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Relativistic quantum mechanics
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<h2 id="combining-special-relativity-and-quantum-mechanics">Combining special relativity and quantum mechanics</h2> <p>One approach is to modify the Schrödinger picture to be consistent with special relativity.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>A <a href="/facts/Mathematical_formulation_of_quantum_mechanics/fRLw4sFU">postulate of quantum mechanics</a> is that the <a href="/facts/Time_evolution/8XxTc0fN">time evolution</a> of any quantum system is given by the <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a>: </p> i ℏ ∂ ∂ t ψ = H ^ ψ {\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi } <p>using a suitable <a href="/facts/Hamiltonian_operator/5XwK2HmD">Hamiltonian operator</a> <i>Ĥ</i> corresponding to the system. The solution is a <a href="/facts/Complex_number/2w7mqI7e">complex</a>-valued <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a> <i>ψ</i>(r, <i>t</i>), a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of the <a href="/facts/Three-dimensional_space/KAqKWUbS">3D</a> <a href="/facts/Position_vector/Tpk5d9f2">position vector</a> r of the particle at time <i>t</i>, describing the behavior of the system. </p><p>Every particle has a non-negative <a href="/facts/Spin_quantum_number/5Nh2H8gr">spin quantum number</a> <i>s</i>. The number 2<i>s</i> is an integer, odd for <a href="/facts/Fermion/LEckdDsR">fermions</a> and even for <a href="/facts/Boson/XtPFcjb5">bosons</a>. Each <i>s</i> has 2<i>s</i> + 1 <i>z</i>-projection quantum numbers; <i>σ</i> = <i>s</i>, <i>s</i> − 1, ... , −<i>s</i> + 1, −<i>s</i>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> This is an additional discrete variable the wavefunction requires; <i>ψ</i>(r, <i>t</i>, <i>σ</i>). </p><p>Historically, in the early 1920s <a href="/facts/Wolfgang_Pauli/Ugk4ubXk">Pauli</a>, <a href="/facts/Ralph_Kronig/e90Wt6F5">Kronig</a>, <a href="/facts/George_Uhlenbeck/IVKgKcVF">Uhlenbeck</a> and <a href="/facts/Samuel_Goudsmit/C5UszfUx">Goudsmit</a> were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the <a href="/facts/Pauli_exclusion_principle/lsBKGeHD">Pauli exclusion principle</a> (1925) and the more general <a href="/facts/Spin%E2%80%93statistics_theorem/j2qqVpez">spin–statistics theorem</a> (1939) due to <a href="/facts/Markus_Fierz/N716eY5I">Fierz</a>, rederived by Pauli a year later. This is the explanation for a diverse range of <a href="/facts/Subatomic_particle/tMEzcTWQ">subatomic particle</a> behavior and phenomena: from the <a href="/facts/Electronic_configuration/TQjRr22P">electronic configurations</a> of atoms, nuclei (and therefore all <a href="/facts/Chemical_element/0jducgWD">elements</a> on the <a href="/facts/Periodic_table/umdGIfUb">periodic table</a> and their <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>), to the quark configurations and <a href="/facts/Colour_charge/aWS1xsCR">colour charge</a> (hence the properties of <a href="/facts/Baryon/QFwAUWK2">baryons</a> and <a href="/facts/Meson/9R0kIc1q">mesons</a>). </p><p>A fundamental prediction of special relativity is the relativistic <a href="/facts/Energy%E2%80%93momentum_relation/HPXngJQF">energy–momentum relation</a>; for a particle of <a href="/facts/Rest_mass/aRloR0nW">rest mass</a> <i>m</i>, and in a particular <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> with <a href="/facts/Energy/wd8PrQM7">energy</a> <i>E</i> and 3-<a href="/facts/Momentum/TTkGv5eY">momentum</a> p with <a href="/facts/Norm_(mathematics)/xIbR4uE1">magnitude</a> in terms of the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> p = p ⋅ p {\displaystyle p={\sqrt {\mathbf {p} \cdot \mathbf {p} }}} , it is:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> E 2 = c 2 p ⋅ p + ( m c 2 ) 2 . {\displaystyle E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(mc^{2})^{2}\,.} <p>These equations are used together with the <a href="/facts/Energy_operator/jbdzxwIr">energy</a> and <a href="/facts/Momentum_operator/nPx4UL3k">momentum</a> <a href="/facts/Operator_(physics)/rCJw8UFz">operators</a>, which are respectively: </p> E ^ = i ℏ ∂ ∂ t , p ^ = − i ℏ ∇ , {\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,,\quad {\hat {\mathbf {p} }}=-i\hbar \nabla \,,} <p>to construct a <a href="/facts/Relativistic_wave_equation/UVF8oBjd">relativistic wave equation</a> (RWE): a <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> consistent with the energy–momentum relation, and is solved for <i>ψ</i> 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 <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> 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). </p><p>The <a href="/facts/Heisenberg_picture/Y5XwceMA">Heisenberg picture</a> is another formulation of QM, in which case the wavefunction <i>ψ</i> is <i>time-independent</i>, and the operators <i>A</i>(<i>t</i>) contain the time dependence, governed by the equation of motion: </p> d d t A = 1 i ℏ [ A , H ^ ] + ∂ ∂ t A , {\displaystyle {\frac {d}{dt}}A={\frac {1}{i\hbar }}[A,{\hat {H}}]+{\frac {\partial }{\partial t}}A\,,} <p>This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>Historically, around 1926, <a href="/facts/Erwin_Schr%C3%B6dinger/wCprepF2">Schrödinger</a> and <a href="/facts/Werner_Heisenberg/n4qzYvj4">Heisenberg</a> show that wave mechanics and <a href="/facts/Matrix_mechanics/nJxC4pKP">matrix mechanics</a> are equivalent, later furthered by Dirac using <a href="/facts/Transformation_theory_(quantum_mechanics)/MngdLH1r">transformation theory</a>. </p><p>A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to apply <a href="/facts/Representations_of_the_Lorentz_group/XJRuPVkL">representations of the Lorentz group</a>. </p> <h3>Space and time</h3> <p>In <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a> 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 <a href="/facts/Many_particle_system/5goGsW5x">many particle system</a> <i>ψ</i>(r1, r2, r3, ..., <i>t</i>, <i>σ</i>1, <i>σ</i>2, <i>σ</i>3...). </p><p>In <a href="/facts/Relativistic_mechanics/QQ2X466d">relativistic mechanics</a>, the <a href="/facts/Coordinate_system/Mtd0q4SP">spatial coordinates</a> and <a href="/facts/Coordinate_time/KcXozga6">coordinate time</a> are <i>not</i> absolute; any two observers moving relative to each other can measure different locations and times of <a href="/facts/Event_(relativity)/qRgszk7U">events</a>. The position and time coordinates combine naturally into a <a href="/facts/Four-vector/89GpELMA">four-dimensional spacetime position</a> X = (<i>ct</i>, r) corresponding to events, and the energy and 3-momentum combine naturally into the <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a> P = (<i>E</i>/<i>c</i>, p) of a dynamic particle, as measured in <i>some</i> <a href="/facts/Frame_of_reference/15yYfB9M">reference frame</a>, change according to a <a href="/facts/Lorentz_transformation/jotJL06G">Lorentz transformation</a> 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. </p><p>Under a proper <a href="/facts/Orthochronous/t1APc2Ye">orthochronous</a> Lorentz transformation (r, <i>t</i>) → Λ(r, <i>t</i>) in <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a>, all one-particle quantum states <i>ψσ</i> locally transform under some <a href="/facts/Representation_theory_of_the_Lorentz_group/XJRuPVkL">representation</a> <i>D</i> of the <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a>:<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> ψ σ ( r , t ) → D ( Λ ) ψ σ ( Λ − 1 ( r , t ) ) {\displaystyle \psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))} <p>where <i>D</i>(Λ) is a finite-dimensional representation, in other words a (2<i>s</i> + 1)×(2<i>s</i> + 1) <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> . Again, <i>ψ</i> is thought of as a <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> containing components with the (2<i>s</i> + 1) allowed values of <i>σ</i>. The <a href="/facts/Quantum_number/AByP6F0y">quantum numbers</a> <i>s</i> and <i>σ</i> as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of <i>σ</i> may occur more than once depending on the representation. </p> <p class="note">Further information: <a href="/facts/Generator_(mathematics)/Jo1eBRbD">Generator (mathematics)</a>, <a href="/facts/Group_theory/NfowcI5H">group theory</a>, <a href="/facts/Representation_theory_of_the_Lorentz_group/XJRuPVkL">Representation theory of the Lorentz group</a>, and <a href="/facts/Symmetries_in_quantum_mechanics/r0CYzmzs">symmetries in quantum mechanics</a></p> <h3>Non-relativistic and relativistic Hamiltonians</h3> <p class="note">Main article: <a href="/facts/Hamiltonian_operator/5XwK2HmD">Hamiltonian operator</a></p> <p>The <a href="/facts/Hamiltonian_mechanics/2ReerlNA">classical Hamiltonian</a> for a particle in a <a href="/facts/Scalar_potential/lXKtCr1c">potential</a> is the <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a> p·p/2<i>m</i> plus the <a href="/facts/Potential_energy/I03KLbmn">potential energy</a> <i>V</i>(r, <i>t</i>), with the corresponding quantum operator in the <a href="/facts/Schr%C3%B6dinger_picture/4VK89MbE">Schrödinger picture</a>: </p> H ^ = p ^ ⋅ p ^ 2 m + V ( r , t ) {\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)} <p>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 <i>and</i> momentum leading to difficulties. Naively setting: </p> H ^ = E ^ = c 2 p ^ ⋅ p ^ + ( m c 2 ) 2 ⇒ i ℏ ∂ ∂ t ψ = c 2 p ^ ⋅ p ^ + ( m c 2 ) 2 ψ {\displaystyle {\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 } <p>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 <a href="/facts/Power_series/vYh6sTps">power series</a> before the momentum operator, raised to a power in each term, could act on <i>ψ</i>. As a result of the power series, the space and time <a href="/facts/Derivative_(mathematics)/7Ito70Dh">derivatives</a> are <i>completely asymmetric</i>: 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 <a href="/facts/Quantum_nonlocality/GUA6LlI5">nonlocal</a> and can even <i>violate <a href="/facts/Causality_(physics)/bkoXXhU3">causality</a></i>: if the particle is initially localized at a point r0 so that <i>ψ</i>(r0, <i>t</i> = 0) is finite and zero elsewhere, then at any later time the equation predicts delocalization <i>ψ</i>(r, <i>t</i>) ≠ 0 everywhere, even for |r| > <i>ct</i> 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 <i>ψ</i>(|r| > <i>ct</i>, <i>t</i>) = 0.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p><p>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 <i>μB</i>, the <a href="/facts/Bohr_magneton/sK2YxXAk">Bohr magneton</a>:<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> μ ^ S = − g μ B ℏ S ^ , | μ S | = − g μ B σ , {\displaystyle {\hat {\boldsymbol {\mu }}}_{S}=-{\frac {g\mu _{B}}{\hbar }}{\hat {\mathbf {S} }}\,,\quad \left|{\boldsymbol {\mu }}_{S}\right|=-g\mu _{B}\sigma \,,} <p>where <i>g</i> is the (spin) <a href="/facts/G-factor_(physics)/kmDsyF5g">g-factor</a> for the particle, and S the <a href="/facts/Spin_operator/4Jvp7e8P">spin operator</a>, so they interact with electromagnetic fields. For a particle in an externally applied <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> B, the interaction term<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> H ^ B = − B ⋅ μ ^ S {\displaystyle {\hat {H}}_{B}=-\mathbf {B} \cdot {\hat {\boldsymbol {\mu }}}_{S}} <p>has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin <i>automatically</i> as a requirement of enforcing the relativistic energy-momentum relation.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p><p>Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including <a href="/facts/Rest_mass/aRloR0nW">rest mass</a> 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 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a>, in which the <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> runs over the spin index <i>σ</i>, so in general a relativistic Hamiltonian: </p> H ^ = H ^ ( r , t , p ^ , S ^ ) {\displaystyle {\hat {H}}={\hat {H}}(\mathbf {r} ,t,{\hat {\mathbf {p} }},{\hat {\mathbf {S} }})} <p>is a function of space, time, and the momentum and spin operators. </p> <h3>The Klein–Gordon and Dirac equations for free particles</h3> <p>Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the <a href="/facts/Klein%E2%80%93Gordon_equation/0a5BRyy3">Klein–Gordon equation</a>:<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p> E ^ 2 ψ = c 2 p ^ ⋅ p ^ ψ + ( m c 2 ) 2 ψ , {\displaystyle {\hat {E}}^{2}\psi =c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}\psi +(mc^{2})^{2}\psi \,,} <p>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 <i>is</i> <a href="/facts/Lorentz_covariance/K5AEuGKg">relativistically invariant</a>, yet this equation alone isn't a sufficient foundation for RQM for at least two reasons: one is that negative-energy states are solutions,<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> 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:<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> </p> ( E ^ − c α ⋅ p ^ − β m c 2 ) ( E ^ + c α ⋅ p ^ + β m c 2 ) ψ = 0 , {\displaystyle \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\,,} <p>where α = (<i>α</i>1, <i>α</i>2, <i>α</i>3) and <i>β</i> are not simply numbers or vectors, but 4 × 4 <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrices</a> that are required to <a href="/facts/Anticommute/N2xrAKH5">anticommute</a> for <i>i</i> ≠ <i>j</i>: </p> α i β = − β α i , α i α j = − α j α i , {\displaystyle \alpha _{i}\beta =-\beta \alpha _{i},\quad \alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,,} <p>and square to the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>: </p> α i 2 = β 2 = I , {\displaystyle \alpha _{i}^{2}=\beta ^{2}=I\,,} <p>so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor: </p> ( E ^ − c α ⋅ p ^ − β m c 2 ) ψ = 0 ⇔ H ^ = c α ⋅ p ^ + β m c 2 {\displaystyle \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}} <p>is the Dirac equation. The other factor is also the Dirac equation, but for a particle of <a href="/facts/Negative_mass/dqGf2T4m">negative mass</a>.