relativistic quantum mechanics

One approach is to modify the Schrödinger picture to be consistent with special relativity.

A postulate of quantum mechanics is that the time evolution of any quantum system is given by the Schrödinger equation:

:$i\backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash psi\; =\backslash hat\{H\}\backslash psi$

using a suitable Hamiltonian operator {{math|Ĥ}} corresponding to the system. The solution is a complex-valued wavefunction {{math|ψ(r, t)}}, a function of the 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|2s}} is an integer, odd for fermions and even for bosons. Each {{math|s}} has {{math|2s + 1}} z-projection quantum numbers; {{math|σ {{=}} s, s − 1, ... , −s + 1, −s}}.Other common notations include {{math|m_s}} and {{math|s_z}} etc., but this would clutter expressions with unnecessary subscripts. The subscripts {{math|σ}} labeling spin values are not to be confused for tensor indices nor the Pauli matrices. This is an additional discrete variable the wavefunction requires; {{math|ψ(r, t, σ)}}.

Historically, in the early 1920s Pauli, Kronig, Uhlenbeck and 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 Fierz, rederived by Pauli a year later. This is the explanation for a diverse range of subatomic particle behavior and phenomena: from the electronic configurations of atoms, nuclei (and therefore all elements on the periodic table and their chemistry), to the quark configurations and colour charge (hence the properties of baryons and mesons).

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 magnitude in terms of the dot product $p\; =\; \backslash sqrt\{\backslash mathbf\{p\}\; \backslash cdot\; \backslash mathbf\{p\}\}$, it is:{{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}}

:$E^2\; =\; c^2\backslash mathbf\{p\}\backslash cdot\backslash mathbf\{p\}\; +\; (mc^2)^2\backslash ,.$

These equations are used together with the energy and momentum operators, which are respectively:

:$\backslash hat\{E\}=i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash ,,\backslash quad\; \backslash hat\{\backslash mathbf\{p\}\}\; =\; -i\backslash hbar\backslash nabla\backslash ,,$

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 derivatives 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:

:$\backslash frac\{d\}\{dt\}A\; =\; \backslash frac\{1\}\{i\backslash hbar\}[A,\backslash hat\{H\}]+\backslash frac\{\backslash partial\}\{\backslash partial\; t\}A\backslash ,,$

This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.{{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}}{{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}}

Historically, around 1926, Schrödinger and Heisenberg show that wave mechanics and matrix mechanics are equivalent, later furthered by Dirac using 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.

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(r₁, r₂, r₃, ..., t, σ₁, σ₂, σ₃...)}}.

In relativistic mechanics, the spatial coordinates and coordinate time are not absolute; any two observers moving relative to each other can measure different locations and times of events. The position and time coordinates combine naturally into a 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 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_σ}} locally transform under some representation {{math|D}} of the Lorentz group:{{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 }};
{{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 }};
{{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 }}

{{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}}

:$\backslash psi\_\backslash sigma(\backslash mathbf\{r\},\; t)\; \backslash rightarrow\; D(\backslash Lambda)\; \backslash psi\_\backslash sigma(\backslash Lambda^\{-1\}(\backslash mathbf\{r\},\; t))$

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

{{Main|Hamiltonian operator}}

The classical Hamiltonian for a particle in a potential is the kinetic energy {{math|p·p/2m}} plus the potential energy {{math|V(r, t)}}, with the corresponding quantum operator in the Schrödinger picture:

:$\backslash hat\{H\}\; =\; \backslash frac\{\backslash hat\{\backslash mathbf\{p\}\}\backslash cdot\backslash hat\{\backslash mathbf\{p\}\}\}\{2m\}\; +\; V(\backslash mathbf\{r\},t)$

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:

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

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 derivatives 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 nonlocal and can even violate causality: if the particle is initially localized at a point {{math|r₀}} so that {{math|ψ(r₀, 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}}.{{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 }}

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_B}}, the Bohr magneton:{{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 }}{{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}}

