3rd | author = B. R. Martin, G.Shaw | series =  Manchester Physics Series|publisher = John Wiley & Sons|year = 2008| page = [https://archive.org/details/particlephysics00mart/page/n24 3]|isbn = 978-0-470-03294-7}}</ref>

Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the [[fine structure]] in the [[Hydrogen spectral series]]. The mysterious underlying property was ''spin''. The first two-dimensional ''spin matrices'' (better known as the [[Pauli matrices]]) were introduced by Pauli in the [[Pauli equation]]; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in [[magnetic field]]s, but this was ''phenomenological''. [[Hermann Weyl|Weyl]] found a relativistic equation in terms of the Pauli matrices; the [[Weyl equation]], for ''massless'' spin-{{sfrac|1|2}} fermions. The problem was resolved by [[Paul Dirac|Dirac]] in the late 1920s, when he furthered the application of equation ({{EquationNote|2}}) to the [[electron]] – by various manipulations he factorized the equation into the form:
{{NumBlk||<math display="block">
\left(\frac{E}{c} - \boldsymbol{\alpha}\cdot\mathbf{p} - \beta mc \right)\left(\frac{E}{c} + \boldsymbol{\alpha}\cdot\mathbf{p} + \beta mc \right)\psi=0 \,,
</math>|{{EquationRef|3A}}}}
and one of these factors is the [[Dirac equation]] (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices {{math|'''α'''}}  and {{math|''β''}} in a relativistic wave equation, and explained the fine structure of hydrogen. The solutions to ({{EquationNote|3A}}) are multi-component [[spinor field]]s, and each component satisfies ({{EquationNote|1}}). A remarkable result of spinor solutions is that half of the components describe a particle while the other half describe an [[antiparticle]]; in this case the electron and [[positron]]. The Dirac equation is now known to apply for all massive [[Spin (physics)|spin-{{sfrac|1|2}}]] [[fermion]]s. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.

Although a landmark in quantum theory, the Dirac equation is only true for spin-{{sfrac|1|2}} fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular – not all physicists were comfortable with the "[[Dirac sea]]" of negative energy states).

=== 1930s–1960s: Relativistic quantum mechanics of higher-spin particles ===

The natural problem became clear: to generalize the Dirac equation to particles with ''any spin''; both fermions and bosons, and in the same equations their [[antiparticle]]s (possible because of the [[spinor]] formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by [[Bartel Leendert van der Waerden|van der Waerden]] in 1929), and ideally with positive energy solutions.<ref name="Esposito"/>

This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of ({{EquationNote|3A}}):
{{NumBlk||<math display="block">
\left(\frac{E}{c} + \boldsymbol{\alpha}\cdot\mathbf{p} - \beta mc \right)\psi=0 \,,
</math>|{{EquationRef|3B}}}}
where {{math|''ψ''}} is a spinor field now with infinitely many components, irreducible to a finite number of [[tensor]]s or spinors, to remove the indeterminacy in sign. The [[matrix (mathematics)|matrices]] {{math|'''''α'''''}} and {{math|''β''}} are infinite-dimensional matrices, related to infinitesimal [[Lorentz transformation]]s. He did not demand that each component of {{EquationNote|3B}} satisfy equation ({{EquationNote|2}}); instead he regenerated the equation using a [[Lorentz covariance|Lorentz-invariant]] [[action (physics)|action]], via the [[principle of least action]], and application of Lorentz group theory.<ref>{{cite journal | author = R. Casalbuoni | year = 2006 | title = Majorana and the Infinite Component Wave Equations | journal = Pos Emc | volume = 2006 | pages = 004 | arxiv = hep-th/0610252| bibcode = 2006hep.th...10252C }}</ref><ref name = "Bekaert, Traubenberg, Valenzuela">{{cite journal |author1=X. Bekaert |author2=M.R. Traubenberg |author3=M. Valenzuela | year = 2009 | title = An infinite supermultiplet of massive higher-spin fields | arxiv = 0904.2533 | doi=10.1088/1126-6708/2009/05/118 | volume=2009 | journal=Journal of High Energy Physics |issue=5 | page=118|bibcode=2009JHEP...05..118B |s2cid=16285006 }}</ref>

Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939) see [[Duffin–Kemmer–Petiau algebra]]. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana's, as spinors were new mathematical tools in the early twentieth century, although Majorana's paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940.<ref name="Esposito"/>

Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors {{math|''A''}} and {{math|''B''}}, symmetric in all indices, for a massive particle of spin {{math|''n'' + {{frac|1|2}}}} for integer {{math|''n''}} (see [[Van der Waerden notation]] for the meaning of the dotted indices):
{{NumBlk||<math display="block">
p_{\gamma\dot{\alpha}}A_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcB_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n}
</math>|{{EquationRef|4A}}}}
{{NumBlk||<math display="block">
p^{\gamma\dot{\alpha}}B_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcA_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} 
</math>|{{EquationRef|4B}}}}
where {{math|''p''}} is the momentum as a covariant spinor operator. For {{math|''n'' {{=}} 0}}, the equations reduce to the coupled Dirac equations and {{math|''A''}} and {{math|''B''}} together transform as the original [[Dirac spinor]]. Eliminating either {{math|''A''}} or {{math|''B''}} shows that {{math|''A''}} and {{math|''B''}} each fulfill ({{EquationNote|1}}).<ref name="Esposito"/> The direct derivation of the Dirac-Pauli-Fierz equations using the Bargmann-Wigner operators is given in.<ref>
{{cite journal
 | last1 = A.P. Isaev
 | last2 = M.A. Podoinitsyn
 | year = 2018
 | title = Two-spinor description of massive particles and relativistic spin projection operators
 | url = https://www.sciencedirect.com/science/article/pii/S0550321318300580
 | journal = Nuclear Physics B
 | volume = 929
 | issue = 
 | pages = 452–484
 | doi = 10.1016/j.nuclphysb.2018.02.013
 | arxiv = 1712.00833
 | bibcode = 2018NuPhB.929..452I
 | s2cid = 59582838
 | access-date = 
}}</ref>

In 1941, Rarita and Schwinger focussed on spin-{{frac|3|2}} particles and derived the [[Rarita–Schwinger equation]], including a [[Lagrangian (field theory)|Lagrangian]] to generate it, and later generalized the equations analogous to spin {{math|''n'' + {{frac|1|2}}}} for integer {{math|''n''}}. In 1945, Pauli suggested Majorana's 1932 paper to [[Homi J. Bhabha|Bhabha]], who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in ({{EquationNote|3A}}) and ({{EquationNote|3B}}) by an arbitrary constant, subject to a set of conditions which the wave functions must obey.<ref>{{cite journal |author1=R.K. Loide |author2=I. Ots |author3=R. Saar | year = 1997 | title = Bhabha relativistic wave equations  | doi=10.1088/0305-4470/30/11/027|bibcode = 1997JPhA...30.4005L | volume=30 | journal=Journal of Physics A: Mathematical and General |issue=11 | pages=4005–4017}}</ref>

Finally, in the year 1948 (the same year as [[Feynman]]'s [[path integral formulation]] was cast), [[Valentine Bargmann|Bargmann]] and [[Eugene Wigner|Wigner]] formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the [[Bargmann–Wigner equations]].<ref name="Esposito"/><ref>{{cite journal|author1=Bargmann, V.|author2=Wigner, E. P.|title=Group theoretical discussion of relativistic wave equations|year=1948|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=34|pages=211–23|issue=5|bibcode = 1948PNAS...34..211B |doi = 10.1073/pnas.34.5.211|pmid=16578292|pmc=1079095|doi-access=free}}</ref> In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by [[H. Joos]] and [[Steven Weinberg]], the [[Joos–Weinberg equation]]. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.<ref name="T Jaroszewicz, P.S Kurzepa"/><ref name="E.A. Jeffery 1978">
{{cite journal | author =E.A. Jeffery
 | year =1978
 | title =Component Minimization of the Bargman–Wigner wavefunction
 | journal =Australian Journal of Physics
 |volume=31 | issue =2
 |pages=137–149
 |bibcode = 1978AuJPh..31..137J
 | doi=10.1071/ph780137| doi-access =free
}}</ref><ref>
{{cite journal
 | author = R.F Guertin
 | year = 1974
 | title = Relativistic hamiltonian equations for any spin
 | journal = Annals of Physics
 | doi=10.1016/0003-4916(74)90180-8
 |bibcode = 1974AnPhy..88..504G
 | volume=88
 | issue = 2
 | pages=504–553
}}</ref>

