View source for Relativistic quantum mechanics

{{Short description|Quantum mechanics taking into account particles near or at the speed of light}}
{{Quantum mechanics|cTopic=Formulations}}

In [[physics]], '''relativistic quantum mechanics''' ('''RQM''') is any [[Poincaré group|Poincaré]] [[Principle of covariance|covariant]] formulation of [[quantum mechanics]] (QM). This theory is applicable to [[massive particle]]s propagating at all [[velocity|velocities]] up to those comparable to the [[speed of light]]&nbsp;''c'', and can accommodate [[massless particle]]s. The theory has application in [[high energy physics]],<ref>{{cite book |author=Perkins, D.H. |title=Introduction to High Energy Physics|publisher=Cambridge University Press |year=2000 |url=https://books.google.com/books?id=e63cNigcmOUC&q=Relativistic+quantum+mechanics&pg=PA19 |isbn=978-0-521-62196-0}}</ref> [[particle physics]] and [[accelerator physics]],<ref name="Martin, Shaw, p 3">{{cite book |first1=B.R. |last1=Martin |first2=G. |last2=Shaw |title=Particle Physics |url=https://archive.org/details/particlephysics00mart |url-access=limited |edition=3rd |series=Manchester Physics Series |publisher=John Wiley & Sons |page=[https://archive.org/details/particlephysics00mart/page/n24 3] |isbn=978-0-470-03294-7|date=2008-12-03 }}</ref> as well as [[atomic physics]], [[chemistry]]<ref>{{cite book |author1=Reiher, M. |author2=Wolf, A. |title=Relativistic Quantum Chemistry |publisher=John Wiley & Sons |year=2009 |url=https://books.google.com/books?id=YwSpxCfsNsEC&q=Relativistic+quantum+mechanics&pg=PA1 
|isbn=978-3-527-62749-3}}</ref> and [[condensed matter physics]].<ref>{{cite book |author=Strange, P. |title=Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics |publisher=Cambridge University Press |year=1998 |url=https://books.google.com/books?id=sdVrBM2w0OwC&q=Relativistic+quantum+mechanics&pg=PR15 |isbn=978-0-521-56583-7}}</ref><ref>{{cite book |author=Mohn, P. |title=Magnetism in the Solid State: An Introduction |page=6 |publisher=Springer |volume=134|series=Springer Series in Solid-State Sciences Series |year=2003 |url=https://books.google.com/books?id=ZgyjojQUyMcC&q=electromagnetic+multipoles+in+relativistic+quantum+mechanics&pg=PA6 |isbn=978-3-540-43183-1}}</ref> ''Non-relativistic quantum mechanics'' refers to the [[mathematical formulation of quantum mechanics]] applied in the context of [[Galilean relativity]], more specifically quantizing the equations of [[classical mechanics]] by replacing dynamical variables by [[operator (physics)|operator]]s. ''Relativistic quantum mechanics'' (RQM) is quantum mechanics applied with [[special relativity]]. Although the earlier formulations, like the [[Schrödinger picture]] and [[Heisenberg picture]] were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.

Key features common to all RQMs include: the prediction of [[antimatter]], [[spin magnetic moment]]s of [[elementary particle|elementary]] [[spin-1/2|spin&nbsp;{{frac|1|2}}]] [[fermion]]s, [[fine structure]], and quantum dynamics of [[charged particle]]s in [[electromagnetic field]]s.<ref name="Martin, Shaw, pp. 5–6">{{cite book |first1=B.R. |last1=Martin |first2=G. |last2=Shaw |title=Particle Physics |url=https://archive.org/details/particlephysics00mart |url-access=limited |edition=3rd |series=Manchester Physics Series |publisher=John Wiley & Sons |pages=[https://archive.org/details/particlephysics00mart/page/n26 5]–6 |isbn=978-0-470-03294-7|date=2008-12-03 }}</ref> The key result is the [[Dirac equation]], from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the [[Hamiltonian operator]] to achieve agreement with experimental observations.

The most successful (and most widely used) RQM is ''relativistic [[quantum field theory]]'' (QFT), in which elementary particles are interpreted as ''field quanta''. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in [[matter creation]] and [[annihilation]].<ref>{{cite book |author=Messiah, A. |year=1981 |title=Quantum Mechanics |publisher=North-Holland Publishing Company |volume=2 |page=875 |url=https://books.google.com/books?id=VR93vUk8d_8C&q=lorentz+group+in+relativistic+quantum+mechanics&pg=PA876 |isbn=978-0-7204-0045-8}}</ref>

In this article, the equations are written in familiar 3D [[vector calculus]] notation and use hats for [[operator (physics)|operators]] (not necessarily in the literature), and where space and time components can be collected, [[tensor index notation]] is shown also (frequently used in the literature), in addition the [[Einstein notation|Einstein summation convention]] is used. [[SI units]] are used here; [[Gaussian units]] and [[natural units]] are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be [[Fourier transform]]ed – see [[position and momentum space]].

==Combining special relativity and quantum mechanics==
One approach is to modify the [[Schrödinger picture]] to be consistent with special relativity.<ref name="Martin, Shaw, p 3"/>

A [[mathematical formulation of quantum mechanics|postulate of quantum mechanics]] is that the [[time evolution]] of any quantum system is given by the [[Schrödinger equation]]:

:<math>i\hbar \frac{\partial}{\partial t}\psi =\hat{H}\psi</math>

using a suitable [[Hamiltonian operator]] {{math|''Ĥ''}} corresponding to the system. The solution is a [[complex number|complex]]-valued [[wavefunction]] {{math|''ψ''('''r''', ''t'')}}, a [[function (mathematics)|function]] of the [[three-dimensional space|3D]] [[position vector]] {{math|'''r'''}} of the particle at time {{math|''t''}}, describing the behavior of the system.

Every particle has a non-negative [[spin quantum number]] {{math|''s''}}. The number {{math|2''s''}} is an integer, odd for [[fermion]]s and even for [[boson]]s. Each {{math|''s''}} has {{math|2''s'' + 1}} ''z''-projection quantum numbers; {{math|''σ''&nbsp;{{=}}&nbsp;''s'', ''s''&nbsp;&minus;&nbsp;1, ... , &minus;''s''&nbsp;+&nbsp;1, &minus;''s''}}.<ref group=lower-alpha>Other common notations include {{math|''m<sub>s</sub>''}} and {{math|''s<sub>z</sub>''}} etc., but this would clutter expressions with unnecessary subscripts. The subscripts {{math|''σ''}} labeling spin values are not to be confused for [[tensor index notation|tensor indices]] nor the [[Pauli matrices]].</ref> This is an additional discrete variable the wavefunction requires; {{math|''ψ''('''r''',&nbsp;''t'',&nbsp;''σ'')}}.

Historically, in the early 1920s [[Wolfgang Pauli|Pauli]], [[Ralph Kronig|Kronig]], [[George Uhlenbeck|Uhlenbeck]] and [[Samuel Goudsmit|Goudsmit]] were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the [[Pauli exclusion principle]] (1925) and the more general [[spin–statistics theorem]] (1939) due to [[Markus Fierz|Fierz]], rederived by Pauli a year later. This is the explanation for a diverse range of [[subatomic particle]] behavior and phenomena: from the [[electronic configuration]]s of atoms, nuclei (and therefore all [[Chemical element|element]]s on the [[periodic table]] and their [[chemistry]]), to the quark configurations and [[colour charge]] (hence the properties of [[baryon]]s and [[meson]]s).

A fundamental prediction of special relativity is the relativistic [[energy–momentum relation]]; for a particle of [[rest mass]] {{math|''m''}}, and in a particular [[frame of reference]] with [[energy]] {{math|''E''}} and 3-[[momentum]] {{math|'''p'''}} with [[Norm (mathematics)|magnitude]] in terms of the [[dot product]] <math>p = \sqrt{\mathbf{p} \cdot \mathbf{p}}</math>, it is:<ref>{{cite book|title=Dynamics and Relativity|url=https://archive.org/details/dynamicsrelativi00fors|url-access=limited|author1=Forshaw, J.R. |author2=Smith, A.G. |series=Manchester Physics Series |publisher=John Wiley & Sons |year=2009 |pages=[https://archive.org/details/dynamicsrelativi00fors/page/n272 258]–259 |isbn=978-0-470-01460-8}}</ref>

:<math>E^2 = c^2\mathbf{p}\cdot\mathbf{p} + (mc^2)^2\,.</math>

These equations are used together with the [[energy operator|energy]] and [[momentum operator|momentum]] [[operator (physics)#Operators in quantum mechanics|operator]]s, which are respectively:

:<math>\hat{E}=i\hbar\frac{\partial}{\partial t}\,,\quad \hat{\mathbf{p}} = -i\hbar\nabla\,,</math>

to construct a [[relativistic wave equation]] (RWE): a [[partial differential equation]] consistent with the energy–momentum relation, and is solved for {{math|''ψ''}} to predict the quantum dynamics of the particle. For space and time to be placed on equal footing, as in relativity, the orders of space and time [[partial derivative]]s should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified. This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a differential equation).

The [[Heisenberg picture]] is another formulation of QM, in which case the wavefunction {{math|''ψ''}} is ''time-independent'', and the operators {{math|''A''(''t'')}} contain the time dependence, governed by the equation of motion:

:<math>\frac{d}{dt}A = \frac{1}{i\hbar}[A,\hat{H}]+\frac{\partial}{\partial t}A\,,</math>

This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.<ref>{{cite book |author=Greiner, W. |page=70 |year=2000 |edition=3rd |title=Relativistic Quantum Mechanics. Wave Equations |publisher=Springer |isbn=978-3-540-67457-3 |url=https://books.google.com/books?id=2DAInxwvlHYC&q=Relativistic+quantum+mechanics&pg=PA1}}</ref><ref>{{cite news |author=Wachter, A. |page=34 |title=Relativistic quantum mechanics |publisher=Springer |year=2011 |url=https://books.google.com/books?id=NjZogv2yFzAC&q=armin+wachter+relativistic+quantum+mechanics |isbn=978-90-481-3645-2}}</ref>

Historically, around 1926, [[Erwin Schrödinger|Schrödinger]] and [[Werner Heisenberg|Heisenberg]] show that wave mechanics and [[matrix mechanics]] are equivalent, later furthered by Dirac using [[Transformation theory (quantum mechanics)|transformation theory]].

A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to apply [[representations of the Lorentz group]].

===Space and time===
In [[classical mechanics]] and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space. Thus in non-relativistic QM one has for a [[many particle system]] {{math|''ψ''('''r'''<sub>1</sub>, '''r'''<sub>2</sub>, '''r'''<sub>3</sub>, ..., ''t'', ''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ''σ''<sub>3</sub>...)}}.