:$\backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S\; =\; -\; \backslash frac\{g\backslash mu\_B\}\{\backslash hbar\}\backslash hat\{\backslash mathbf\{S\}\}\backslash ,,\backslash quad\; \backslash left|\backslash boldsymbol\{\backslash mu\}\_S\backslash right|\; =\; -\; g\backslash mu\_B\; \backslash sigma\backslash ,,$

where {{math|g}} is the (spin) g-factor for the particle, and {{math|S}} the spin operator, so they interact with electromagnetic fields. For a particle in an externally applied magnetic field {{math|B}}, the interaction term{{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}}

:$\backslash hat\{H\}\_B\; =\; -\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S$

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.{{cite book|title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=425 |isbn=978-0-13-146100-0}}

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 matrices, in which the matrix multiplication runs over the spin index {{math|σ}}, so in general a relativistic Hamiltonian:

:$\backslash hat\{H\}\; =\; \backslash hat\{H\}(\backslash mathbf\{r\},\; t,\; \backslash hat\{\backslash mathbf\{p\}\},\; \backslash hat\{\backslash mathbf\{S\}\})$

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

Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the Klein–Gordon equation:{{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}}

:$\backslash hat\{E\}^2\; \backslash psi\; =\; c^2\backslash hat\{\backslash mathbf\{p\}\}\backslash cdot\backslash hat\{\backslash mathbf\{p\}\}\backslash psi\; +\; (mc^2)^2\backslash psi\; \backslash ,,$

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 relativistically invariant, yet this equation alone isn't a sufficient foundation for RQM for at least two reasons: one is that negative-energy states are solutions,{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=415 |isbn=978-0-13-146100-0}} 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:{{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}}{{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}}

:$$

\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 \,,

where {{math|α {{=}} (α₁, α₂, α₃)}} and {{math|β}} are not simply numbers or vectors, but 4 × 4 Hermitian matrices that are required to anticommute for {{math|i ≠ j}}:

:$\backslash alpha\_i\; \backslash beta\; =\; -\; \backslash beta\; \backslash alpha\_i,\; \backslash quad\; \backslash alpha\_i\backslash alpha\_j\; =\; -\; \backslash alpha\_j\backslash alpha\_i\; \backslash ,,$

and square to the identity matrix:

:$\backslash alpha\_i^2\; =\; \backslash beta^2\; =\; I\; \backslash ,,$

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

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

is the Dirac equation. The other factor is also the Dirac equation, but for a particle of negative mass. 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 + cα · p + βmc²}}, 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,{{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}} so Dirac postulated that negative energy states are always occupied, because according to the Pauli principle, electronic transitions from positive to negative energy levels in atoms would be forbidden. See Dirac sea for details.

In non-relativistic quantum mechanics, the square modulus of the wavefunction {{math|ψ}} gives the probability density function {{math|ρ {{=}} {{!}}ψ{{!}}²}}. 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 functions of space and time. The Dirac equation does:{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=423 |isbn=978-0-13-146100-0}}

:$\backslash rho=\backslash psi^\backslash dagger\; \backslash psi,\; \backslash quad\; \backslash mathbf\{j\}\; =\; \backslash psi^\backslash dagger\; \backslash gamma^0\; \backslash boldsymbol\{\backslash gamma\}\; \backslash psi\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; J^\backslash mu\; =\; \backslash psi^\backslash dagger\; \backslash gamma^0\; \backslash gamma^\backslash mu\; \backslash psi$

where the dagger denotes the Hermitian adjoint (authors usually write {{math|{{overline|ψ}} {{=}} ψ^†γ⁰}} for the Dirac adjoint) and {{math|J^μ}} is the probability four-current, while the Klein–Gordon equation does not:{{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}}

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

where {{math|∂^μ}} 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. The density and current of electric charge always satisfy a continuity equation:

:$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash nabla\backslash cdot\backslash mathbf\{J\}\; =\; 0\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash partial\_\backslash mu\; J^\backslash mu\; =\; 0\; \backslash ,,$

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.