<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> 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 <i>E</i> + <i>c</i>α · p + <i>βmc</i>2, and comparison with the KG equation determines the constraints on α and <i>β</i>. The positive mass equation can continue to be used without loss of continuity. The matrices multiplying <i>ψ</i> 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,<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> so Dirac postulated that negative energy states are always occupied, because according to the <a href="/facts/Pauli_exclusion_principle/lsBKGeHD">Pauli principle</a>, <a href="/facts/Electronic_transition/zboUt1QR">electronic transitions</a> from positive to negative energy levels in <a href="/facts/Atom/FoQ4kGN5">atoms</a> would be forbidden. See <a href="/facts/Dirac_sea/x40KMcu3">Dirac sea</a> for details. </p> <h3>Densities and currents</h3> <p>In non-relativistic quantum mechanics, the <a href="/facts/Square_modulus/Jzpq0R0z">square modulus</a> of the <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a> <i>ψ</i> gives the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> <i>ρ</i> = |<i>ψ</i>|2. This is the <a href="/facts/Copenhagen_interpretation/DvMFrYJa">Copenhagen interpretation</a>, circa 1927. In RQM, while <i>ψ</i>(r, <i>t</i>) is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density <i>ρ</i> or <a href="/facts/Probability_current/lAl6cWU2">probability current</a> j (really meaning <i>probability current density</i>) because they are <i>not</i> <a href="/facts/Positive-definite_function/RKqaCfp9">positive-definite functions</a> of space and time. The <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> does:<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> </p> ρ = ψ † ψ , j = ψ † γ 0 γ ψ ⇌ J μ = ψ † γ 0 γ μ ψ {\displaystyle \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 } <p>where the dagger denotes the <a href="/facts/Hermitian_adjoint/gN6RHphd">Hermitian adjoint</a> (authors usually write <i>ψ</i> = <i>ψ</i>†<i>γ</i>0 for the <a href="/facts/Dirac_adjoint/7NxDExM0">Dirac adjoint</a>) and <i>Jμ</i> is the <a href="/facts/Probability_current/lAl6cWU2">probability four-current</a>, while the <a href="/facts/Klein%E2%80%93Gordon_equation/0a5BRyy3">Klein–Gordon equation</a> does not:<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> </p> ρ = i ℏ 2 m c 2 ( ψ ∗ ∂ ψ ∂ t − ψ ∂ ψ ∗ ∂ t ) , j = − i ℏ 2 m ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) ⇌ J μ = i ℏ 2 m ( ψ ∗ ∂ μ ψ − ψ ∂ μ ψ ∗ ) {\displaystyle \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 ^{*})} <p>where ∂<i>μ</i> is the <a href="/facts/Four-gradient/dkVV0lnY">four-gradient</a>. Since the initial values of both <i>ψ</i> and ∂<i>ψ</i>/∂<i>t</i> may be freely chosen, the density can be negative. </p><p>Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as <a href="/facts/Charge_density/xFDPXUDM">charge density</a> and <a href="/facts/Current_density/IlUKSe2B">current density</a> when multiplied by <a href="/facts/Electric_charge/xxsBfmlB">electric charge</a>. Then, the wavefunction <i>ψ</i> is not a wavefunction at all, but reinterpreted as a <i>field</i>.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> The density and current of electric charge always satisfy a <a href="/facts/Continuity_equation/GVav2Vet">continuity equation</a>: </p> ∂ ρ ∂ t + ∇ ⋅ J = 0 ⇌ ∂ μ J μ = 0 , {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0\quad \rightleftharpoons \quad \partial _{\mu }J^{\mu }=0\,,} <p>as charge is a <a href="/facts/Conserved_quantity/B1I6D5Ip">conserved quantity</a>. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions. </p> <h2 id="spin-and-electromagnetically-interacting-particles">Spin and electromagnetically interacting particles</h2> <p>Including interactions in RWEs is generally difficult. <a href="/facts/Minimal_coupling/wFh0TmCk">Minimal coupling</a> is a simple way to include the electromagnetic interaction. For one charged particle of <a href="/facts/Electric_charge/xxsBfmlB">electric charge</a> <i>q</i> in an electromagnetic field, given by the <a href="/facts/Magnetic_vector_potential/KJGSgMYb">magnetic vector potential</a> A(r, <i>t</i>) defined by the magnetic field B = ∇ × A, and <a href="/facts/Electric_scalar_potential/Nmqm6hm5">electric scalar potential</a> <i>ϕ</i>(r, <i>t</i>), this is:<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p> E ^ → E ^ − q ϕ , p ^ → p ^ − q A ⇌ P ^ μ → P ^ μ − q A μ {\displaystyle {\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 }-qA_{\mu }} <p>where <i>Pμ</i> is the <a href="/facts/Four-momentum/t1F5Lmbr">four-momentum</a> that has a corresponding <a href="/facts/4-momentum_operator/nPx4UL3k">4-momentum operator</a>, and <i>Aμ</i> the <a href="/facts/Four-potential/S8gTFNAt">four-potential</a>. In the following, the non-relativistic limit refers to the limiting cases: </p> E − e ϕ ≈ m c 2 , p ≈ m v , {\displaystyle E-e\phi \approx mc^{2}\,,\quad \mathbf {p} \approx m\mathbf {v} \,,} <p>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. </p> <h3>Spin 0</h3> <p>In RQM, the KG equation admits the minimal coupling prescription; </p> ( E ^ − q ϕ ) 2 ψ = c 2 ( p ^ − q A ) 2 ψ + ( m c 2 ) 2 ψ ⇌ [ ( P ^ μ − q A μ ) ( P ^ μ − q A μ ) − ( m c ) 2 ] ψ = 0. {\displaystyle {({\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 }-qA_{\mu })}{({\hat {P}}^{\mu }-qA^{\mu })}-{(mc)}^{2}\right]\psi =0.} <p>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 <i>irreducible</i> one-dimensional scalar <a href="/facts/Representation_theory_of_the_Lorentz_group/XJRuPVkL">(0,0)</a> representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of (0,0) representations. Solutions that do not belong to the irreducible (0,0) representation will have two or more <i>independent</i> 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 1/2, see below. Thus if a system satisfies the KG equation <i>only</i>, it can only be interpreted as a system with zero spin. </p><p>The electromagnetic field is treated classically according to <a href="/facts/Maxwell%27s_equations/LfJ1drHS">Maxwell's equations</a> 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 <i>π</i>-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. </p><p>The KG equation is applicable to spinless charged <a href="/facts/Boson/XtPFcjb5">bosons</a> in an external electromagnetic potential.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> As such, the equation cannot be applied to the description of atoms, since the electron is a spin 1/2 particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> </p> ( i ℏ ∂ ∂ t − q ϕ ) ψ = 1 2 m ( p ^ − q A ) 2 ψ ⇔ H ^ = 1 2 m ( p ^ − q A ) 2 + q ϕ . {\displaystyle \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 .} <h3>Spin 1/2</h3> <p class="note">Main article: <a href="/facts/Spin-1/2/3I929M4X">spin-1/2</a></p> <p>Non relativistically, spin was <i><a href="/facts/Phenomenological_model/z8vypElI">phenomenologically</a></i> introduced in the <a href="/facts/Pauli_equation/gdPIf0cw">Pauli equation</a> by <a href="/facts/Wolfgang_Pauli/Ugk4ubXk">Pauli</a> in 1927 for particles in an <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a>: </p> ( i ℏ ∂ ∂ t − q ϕ ) ψ = [ 1 2 m ( σ ⋅ ( p − q A ) ) 2 ] ψ ⇔ H ^ = 1 2 m ( σ ⋅ ( p − q A ) ) 2 + q ϕ {\displaystyle \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 } <p>by means of the 2 × 2 <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a>, and <i>ψ</i> is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component <a href="/facts/Spinor_field/z88hf7ps">spinor field</a>: </p> ψ = ( ψ ↑ ψ ↓ ) {\displaystyle \psi ={\begin{pmatrix}\psi _{\uparrow }\\\psi _{\downarrow }\end{pmatrix}}} <p>where the subscripts ↑ and ↓ refer to the "spin up" (<i>σ</i> = +1/2) and "spin down" (<i>σ</i> = −1/2) states.