In [[relativistic mechanics]], the [[Coordinate system|spatial coordinates]] and [[coordinate time]] are ''not'' absolute; any two observers moving relative to each other can measure different locations and times of [[Event (relativity)|event]]s. The position and time coordinates combine naturally into a [[four-vector#Four-position|four-dimensional spacetime position]] {{math|'''X''' {{=}} (''ct'', '''r''')}} corresponding to events, and the energy and 3-momentum combine naturally into the [[four-momentum]] {{math|'''P''' {{=}} (''E''/''c'', '''p''')}} of a dynamic particle, as measured in ''some'' [[frame of reference|reference frame]], change according to a [[Lorentz transformation]] as one measures in a different frame boosted and/or rotated relative the original frame in consideration. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.

Under a proper [[orthochronous]] [[Lorentz transformation]] {{math|('''r''', ''t'') → Λ('''r''', ''t'')}} in [[Minkowski space]], all one-particle quantum states {{math|''ψ<sub>σ</sub>''}} 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/weinberg1.pdf |access-date=2014-08-24 |archive-date=2020-12-04 |archive-url=https://web.archive.org/web/20201204053451/http://theory.fi.infn.it/becattini/files/weinberg1.pdf |url-status=dead }};<br/> {{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=2013-04-14 |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 }};<br/> {{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=2013-04-14 |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>{{cite arXiv |author=Masakatsu, K.  |year=2012 |title=Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation |eprint=1208.0644 |class=gr-qc}}</ref>

:<math>\psi_\sigma(\mathbf{r}, t) \rightarrow D(\Lambda) \psi_\sigma(\Lambda^{-1}(\mathbf{r}, t)) </math>

where {{math|''D''(Λ)}} is a finite-dimensional representation, in other words a {{math|(2''s'' + 1)×(2''s'' + 1)}} [[square matrix]] . Again, {{math|''ψ''}} is thought of as a [[column vector]] containing components with the {{math|(2''s'' + 1)}} allowed values of {{math|''σ''}}. The [[quantum number]]s {{math|''s''}} 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.

{{further|Generator (mathematics)|group theory|Representation theory of the Lorentz group|symmetries in quantum mechanics}}

===Non-relativistic and relativistic Hamiltonians===
{{Main|Hamiltonian operator}}

The [[Hamiltonian mechanics|classical Hamiltonian]] for a particle in a [[Scalar potential|potential]]<!--Potential energy is linked later in the sentence--> is the [[kinetic energy]] {{math|'''p'''·'''p'''/2''m''}} plus the [[potential energy]] {{math|''V''('''r''', ''t'')}}, with the corresponding quantum operator in the [[Schrödinger picture]]:

:<math>\hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r},t) </math>

and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energy ''and'' momentum leading to difficulties. Naively setting:

:<math>\hat{H} = \hat{E} = \sqrt{c^2 \hat{\mathbf{p}}\cdot\hat{\mathbf{p}} + (mc^2)^2} \quad \Rightarrow \quad i\hbar\frac{\partial}{\partial t}\psi = \sqrt{c^2 \hat{\mathbf{p}}\cdot \hat{\mathbf{p}} + (mc^2)^2} \, \psi</math>

is not helpful for several reasons. The square root of the operators cannot be used as it stands; it would have to be expanded in a [[power series]] before the momentum operator, raised to a power in each term, could act on {{math|''ψ''}}. As a result of the power series, the space and time [[derivative (mathematics)|derivative]]s are ''completely asymmetric'': infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. Another problem, less obvious and more severe, is that it can be shown to be [[Quantum nonlocality|nonlocal]] and can even ''violate [[Causality (physics)|causality]]'': if the particle is initially localized at a point {{math|'''r'''<sub>0</sub>}} so that {{math|''ψ''('''r'''<sub>0</sub>, ''t'' {{=}} 0)}} is finite and zero elsewhere, then at any later time the equation predicts delocalization {{math|''ψ''('''r''', ''t'') ≠ 0}} everywhere, even for {{math|{{!}}'''r'''{{!}} > ''ct''}} which means the particle could arrive at a point before a pulse of light could. This would have to be remedied by the additional constraint {{math|''ψ''({{math|{{!}}'''r'''{{!}} > ''ct''}}, ''t'') {{=}} 0}}.<ref name="Parker 1994">{{cite book |pages=[https://archive.org/details/mcgrawhillencycl1993park/page/1193 1193–1194] |author=Parker, C.B. |title=McGraw Hill Encyclopaedia of Physics |publisher=McGraw Hill |edition=2nd |year=1994 |isbn=978-0-07-051400-3 |url=https://archive.org/details/mcgrawhillencycl1993park/page/1193 }}</ref>

There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrödinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units of {{math|''μ<sub>B</sub>''}}, the [[Bohr magneton]]:<ref>{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |edition=2nd |page=[https://archive.org/details/quantumphysicsof00eisb/page/274 274] |author1=Resnick, R. |author2=Eisberg, R. |publisher=John Wiley & Sons |year=1985 |isbn=978-0-471-87373-0 |url=https://archive.org/details/quantumphysicsof00eisb/page/274 }}</ref><ref>{{cite book |author1=Landau, L.D. |author2=Lifshitz, E.M. |title=Quantum Mechanics Non-Relativistic Theory |volume=3 |page=455 |publisher=Elsevier |year=1981 |url=https://books.google.com/books?id=SvdoN3k8EysC&q=magnetic+moments+in+relativistic+quantum+mechanics&pg=PA455 |isbn=978-0-08-050348-6}}</ref>

:<math>\hat{\boldsymbol{\mu}}_S = - \frac{g\mu_B}{\hbar}\hat{\mathbf{S}}\,,\quad \left|\boldsymbol{\mu}_S\right| = - g\mu_B \sigma\,,</math>

where {{math|''g''}} is the (spin) [[g-factor (physics)|g-factor]] for the particle, and {{math|'''S'''}} the [[spin operator]], so they interact with [[electromagnetic field]]s. For a particle in an externally applied [[magnetic field]] {{math|'''B'''}}, the interaction term<ref name="Schuam p 181">{{cite book |author1=Peleg, Y. |author2=Pnini, R. |author3=Zaarur, E. |author4=Hecht, E. |year=2010 |edition=2nd |title=Quantum Mechanics |series=Shaum's outlines |publisher=McGraw–Hill |page=181 |isbn=978-0-07-162358-2}}</ref>

:<math>\hat{H}_B = - \mathbf{B} \cdot \hat{\boldsymbol{\mu}}_S </math>

has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin ''automatically'' as a requirement of enforcing the relativistic energy-momentum relation.<ref>{{cite book|title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=425 |isbn=978-0-13-146100-0}}</ref>

Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including [[rest mass]] and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form of [[matrix (mathematics)|matrices]], in which the [[matrix multiplication]] runs over the spin index {{math|''σ''}}, so in general a relativistic Hamiltonian:

:<math>\hat{H} = \hat{H}(\mathbf{r}, t, \hat{\mathbf{p}}, \hat{\mathbf{S}})</math>

is a function of space, time, and the momentum and spin operators.

===The Klein–Gordon and Dirac equations for free particles===
Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the [[Klein–Gordon equation]]:<ref>{{cite news |author=Wachter, A. |page=5 |title=Relativistic quantum mechanics |publisher=Springer |year=2011 |url=https://books.google.com/books?id=NjZogv2yFzAC&q=electromagnetic+multipoles+in+relativistic+quantum+mechanics&pg=PA367 |isbn=978-90-481-3645-2}}</ref>

:<math>\hat{E}^2 \psi = c^2\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}\psi + (mc^2)^2\psi \,,</math>

and was discovered by many people because of the straightforward way of obtaining it, notably by Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation.  This ''is'' [[Lorentz covariance|relativistically invariant]], yet this equation alone isn't a sufficient foundation for RQM for a at least two reasons: one is that negative-energy states are solutions,<ref name="Martin, Shaw, p 3"/><ref>{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=415 |isbn=978-0-13-146100-0}}</ref> another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form:<ref name="Penrose 2005, p 620–621">{{cite book |author=Penrose, R. |title=The Road to Reality |publisher=Vintage Books |pages=620–621 |year=2005 |isbn=978-0-09-944068-0 |title-link=The Road to Reality}}</ref><ref>{{Cite book |title=Physics of Atoms and Molecules |author=Bransden, B.H. |author2=Joachain, C.J. |year=1983 |publisher=Prentice Hall |edition=1st |page=634 |isbn=978-0-582-44401-0}}</ref>

:<math>
\left(\hat{E} - c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} - \beta mc^2 \right)\left(\hat{E} + c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2 \right)\psi=0 \,,
</math>

where {{math|'''α''' {{=}} (''α''<sub>1</sub>, ''α''<sub>2</sub>, ''α''<sub>3</sub>)}} and {{math|''β''}} are not simply numbers or vectors, but 4 × 4 [[Hermitian matrix|Hermitian matrices]] that are required to [[anticommute]] for {{math|''i'' ≠ ''j''}}:

:<math>\alpha_i \beta = - \beta \alpha_i, \quad \alpha_i\alpha_j = - \alpha_j\alpha_i  \,,</math>

and square to the [[identity matrix]]:

:<math> \alpha_i^2 = \beta^2 = I \,, </math>

so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor:

:<math>\left(\hat{E} - c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} - \beta mc^2 \right)\psi=0 \quad \Leftrightarrow \quad \hat{H} = c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2</math>

is the [[Dirac equation]]. The other factor is also the Dirac equation, but for a particle of [[negative mass]].<ref name="Penrose 2005, p 620–621"/> Each factor is relativistically invariant. The reasoning can be done the other way round: propose the Hamiltonian in the above form, as Dirac did in 1928, then pre-multiply the equation by the other factor of operators {{math|''E'' + ''c'''''α''' · '''p''' + ''βmc''<sup>2</sup>}}, and comparison with the KG equation determines the constraints on {{math|'''α'''}} and {{math|''β''}}. The positive mass equation can continue to be used without loss of continuity. The matrices multiplying {{math|''ψ''}} suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions,<ref name="Martin, Shaw, pp. 5–6" /><ref>{{cite book |author=Grandy, W.T. |title=Relativistic quantum mechanics of leptons and fields |page=54 |publisher=Springer |year=1991 |url=https://books.google.com/books?id=BPCFI4yFMbcC&q=magnetic+moments+in+relativistic+quantum+mechanics&pg=PA67 |isbn=978-0-7923-1049-5}}</ref> so Dirac postulated that negative energy states are always occupied, because according to the [[Pauli exclusion principle|Pauli principle]], [[electronic transition]]s from positive to negative energy levels in [[atom]]s would be forbidden. See [[Dirac sea]] for details.