=== 1960s–present ===

The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.<ref name = "Bekaert, Traubenberg, Valenzuela"/>

== 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>

== Constructing RWEs ==

=== Using 4-vectors and the energy–momentum relation ===
{{main|Four vector|Energy–momentum relation}}

Start with the standard [[special relativity]] (SR) 4-vectors
* [[4-position]] <math>X^\mu = \mathbf{X} = (ct,\vec{\mathbf{x}})</math>
* [[4-velocity]] <math>U^\mu = \mathbf{U} = \gamma(c,\vec{\mathbf{u}})</math>
* [[4-momentum]] <math>P^\mu = \mathbf{P} = \left(\frac{E}{c},\vec{\mathbf{p}}\right)</math>
* [[4-wavevector]] <math>K^\mu = \mathbf{K} = \left(\frac{\omega}{c},\vec{\mathbf{k}}\right)</math>
* [[4-gradient]] <math>\partial^\mu = \mathbf{\partial} = \left(\frac{\partial_t}{c},-\vec{\mathbf{\nabla}}\right)</math>

Note that each 4-vector is related to another by a [[Lorentz scalar]]:
* <math>\mathbf{U} = \frac{d}{d\tau} \mathbf{X}</math>, where <math>\tau</math> is the [[proper time]]
* <math>\mathbf{P} = m_0 \mathbf{U}</math>, where <math>m_0</math> is the [[rest mass]]
* <math>\mathbf{K} = (1/\hbar) \mathbf{P}</math>, which is the [[4-vector]] version of the [[Planck–Einstein relation]] & the [[de Broglie]] [[matter wave]] relation
* <math>\mathbf{\partial} = -i \mathbf{K}</math>, which is the [[4-gradient]] version of [[complex-valued]] [[plane waves]]

Now, just apply the standard Lorentz scalar product rule to each one:
* <math>\mathbf{U} \cdot \mathbf{U} = (c)^2</math>
* <math>\mathbf{P} \cdot \mathbf{P} = (m_0 c)^2</math>
* <math>\mathbf{K} \cdot \mathbf{K} = \left(\frac{m_0 c}{\hbar}\right)^2</math>
* <math>\mathbf{\partial} \cdot \mathbf{\partial} = \left(\frac{-i m_0 c}{\hbar}\right)^2 = -\left(\frac{m_0 c}{\hbar}\right)^2</math>
The last equation is a fundamental quantum relation.

When applied to a Lorentz scalar field <math>\psi</math>, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations.
* <math>\left[\mathbf{\partial} \cdot \mathbf{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0</math>: in 4-vector format
* <math>\left[\partial_\mu \partial^\mu + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0</math>: in tensor format
* <math>\left[(\hbar \partial_{\mu} + i m_0 c)(\hbar \partial^{\mu} -i m_0 c)\right]\psi = 0</math>: in factored tensor format

The [[Schrödinger equation]] is the low-velocity [[Limiting case (mathematics)|limiting case]] ({{math|''v'' ≪ ''c''}}) of the Klein–Gordon equation.

When the relation is applied to a four-vector field <math>A^\mu</math> instead of a Lorentz scalar field <math>\psi</math>, then one gets the [[Proca equation]] (in [[Lorenz gauge]]):
<math display="block">\left[\mathbf{\partial} \cdot \mathbf{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]A^\mu = 0</math>

If the rest mass term is set to zero (light-like particles), then this gives the free [[Maxwell equation]] (in [[Lorenz gauge]])
<math display="block">[\mathbf{\partial} \cdot \mathbf{\partial}]A^\mu = 0</math>