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> </p><p>In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above; </p> ( i ℏ ∂ ∂ t − q ϕ ) ψ = γ 0 [ c γ ⋅ ( p ^ − q A ) − m c 2 ] ψ ⇌ [ γ μ ( P ^ μ − q A μ ) − m c 2 ] ψ = 0 {\displaystyle \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 }-qA_{\mu })-mc^{2}\right]\psi =0} <p>and was the first equation to accurately <i>predict</i> spin, a consequence of the 4 × 4 <a href="/facts/Gamma_matrices/C2y9Aumy">gamma matrices</a> <i>γ</i>0 = <i>β</i>, γ = (<i>γ</i>1, <i>γ</i>2, <i>γ</i>3) = <i>β</i>α = (<i>βα</i>1, <i>βα</i>2, <i>βα</i>3). 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 <i>ψ</i> is a four-component spinor field, which is conventionally split into two two-component spinors in the form:<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> </p> ψ = ( ψ + ψ − ) = ( ψ + ↑ ψ + ↓ ψ − ↑ ψ − ↓ ) {\displaystyle \psi ={\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}\psi _{+\uparrow }\\\psi _{+\downarrow }\\\psi _{-\uparrow }\\\psi _{-\downarrow }\end{pmatrix}}} <p>The 2-spinor <i>ψ</i>+ corresponds to a particle with 4-momentum (<i>E</i>, p) and charge <i>q</i> and two spin states (<i>σ</i> = ±1/2, as before). The other 2-spinor <i>ψ</i>− corresponds to a similar particle with the same mass and spin states, but <i>negative</i> 4-momentum −(<i>E</i>, p) and <i>negative</i> charge −<i>q</i>, that is, negative energy states, <a href="/facts/T-symmetry/Tgf2LJMc">time-reversed</a> momentum, and <a href="/facts/C-symmetry/YJ1vjdfD">negated charge</a>. This was the first interpretation and prediction of a particle and <i>corresponding <a href="/facts/Antiparticle/7Ic43ThI">antiparticle</a></i>. See <a href="/facts/Dirac_spinor/IDhhu8Eu">Dirac spinor</a> and <a href="/facts/Bispinor/wm8UNWxJ">bispinor</a> for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> for how). When applied a one-electron atom or ion, setting A = 0 and <i>ϕ</i> to the appropriate electrostatic potential, additional relativistic terms include the <a href="/facts/Spin%E2%80%93orbit_interaction/Pv6maL1S">spin–orbit interaction</a>, electron <a href="/facts/Gyromagnetic_ratio/Afg6mY6M">gyromagnetic ratio</a>, and <a href="/facts/Darwin_term/6lejlmN2">Darwin term</a>. In ordinary QM these terms have to be put in by hand and treated using <a href="/facts/Perturbation_theory/JWrNgbVm">perturbation theory</a>. The positive energies do account accurately for the fine structure. </p><p>Within RQM, for massless particles the Dirac equation reduces to: </p> ( E ^ c + σ ⋅ p ^ ) ψ + = 0 , ( E ^ c − σ ⋅ p ^ ) ψ − = 0 ⇌ σ μ P ^ μ ψ + = 0 , σ μ P ^ μ ψ − = 0 , {\displaystyle \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\,,} <p>the first of which is the <a href="/facts/Weyl_equation/8KhWTFaM">Weyl equation</a>, a considerable simplification applicable for massless <a href="/facts/Neutrino/yGWVfSg2">neutrinos</a>.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> This time there is a 2 × 2 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix <i>σ</i>0 which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives). </p><p>The Pauli and gamma matrices were introduced here, in theoretical physics, rather than <a href="/facts/Pure_mathematics/V29nQC1C">pure mathematics</a> itself. They have applications to <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> and to the <a href="/facts/SO(2)/kuzdWtHo">SO(2)</a> and <a href="/facts/SO(3)/yXbKPzhT">SO(3)</a> <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>, because they satisfy the important <a href="/facts/Commutator/eYxzFmFY">commutator</a> [ , ] and <a href="/facts/Commutator/eYxzFmFY">anticommutator</a> [ , ]+ relations respectively: </p> [ σ a , σ b ] = 2 i ε a b c σ c , [ σ a , σ b ] + = 2 δ a b σ 0 {\displaystyle \left[\sigma _{a},\sigma _{b}\right]=2i\varepsilon _{abc}\sigma _{c}\,,\quad \left[\sigma _{a},\sigma _{b}\right]_{+}=2\delta _{ab}\sigma _{0}} <p>where <i>εabc</i> is the <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>. The gamma matrices form <a href="/facts/Basis_(linear_algebra)/89IPoN6c">bases</a> in <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>, and have a connection to the components of the flat spacetime <a href="/facts/Minkowski_metric/vDkfkQJn">Minkowski metric</a> <i>ηαβ</i> in the anticommutation relation: </p> [ γ α , γ β ] + = γ α γ β + γ β γ α = 2 η α β , {\displaystyle \left[\gamma ^{\alpha },\gamma ^{\beta }\right]_{+}=\gamma ^{\alpha }\gamma ^{\beta }+\gamma ^{\beta }\gamma ^{\alpha }=2\eta ^{\alpha \beta }\,,} <p>(This can be extended to <a href="/facts/Curved_space/93NBB86g">curved spacetime</a> by introducing <a href="/facts/Cartan_formalism_(physics)/jyEhSaHM">vierbeins</a>, but is not the subject of special relativity). </p><p>In 1929, the <a href="/facts/Breit_equation/Hsh5sAcK">Breit equation</a> was found to describe two or more electromagnetically interacting massive spin 1/2 fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum <a href="/facts/Many-particle_system/n2rIxgwv">many-particle system</a>. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums. </p> <h3>Helicity and chirality</h3> <p class="note">Main articles: <a href="/facts/Helicity_(particle_physics)/vvNpFcFE">Helicity (particle physics)</a> and <a href="/facts/Chirality_(physics)/H8Xbcvzt">Chirality (physics)</a></p> <p class="note">See also: <a href="/facts/Spin_polarization/S36kxcj2">spin polarization</a></p> <p>The <a href="/facts/Helicity_(particle_physics)/vvNpFcFE">helicity operator</a> is defined by; </p> h ^ = S ^ ⋅ p ^ | p | = S ^ ⋅ c p ^ E 2 − ( m 0 c 2 ) 2 {\displaystyle {\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_{0}c^{2})^{2}}}}} <p>where p is the momentum operator, S the spin operator for a particle of spin <i>s</i>, <i>E</i> is the total energy of the particle, and <i>m</i>0 its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> 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. </p><p>An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin 1/2 operator on the 3-momentum (times <i>c</i>), σ · <i>c</i> p, which is the helicity (for the spin 1/2 case) times E 2 − ( m 0 c 2 ) 2 {\displaystyle {\sqrt {E^{2}-(m_{0}c^{2})^{2}}}} . </p><p>For massless particles the helicity simplifies to: </p> h ^ = S ^ ⋅ c p ^ E {\displaystyle {\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{E}}} <h3>Higher spins</h3> <p>The Dirac equation can only describe particles of spin 1/2. Beyond the Dirac equation, RWEs have been applied to <a href="/facts/Free_particle/Fjbx90yW">free particles</a> of various spins. In 1936, Dirac extended his equation to all fermions, three years later <a href="/facts/Markus_Fierz/N716eY5I">Fierz</a> and Pauli rederived the same equation.<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> The <a href="/facts/Bargmann%E2%80%93Wigner_equations/Nwb9aklp">Bargmann–Wigner equations</a> were found in 1948 using Lorentz group theory, applicable for all free particles with any spin.