===Densities and currents===
In non-relativistic quantum mechanics, the [[square modulus]] of the [[wavefunction]] {{math|''ψ''}} gives the [[probability density function]] {{math|''ρ'' {{=}} {{!}}''ψ''{{!}}<sup>2</sup>}}. This is the [[Copenhagen interpretation]], circa 1927. In RQM, while {{math|''ψ''('''r''', ''t'')}} is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density {{math|''ρ''}} or [[probability current]] {{math|'''j'''}} (really meaning ''probability current density'') because they are ''not'' [[positive-definite function]]s of space and time. The [[Dirac equation]] does:<ref>{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=423 |isbn=978-0-13-146100-0}}</ref>

:<math>\rho=\psi^\dagger \psi, \quad \mathbf{j} = \psi^\dagger \gamma^0 \boldsymbol{\gamma} \psi \quad \rightleftharpoons \quad J^\mu = \psi^\dagger \gamma^0 \gamma^\mu \psi </math>

where the dagger denotes the [[Hermitian adjoint]] (authors usually write {{math|{{overline|''ψ''}} {{=}} ''ψ''<sup>†</sup>''γ''<sup>0</sup>}} for the [[Dirac adjoint]]) and {{math|''J<sup>μ</sup>''}} is the [[Probability current#Definition (relativistic 4-current)|probability four-current]], while the [[Klein–Gordon equation]] does not:<ref>{{cite book|title=Quantum Field Theory |url=https://archive.org/details/quantumfieldtheo00mcma_095 |url-access=limited |series=Demystified |author=McMahon, D.  |publisher=McGraw Hill |page=[https://archive.org/details/quantumfieldtheo00mcma_095/page/n132 114] |year=2008 |isbn=978-0-07-154382-8}}</ref>

:<math>\rho = \frac{i\hbar}{2mc^2}\left(\psi^{*}\frac{\partial \psi}{\partial t} - \psi \frac{\partial \psi^*}{\partial t}\right)\, ,\quad \mathbf{j} = -\frac{i\hbar}{2m}\left(\psi^* \nabla \psi - \psi \nabla \psi^*\right)  \quad \rightleftharpoons  \quad J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*) </math>

where {{math|∂<sup>''μ''</sup>}} is the [[four-gradient]]. Since the initial values of both {{math|''ψ''}} and {{math|∂''ψ''/∂''t''}} may be freely chosen, the density can be negative.

Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as [[charge density]] and [[current density]] when multiplied by [[electric charge]]. Then, the wavefunction {{math|''ψ''}} is not a wavefunction at all, but reinterpreted as a ''field''.<ref name="Parker 1994"/> The density and current of electric charge always satisfy a [[continuity equation]]:

:<math>\frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf{J} = 0 \quad \rightleftharpoons  \quad \partial_\mu J^\mu = 0 \,, </math>

as charge is a [[conserved quantity]]. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions.

==Spin and electromagnetically interacting particles==
Including interactions in RWEs is generally difficult. [[Minimal coupling]] is a simple way to include the electromagnetic interaction. For one charged particle of [[electric charge]] {{math|''q''}} in an electromagnetic field, given by the [[magnetic vector potential]] {{math|'''A'''('''r''', ''t'')}} defined by the magnetic field {{math|'''B''' {{=}} &nabla; &times; '''A'''}}, and [[electric scalar potential]] {{math|''ϕ''('''r''', ''t'')}}, this is:<ref>{{Cite book|title=Physics of Atoms and Molecules |author1=Bransden, B.H. |author2=Joachain, C.J. |year=1983 |publisher=Prentice Hall |edition=1st |pages=632–635 |isbn=978-0-582-44401-0}}</ref>

:<math>\hat{E} \rightarrow \hat{E} - q\phi \,, \quad \hat{\mathbf{p}}\rightarrow \hat{\mathbf{p}} - q \mathbf{A} \quad \rightleftharpoons \quad \hat{P}_\mu \rightarrow \hat{P}_\mu -q A_\mu</math>

where {{math|''P<sub>μ</sub>''}} is the [[four-momentum]] that has a corresponding [[4-momentum operator]], and {{math|''A<sub>μ</sub>''}} the [[four-potential]]. In the following, the non-relativistic limit refers to the limiting cases:

:<math>E - e\phi \approx mc^2\,,\quad \mathbf{p} \approx m \mathbf{v}\,,</math>

that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.

===Spin 0===
In RQM, the KG equation admits the minimal coupling prescription;

:<math>{(\hat{E} - q\phi)}^2 \psi = c^2{(\hat{\mathbf{p}} - q \mathbf{A})}^2\psi + (mc^2)^2\psi \quad \rightleftharpoons \quad \left[{(\hat{P}_\mu - q A_\mu)}{(\hat{P}^\mu - q A^\mu)} - {(mc)}^2 \right] \psi = 0.</math>

In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under the ''irreducible'' one-dimensional scalar {{math|[[Representation theory of the Lorentz group|(0,0)]]}} representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of {{math|(0,0)}} representations. Solutions that do not belong to the irreducible {{math|(0,0)}} representation will have two or more ''independent'' components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for spin&nbsp;{{sfrac|1|2}}, see below. Thus if a system satisfies the KG equation ''only'', it can only be interpreted as a system with zero spin.

The electromagnetic field is treated classically according to [[Maxwell's equations]] and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as the ''π''-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions.

The KG equation is applicable to spinless charged [[boson]]s in an external electromagnetic potential.<ref name="Martin, Shaw, p 3"/> As such, the equation cannot be applied to the description of atoms, since the electron is a spin&nbsp;{{sfrac|1|2}} particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:<ref name="Schuam p 181"/>

:<math>\left ( i\hbar \frac{\partial}{\partial t}- q\phi\right) \psi = \frac{1}{2m}{(\hat{\mathbf{p}} - q \mathbf{A})}^2 \psi \quad \Leftrightarrow \quad \hat{H} = \frac{1}{2m}{(\hat{\mathbf{p}} - q \mathbf{A})}^2 + q\phi.</math>

===Spin {{sfrac|1|2}}===
{{Main|spin-1/2}}

Non relativistically, spin was ''[[phenomenological model|phenomenologically]]'' introduced in the [[Pauli equation]] by [[Wolfgang Pauli|Pauli]] in 1927 for particles in an [[electromagnetic field]]:

:<math>\left(i \hbar \frac{\partial}{\partial t} - q \phi  \right) \psi = \left[ \frac{1}{2m}{(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))}^2 \right] \psi \quad \Leftrightarrow \quad \hat{H} = \frac{1}{2m}{(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))}^2 + q \phi </math>

by means of the 2&thinsp;&times;&thinsp;2 [[Pauli matrices]], and {{math|''&psi;''}} is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component [[spinor field]]:

:<math>\psi=\begin{pmatrix}\psi_{\uparrow} \\ \psi_{\downarrow} \end{pmatrix}</math>

where the subscripts ↑ and ↓ refer to the "spin up" ({{math|''σ'' {{=}} +{{sfrac|1|2}}}}) and "spin down" ({{math|''σ'' {{=}} &minus;{{sfrac|1|2}}}}) states.<ref group=lower-alpha>This spinor notation is not necessarily standard; the literature usually writes <math>\psi=\begin{pmatrix} u^1 \\ u^2 \end{pmatrix}</math> or <math>\psi=\begin{pmatrix} \chi \\ \eta \end{pmatrix}</math> etc., but in the context of spin&nbsp;{{sfrac|1|2}}, this informal identification is commonly made.</ref>

In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;

:<math>\left(i \hbar \frac{\partial}{\partial t} -q\phi \right)\psi = \gamma^0 \left[ c\boldsymbol{\gamma}\cdot{(\hat{\mathbf{p}} - q\mathbf{A})} - mc^2 \right] \psi \quad \rightleftharpoons \quad \left[\gamma^\mu (\hat{P}_\mu - q A_\mu) - mc^2 \right]\psi = 0</math>

and was the first equation to accurately ''predict'' spin, a consequence of the 4 &times; 4 [[gamma matrices]] {{math|''γ''<sup>0</sup> {{=}} ''β'', '''γ''' {{=}} (''γ''<sub>1</sub>, ''γ''<sub>2</sub>, ''γ''<sub>3</sub>) {{=}} ''β'''''α''' {{=}} (''βα''<sub>1</sub>, ''βα''<sub>2</sub>, ''βα''<sub>3</sub>)}}. There is a 4 × 4 [[identity matrix]] pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Here {{math|''&psi;''}} is a four-component spinor field, which is conventionally split into two two-component spinors in the form:<ref group=lower-alpha>Again this notation is not necessarily standard, the more advanced literature usually writes

:<math>\psi=\begin{pmatrix}u \\ v \end{pmatrix} = \begin{pmatrix} u^1 \\ u^2 \\ v^1 \\ v^2 \end{pmatrix} </math> etc.,

but here we show informally the correspondence of energy, helicity, and spin states.</ref>

:<math>\psi=\begin{pmatrix}\psi_{+} \\ \psi_{-} \end{pmatrix} = \begin{pmatrix}\psi_{+\uparrow} \\ \psi_{+\downarrow} \\ \psi_{-\uparrow}  \\ \psi_{-\downarrow} \end{pmatrix} </math>

The 2-spinor {{math|''ψ''<sub>+</sub>}} corresponds to a particle with 4-momentum {{math|(''E'', '''p''')}} and charge {{math|''q''}} and two spin states ({{math|''σ'' {{=}} ±{{sfrac|1|2}}}}, as before). The other 2-spinor {{math|''ψ''<sub>−</sub>}} corresponds to a similar particle with the same mass and spin states, but ''negative'' 4-momentum {{math|−(''E'', '''p''')}} and ''negative'' charge {{math|−''q''}}, that is, negative energy states, [[T-symmetry|time-reversed]] momentum, and [[C-symmetry|negated charge]]. This was the first interpretation and prediction of a particle and ''corresponding [[antiparticle]]''. See [[Dirac spinor]] and [[bispinor]] for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see [[Dirac equation#Comparison with the Pauli theory|Dirac equation]] for how). When applied a one-electron atom or ion, setting {{math|'''A''' {{=}} '''0'''}} and {{math|''ϕ''}} to the appropriate electrostatic potential, additional relativistic terms include the [[spin–orbit interaction]], electron [[gyromagnetic ratio]], and [[Darwin term]]. In ordinary QM these terms have to be put in by hand and treated using [[perturbation theory]]. The positive energies do account accurately for the fine structure.

Within RQM, for massless particles the Dirac equation reduces to:

:<math> \left(\frac{\hat{E}}{c} + \boldsymbol{\sigma}\cdot \hat{\mathbf{p}} \right) \psi_{+} = 0 \,,\quad \left(\frac{\hat{E}}{c} - \boldsymbol{\sigma}\cdot \hat{\mathbf{p}} \right) \psi_{-} = 0 \quad \rightleftharpoons \quad \sigma^\mu \hat{P}_\mu \psi_{+} = 0\,,\quad \sigma_\mu \hat{P}^\mu \psi_{-} = 0\,,</math>

the first of which is the [[Weyl equation]], a considerable simplification applicable for massless [[neutrino]]s.<ref name="C.B. 1994">{{cite book|author=Parker, C.B.|year=1994|title=McGraw Hill Encyclopaedia of Physics|edition=2nd|publisher=McGraw Hill|page=[https://archive.org/details/mcgrawhillencycl1993park/page/1194 1194]|isbn=978-0-07-051400-3|url=https://archive.org/details/mcgrawhillencycl1993park/page/1194}}.</ref> This time there is a 2 × 2 [[identity matrix]] pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix {{math|''σ''<sub>0</sub>}} which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).