=== Representations of the Lorentz group ===

Under a proper [[orthochronous]] Lorentz transformation {{math|''x'' → Λ''x''}} in [[Minkowski space]], all one-particle quantum states {{math|''ψ''<sup>''j''</sup><sub>''σ''</sub>}} of spin {{math|''j''}} with spin z-component {{math|''σ''}} locally transform under some [[Representation theory of the Lorentz group|representation]] {{math|''D''}} of the [[Lorentz group]]:<ref name="Weinberg">{{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=133|pages=B1318–B1332|year=1964|doi=10.1103/PhysRev.133.B1318|title=Feynman Rules ''for Any'' spin|issue=5B|bibcode=1964PhRv..133.1318W|url=http://theory.fi.infn.it/becattini/files/weinberg3.pdf|access-date=2016-12-29|archive-date=2022-03-25|archive-url=https://web.archive.org/web/20220325020742/http://theory.fi.infn.it/becattini/files/weinberg3.pdf|url-status=dead}}; {{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=134|pages=B882–B896|year=1964|doi=10.1103/PhysRev.134.B882|title=Feynman Rules ''for Any'' spin. II. Massless Particles|issue=4B|bibcode=1964PhRv..134..882W|url=http://theory.fi.infn.it/becattini/files/weinberg2.pdf|access-date=2016-12-29|archive-date=2022-03-09|archive-url=https://web.archive.org/web/20220309040610/http://theory.fi.infn.it/becattini/files/weinberg2.pdf|url-status=dead}}; {{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=181|pages=1893–1899|year=1969|doi=10.1103/PhysRev.181.1893|title=Feynman Rules ''for Any'' spin. III|issue=5|bibcode=1969PhRv..181.1893W|url=http://theory.fi.infn.it/becattini/files/weinberg3.pdf|access-date=2016-12-29|archive-date=2022-03-25|archive-url=https://web.archive.org/web/20220325020742/http://theory.fi.infn.it/becattini/files/weinberg3.pdf|url-status=dead}}</ref><ref name="Kenmoku">
{{cite arXiv
 | author = K. Masakatsu
 | year = 2012
 | title = Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation
 | class = gr-qc
 | eprint = 1208.0644
}}</ref>
<math display="block">\psi(x) \rightarrow D(\Lambda) \psi(\Lambda^{-1}x) </math>
where {{math|''D''(Λ)}} is some finite-dimensional representation, i.e. a matrix. Here {{math|''ψ''}} is thought of as a [[column vector]] containing components with the allowed values of {{math|''σ''}}. The [[quantum number]]s {{math|''j''}} and {{math|''σ''}} as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of {{math|''σ''}} may occur more than once depending on the representation. Representations with several possible values for {{math|''j''}} are considered below.

The [[Representation theory#Subrepresentations, quotients, and irreducible representations|irreducible representation]]s are labeled by a pair of half-integers or integers {{math|(''A'', ''B'')}}. From these all other representations can be built up using a variety of standard methods, like taking [[tensor product]]s and [[direct sum]]s. In particular, [[space-time]] itself constitutes a [[4-vector]] representation {{math|({{sfrac|1|2}}, {{sfrac|1|2}})}} so that {{math|Λ ∈ ''D''<sup>(1/2, 1/2)</sup>}}. To put this into context; [[Dirac spinor]]s transform under the {{math|({{sfrac|1|2}}, 0) ⊕ (0, {{sfrac|1|2}})}} representation. In general, the {{math|(''A'', ''B'')}} representation space has [[Linear subspace|subspace]]s that under the [[subgroup]] of spatial [[rotation]]s, [[SO(3)]], transform irreducibly like objects of spin ''j'', where each allowed value:
<math display="block">j = A + B, A + B - 1, \dots, |A - B|,</math>
occurs exactly once.<ref>
{{citation
 | last = Weinberg
 | first = S
 | page = [https://archive.org/details/quantumtheoryoff00stev/page/ <!---please insert, in 1995 edition it's 231--->]
 | year = 2002
 | chapter = 5
 | title = The Quantum Theory of Fields, vol I
 | publisher = Cambridge University Press
 | isbn = 0-521-55001-7
 | chapter-url = https://archive.org/details/quantumtheoryoff00stev/page/
}}</ref> In general, ''tensor products of irreducible representations'' are reducible; they decompose as direct sums of irreducible representations.