<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> Considering the factorization of the KG equation above, and more rigorously by <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a> theory, it becomes apparent to introduce spin in the form of matrices. </p><p>The wavefunctions are multicomponent <a href="/facts/Spinor_field/z88hf7ps">spinor fields</a>, which can be represented as <a href="/facts/Column_vector/pPuRr9Xk">column vectors</a> of <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> of space and time: </p> ψ ( r , t ) = [ ψ σ = s ( r , t ) ψ σ = s − 1 ( r , t ) ⋮ ψ σ = − s + 1 ( r , t ) ψ σ = − s ( r , t ) ] ⇌ ψ ( r , t ) † = [ ψ σ = s ( r , t ) ⋆ ψ σ = s − 1 ( r , t ) ⋆ ⋯ ψ σ = − s + 1 ( r , t ) ⋆ ψ σ = − s ( r , t ) ⋆ ] {\displaystyle \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}}} <p>where the expression on the right is the <a href="/facts/Hermitian_conjugate/gN6RHphd">Hermitian conjugate</a>. For a <i>massive</i> particle of spin <i>s</i>, there are 2<i>s</i> + 1 components for the particle, and another 2<i>s</i> + 1 for the corresponding <a href="/facts/Antiparticle/7Ic43ThI">antiparticle</a> (there are 2<i>s</i> + 1 possible <i>σ</i> values in each case), altogether forming a 2(2<i>s</i> + 1)-component spinor field: </p> ψ ( r , t ) = [ ψ + , σ = s ( r , t ) ψ + , σ = s − 1 ( r , t ) ⋮ ψ + , σ = − s + 1 ( r , t ) ψ + , σ = − s ( r , t ) ψ − , σ = s ( r , t ) ψ − , σ = s − 1 ( r , t ) ⋮ ψ − , σ = − s + 1 ( r , t ) ψ − , σ = − s ( r , t ) ] ⇌ ψ ( r , t ) † [ ψ + , σ = s ( r , t ) ⋆ ψ + , σ = s − 1 ( r , t ) ⋆ ⋯ ψ − , σ = − s ( r , t ) ⋆ ] {\displaystyle \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}}} <p>with the + subscript indicating the particle and − subscript for the antiparticle. However, for <i>massless</i> particles of spin <i>s</i>, there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +<i>s</i> and the other for the antiparticle in the opposite helicity state corresponding to −<i>s</i>: </p> ψ ( r , t ) = ( ψ + ( r , t ) ψ − ( r , t ) ) {\displaystyle \psi (\mathbf {r} ,t)={\begin{pmatrix}\psi _{+}(\mathbf {r} ,t)\\\psi _{-}(\mathbf {r} ,t)\end{pmatrix}}} <p>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, <a href="/facts/%C3%89lie_Cartan/Uvhkp66S">Élie Cartan</a> found the most general form of <a href="/facts/Spinor/sKFNnEwP">spinors</a> in 1913, prior to the spinors revealed in the RWEs following the year 1927. </p><p>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.<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> For spin greater than <i>ħ</i>/2, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (<a href="/facts/Electric_dipole_moment/ou2LDUVg">electric dipole moments</a> and <a href="/facts/Magnetic_dipole_moment/VpAQKJtr">magnetic dipole moments</a>) allowed by the <a href="/facts/Spin_quantum_number/5Nh2H8gr">spin quantum number</a> are arbitrary. (Theoretically, <a href="/facts/Magnetic_charge/R1TSbfMY">magnetic charge</a> would contribute also). For example, the spin 1/2 case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible.<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> For more on this topic, see <a href="/facts/Multipole_expansion/Kao2WRBA">multipole expansion</a> and (for example) Cédric Lorcé (2009).<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a><a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> </p> <h2 id="velocity-operator">Velocity operator</h2> <p>The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition p = <i>m</i> v, and substituting quantum operators in the usual way:<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> </p> v ^ = 1 m p ^ {\displaystyle {\hat {\mathbf {v} }}={\frac {1}{m}}{\hat {\mathbf {p} }}} <p>which has eigenvalues that take <i>any</i> value. In RQM, the Dirac theory, it is: </p> v ^ = i ℏ [ H ^ , r ^ ] {\displaystyle {\hat {\mathbf {v} }}={\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\mathbf {r} }}\right]} <p>which must have eigenvalues between ±<i>c</i>. See <a href="/facts/Foldy%E2%80%93Wouthuysen_transformation/5KT6II50">Foldy–Wouthuysen transformation</a> for more theoretical background. </p> <h2 id="relativistic-quantum-lagrangians">Relativistic quantum Lagrangians</h2> <p>The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for <i>ψ</i>. An equivalent alternative is to determine a <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> (really meaning <i><a href="/facts/Lagrangian_density/p9BCUz57">Lagrangian density</a></i>), then generate the differential equation by the <a href="/facts/Classical_field_theory/7HuKY1qu">field-theoretic Euler–Lagrange equation</a>: </p> ∂ μ ( ∂ L ∂ ( ∂ μ ψ ) ) − ∂ L ∂ ψ = 0 {\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0\,} <p>For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a> </p> L = ψ ¯ ( γ μ P μ − m c ) ψ {\displaystyle {\mathcal {L}}={\overline {\psi }}(\gamma ^{\mu }P_{\mu }-mc)\psi } <p>and Klein–Gordon Lagrangian is: </p> L = − ℏ 2 m η μ ν ∂ μ ψ ∗ ∂ ν ψ − m c 2 ψ ∗ ψ . {\displaystyle {\mathcal {L}}=-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }\psi ^{*}\partial _{\nu }\psi -mc^{2}\psi ^{*}\psi \,.} <p>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 <i>ψ</i> is the subject of QFT rather than RQM: Feynman's <a href="/facts/Path_integral_formulation/6lt2MHkW">path integral formulation</a> uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> </p> <h2 id="relativistic-quantum-angular-momentum">Relativistic quantum angular momentum</h2> <p>In non-relativistic QM, the <a href="/facts/Angular_momentum_operator/d55sXn5u">angular momentum operator</a> is formed from the classical <a href="/facts/Pseudovector/c3b5GgWD">pseudovector</a> definition L = r × p. In RQM, the position and momentum operators are inserted directly where they appear in the orbital <a href="/facts/Relativistic_angular_momentum/IlhzVW01">relativistic angular momentum</a> tensor defined from the four-dimensional position and momentum of the particle, equivalently a <a href="/facts/Bivector/PBRcVE8q">bivector</a> in the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> formalism:<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a><a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> </p> M α β = X α P β − X β P α = 2 X [ α P β ] ⇌ M = X ∧ P , {\displaystyle M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }=2X^{[\alpha }P^{\beta ]}\quad \rightleftharpoons \quad \mathbf {M} =\mathbf {X} \wedge \mathbf {P} \,,} <p>which are six components altogether: three are the non-relativistic 3-orbital angular momenta; <i>M</i>12 = <i>L</i>3, <i>M</i>23 = <i>L</i>1, <i>M</i>31 = <i>L</i>2, and the other three <i>M</i>01, <i>M</i>02, <i>M</i>03 are boosts of the <a href="/facts/Centre_of_mass/n4Lr9GAz">centre of mass</a> of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass <i>m</i>, the <i>total</i> angular momentum tensor is: </p> J α β = 2 X [ α P β ] + 1 m 2 ε α β γ δ W γ p δ ⇌ J = X ∧ P + 1 m 2 ⋆ ( W ∧ P ) {\displaystyle 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} )} <p>where the star denotes the <a href="/facts/Hodge_dual/AH6tmHhU">Hodge dual</a>, and </p> W α = 1 2 ε α β γ δ M β γ p δ ⇌ W = ⋆ ( M ∧ P ) {\displaystyle 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} )} <p>is the <a href="/facts/Pauli%E2%80%93Lubanski_pseudovector/r8Tnsnbc">Pauli–Lubanski pseudovector</a>.