The Pauli and gamma matrices were introduced here, in theoretical physics, rather than [[pure mathematics]] itself. They have applications to [[quaternion]]s and to the [[SO(2)]] and [[SO(3)]] [[Lie group]]s, because they satisfy the important [[commutator]] [&nbsp;,&nbsp;] and [[Commutator#Anticommutator|anticommutator]] [&nbsp;,&nbsp;]<sub>+</sub> relations respectively:

:<math>\left[\sigma_a, \sigma_b \right] = 2i \varepsilon_{abc} \sigma_c \,, \quad \left[\sigma_a, \sigma_b \right]_{+} = 2\delta_{ab}\sigma_0</math>

where {{math|''ε<sub>abc</sub>''}} is the [[three-dimensional]] [[Levi-Civita symbol]]. The gamma matrices form [[basis (linear algebra)|bases]] in [[Clifford algebra]], and have a connection to the components of the flat spacetime [[Minkowski metric]] {{math|''η<sup>αβ</sup>''}} in the anticommutation relation:

:<math>\left[\gamma^\alpha,\gamma^\beta\right]_{+} = \gamma^\alpha\gamma^\beta + \gamma^\beta\gamma^\alpha = 2\eta^{\alpha\beta}\,,</math>

(This can be extended to [[curved space]]time by introducing [[Cartan formalism (physics)|vierbein]]s, but is not the subject of special relativity).

In 1929, the [[Breit equation]] was found to describe two or more electromagnetically interacting massive spin&nbsp;{{sfrac|1|2}} fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum [[many-particle system]]. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.

===Helicity and chirality===
{{Main|Helicity (particle physics)|Chirality (physics)}}
{{See also|spin polarization}}

The [[Helicity (particle physics)|helicity operator]] is defined by;

:<math>\hat{h} = \hat{\mathbf{S}}\cdot \frac{\hat{\mathbf{p}}}{|\mathbf{p}|} = \hat{\mathbf{S}} \cdot \frac{c\hat{\mathbf{p}}}{\sqrt{E^2 - (m_0c^2)^2}}</math>

where '''p''' is the momentum operator, '''S''' the spin operator for a particle of spin ''s'', ''E'' is the total energy of the particle, and ''m''<sub>0</sub> its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.<ref>{{cite book |title=Supersymmetry |author=Labelle, P. |series=Demystified |publisher=McGraw-Hill |year=2010 |isbn=978-0-07-163641-4}}</ref> Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.

An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin&nbsp;{{sfrac|1|2}} operator on the 3-momentum (times ''c''), {{math|'''σ''' · ''c'' '''p'''}}, which is the helicity (for the spin&nbsp;{{sfrac|1|2}} case) times <math>\sqrt{E^2 - (m_0c^2)^2}</math>.

For massless particles the helicity simplifies to:

:<math>\hat{h}  = \hat{\mathbf{S}} \cdot \frac{c\hat{\mathbf{p}}}{E} </math>

===Higher spins===
The Dirac equation can only describe particles of spin&nbsp;{{sfrac|1|2}}. Beyond the Dirac equation, RWEs have been applied to [[free particle]]s of various spins. In 1936, Dirac extended his equation to all fermions, three years later [[Markus Fierz|Fierz]] and Pauli rederived the same equation.<ref>{{cite journal | author=Esposito, S.  |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 |issue=6 |journal=Annals of Physics |pages=1617–1644 |bibcode=2012AnPhy.327.1617E|s2cid=119147261 }}</ref> The [[Bargmann–Wigner equations]] were found in 1948 using Lorentz group theory, applicable for all free particles with any spin.<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><ref>{{cite journal |author=Wigner, E. |year=1937 |title=On Unitary Representations Of The Inhomogeneous Lorentz Group |journal=Annals of Mathematics |volume=40 |pages=149–204 |number=1 |url=http://courses.theophys.kth.se/SI2390/wigner_1939.pdf |doi=10.2307/1968551 |bibcode=1939AnMat..40..149W |jstor=1968551 |s2cid=121773411 |access-date=2013-04-14 |archive-date=2015-10-04 |archive-url=https://web.archive.org/web/20151004025027/http://courses.theophys.kth.se/SI2390/wigner_1939.pdf |url-status=dead }}</ref> Considering the factorization of the KG equation above, and more rigorously by [[Lorentz group]] theory, it becomes apparent to introduce spin in the form of matrices.

The wavefunctions are multicomponent [[spinor field]]s, which can be represented as [[column vector]]s of [[function (mathematics)|function]]s of space and time:

:<math>\psi(\mathbf{r},t) = \begin{bmatrix} \psi_{\sigma=s}(\mathbf{r},t) \\ \psi_{\sigma=s - 1}(\mathbf{r},t) \\ \vdots \\ \psi_{\sigma=-s + 1}(\mathbf{r},t) \\ \psi_{\sigma=-s}(\mathbf{r},t) \end{bmatrix}\quad\rightleftharpoons\quad {\psi(\mathbf{r},t)}^\dagger = \begin{bmatrix} {\psi_{\sigma=s}(\mathbf{r},t)}^\star & {\psi_{\sigma=s - 1}(\mathbf{r},t)}^\star & \cdots & {\psi_{\sigma=-s + 1}(\mathbf{r},t)}^\star & {\psi_{\sigma=-s}(\mathbf{r},t)}^\star \end{bmatrix}</math>

where the expression on the right is the [[Hermitian conjugate]]. For a ''massive'' particle of spin {{math|''s''}}, there are {{math|2''s'' + 1}} components for the particle, and another {{math|2''s'' + 1}} for the corresponding [[antiparticle]] (there are {{math|2''s'' + 1}} possible {{math|''σ''}} values in each case), altogether forming a {{math|2(2''s'' + 1)}}-component spinor field:

:<math>\psi(\mathbf{r},t) = \begin{bmatrix} \psi_{+,\,\sigma=s}(\mathbf{r},t) \\ \psi_{+,\,\sigma=s - 1}(\mathbf{r},t) \\ \vdots \\ \psi_{+,\,\sigma=-s + 1}(\mathbf{r},t) \\ \psi_{+,\,\sigma=-s}(\mathbf{r},t) \\ \psi_{-,\,\sigma=s}(\mathbf{r},t) \\ \psi_{-,\,\sigma=s - 1}(\mathbf{r},t) \\ \vdots \\ \psi_{-,\,\sigma=-s + 1}(\mathbf{r},t) \\ \psi_{-,\,\sigma=-s}(\mathbf{r},t) \end{bmatrix}\quad\rightleftharpoons\quad {\psi(\mathbf{r},t)}^\dagger\begin{bmatrix} {\psi_{+,\,\sigma=s}(\mathbf{r},t)}^\star & {\psi_{+,\,\sigma=s - 1}(\mathbf{r},t)}^\star & \cdots & {\psi_{-,\,\sigma=-s}(\mathbf{r},t)}^\star \end{bmatrix} </math>

with the + subscript indicating the particle and − subscript for the antiparticle. However, for ''massless'' particles of spin ''s'', there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +''s'' and the other for the antiparticle in the opposite helicity state corresponding to &minus;''s'':

:<math>\psi(\mathbf{r},t) = \begin{pmatrix} \psi_{+}(\mathbf{r},t) \\ \psi_{-}(\mathbf{r},t) \end{pmatrix}</math>

According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically, [[Élie Cartan]] found the most general form of [[spinor]]s in 1913, prior to the spinors revealed in the RWEs following the year 1927.

For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies.<ref>{{cite journal |author1=Jaroszewicz, T. |author2=Kurzepa, P.S |year=1992 |title=Geometry of spacetime propagation of spinning particles |journal=Annals of Physics|doi=10.1016/0003-4916(92)90176-M |bibcode=1992AnPhy.216..226J |volume=216 |issue=2 |pages=226–267}}</ref> For spin greater than {{sfrac|''ħ''|2}}, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments ([[electric dipole moment]]s and [[magnetic dipole moment]]s) allowed by the [[spin quantum number]] are arbitrary. (Theoretically, [[magnetic charge]] would contribute also). For example, the spin&nbsp;{{sfrac|1|2}} case only allows a magnetic dipole, but for spin&nbsp;1 particles magnetic quadrupoles and electric dipoles are also possible.<ref name="C.B. 1994"/> For more on this topic, see [[multipole expansion]] and (for example) Cédric Lorcé (2009).<ref>{{cite arXiv |first=Cédric |last=Lorcé |year=2009 |title=Electromagnetic Properties for Arbitrary Spin Particles: Part 1 − Electromagnetic Current and Multipole Decomposition |eprint=0901.4199 |class=hep-ph}}</ref><ref>{{cite journal |first=Cédric |last=Lorcé |year=2009 |title=Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities |journal=Physical Review D |volume=79 |issue=11 |page=113011 |arxiv=0901.4200 |doi=10.1103/PhysRevD.79.113011 |bibcode=2009PhRvD..79k3011L|s2cid=17801598 }}</ref>

==Velocity operator==
The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition {{math|'''p''' {{=}} ''m'' '''v'''}}, and substituting quantum operators in the usual way:<ref>{{cite book |author=Strange, P. |title=Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics |page=206 |publisher=Cambridge University Press |year=1998|url=https://books.google.com/books?id=sdVrBM2w0OwC&q=velocity+operator+in+relativistic+quantum+mechanics&pg=PA208 |isbn=978-0-521-56583-7}}</ref>

:<math>\hat{\mathbf{v}} = \frac{1}{m}\hat{\mathbf{p}}</math>

which has eigenvalues that take ''any'' value. In RQM, the Dirac theory, it is:

:<math>\hat{\mathbf{v}} = \frac{i}{\hbar}\left[\hat{H},\hat{\mathbf{r}}\right]</math>

which must have eigenvalues between ±''c''. See [[Foldy–Wouthuysen transformation]] for more theoretical background.