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 {{=}} ∇ × A}}, and electric scalar potential {{math|ϕ(r, t)}}, this is:{{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}}

:$\backslash hat\{E\}\; \backslash rightarrow\; \backslash hat\{E\}\; -\; q\backslash phi\; \backslash ,,\; \backslash quad\; \backslash hat\{\backslash mathbf\{p\}\}\backslash rightarrow\; \backslash hat\{\backslash mathbf\{p\}\}\; -\; q\; \backslash mathbf\{A\}\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash hat\{P\}\_\backslash mu\; \backslash rightarrow\; \backslash hat\{P\}\_\backslash mu\; -q\; A\_\backslash mu$

where {{math|P_μ}} is the four-momentum that has a corresponding 4-momentum operator, and {{math|A_μ}} the four-potential. In the following, the non-relativistic limit refers to the limiting cases:

:$E\; -\; e\backslash phi\; \backslash approx\; mc^2\backslash ,,\backslash quad\; \backslash mathbf\{p\}\; \backslash approx\; m\; \backslash mathbf\{v\}\backslash ,,$

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.

In RQM, the KG equation admits the minimal coupling prescription;

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

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|(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 {{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 bosons in an external electromagnetic potential. As such, the equation cannot be applied to the description of atoms, since the electron is a spin {{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:

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

{{Main|spin-1/2}}

Non relativistically, spin was phenomenologically introduced in the Pauli equation by Pauli in 1927 for particles in an electromagnetic field:

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

by means of the 2 × 2 Pauli matrices, and {{math|ψ}} is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component spinor field:

:$\backslash psi=\backslash begin\{pmatrix\}\backslash psi\_\{\backslash uparrow\}\; \backslash \backslash \; \backslash psi\_\{\backslash downarrow\}\; \backslash end\{pmatrix\}$

where the subscripts ↑ and ↓ refer to the "spin up" ({{math|σ {{=}} +{{sfrac|1|2}}}}) and "spin down" ({{math|σ {{=}} −{{sfrac|1|2}}}}) states.This spinor notation is not necessarily standard; the literature usually writes $\backslash psi=\backslash begin\{pmatrix\}\; u^1\; \backslash \backslash \; u^2\; \backslash end\{pmatrix\}$ or $\backslash psi=\backslash begin\{pmatrix\}\; \backslash chi\; \backslash \backslash \; \backslash eta\; \backslash end\{pmatrix\}$ etc., but in the context of spin {{sfrac|1|2}}, this informal identification is commonly made.

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

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

and was the first equation to accurately predict spin, a consequence of the 4 × 4 gamma matrices {{math|γ⁰ {{=}} β, γ {{=}} (γ₁, γ₂, γ₃) {{=}} βα {{=}} (βα₁, βα₂, βα₃)}}. 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|ψ}} is a four-component spinor field, which is conventionally split into two two-component spinors in the form:Again this notation is not necessarily standard, the more advanced literature usually writes

:$\backslash psi=\backslash begin\{pmatrix\}u\; \backslash \backslash \; v\; \backslash end\{pmatrix\}\; =\; \backslash begin\{pmatrix\}\; u^1\; \backslash \backslash \; u^2\; \backslash \backslash \; v^1\; \backslash \backslash \; v^2\; \backslash end\{pmatrix\}$ etc.,

but here we show informally the correspondence of energy, helicity, and spin states.

:$\backslash psi=\backslash begin\{pmatrix\}\backslash psi\_\{+\}\; \backslash \backslash \; \backslash psi\_\{-\}\; \backslash end\{pmatrix\}\; =\; \backslash begin\{pmatrix\}\backslash psi\_\{+\backslash uparrow\}\; \backslash \backslash \; \backslash psi\_\{+\backslash downarrow\}\; \backslash \backslash \; \backslash psi\_\{-\backslash uparrow\}\; \backslash \backslash \; \backslash psi\_\{-\backslash downarrow\}\; \backslash end\{pmatrix\}$

The 2-spinor {{math|ψ₊}} 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₋}} 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, time-reversed momentum, and 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 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:

:$\backslash left(\backslash frac\{\backslash hat\{E\}\}\{c\}\; +\; \backslash boldsymbol\{\backslash sigma\}\backslash cdot\; \backslash hat\{\backslash mathbf\{p\}\}\; \backslash right)\; \backslash psi\_\{+\}\; =\; 0\; \backslash ,,\backslash quad\; \backslash left(\backslash frac\{\backslash hat\{E\}\}\{c\}\; -\; \backslash boldsymbol\{\backslash sigma\}\backslash cdot\; \backslash hat\{\backslash mathbf\{p\}\}\; \backslash right)\; \backslash psi\_\{-\}\; =\; 0\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash sigma^\backslash mu\; \backslash hat\{P\}\_\backslash mu\; \backslash psi\_\{+\}\; =\; 0\backslash ,,\backslash quad\; \backslash sigma\_\backslash mu\; \backslash hat\{P\}^\backslash mu\; \backslash psi\_\{-\}\; =\; 0\backslash ,,$

the first of which is the Weyl equation, a considerable simplification applicable for massless neutrinos.{{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}}. 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|σ₀}} 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 quaternions and to the SO(2) and SO(3) Lie groups, because they satisfy the important commutator [ , ] and anticommutator [ , ]₊ relations respectively:

:$\backslash left[\backslash sigma\_a,\; \backslash sigma\_b\; \backslash right]\; =\; 2i\; \backslash varepsilon\_\{abc\}\; \backslash sigma\_c\; \backslash ,,\; \backslash quad\; \backslash left[\backslash sigma\_a,\; \backslash sigma\_b\; \backslash right]\_\{+\}\; =\; 2\backslash delta\_\{ab\}\backslash sigma\_0$

where {{math|ε_abc}} is the three-dimensional Levi-Civita symbol. The gamma matrices form bases in Clifford algebra, and have a connection to the components of the flat spacetime Minkowski metric {{math|η^αβ}} in the anticommutation relation:

:$\backslash left[\backslash gamma^\backslash alpha,\backslash gamma^\backslash beta\backslash right]\_\{+\}\; =\; \backslash gamma^\backslash alpha\backslash gamma^\backslash beta\; +\; \backslash gamma^\backslash beta\backslash gamma^\backslash alpha\; =\; 2\backslash eta^\{\backslash alpha\backslash beta\}\backslash ,,$

(This can be extended to curved spacetime by introducing vierbeins, but is not the subject of special relativity).

In 1929, the Breit equation was found to describe two or more electromagnetically interacting massive spin {{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.

{{Main|Helicity (particle physics)|Chirality (physics)}}

{{See also|spin polarization}}

The helicity operator is defined by;

:$\backslash hat\{h\}\; =\; \backslash hat\{\backslash mathbf\{S\}\}\backslash cdot\; \backslash frac\{\backslash hat\{\backslash mathbf\{p\}\}\}$

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 m₀ its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.{{cite book |title=Supersymmetry |author=Labelle, P. |series=Demystified |publisher=McGraw-Hill |year=2010 |isbn=978-0-07-163641-4}} 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 {{sfrac|1|2}} operator on the 3-momentum (times c), {{math|σ · c p}}, which is the helicity (for the spin {{sfrac|1|2}} case) times $\backslash sqrt\{E^2\; -\; (m\_0c^2)^2\}$.

For massless particles the helicity simplifies to:

:$\backslash hat\{h\}\; =\; \backslash hat\{\backslash mathbf\{S\}\}\; \backslash cdot\; \backslash frac\{c\backslash hat\{\backslash mathbf\{p\}\}\}\{E\}$

The Dirac equation can only describe particles of spin {{sfrac|1|2}}. Beyond the Dirac equation, RWEs have been applied to free particles of various spins. In 1936, Dirac extended his equation to all fermions, three years later Fierz and Pauli rederived the same equation.{{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 }} The Bargmann–Wigner equations were found in 1948 using Lorentz group theory, applicable for all free particles with any spin.{{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 }}{{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 }} 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 fields, which can be represented as column vectors of functions of space and time:

:

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

:

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 −s:

:$\backslash psi(\backslash mathbf\{r\},t)\; =\; \backslash begin\{pmatrix\}\; \backslash psi\_\{+\}(\backslash mathbf\{r\},t)\; \backslash \backslash \; \backslash psi\_\{-\}(\backslash mathbf\{r\},t)\; \backslash end\{pmatrix\}$

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 spinors 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.{{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}} 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 moments and magnetic dipole moments) allowed by the spin quantum number are arbitrary. (Theoretically, magnetic charge would contribute also). For example, the spin {{sfrac|1|2}} case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible. For more on this topic, see multipole expansion and (for example) Cédric Lorcé (2009).{{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}}{{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 }}