<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a> For more on relativistic spin, see (for example) Troshin & Tyurin (1994).<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a> </p> <h3>Thomas precession and spin–orbit interactions</h3> <p>In 1926, the <a href="/facts/Thomas_precession/xswGe6Jc">Thomas precession</a> is discovered: relativistic corrections to the spin of elementary particles with application in the <a href="/facts/Spin%E2%80%93orbit_interaction/Pv6maL1S">spin–orbit interaction</a> of atoms and rotation of macroscopic objects.<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a> In 1939 Wigner derived the Thomas precession. </p><p>In <a href="/facts/Classical_electromagnetism_and_special_relativity/0UEQ840t">classical electromagnetism and special relativity</a>, an electron moving with a velocity v through an electric field E but not a magnetic field B, will in its own frame of reference experience a <a href="/facts/Lorentz_transformation/jotJL06G">Lorentz-transformed</a> magnetic field B′: </p> B ′ = E × v c 2 1 − ( v / c ) 2 . {\displaystyle \mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}{\sqrt {1-\left(v/c\right)^{2}}}}}\,.} <p>In the non-relativistic limit <i>v</i> << <i>c</i>: </p> B ′ = E × v c 2 , {\displaystyle \mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\,,} <p>so the non-relativistic spin interaction Hamiltonian becomes:<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a> </p> H ^ = − B ′ ⋅ μ ^ S = − ( B + E × v c 2 ) ⋅ μ ^ S , {\displaystyle {\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}\,,} <p>where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order (<i>v/c</i>)², but this disagrees with experimental atomic spectra by a factor of 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 <a href="/facts/Rotating_frame_of_reference/X4ngwkz5">rotating frame of reference</a>, and this additional precession of the electron is called the <i>Thomas precession</i>. It can be shown<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a> 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: </p> H ^ = − B ′ ⋅ μ ^ S = − ( B + E × v 2 c 2 ) ⋅ μ ^ S . {\displaystyle {\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}\,.} <p>In the case of RQM, the factor of 1⁄2 is predicted by the Dirac equation.<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a> </p> <h2 id="history">History</h2> <p>The events which led to and established RQM, and the continuation beyond into <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a> (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985),<a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a> and <a href="/facts/Peter_Atkins/mB5XLbDy">P.W Atkins</a> (1974)<a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a>]. 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 <i>necessary</i> component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found <a href="/facts/Atomic_physics/7Jlm4I7k">atomic physics</a>, <a href="/facts/Nuclear_physics/fzQbt1aF">nuclear physics</a>, and <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>; by considering <a href="/facts/Spectroscopy/mj71FuNK">spectroscopy</a>, <a href="/facts/Diffraction/Mk80xXHT">diffraction</a> and <a href="/facts/Scattering/nMou7lEH">scattering</a> of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin. </p> <h3>Relativistic description of particles in quantum phenomena</h3> <p><a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> in 1905 explained of the <a href="/facts/Photoelectric_effect/gIDcdm5U">photoelectric effect</a>; a particle description of light as <a href="/facts/Photons/mFNhAEzA">photons</a>. In 1916, <a href="/facts/Arnold_Sommerfeld/cJtpVHvn">Sommerfeld</a> explains <a href="/facts/Fine_structure/6lejlmN2">fine structure</a>; the splitting of the <a href="/facts/Spectral_line/HuI4Bj5f">spectral lines</a> of <a href="/facts/Atoms/FoQ4kGN5">atoms</a> due to first order relativistic corrections. The <a href="/facts/Compton_effect/Q98Ty8rz">Compton effect</a> of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering. <a href="/facts/Louis_de_Broglie/v696wFwH">de Broglie</a> extends <a href="/facts/Wave%E2%80%93particle_duality/Au3UlcwQ">wave–particle duality</a> to <a href="/facts/Matter/osyc04UN">matter</a>: the <a href="/facts/De_Broglie_relations/Rpszb8f1">de Broglie relations</a>, which are consistent with special relativity and quantum mechanics. By 1927, <a href="/facts/Clinton_Davisson/q9V3p4RQ">Davisson</a> and <a href="/facts/Lester_Germer/xhwA4oxz">Germer</a> and separately <a href="/facts/George_Paget_Thomson/Mmk3oA0A">G. Thomson</a> successfully diffract electrons, providing experimental evidence of wave-particle duality. </p> <h3>Experiments</h3> <ul><li>1897 <a href="/facts/J._J._Thomson/Y4jyndJF">J. J. Thomson</a> discovers the <a href="/facts/Electron/L0KBo5cW">electron</a> and measures its <a href="/facts/Mass-to-charge_ratio/QaKpLt6B">mass-to-charge ratio</a>. Discovery of the <a href="/facts/Zeeman_effect/A6NMlp2D">Zeeman effect</a>: the splitting a <a href="/facts/Spectral_line/HuI4Bj5f">spectral line</a> into several components in the presence of a static magnetic field.</li> <li>1908 <a href="/facts/Robert_Andrews_Millikan/rrWMCh2y">Millikan</a> measures the charge on the electron and finds experimental evidence of its quantization, in the <a href="/facts/Oil_drop_experiment/C2BBYkKh">oil drop experiment</a>.</li> <li>1911 <a href="/facts/Alpha_particle/dzNMs9WP">Alpha particle</a> scattering in the <a href="/facts/Geiger%E2%80%93Marsden_experiment/cmgI1Ha9">Geiger–Marsden experiment</a>, led by <a href="/facts/Ernest_Rutherford/hjjMEskl">Rutherford</a>, showed that atoms possess an internal structure: the <a href="/facts/Atomic_nucleus/orpIm5Xl">atomic nucleus</a>.<a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a></li> <li>1913 The <a href="/facts/Stark_effect/NRLArv1u">Stark effect</a> is discovered: splitting of spectral lines due to a static <a href="/facts/Electric_field/tt4MooBs">electric field</a> (compare with the Zeeman effect).</li> <li>1922 <a href="/facts/Stern%E2%80%93Gerlach_experiment/PCWPE56v">Stern–Gerlach experiment</a>: experimental evidence of spin and its quantization.</li> <li>1924 <a href="/facts/Edmund_Clifton_Stoner/30kenmeg">Stoner</a> studies splitting of <a href="/facts/Energy_level/66x51Khc">energy levels</a> in <a href="/facts/Magnetic_field/Ta8Ev588">magnetic fields</a>.</li> <li>1932 Experimental discovery of the <a href="/facts/Neutron/oRCM1geb">neutron</a> by <a href="/facts/James_Chadwick/nwStNAzI">Chadwick</a>, and <a href="/facts/Positron/HgV4zBzA">positrons</a> by <a href="/facts/Carl_David_Anderson/rIMqD9ax">Anderson</a>, confirming the theoretical prediction of positrons.</li> <li>1958 Discovery of the <a href="/facts/M%C3%B6ssbauer_effect/Q3PU4Kaq">Mössbauer effect</a>: resonant and recoil-free emission and absorption of <a href="/facts/Gamma_radiation/5Drf913J">gamma radiation</a> by atomic nuclei bound in a solid, useful for accurate measurements of <a href="/facts/Gravitational_redshift/cEFgB3Ns">gravitational redshift</a> and <a href="/facts/Time_dilation/fYj7FYbK">time dilation</a>, and in the analysis of nuclear electromagnetic moments in <a href="/facts/Hyperfine_interaction/aqj0dMut">hyperfine interactions</a>.