==Relativistic quantum Lagrangians==
The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for {{math|''ψ''}}. An equivalent alternative is to determine a [[Lagrangian (field theory)|Lagrangian]] (really meaning ''[[Lagrangian density]]''), then generate the differential equation by the [[Classical field theory#Relativistic field theory|field-theoretic Euler–Lagrange equation]]:

:<math> \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0 \,</math>

For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:<ref>{{cite book |title=Supersymmetry |url=https://archive.org/details/supersymmetrydem00labe |url-access=limited |author=Labelle, P. |series=Demystified |publisher=McGraw-Hill |page=[https://archive.org/details/supersymmetrydem00labe/page/n28 14] |year=2010 |isbn=978-0-07-163641-4}}</ref>

:<math>\mathcal{L} = \overline{\psi}(\gamma^\mu P_\mu - mc)\psi</math>

and Klein–Gordon Lagrangian is:

:<math>\mathcal{L} = - \frac{\hbar^2}{m} \eta^{\mu \nu} \partial_{\mu}\psi^{*} \partial_{\nu}\psi - m c^2 \psi^{*} \psi\,.</math>

This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of {{math|''ψ''}} is the subject of QFT rather than RQM: Feynman's [[path integral formulation]] uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).<ref>{{cite book |author-link=Steven Weinberg |first=S. |last=Weinberg |year=1995 |title=The Quantum Theory of Fields |volume=1 |publisher=Cambridge University Press |isbn=978-0-521-55001-7 |url=https://archive.org/details/quantumtheoryoff00stev }}</ref>

==Relativistic quantum angular momentum==
In non-relativistic QM, the [[angular momentum operator]] is formed from the classical [[pseudovector]] definition {{math|'''L''' {{=}} '''r''' × '''p'''}}. In RQM, the position and momentum operators are inserted directly where they appear in the orbital [[relativistic angular momentum]] tensor defined from the four-dimensional position and momentum of the particle, equivalently a [[bivector]] in the [[exterior algebra]] formalism:<ref>{{cite book |author=Penrose, R. |title=The Road to Reality |publisher=Vintage Books |pages=437, 566–569 |year=2005 |isbn=978-0-09-944068-0 |title-link=The Road to Reality}}</ref><ref group=lower-alpha>Some authors, including Penrose, use ''Latin'' letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.</ref>

:<math>M^{\alpha\beta} = X^\alpha P^\beta - X^\beta P^\alpha = 2 X^{[\alpha} P^{\beta]} \quad \rightleftharpoons \quad \mathbf{M} = \mathbf{X}\wedge\mathbf{P}\,,</math>

which are six components altogether: three are the non-relativistic 3-orbital angular momenta; {{math|''M''<sup>12</sup> {{=}} ''L''<sup>3</sup>}}, {{math|''M''<sup>23</sup> {{=}} ''L''<sup>1</sup>}}, {{math|''M''<sup>31</sup> {{=}} ''L''<sup>2</sup>}}, and the other three {{math|''M''<sup>01</sup>}}, {{math|''M''<sup>02</sup>}}, {{math|''M''<sup>03</sup>}} are boosts of the [[centre of mass]] of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass {{math|''m''}}, the ''total'' angular momentum tensor is:

:<math>J^{\alpha\beta} = 2X^{[\alpha} P^{\beta]} + \frac{1}{m^2}\varepsilon^{\alpha \beta \gamma \delta} W_\gamma p_\delta \quad \rightleftharpoons \quad \mathbf{J} = \mathbf{X}\wedge\mathbf{P} + \frac{1}{m^2}\star(\mathbf{W}\wedge\mathbf{P})</math>

where the star denotes the [[Hodge dual]], and

:<math>W_\alpha =\frac{1}{2}\varepsilon_{\alpha \beta \gamma \delta}M^{\beta \gamma}p^\delta \quad \rightleftharpoons \quad \mathbf{W} = \star(\mathbf{M}\wedge\mathbf{P})</math>

is the [[Pauli–Lubanski pseudovector]].<ref>{{cite book |title=Quantum Field Theory |author=Ryder, L.H. |publisher=Cambridge University Press |edition=2nd |isbn=978-0-521-47814-4 |year=1996 |page=62 |url=https://books.google.com/books?id=nnuW_kVJ500C&q=pauli-lubanski+pseudovector&pg=PA62}}</ref> For more on relativistic spin, see (for example) Troshin & Tyurin (1994).<ref>{{cite book |author1=Troshin, S.M. |author2=Tyurin, N.E. |year=1994 |publisher=World Scientific |title=Spin phenomena in particle interactions|url=https://books.google.com/books?id=AU2DV1hKpuoC&q=pauli-lubanski+pseudovector&pg=PA9 |isbn=978-981-02-1692-4|bibcode=1994sppi.book.....T }}</ref>

===Thomas precession and spin–orbit interactions===
In 1926, the [[Thomas precession]] is discovered: relativistic corrections to the spin of elementary particles with application in the [[spin–orbit interaction]] of atoms and rotation of macroscopic objects.<ref>{{cite book |author1-link=Charles W. Misner |first1=C.W. |last1=Misner |author2-link=Kip S. Thorne |first2=K.S. |last2=Thorne |author3-link=John A. Wheeler |first3=J.A. |last3=Wheeler |title=Gravitation |page=[https://archive.org/details/gravitation00cwmi/page/n1178 1146] |isbn=978-0-7167-0344-0 |title-link=Gravitation (book) |date=1973-09-15 |publisher=Macmillan |df=dmy-all}}</ref><ref>{{cite book |author1=Ciufolini, I. |author2=Matzner, R.R.A. |title=General relativity and John Archibald Wheeler |page=329 |publisher=Springer |year=2010 |url=https://books.google.com/books?id=v0pSfo8vrtsC&q=thomas+precession+relativistic+quantum+mechanics&pg=PA329 |isbn=978-90-481-3735-0}}</ref> In 1939 Wigner derived the Thomas precession.

In [[classical electromagnetism and special relativity#The E and B fields|classical electromagnetism and special relativity]], an electron moving with a velocity {{math|'''v'''}} through an electric field {{math|'''E'''}} but not a magnetic field {{math|'''B'''}}, will in its own frame of reference experience a [[Lorentz transformation|Lorentz-transformed]] magnetic field {{math|'''B&prime;'''}}:

:<math>\mathbf{B}' = \frac{\mathbf{E} \times \mathbf{v}}{c^2\sqrt{1- \left(v/c\right)^2}} \,.</math>

In the non-relativistic limit {{math|''v'' << ''c''}}:

:<math>\mathbf{B}' = \frac{\mathbf{E} \times \mathbf{v}}{c^2} \,,</math>

so the non-relativistic spin interaction Hamiltonian becomes:<ref name="Kroemer">{{cite journal |author=Kroemer, H. |year=2003 |title=The Thomas precession factor in spin–orbit interaction |doi=10.1119/1.1615526 |url=http://www.ece.ucsb.edu/faculty/Kroemer/pubs/13_04Thomas.pdf|arxiv = physics/0310016 |bibcode=2004AmJPh..72...51K |volume=72 |issue=1 |journal=American Journal of Physics |pages=51–52|s2cid=119533324 }}</ref>

:<math>\hat{H} = - \mathbf{B}'\cdot \hat{\boldsymbol{\mu}}_S = -\left(\mathbf{B} + \frac{\mathbf{E} \times \mathbf{v}}{c^2} \right) \cdot \hat{\boldsymbol{\mu}}_S \,, </math>

where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order {{math|(''v/c'')²}}, but this disagrees with experimental atomic spectra by a factor of {{frac|1|2}}. It was pointed out by L.&nbsp;Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in a [[rotating frame of reference]], and this additional precession of the electron is called the ''Thomas precession''. It can be shown<ref>{{cite book |author=Jackson, J.D. |author-link=John David Jackson (physicist) |page=[https://archive.org/details/classicalelectro00jack_449/page/n547 548] |year=1999 |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |edition=3rd |publisher=Wiley |isbn=978-0-471-30932-1}}</ref> that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is:

:<math>\hat{H} = - \mathbf{B}'\cdot \hat{\boldsymbol{\mu}}_S = -\left(\mathbf{B} + \frac{\mathbf{E} \times \mathbf{v}}{2c^2} \right) \cdot \hat{\boldsymbol{\mu}}_S \,.</math>

In the case of RQM, the factor of {{frac|1|2}} is predicted by the Dirac equation.<ref name="Kroemer"/>

==History==
The events which led to and established RQM, and the continuation beyond into [[quantum electrodynamics]] (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),<ref>{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles |edition=2nd |pages=[https://archive.org/details/quantumphysicsof00eisb/page/57 57, 114–116, 125–126, 272] |author1=Resnick, R. |author2=Eisberg, R. |publisher=John Wiley & Sons |year=1985 |isbn=978-0-471-87373-0 |url=https://archive.org/details/quantumphysicsof00eisb/page/57 }}</ref> and [[Peter Atkins|P.W Atkins]] (1974)<ref>{{cite book |title=Quanta: A handbook of concepts |author-link=Peter Atkins |first=P.W. |last=Atkins |publisher=Oxford University Press |pages=168–169, 176, 263, 228 |year=1974 |isbn=978-0-19-855493-6}}</ref>]. More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a ''necessary'' component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found [[atomic physics]], [[nuclear physics]], and [[particle physics]]; by considering [[spectroscopy]], [[diffraction]] and [[scattering]] of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.

===Relativistic description of particles in quantum phenomena===
[[Albert Einstein]] in 1905 explained of the [[photoelectric effect]]; a particle description of light as [[photons]]. In 1916, [[Arnold Sommerfeld|Sommerfeld]] explains [[fine structure]]; the splitting of the [[spectral line]]s of [[atoms]] due to first order relativistic corrections. The [[Compton effect]] of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering. [[Louis de Broglie|de Broglie]] extends [[wave–particle duality]] to [[matter]]: the [[de Broglie relations]], which are consistent with special relativity and quantum mechanics. By 1927, [[Clinton Davisson|Davisson]] and [[Lester Germer|Germer]] and separately [[George Paget Thomson|G. Thomson]] successfully diffract electrons, providing experimental evidence of wave-particle duality.