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:{{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}}

:$\backslash hat\{\backslash mathbf\{v\}\}\; =\; \backslash frac\{1\}\{m\}\backslash hat\{\backslash mathbf\{p\}\}$

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

:$\backslash hat\{\backslash mathbf\{v\}\}\; =\; \backslash frac\{i\}\{\backslash hbar\}\backslash left[\backslash hat\{H\},\backslash hat\{\backslash mathbf\{r\}\}\backslash right]$

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

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 (really meaning Lagrangian density), then generate the differential equation by the field-theoretic Euler–Lagrange equation:

:$\backslash partial\_\backslash mu\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\; \backslash partial\_\backslash mu\; \backslash psi\; )\}\; \backslash right)\; -\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; \backslash psi\}\; =\; 0\; \backslash ,$

For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:{{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}}

:$\backslash mathcal\{L\}\; =\; \backslash overline\{\backslash psi\}(\backslash gamma^\backslash mu\; P\_\backslash mu\; -\; mc)\backslash psi$

and Klein–Gordon Lagrangian is:

:$\backslash mathcal\{L\}\; =\; -\; \backslash frac\{\backslash hbar^2\}\{m\}\; \backslash eta^\{\backslash mu\; \backslash nu\}\; \backslash partial\_\{\backslash mu\}\backslash psi^\{*\}\; \backslash partial\_\{\backslash nu\}\backslash psi\; -\; m\; c^2\; \backslash psi^\{*\}\; \backslash psi\backslash ,.$

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).{{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 }}

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:{{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}}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.

:$M^\{\backslash alpha\backslash beta\}\; =\; X^\backslash alpha\; P^\backslash beta\; -\; X^\backslash beta\; P^\backslash alpha\; =\; 2\; X^\{[\backslash alpha\}\; P^\{\backslash beta]\}\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash mathbf\{M\}\; =\; \backslash mathbf\{X\}\backslash wedge\backslash mathbf\{P\}\backslash ,,$

which are six components altogether: three are the non-relativistic 3-orbital angular momenta; {{math|M¹² {{=}} L³}}, {{math|M²³ {{=}} L¹}}, {{math|M³¹ {{=}} L²}}, and the other three {{math|M⁰¹}}, {{math|M⁰²}}, {{math|M⁰³}} 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:

:$J^\{\backslash alpha\backslash beta\}\; =\; 2X^\{[\backslash alpha\}\; P^\{\backslash beta]\}\; +\; \backslash frac\{1\}\{m^2\}\backslash varepsilon^\{\backslash alpha\; \backslash beta\; \backslash gamma\; \backslash delta\}\; W\_\backslash gamma\; p\_\backslash delta\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash mathbf\{J\}\; =\; \backslash mathbf\{X\}\backslash wedge\backslash mathbf\{P\}\; +\; \backslash frac\{1\}\{m^2\}\backslash star(\backslash mathbf\{W\}\backslash wedge\backslash mathbf\{P\})$

where the star denotes the Hodge dual, and

:$W\_\backslash alpha\; =\backslash frac\{1\}\{2\}\backslash varepsilon\_\{\backslash alpha\; \backslash beta\; \backslash gamma\; \backslash delta\}M^\{\backslash beta\; \backslash gamma\}p^\backslash delta\; \backslash quad\; \backslash rightleftharpoons\; \backslash quad\; \backslash mathbf\{W\}\; =\; \backslash star(\backslash mathbf\{M\}\backslash wedge\backslash mathbf\{P\})$

is the Pauli–Lubanski pseudovector.{{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}} For more on relativistic spin, see (for example) Troshin & Tyurin (1994).{{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 }}

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.{{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}}{{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}} In 1939 Wigner derived the Thomas precession.