The representations {{math|''D''<sup>(''j'', 0)</sup>}} and {{math|''D''<sup>(0, ''j'')</sup>}} can each separately represent particles of spin {{math|''j''}}. A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.

== Non-linear equations ==
{{further|Non-linear differential equation}}

There are equations which have solutions that do not satisfy the superposition principle.

=== Nonlinear gauge fields ===
* [[Yang–Mills theory|Yang–Mills equation]]: describes a non-abelian gauge field
* [[Yang–Mills–Higgs equations]]: describes a non-abelian gauge field coupled with a massive spin-0 particle

=== Spin 2 ===
* [[Einstein field equations]]: describe interaction of matter with the [[gravitational field]] (massless spin-2 field): <math display="block">R_{\mu \nu} - \frac{1}{2} g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = \frac{8 \pi G}{c^4} T_{\mu \nu}</math> The solution is a [[metric tensor|metric]] [[tensor field]], rather than a wave function.

== See also ==
* [[List of equations in nuclear and particle physics]]
* [[List of equations in quantum mechanics]]
* [[Lorentz transformation]]
* [[Mathematical descriptions of the electromagnetic field]]
** [[Quantization of the electromagnetic field]]
* [[Minimal coupling]]
* [[Scalar field theory]]
* [[Status of special relativity]]

== References ==
{{reflist|30em}}

== Further reading ==
{{refbegin}}
* {{cite book|title=Encyclopaedia of Physics|edition=2nd|author1=R.G. Lerner|author1-link=Rita G. Lerner|author2=G.L. Trigg|publisher=VHC publishers|year=1991|isbn=0-89573-752-3|url=https://archive.org/details/encyclopediaofph00lern}}
* {{cite book|title=McGraw Hill Encyclopaedia of Physics|edition=2nd|author=C.B. Parker|year=1994|publisher=McGraw-Hill |isbn=0-07-051400-3|url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book|title=The Cambridge Handbook of Physics Formulas|author=G. Woan, Cambridge University Press|year=2010|publisher=Cambridge University Press |isbn=978-0-521-57507-2|url-access=registration|url=https://archive.org/details/cambridgehandboo0000woan}}
* {{cite book|title=Relativity DeMystified|author=D. McMahon|publisher=Mc Graw Hill (USA)|year=2006|isbn=0-07-145545-0}}
* {{cite book|title=Gravitation|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman|year=1973|isbn=0-7167-0344-0}}
* {{cite book|title=Particle Physics (Manchester series)|edition=2nd|author1=B.R. Martin|author2=G. Shaw|publisher=John Wiley & Sons|year=2008|isbn=978-0-470-03294-7}}
* {{cite book|title=Supersymmetry|author=P. Labelle, Demystified|publisher=McGraw-Hill (USA)|year=2010|isbn=978-0-07-163641-4}}
* {{cite book|title=Physics of Atoms and Molecules|author1=B.H. Bransden|author2=C.J. Joachain|publisher=Longman|year=1983|isbn=0-582-44401-2}}
* {{cite book|title=Quantum Mechanics|author=E. Abers|publisher=Addison Wesley|year=2004|isbn=978-0-13-146100-0}}
* {{cite book|title=Quantum Field Theory|author=D. McMahon|publisher=Mc Graw Hill (USA)|year=2008|isbn=978-0-07-154382-8}}
* {{cite journal
 | author = M. Pillin
 | year = 1994
 | title = q-Deformed Relativistic Wave Equations
 | arxiv = hep-th/9310097
 | doi=10.1063/1.530487
 | volume=35
 | journal=Journal of Mathematical Physics
 | issue = 6
 | pages=2804–2817
 | bibcode=1994JMP....35.2804P
 | s2cid = 5919588
}}
{{refend}}

[[Category:Equations of physics]]
[[Category:Quantum field theory]]
[[Category:Quantum mechanics]]
[[Category:Special relativity|Wave equations]]
[[Category:Waves]]