===Experiments===
* 1897 [[J. J. Thomson]] discovers the [[electron]] and measures its [[mass-to-charge ratio]]. Discovery of the [[Zeeman effect]]: the splitting a [[spectral line]] into several components in the presence of a static magnetic field.
* 1908 [[Robert Andrews Millikan|Millikan]] measures the charge on the electron and finds experimental evidence of its quantization, in the [[oil drop experiment]].
* 1911 [[Alpha particle]] scattering in the [[Geiger–Marsden experiment]], led by [[Ernest Rutherford|Rutherford]], showed that atoms possess an internal structure: the [[atomic nucleus]].<ref>{{cite book |author=Krane, K.S. |year=1988 |title=Introductory Nuclear Physics |url=https://archive.org/details/introductorynucl00kran |url-access=limited |publisher=John Wiley & Sons |pages=[https://archive.org/details/introductorynucl00kran/page/n418 396]–405 |isbn=978-0-471-80553-3}}</ref>
* 1913 The [[Stark effect]] is discovered: splitting of spectral lines due to a static [[electric field]] (compare with the Zeeman effect). 
* 1922 [[Stern–Gerlach experiment]]: experimental evidence of spin and its quantization.
* 1924 [[Edmund Clifton Stoner|Stoner]] studies splitting of [[energy level]]s in [[magnetic field]]s.
* 1932 Experimental discovery of the [[neutron]] by [[James Chadwick|Chadwick]], and [[positron]]s by [[Carl David Anderson|Anderson]], confirming the theoretical prediction of positrons.
* 1958 Discovery of the [[Mössbauer effect]]: resonant and recoil-free emission and absorption of [[gamma radiation]] by atomic nuclei bound in a solid, useful for accurate measurements of [[gravitational redshift]] and [[time dilation]], and in the analysis of nuclear electromagnetic moments in [[hyperfine interaction]]s.<ref>{{cite book |author=Krane, K.S. |year=1988 |title=Introductory Nuclear Physics |url=https://archive.org/details/introductorynucl00kran |url-access=limited |publisher=John Wiley & Sons |pages=[https://archive.org/details/introductorynucl00kran/page/n383 361]–370 |isbn=978-0-471-80553-3}}</ref>

===Quantum non-locality and relativistic locality===
In 1935, Einstein, [[Nathan Rosen|Rosen]], [[Boris Podolsky|Podolsky]] published a paper<ref>{{cite journal |title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? |author1=Einstein, A. |author2=Podolsky, B. |author3=Rosen, N. |year=1935 |doi=10.1103/PhysRev.47.777 |journal=Phys. Rev. |volume=47 |issue=10 |bibcode=1935PhRv...47..777E |pages=777–780|url=https://cds.cern.ch/record/1060284/files/PhysRev.48.696.pdf |doi-access=free }}</ref> concerning [[quantum entanglement]] of particles, questioning [[quantum nonlocality]] and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceed ''c''). QM does ''not'' violate SR.<ref>{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=192 |isbn=978-0-13-146100-0}}</ref><ref>{{cite book |author=Penrose, R. |title=The Road to Reality| publisher=Vintage Books |year=2005 |isbn=978-0-09-944068-0 |title-link=The Road to Reality}} ''Chapter '''23''': The entangled quantum world''</ref> In 1959, [[David Bohm|Bohm]] and [[Yakir Aharonov|Aharonov]] publish a paper<ref>
{{cite journal |author1=Aharonov, Y. |author2=Bohm, D. |year=1959 |title=Significance of electromagnetic potentials in quantum theory |journal=[[Physical Review]] |volume=115 |issue=3 |pages=485–491 |doi=10.1103/PhysRev.115.485 |bibcode=1959PhRv..115..485A|doi-access=free }}</ref> on the [[Aharonov–Bohm effect]], questioning the status of electromagnetic potentials in QM. The [[electromagnetic field tensor|EM field tensor]] and [[Electromagnetic four-potential|EM 4-potential]] formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964, [[Bell's theorem]] was published in a paper on the EPR paradox,<ref>{{cite journal |last=Bell |first=John |year=1964 |title=On the Einstein Podolsky Rosen Paradox |url=http://homepages.physik.uni-muenchen.de/~vondelft/Lehre/09qm/lec21-22-BellInequalities/Bell1964.pdf |journal=[[Physics (American Physical Society journal)|Physics]] |volume=1 |issue=3 |pages=195–200
|doi=10.1103/PhysicsPhysiqueFizika.1.195|doi-access=free }}</ref> showing that QM cannot be derived from [[local hidden-variable theory|local hidden-variable theories]] if locality is to be maintained.

===The Lamb shift===
{{Main|Lamb shift}}

In 1947, the Lamb shift was discovered: a small difference in the <sup>2</sup>''S''<sub>{{frac|1|2}}</sub> and <sup>2</sup>''P''<sub>{{frac|1|2}}</sub> levels of hydrogen, due to the interaction between the electron and vacuum. [[Willis Lamb|Lamb]] and [[Robert Retherford|Retherford]] experimentally measure stimulated radio-frequency transitions the <sup>2</sup>''S''<sub>{{frac|1|2}}</sub> and <sup>2</sup>''P''<sub>{{frac|1|2}}</sub> hydrogen levels by [[microwave]] radiation.<ref>{{cite journal|title=Fine Structure of the Hydrogen Atom by a Microwave Method |first1=Willis E. |last1=Lamb |author1-link=Willis Lamb |last2=Retherford |first2=Robert C. |journal=[[Physical Review]] |volume=72 |issue=3 |pages=241–243 |year=1947 |doi=10.1103/PhysRev.72.241 |bibcode=1947PhRv...72..241L|doi-access=free }}</ref> An explanation of the Lamb shift is presented by [[Hans Bethe|Bethe]]. Papers on the effect were published in the early 1950s.<ref><!--- IF FOR ANY REASON THIS REF IS DELETED - MAKE SURE IT'S IN THE MAIN ARTICLE FIRST. --->{{cite journal |author1=Lamb, W.E. Jr. |author2=Retherford, R.C. |name-list-style=amp |year=1950 |title=Fine Structure of the Hydrogen Atom. Part I |journal=Phys. Rev. |volume=79 |issue=4 |pages=549–572 |bibcode=1950PhRv...79..549L |doi=10.1103/PhysRev.79.549}} <br/>
{{cite journal |author1=Lamb, W.E. Jr. |author2=Retherford, R.C. |name-list-style=amp |year=1951 |title=Fine Structure of the Hydrogen Atom. Part II |journal=Phys. Rev. |volume=81 |issue=2 |pages=222–232 |bibcode=1951PhRv...81..222L |doi=10.1103/PhysRev.81.222}}{{cite journal |author=Lamb, W.E. Jr. |year=1952 |title=Fine Structure of the Hydrogen Atom. III |journal=Phys. Rev. |volume=85 |issue=2|pages=259–276 |bibcode=1952PhRv...85..259L |doi=10.1103/PhysRev.85.259|pmid=17775407 }} <br/>
{{cite journal |author1=Lamb, W.E. Jr. |author2=Retherford, R.C. |name-list-style=amp |year=1952 |title=Fine Structure of the Hydrogen Atom. IV |journal=Phys. Rev. |volume=86 |issue=6 |pages=1014–1022 |bibcode=1952PhRv...86.1014L |doi=10.1103/PhysRev.86.1014|pmid=17775407 }} <br/>
{{cite journal |author1=Triebwasser, S. |author2=Dayhoff, E.S. |author3=Lamb, W.E. Jr. |name-list-style=amp |year=1953|title=Fine Structure of the Hydrogen Atom. V |journal=Phys. Rev. |volume=89 |issue=1 |pages=98–106 |bibcode=1953PhRv...89...98T |doi=10.1103/PhysRev.89.98}}</ref>

===Development of quantum electrodynamics===
* 1927 [[Paul Dirac|Dirac]] establishes the field of QED, also coining the term "quantum electrodynamics".<ref>{{Cite journal |last=Dirac |first=Paul |date=1927-03-01 |title=The quantum theory of the emission and absorption of radiation |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=114 |issue=767 |pages=243–265 |doi=10.1098/rspa.1927.0039 |issn=0950-1207|doi-access=free }}</ref>
* 1943 [[Sin-Itiro Tomonaga|Tomonaga]] begins work on [[renormalization]], influential in QED.
* 1947 [[Julian Schwinger|Schwinger]] calculates the [[Anomalous magnetic dipole moment#Anomalous magnetic moment of the electron|anomalous magnetic moment of the electron]]. [[Polykarp Kusch|Kusch]] measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.

==See also==
{{Col-begin}}
{{Col-break}}

===Atomic physics and chemistry===
* [[Relativistic quantum chemistry]]
* [[Breit equation]]
* [[Electron spin resonance]]
* [[Fine-structure constant]]

===Mathematical physics===
* [[Quantum spacetime]]
* [[Spin connection]]
* [[Spinor bundle]]
* [[Dirac equation in the algebra of physical space]]
* [[Casimir invariant]]
* [[Casimir operator]]
* [[Wigner D-matrix]]

{{Col-break}}

===Particle physics and quantum field theory===
* [[Zitterbewegung]]
* [[Two-body Dirac equations]]
* [[Relativistic Heavy Ion Collider]]
* [[Symmetry (physics)]]
* [[Parity (physics)|Parity]]
* [[CPT invariance]]
* [[Chirality (physics)#Chirality and helicity|Chirality (physics)]]
* [[Standard model]]
* [[Gauge theory]]
* [[Tachyon]]
* [[Modern searches for Lorentz violation]]