In 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-transformed magnetic field {{math|B′}}:

:$\backslash mathbf\{B\}\text{'}\; =\; \backslash frac\{\backslash mathbf\{E\}\; \backslash times\; \backslash mathbf\{v\}\}\{c^2\backslash sqrt\{1-\; \backslash left(v/c\backslash right)^2\}\}\; \backslash ,.$

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

:$\backslash mathbf\{B\}\text{'}\; =\; \backslash frac\{\backslash mathbf\{E\}\; \backslash times\; \backslash mathbf\{v\}\}\{c^2\}\; \backslash ,,$

so the non-relativistic spin interaction Hamiltonian becomes:{{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 }}

:$\backslash hat\{H\}\; =\; -\; \backslash mathbf\{B\}\text{'}\backslash cdot\; \backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S\; =\; -\backslash left(\backslash mathbf\{B\}\; +\; \backslash frac\{\backslash mathbf\{E\}\; \backslash times\; \backslash mathbf\{v\}\}\{c^2\}\; \backslash right)\; \backslash cdot\; \backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S\; \backslash ,,$

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. 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{{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}} 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:

:$\backslash hat\{H\}\; =\; -\; \backslash mathbf\{B\}\text{'}\backslash cdot\; \backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S\; =\; -\backslash left(\backslash mathbf\{B\}\; +\; \backslash frac\{\backslash mathbf\{E\}\; \backslash times\; \backslash mathbf\{v\}\}\{2c^2\}\; \backslash right)\; \backslash cdot\; \backslash hat\{\backslash boldsymbol\{\backslash mu\}\}\_S\; \backslash ,.$

In the case of RQM, the factor of {{frac|1|2}} is predicted by the Dirac equation.

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),{{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 }} and P.W Atkins (1974){{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}}]. 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.

Albert Einstein in 1905 explained of the photoelectric effect; a particle description of light as photons. In 1916, Sommerfeld explains fine structure; the splitting of the spectral lines 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. de Broglie extends wave–particle duality to matter: the de Broglie relations, which are consistent with special relativity and quantum mechanics. By 1927, Davisson and Germer and separately G. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.

• 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 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 Rutherford, showed that atoms possess an internal structure: the atomic nucleus.{{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}}
• 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 Stoner studies splitting of energy levels in magnetic fields.
• 1932 Experimental discovery of the neutron by Chadwick, and positrons by 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 interactions.{{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}}

In 1935, Einstein, Rosen, Podolsky published a paper{{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 }} 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.{{cite book |title=Quantum Mechanics |author=Abers, E. |publisher=Addison Wesley |year=2004 |page=192 |isbn=978-0-13-146100-0}}{{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 In 1959, Bohm and Aharonov publish a paper

{{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 }} on the Aharonov–Bohm effect, questioning the status of electromagnetic potentials in QM. The EM field tensor and 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,{{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 |volume=1 |issue=3 |pages=195–200

|doi=10.1103/PhysicsPhysiqueFizika.1.195|doi-access=free }} showing that QM cannot be derived from local hidden-variable theories if locality is to be maintained.

{{Main|Lamb shift}}

In 1947, the Lamb shift was discovered: a small difference in the ²S_{{frac|1|2}} and ²P_{{frac|1|2}} levels of hydrogen, due to the interaction between the electron and vacuum. Lamb and Retherford experimentally measure stimulated radio-frequency transitions the ²S_{{frac|1|2}} and ²P_{{frac|1|2}} hydrogen levels by microwave radiation.{{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 }} An explanation of the Lamb shift is presented by Bethe. Papers on the effect were published in the early 1950s.{{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}}

{{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 }}

{{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 }}

{{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}}

• 1927 Dirac establishes the field of QED, also coining the term "quantum electrodynamics".{{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 }}
• 1943 Tomonaga begins work on renormalization, influential in QED.
• 1947 Schwinger calculates the anomalous magnetic moment of the electron. Kusch measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.

{{Col-begin}}

{{Col-break}}

• Relativistic quantum chemistry
• Breit equation
• Electron spin resonance
• Fine-structure constant

• Quantum spacetime
• Spin connection
• Spinor bundle
• Dirac equation in the algebra of physical space
• Casimir invariant
• Casimir operator
• Wigner D-matrix

{{Col-break}}

• Zitterbewegung
• Two-body Dirac equations
• Relativistic Heavy Ion Collider
• Symmetry (physics)
• Parity
• CPT invariance
• Chirality (physics)
• Standard model
• Gauge theory
• Tachyon
• Modern searches for Lorentz violation

{{col-end}}

{{notelist|1}}

{{reflist}}

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• {{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