Editing Relativistic wave equations (section)

== 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.