{{col-end}}

==Footnotes==
{{notelist|1}}

==References==
{{reflist}}

===Selected books===
<!---PLEASE PLEASE PLEASE DO NOT DELETE ANYTHING AS "IRRELEVANT/OUTDATED" FROM PERSONAL PREFERENCES IN THIS REF SECTION - MOST OF THE BOOKS WILL BE MOVED INTO THE ARTICLE AT SOME POINT--->
* {{cite book |title=Principles of Quantum Mechanics |author-link=Paul Dirac |first=P.A.M. |last=Dirac |edition=4th |publisher=Clarendon Press |year=1981 |url=https://books.google.com/books?id=XehUpGiM6FIC&q=principles+of+quantum+mechanics+dirac+4th+edition|isbn=978-0-19-852011-5}}
* {{cite book |author=Dirac, P.A.M. |title=Lectures on Quantum Mechanics |publisher=Courier Dover Publications |year=1964 |url=https://books.google.com/books?id=GVwzb1rZW9kC&q=Relativistic+quantum+mechanics|isbn=978-0-486-41713-4}}
* {{Cite book |title=The Dirac Equation |author=Thaller, B. |year=2010 |publisher=Springer |isbn=978-3-642-08134-7|url=https://books.google.com/books?id=822rcQAACAAJ&q=dirac+equation+springer}}
* {{Cite book |title=General Principles of Quantum Mechanics |author-link=Wolfgang Pauli |first=W. |last=Pauli |year=1980 |publisher=Springer |isbn=978-3-540-09842-3 |url=https://books.google.com/books?id=93sfAQAAMAAJ&q=General+Principles+of+Quantum+Mechanics}}
* {{cite book |first=E. |last=Merzbacher |year=1998 |edition=3rd |title=Quantum Mechanics |publisher=John Wiley & Sons |isbn =978-0-471-88702-7}}
* {{cite book |first=A. |last=Messiah |year=1961 |title=Quantum Mechanics |publisher=John Wiley & Sons |volume=1 |isbn=978-0-471-59766-7}}
* {{cite book |first1=J.D. |last1=Bjorken |first2=S.D. |last2=Drell |year=1964 |title=Relativistic Quantum Mechanics (Pure & Applied Physics) |publisher=McGraw-Hill |isbn=978-0-07-005493-6}}
* {{cite book |author1=Feynman, R.P. |author2=Leighton, R.B. |author3=Sands, M. |year=1965 |title=Feynman Lectures on Physics |publisher=Addison-Wesley |volume=3 |isbn=978-0-201-02118-9 |url=https://feynmanlectures.caltech.edu/III_toc.html}}
* {{cite book |author=Schiff, L.I. |year=1968 |edition=3rd |title=Quantum Mechanics |url=https://archive.org/details/quantummechanics0000schi |url-access=registration |publisher=McGraw-Hill}}
* {{cite book |author-link=Freeman Dyson |first=F. |last=Dyson |edition=2nd |year=2011 |title=Advanced Quantum Mechanics |publisher=World Scientific |isbn=978-981-4383-40-0}}
* {{cite book |author=Clifton, R.K. |title=Perspectives on Quantum Reality: Non-Relativistic, Relativistic, and Field-Theoretic|publisher=Springer |year=2011 |url=https://books.google.com/books?id=TTKacQAACAAJ&q=Relativistic+quantum+mechanics |isbn=978-90-481-4643-7}}
* {{cite book |author1-link=Claude Cohen-Tannoudji |first1=C. |last1=Tannoudji |first2=B. |last2=Diu |first3=F. |last3=Laloë |year=1977 |title=Quantum Mechanics |publisher=Wiley VCH |volume=1 |isbn=978-0-471-16433-3}}
* {{cite book |author1-link=Claude Cohen-Tannoudji |first1=C. |last1=Tannoudji |first2=B. |last2=Diu |first3=F. |last3=Laloë |year=1977 |title=Quantum Mechanics |publisher=Wiley VCH |volume=2 |isbn=978-0-471-16435-7}}
* {{cite book |author=Rae, A.I.M. |year=2008| title = Quantum Mechanics |edition=5th|publisher=Taylor & Francis |volume=2 |isbn=978-1-58488-970-0 |url=https://books.google.com/books?id=YDhHAQAAIAAJ&q=quantum+mechanics+Alastair+Rae+5th+edition}}
* {{cite book |author=Pilkuhn, H. |edition=2nd |year = 2005 |series=Texts and Monographs in Physics Series |title = Relativistic Quantum Mechanics |publisher=Springer |isbn=978-3-540-28522-9 |url=https://books.google.com/books?id=jEW6fY6-iVcC&q=Relativistic+quantum+mechanics&pg=PA1}}
* {{cite book |author=Parthasarathy, R. |title=Relativistic quantum mechanics |publisher=Alpha Science International |year=2010 |url=https://books.google.com/books?id=1t0WAQAAMAAJ&q=relativistic+quantum+mechanics |isbn=978-1-84265-573-3}}
* {{cite book |first1=U. |last1=Kaldor |first2=S. |last2=Wilson |title=Theoretical Chemistry and Physics of Heavy and Superheavy Elements
|publisher=Springer |year=2003 |isbn=978-1-4020-1371-3 |url=https://books.google.com/books?id=0xcAM5BzS-wC&q=Relativistic_quantum_mechanics}}
* {{cite book |author=Thaller, B. |title=Advanced visual quantum mechanics |publisher=Springer |year=2005|url=https://books.google.com/books?id=iq1Gi6hmTRAC&q=Relativistic+quantum+mechanics |isbn=978-0-387-27127-9|bibcode=2005avqm.book.....T }}
* {{cite book |first1=H.P. |last1=Breuer |first2=F. |last2=Petruccione |title=Relativistic Quantum Measurement and Decoherence |publisher=Springer |year=2000 |url=https://archive.org/details/springer_10.1007-3-540-45369-5 |quote=Relativistic quantum mechanics. |isbn=978-3-540-41061-4}}
* {{cite book |author=Shepherd, P.J. |title=A Course in Theoretical Physics |publisher=John Wiley & Sons |year=2013 |url=https://books.google.com/books?id=CoV4JemLm40C&q=A+course+in+theoretical+physics |isbn=978-1-118-51692-8}}
*{{cite book |author1-link=Hans Bethe |first1=H.A. |last1=Bethe |first2=R.W. |last2=Jackiw |title=Intermediate Quantum Mechanics |publisher=Addison-Wesley |year=1997 |url=https://books.google.com/books?id=7s5gog6O0fkC&q=magnetic+moments+in+relativistic+quantum+mechanics |isbn=978-0-201-32831-8 }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}
*{{cite book |author=Heitler, W. |title=The Quantum Theory of Radiation |publisher=Courier Dover Publications |year=1954 |edition=3rd |url=https://books.google.com/books?id=L7w7UpecbKYC&q=magnetic+moments+in+relativistic+quantum+mechanics |isbn=978-0-486-64558-2}}
*{{cite book |author1=Gottfried, K. |author2=Yan, T. |edition=2nd |title=Quantum Mechanics: Fundamentals |page=245 |publisher=Springer |year=2003 |url=https://books.google.com/books?id=8gFX-9YcvIYC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA245 |isbn=978-0-387-95576-6}}
*{{cite book |author=Schwabl, F.|title=Quantum Mechanics |page=220 |publisher=Springer |year=2010 |url=https://books.google.com/books?id=pTHb4NK2eZcC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA220 |isbn=978-3-540-71933-5}}
*{{cite book |author=Sachs, R.G. |title=The Physics of Time Reversal |edition=2nd |publisher=University of Chicago Press |year=1987 |url=https://archive.org/details/physicsoftimerev0000sach |url-access=registration |page=[https://archive.org/details/physicsoftimerev0000sach/page/280 280] |quote=hyperfine structure in relativistic quantum mechanics. |isbn=978-0-226-73331-9}}

===Group theory in quantum physics===
*{{cite book |author-link=Hermann Weyl |first=H. |last=Weyl |title=The theory of groups and quantum mechanics |page=[https://archive.org/details/theoryofgroupsqu1950weyl/page/203 203] |publisher=Courier Dover Publications |year=1950 |url=https://archive.org/details/theoryofgroupsqu1950weyl |url-access=registration |quote=magnetic moments in relativistic quantum mechanics. |isbn=9780486602691}}
*{{cite book |author=Tung, W.K. |title=Group Theory in Physics
 |publisher=World Scientific |year=1985 |url=https://books.google.com/books?id=O89tgpOBO04C&q=group+theory+in+physics |isbn=978-9971-966-56-0}}
*{{cite book |author=Heine, V. |title=Group Theory in Quantum Mechanics: An Introduction to Its Present Usage |publisher=Courier Dover Publications |year=1993 |url=https://books.google.com/books?id=NayFD34uEu0C&q=lorentz+group+in+relativistic+quantum+mechanics&pg=PA363 |isbn=978-0-486-67585-5}}

===Selected papers===
<!---PLEASE PLEASE PLEASE DO NOT DELETE ANYTHING AS "IRRELEVANT/OUTDATED" FROM PERSONAL PREFERENCES IN THIS REF SECTION - MOST OF THE PAPERS WILL BE MOVED INTO THE ARTICLE AT SOME POINT--->
* {{cite journal |author-link=Paul Dirac |first=P.A.M. |last=Dirac |year=1932 |title=Relativistic Quantum Mechanics |journal=[[Proceedings of the Royal Society A]] |volume=136 |issue=829 |doi=10.1098/rspa.1932.0094 |bibcode = 1932RSPSA.136..453D |pages=453–464|doi-access=free }}
* {{cite news |author-link=Wolfgang Pauli |first=W. |last=Pauli |year=1945 |title=Exclusion principle and quantum mechanics
|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.pdf}}
* {{cite journal |author=Antoine, J.P. |year=2004 |title= Relativistic Quantum Mechanics |volume=37 |issue=4 |journal=J. Phys. A |doi=10.1088/0305-4470/37/4/B01 |bibcode=2004JPhA...37.1463P |pages=1465 |citeseerx=10.1.1.499.2793}}
* {{cite news |title=Relativistic quantum mechanics of supersymmetric particles |author1=Henneaux, M. |author2=Teitelboim, C. |volume=143 |year=1982}}
* {{cite journal |journal=Phys. Rev. A |volume=34 |issue=3|year=1986 |title=Parametrizing relativistic quantum mechanics |author=Fanchi, J.R. |doi=10.1103/PhysRevA.34.1677 |bibcode=1986PhRvA..34.1677F |pages=1677–1681|pmid=9897446 }}
* {{cite journal |author=Ord, G.N. |year=1983 |title=Fractal space-time: a geometric analogue of relativistic quantum mechanics |volume=16 |issue=9 |journal=J. Phys. A |doi=10.1088/0305-4470/16/9/012 |bibcode=1983JPhA...16.1869O |pages=1869–1884}}
* {{cite journal |year=1982 |title=Relativistic quantum mechanics of particles with direct interactions |author1=Coester, F. |author2=Polyzou, W.N. |journal=Phys. Rev. D |doi=10.1103/PhysRevD.26.1348 |volume=26 |issue=6 |pages=1348–1367 |bibcode=1982PhRvD..26.1348C}}
* {{cite journal |author1=Mann, R.B. |author2=Ralph, T.C. |year=2012 |journal=Class. Quantum Grav. |doi=10.1088/0264-9381/29/22/220301 |title= Relativistic quantum information |volume=29 |issue=22 |bibcode=2012CQGra..29v0301M |pages=220301|s2cid=123341332 }}
* {{cite journal |author=Low, S.G. |year=1997 |journal=J. Math. Phys. |doi=10.1088/0264-9381/29/22/220301 |title=Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3) |volume=38 |issue=22 |arxiv=physics/9703008 |bibcode=2012CQGra..29v0301M |pages=2197–2209|s2cid=123341332 }}
* {{cite journal |author1=Fronsdal, C. |author2=Lundberg, L.E. |year=1997 |journal=Phys. Rev. D |doi=10.1103/PhysRevD.1.3247|title= Relativistic Quantum Mechanics of Two Interacting Particles|volume=1 |issue=12 |arxiv=physics/9703008 |bibcode=1970PhRvD...1.3247F |pages=3247–3258}}
*{{cite journal |author1=Bordovitsyn, V.A. |author2=Myagkii, A.N. |title=Spin–orbital motion and Thomas precession in the classical and quantum theories |doi=10.1119/1.1615526 |url=https://cds.cern.ch/record/488497/files/0102107.pdf |arxiv=physics/0310016 |bibcode=2004AmJPh..72...51K |volume=72 |journal=American Journal of Physics |pages=51–52 |year=2004|issue=1 |s2cid=119533324 }}
*{{cite journal |author=Rȩbilas, K. |year=2013 |title=Comment on 'Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession' |journal=Eur. J. Phys. |volume=34 |issue=3 |doi=10.1088/0143-0807/34/3/L55|bibcode = 2013EJPh...34L..55R |pages=L55–L61|s2cid=122527454 }}
*{{cite journal |author=Corben, H.C. |year=1993 |page=551 |title=Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession |journal=Am. J. Phys. |volume=61 |issue=6 |doi=10.1119/1.17207 |bibcode=1993AmJPh..61..551C }}