<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a></li></ul> <h3>Quantum non-locality and relativistic locality</h3> <p>In 1935, Einstein, <a href="/facts/Nathan_Rosen/0ks08uEe">Rosen</a>, <a href="/facts/Boris_Podolsky/aHp6zv1f">Podolsky</a> published a paper<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a> concerning <a href="/facts/Quantum_entanglement/afzslwN0">quantum entanglement</a> of particles, questioning <a href="/facts/Quantum_nonlocality/GUA6LlI5">quantum nonlocality</a> 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 <i>c</i>). QM does <i>not</i> violate SR.<a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a><a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a> In 1959, <a href="/facts/David_Bohm/c43gKR8Q">Bohm</a> and <a href="/facts/Yakir_Aharonov/SuddhMRJ">Aharonov</a> publish a paper<a class="footnote-ref" id="fnref:66" href="#fn:66"><sup>66</sup></a> on the <a href="/facts/Aharonov%E2%80%93Bohm_effect/wQIXNPUM">Aharonov–Bohm effect</a>, questioning the status of electromagnetic potentials in QM. The <a href="/facts/Electromagnetic_field_tensor/MWYuKLPy">EM field tensor</a> and <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">EM 4-potential</a> 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, <a href="/facts/Bell%27s_theorem/lg1V6kpk">Bell's theorem</a> was published in a paper on the EPR paradox,<a class="footnote-ref" id="fnref:67" href="#fn:67"><sup>67</sup></a> showing that QM cannot be derived from <a href="/facts/Local_hidden-variable_theory/wXo2mmvp">local hidden-variable theories</a> if locality is to be maintained. </p> <h3>The Lamb shift</h3> <p class="note">Main article: <a href="/facts/Lamb_shift/p4rqJV2j">Lamb shift</a></p> <p>In 1947, the Lamb shift was discovered: a small difference in the 2<i>S</i>1⁄2 and 2<i>P</i>1⁄2 levels of hydrogen, due to the interaction between the electron and vacuum. <a href="/facts/Willis_Lamb/srfCyHko">Lamb</a> and <a href="/facts/Robert_Retherford/N12oAFeG">Retherford</a> experimentally measure stimulated radio-frequency transitions the 2<i>S</i>1⁄2 and 2<i>P</i>1⁄2 hydrogen levels by <a href="/facts/Microwave/HmVSN31Y">microwave</a> radiation.<a class="footnote-ref" id="fnref:68" href="#fn:68"><sup>68</sup></a> An explanation of the Lamb shift is presented by <a href="/facts/Hans_Bethe/X6bkxRRI">Bethe</a>. Papers on the effect were published in the early 1950s.<a class="footnote-ref" id="fnref:69" href="#fn:69"><sup>69</sup></a> </p> <h3>Development of quantum electrodynamics</h3> <ul><li>1927 <a href="/facts/Paul_Dirac/6bt5yRdk">Dirac</a> establishes the field of QED, also coining the term "quantum electrodynamics".<a class="footnote-ref" id="fnref:70" href="#fn:70"><sup>70</sup></a></li> <li>1943 <a href="/facts/Sin-Itiro_Tomonaga/2wqqyq8g">Tomonaga</a> begins work on <a href="/facts/Renormalization/B4PdTB4o">renormalization</a>, influential in QED.</li> <li>1947 <a href="/facts/Julian_Schwinger/BdepoI1r">Schwinger</a> calculates the <a href="/facts/Anomalous_magnetic_dipole_moment/s8fVr0An">anomalous magnetic moment of the electron</a>. <a href="/facts/Polykarp_Kusch/1D2GLXbW">Kusch</a> measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.</li></ul> <h2 id="see-also">See also</h2> <table><tbody><tr><td><h3>Atomic physics and chemistry</h3><ul><li><a href="/facts/Relativistic_quantum_chemistry/8A1se1CS">Relativistic quantum chemistry</a></li><li><a href="/facts/Breit_equation/Hsh5sAcK">Breit equation</a></li><li><a href="/facts/Electron_spin_resonance/TDHV8c9e">Electron spin resonance</a></li><li><a href="/facts/Fine-structure_constant/zyzpmTJV">Fine-structure constant</a></li></ul><h3>Mathematical physics</h3><ul><li><a href="/facts/Quantum_spacetime/ExKmzFzQ">Quantum spacetime</a></li><li><a href="/facts/Spin_connection/GdnYxk3f">Spin connection</a></li><li><a href="/facts/Spinor_bundle/xjXIskf5">Spinor bundle</a></li><li><a href="/facts/Dirac_equation_in_the_algebra_of_physical_space/CaB86ojr">Dirac equation in the algebra of physical space</a></li><li><a href="/facts/Casimir_invariant/rG71qcFb">Casimir invariant</a></li><li><a href="/facts/Casimir_operator/rG71qcFb">Casimir operator</a></li><li><a href="/facts/Wigner_D-matrix/bU1B3BDX">Wigner D-matrix</a></li></ul></td><td><h3>Particle physics and quantum field theory</h3><ul><li><a href="/facts/Zitterbewegung/khtLRolH">Zitterbewegung</a></li><li><a href="/facts/Two-body_Dirac_equations/MthD2kvx">Two-body Dirac equations</a></li><li><a href="/facts/Relativistic_Heavy_Ion_Collider/MhxfcI5h">Relativistic Heavy Ion Collider</a></li><li><a href="/facts/Symmetry_(physics)/gJmgsooI">Symmetry (physics)</a></li><li><a href="/facts/Parity_(physics)/emSbJ4hD">Parity</a></li><li><a href="/facts/CPT_invariance/JCH2z2j2">CPT invariance</a></li><li><a href="/facts/Chirality_(physics)/H8Xbcvzt">Chirality (physics)</a></li><li><a href="/facts/Standard_model/LhVGf9ow">Standard model</a></li><li><a href="/facts/Gauge_theory/fSW3gggv">Gauge theory</a></li><li><a href="/facts/Tachyon/bqvlsfTa">Tachyon</a></li><li><a href="/facts/Modern_searches_for_Lorentz_violation/hW38zxnH">Modern searches for Lorentz violation</a></li></ul></td></tr></tbody></table> <h2 id="footnotes">Footnotes</h2> <h3>Selected books</h3> <ul><li><a href="/facts/Paul_Dirac/6bt5yRdk">Dirac, P.A.M.</a> (1981). <a href="https://books.google.com/books?id=XehUpGiM6FIC&q=principles+of+quantum+mechanics+dirac+4th+edition"><i>Principles of Quantum Mechanics</i></a> (4th ed.). Clarendon Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-852011-5.</li> <li>Dirac, P.A.M. (1964). <a href="https://books.google.com/books?id=GVwzb1rZW9kC&q=Relativistic+quantum+mechanics"><i>Lectures on Quantum Mechanics</i></a>. Courier Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-41713-4.</li> <li>Thaller, B. (2010). <a href="https://books.google.com/books?id=822rcQAACAAJ&q=dirac+equation+springer"><i>The Dirac Equation</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-08134-7.</li> <li><a href="/facts/Wolfgang_Pauli/Ugk4ubXk">Pauli, W.</a> (1980). <a href="https://books.google.com/books?id=93sfAQAAMAAJ&q=General+Principles+of+Quantum+Mechanics"><i>General Principles of Quantum Mechanics</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09842-3.</li> <li>Merzbacher, E. (1998). <i>Quantum Mechanics</i> (3rd ed.). John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-88702-7.</li> <li>Messiah, A. (1961). <i>Quantum Mechanics</i>. Vol. 1. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-59766-7.</li> <li>Bjorken, J.D.; Drell, S.D. (1964). <i>Relativistic Quantum Mechanics (Pure & Applied Physics)</i>. McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-005493-6.</li> <li>Feynman, R.P.; Leighton, R.B.; Sands, M. (1965). <a href="https://feynmanlectures.caltech.edu/III_toc.html"><i>Feynman Lectures on Physics</i></a>. Vol. 3. Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-02118-9.</li> <li>Schiff, L.I. (1968). <a href="https://archive.org/details/quantummechanics0000schi"><i>Quantum Mechanics</i></a> (3rd ed.). McGraw-Hill.