==Further reading==

===Relativistic quantum mechanics and field theory===
<!---PLEASE PLEASE PLEASE DO NOT DELETE ANYTHING AS "IRRELEVANT/OUTDATED" FROM PERSONAL PREFERENCES IN THIS REF SECTION - MOST OF THE BOOKS MAY BE MOVED INTO THE ARTICLE AT SOME POINT, BUT THEY SERVE AS GOOD ENTRY POINTS INTO QFT AND HAVE SOME SCOPE FOR RQM--->
* {{cite book |isbn=978-1-139-50432-4 |author=Ohlsson, T.|author-link= Tommy Ohlsson |title=Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory |publisher=Cambridge University Press |year=2011 |page=10 |url=https://books.google.com/books?id=hRavtAW5EFcC&q=pauli-lubanski+pseudovector&pg=PA11}}
*{{cite book |author1=Aitchison, I.J.R. |author2=Hey, A.J.G. |title=Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED |volume=1 |edition=3rd |publisher=CRC Press |year=2002 |url=https://books.google.com/books?id=n7k_QS4Hb0YC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA50 |isbn=978-0-8493-8775-3}}
* {{cite book |author=Griffiths, D. |title=Introduction to Elementary Particles |publisher=John Wiley & Sons |year=2008 |url=https://books.google.com/books?id=Wb9DYrjcoKAC&q=hyperfine+structure+in+relativistic+quantum+mechanics 
|isbn=978-3-527-61847-7}}
* {{cite book |title=Relativistic quantum mechanics and introduction to quantum field theory |url=https://books.google.com/books?id=tTJHB5hepQUC&q=relativistic+quantum+mechanics |author=Capri, Anton Z. |year=2002 |publisher=World Scientific |isbn=978-981-238-137-8|bibcode = 2002rqmi.book.....C}}
* {{cite book |title=Relativistic quantum mechanics and quantum fields |url=https://books.google.com/books?id=gJR_gVU52NkC&q=relativistic+quantum+mechanics|author1-link=Wu Ta-You|author2-link=Woei-Yann Pauchy Hwang |author1=Wu, Ta-you |author2=Hwang, W.Y. Pauchy |year=1991 |publisher=World Scientific|isbn=978-981-02-0608-6}}
* {{cite book |title=Elementary particle physics, Quantum Field Theory |author=Nagashima, Y. |volume=1 |year=2010 |isbn=978-3-527-40962-4}}
* {{cite book |author1=Bjorken, J.D. |author2=Drell, S.D. |year=1965 |title=Relativistic Quantum Fields (Pure & Applied Physics) |publisher=McGraw-Hill |isbn=978-0-07-005494-3}}
* {{cite book |author-link=Steven Weinberg |first=S. |last=Weinberg |year=1996 |title=The Quantum Theory of Fields |volume=2 |publisher=Cambridge University Press |isbn=978-0-521-55002-4 |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev }}
* {{cite book |author-link=Steven Weinberg |first=S. |last=Weinberg |year=2000 |title=The Quantum Theory of Fields |volume=3 |publisher=Cambridge University Press |isbn=978-0-521-66000-6 |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev }}
* {{cite book |author=Gross, F. |year=2008 |title=Relativistic Quantum Mechanics and Field Theory |publisher=John Wiley & Sons |isbn=978-3-527-61734-0 |url=https://books.google.com/books?id=o9dXoK_2s6MC&q=fractional+quantum+mechanics+books&pg=PP2}}
* {{cite book |first1=Y.V. |last1=Nazarov |first2=J. |last2=Danon |year=2013 |title=Advanced Quantum Mechanics: A Practical Guide |publisher=Cambridge University Press |isbn=978-0-521-76150-5 |url=https://books.google.com/books?id=BrTP8hEWRGUC&q=Relativistic_quantum_mechanics}}
* {{cite book |title=General Principles of Quantum Field Theory |author=Bogolubov, N.N. |publisher=Springer |edition=2nd |isbn=978-0-7923-0540-8 |year=1989 |page=272 |url=https://books.google.com/books?id=Ef4zDW1V2LkC&q=lamb+shift&pg=PA187
}}
* {{cite book |title=Quantum Field Theory |author1=Mandl, F. |author2=Shaw, G. |publisher=John Wiley & Sons |edition=2nd |isbn=978-0-471-49683-0 |year=2010 |url=https://books.google.com/books?id=7VLMj4AvvicC&q=pauli-lubanski+pseudovector&pg=PA273}}
* {{cite book |author=Lindgren, I. |year=2011 |title=Relativistic Many-body Theory: A New Field-theoretical Approach |publisher=Springer |isbn=978-1-4419-8309-1 |volume=63 |series=Springer series on atomic, optical, and plasma physics |url=https://books.google.com/books?id=noUF83SAl2wC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA9}}
*{{cite book |title=Relativistic Quantum theory of atoms and molecules |publisher=Springer |author=Grant, I.P. |isbn=978-0-387-34671-7 |year=2007 |series=Atomic, optical, and plasma physics |url=https://books.google.com/books?id=yYFCAAAAQBAJ}}

===Quantum theory and applications in general===
<!---PLEASE PLEASE PLEASE DO NOT DELETE ANYTHING AS "IRRELEVANT/OUTDATED" FROM PERSONAL PREFERENCES IN THIS REF SECTION - MOST OF THE BOOKS MAY BE MOVED INTO THE ARTICLE AT SOME POINT, BUT THEY SERVE AS GOOD ENTRY POINTS INTO QFT AND HAVE SOME SCOPE FOR RQM--->
*{{cite book |author1=Aruldhas, G. |author2=Rajagopal, P. |title=Modern Physics |page=395 |publisher=PHI Learning Pvt. Ltd. |year=2005 |url=https://books.google.com/books?id=GnCWJF6TQUwC&q=electromagnetic+moments+for+high-spin+particles&pg=PA395|isbn=978-81-203-2597-5}}
*{{cite book |author=Hummel, R.E. |title=Electronic properties of materials |page=395 |publisher=Springer |year=2011 |url=https://books.google.com/books?id=TsHFou6RftkC&q=electromagnetic+moments+for+high-spin+particles&pg=PA395 |isbn=978-1-4419-8164-6}}
*{{cite book |author=Pavia, D.L. |edition=4th |title=Introduction to Spectroscopy |page=105 |publisher=Cengage Learning |year=2005 |url=https://books.google.com/books?id=FkaNOdwk0FQC&q=magnetic+moments+allowed+by+spin+quantum+number&pg=PA105 |isbn=978-0-495-11478-9}}
*{{cite book |author=Mizutani, U. |title=Introduction to the Electron Theory of Metals |page=387 |publisher=Cambridge University Press |year=2001 |url=https://books.google.com/books?id=zY5z_UGqAcwC&q=magnetic+moments+allowed+by+spin+quantum+number&pg=PA387 |isbn=978-0-521-58709-9}}
*{{cite book |author=Choppin, G.R. |edition=3 |title=Radiochemistry and nuclear chemistry |page=308 |publisher=Butterworth-Heinemann |year=2002 |url=https://books.google.com/books?id=IsAEjPpvyrkC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA308 |isbn=978-0-7506-7463-8}}
*{{cite book |author=Sitenko, A.G. |title=Theory of nuclear reactions |page=443 |publisher=World Scientific |year=1990 |url=https://books.google.com/books?id=IcF5GV7ftT0C&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA443 |isbn=978-9971-5-0482-3}}
*{{cite book |author1=Nolting, W. |author2=Ramakanth, A. |title=Quantum theory of magnetism |publisher=Springer |year=2008 |url=https://books.google.com/books?id=vrcHC9XoHbsC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA748 |isbn=978-3-540-85416-6}}
*{{cite book |author=Luth, H. |title=Quantum Physics in the Nanoworld |page=149 |publisher=Springer |series=Graduate texts in physics |year=2013 |url=https://books.google.com/books?id=9zrbFSHWcCAC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA149 |isbn=978-3-642-31238-0}}
*{{cite book |author=Sattler, K.D. |title=Handbook of Nanophysics: Functional Nanomaterials |pages=40–43 |publisher=CRC Press |year=2010 |url=https://books.google.com/books?id=08BWNlciXx4C&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=SA40-PA3 |isbn=978-1-4200-7553-3}}
*{{cite book |author=Kuzmany, H. |title=Solid-State Spectroscopy |page=256 |publisher=Springer |year=2009 |url=https://books.google.com/books?id=Jg89d0h9SIoC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA256 |isbn=978-3-642-01480-2}}
*{{cite book |author=Reid, J.M. |title=The Atomic Nucleus |edition=2nd |publisher=Manchester University Press |year=1984 |url=https://books.google.com/books?id=rcdRAQAAIAAJ&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA277 |isbn=978-0-7190-0978-5}}
*{{cite book |author=Schwerdtfeger, P. |title=Relativistic Electronic Structure Theory - Fundamentals |page=208 |volume=11 |series=Theoretical and Computational Chemistry |publisher=Elsevier |year=2002 |url=https://books.google.com/books?id=9tO_9Tf6dZgC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA208 |isbn=978-0-08-054046-7}}
*{{cite book |author=Piela, L. |title=Ideas of Quantum Chemistry |page=676 |publisher=Elsevier |year=2006 |url=https://books.google.com/books?id=nbdITbfsP6oC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA676 |isbn=978-0-08-046676-7}}
*{{cite book |title=Quantum (book) |author=Kumar, M. |isbn=978-1-84831-035-3 |year=2009 |title-link=Quantum (book)}}

==External links==
*{{cite book |title=Relativistic Quantum Mechanics, an Introduction |orig-year=2004 |author=Pfeifer, W. |url=http://www.walterpfeifer.ch/index.html |year=2008}}
*{{cite news |title=Relativistic Quantum Mechanics (Lecture Notes) |author=Lukačević, Igor |url=http://www.fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_11_Relativistic_quantum_mechanics.pdf|archive-url=https://web.archive.org/web/20140826114453/http://www.fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_11_Relativistic_quantum_mechanics.pdf |archive-date=2014-08-26 |year=2013}}
*{{cite news |title=Relativistic Quantum Mechanics |publisher=[[University of Cambridge]] |department=[[Cavendish Laboratory]] |url=http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_relqu.pdf}}
*{{cite news |title=Relativistic Quantum Mechanics |publisher=[[University of Glasgow]] |year=2008 |author=Miller, David J. |url=http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf |access-date=2020-11-17 |archive-date=2020-12-19 |archive-url=https://web.archive.org/web/20201219112349/http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf |url-status=dead }}
*{{cite news |author=Swanson, D.G. |year=2007 |title=Quantum Mechanics Foundations and Applications |url=https://archive.org/details/quantummechanics00swan |url-access=limited |page=[https://archive.org/details/quantummechanics00swan/page/n173 160] |location=Alabama, USA |publisher=Taylor & Francis}}
*{{cite web |url=http://mysite.du.edu/~jcalvert/phys/partelec.htm |first=J.B. |last=Calvert |year=2003 |title=The Particle Electron and Thomas Precession}}
*{{cite web |url=http://www.antidogma.ru/english/node58.html |first=S.N. |last=Arteha |title=Spin and the Thomas precession}}

{{Quantum mechanics topics}}

[[Category:Quantum mechanics]]
[[Category:Mathematical physics]]
[[Category:Electromagnetism]]
[[Category:Particle physics]]
[[Category:Atomic physics]]
[[Category:Theory of relativity]]