</li> <li><a href="/facts/Freeman_Dyson/kSRxTh2r">Dyson, F.</a> (2011). <i>Advanced Quantum Mechanics</i> (2nd ed.). World Scientific. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4383-40-0.</li> <li>Clifton, R.K. (2011). <a href="https://books.google.com/books?id=TTKacQAACAAJ&q=Relativistic+quantum+mechanics"><i>Perspectives on Quantum Reality: Non-Relativistic, Relativistic, and Field-Theoretic</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-90-481-4643-7.</li> <li><a href="/facts/Claude_Cohen-Tannoudji/fjs27Pvx">Tannoudji, C.</a>; Diu, B.; Laloë, F. (1977). <i>Quantum Mechanics</i>. Vol. 1. Wiley VCH. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-16433-3.</li> <li><a href="/facts/Claude_Cohen-Tannoudji/fjs27Pvx">Tannoudji, C.</a>; Diu, B.; Laloë, F. (1977). <i>Quantum Mechanics</i>. Vol. 2. Wiley VCH. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-16435-7.</li> <li>Rae, A.I.M. (2008). <a href="https://books.google.com/books?id=YDhHAQAAIAAJ&q=quantum+mechanics+Alastair+Rae+5th+edition"><i>Quantum Mechanics</i></a>. Vol. 2 (5th ed.). Taylor & Francis. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-58488-970-0.</li> <li>Pilkuhn, H. (2005). <a href="https://books.google.com/books?id=jEW6fY6-iVcC&q=Relativistic+quantum+mechanics&pg=PA1"><i>Relativistic Quantum Mechanics</i></a>. Texts and Monographs in Physics Series (2nd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-28522-9.</li> <li>Parthasarathy, R. (2010). <a href="https://books.google.com/books?id=1t0WAQAAMAAJ&q=relativistic+quantum+mechanics"><i>Relativistic quantum mechanics</i></a>. Alpha Science International. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-84265-573-3.</li> <li>Kaldor, U.; Wilson, S. (2003). <a href="https://books.google.com/books?id=0xcAM5BzS-wC&q=Relativistic_quantum_mechanics"><i>Theoretical Chemistry and Physics of Heavy and Superheavy Elements</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4020-1371-3.</li> <li>Thaller, B. (2005). <a href="https://books.google.com/books?id=iq1Gi6hmTRAC&q=Relativistic+quantum+mechanics"><i>Advanced visual quantum mechanics</i></a>. Springer. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005avqm.book.....T">2005avqm.book.....T</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-27127-9.</li> <li>Breuer, H.P.; Petruccione, F. (2000). <a href="https://archive.org/details/springer_10.1007-3-540-45369-5"><i>Relativistic Quantum Measurement and Decoherence</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-41061-4. Relativistic quantum mechanics.</li> <li>Shepherd, P.J. (2013). <a href="https://books.google.com/books?id=CoV4JemLm40C&q=A+course+in+theoretical+physics"><i>A Course in Theoretical Physics</i></a>. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-118-51692-8.</li> <li><a href="/facts/Hans_Bethe/X6bkxRRI">Bethe, H.A.</a>; Jackiw, R.W. (1997). <i>Intermediate Quantum Mechanics</i>. Addison-Wesley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-32831-8.</li> <li>Heitler, W. (1954). <a href="https://books.google.com/books?id=L7w7UpecbKYC&q=magnetic+moments+in+relativistic+quantum+mechanics"><i>The Quantum Theory of Radiation</i></a> (3rd ed.). Courier Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-64558-2.</li> <li>Gottfried, K.; Yan, T. (2003). <a href="https://books.google.com/books?id=8gFX-9YcvIYC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA245"><i>Quantum Mechanics: Fundamentals</i></a> (2nd ed.). Springer. p. 245. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95576-6.</li> <li>Schwabl, F. (2010). <a href="https://books.google.com/books?id=pTHb4NK2eZcC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA220"><i>Quantum Mechanics</i></a>. Springer. p. 220. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-71933-5.</li> <li>Sachs, R.G. (1987). <a href="https://archive.org/details/physicsoftimerev0000sach"><i>The Physics of Time Reversal</i></a> (2nd ed.). University of Chicago Press. p. <a href="https://archive.org/details/physicsoftimerev0000sach/page/280">280</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-226-73331-9. hyperfine structure in relativistic quantum mechanics.</li></ul> <h3>Group theory in quantum physics</h3> <ul><li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, H.</a> (1950). <a href="https://archive.org/details/theoryofgroupsqu1950weyl"><i>The theory of groups and quantum mechanics</i></a>. Courier Dover Publications. p. <a href="https://archive.org/details/theoryofgroupsqu1950weyl/page/203">203</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780486602691. magnetic moments in relativistic quantum mechanics.</li> <li>Tung, W.K. (1985). <a href="https://books.google.com/books?id=O89tgpOBO04C&q=group+theory+in+physics"><i>Group Theory in Physics</i></a>. World Scientific. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-9971-966-56-0.</li> <li>Heine, V. (1993). <a href="https://books.google.com/books?id=NayFD34uEu0C&q=lorentz+group+in+relativistic+quantum+mechanics&pg=PA363"><i>Group Theory in Quantum Mechanics: An Introduction to Its Present Usage</i></a>. Courier Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-67585-5.</li></ul> <h3>Selected papers</h3> <ul><li><a href="/facts/Paul_Dirac/6bt5yRdk">Dirac, P.A.M.</a> (1932). <a href="https://doi.org/10.1098%2Frspa.1932.0094">"Relativistic Quantum Mechanics"</a>. <i><a href="/facts/Proceedings_of_the_Royal_Society_A/qyn2m2ja">Proceedings of the Royal Society A</a></i>. 136 (829): 453–464. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1932RSPSA.136..453D">1932RSPSA.136..453D</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frspa.1932.0094">10.1098/rspa.1932.0094</a>.</li> <li><a href="/facts/Wolfgang_Pauli/Ugk4ubXk">Pauli, W.</a> (1945). <a href="https://www.nobelprize.org/nobel_prizes/physics/laureates/1945/pauli-lecture.pdf">"Exclusion principle and quantum mechanics"</a> (PDF).</li> <li>Antoine, J.P. (2004). "Relativistic Quantum Mechanics". <i>J. Phys. A</i>. 37 (4): 1465. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004JPhA...37.1463P">2004JPhA...37.1463P</a>. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.499.2793">10.1.1.499.2793</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0305-4470%2F37%2F4%2FB01">10.1088/0305-4470/37/4/B01</a>.</li> <li>Henneaux, M.; Teitelboim, C. (1982). "Relativistic quantum mechanics of supersymmetric particles". Vol. 143.</li> <li>Fanchi, J.R. (1986). "Parametrizing relativistic quantum mechanics". <i>Phys. Rev. A</i>. 34 (3): 1677–1681. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1986PhRvA..34.1677F">1986PhRvA..34.1677F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevA.34.1677">10.1103/PhysRevA.34.1677</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/9897446">9897446</a>.</li> <li>Ord, G.N. (1983). "Fractal space-time: a geometric analogue of relativistic quantum mechanics". <i>J. Phys. A</i>. 16 (9): 1869–1884. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1983JPhA...16.1869O">1983JPhA...16.1869O</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0305-4470%2F16%2F9%2F012">10.1088/0305-4470/16/9/012</a>.</li> <li>Coester, F.; Polyzou, W.N. (1982). "Relativistic quantum mechanics of particles with direct interactions". <i>Phys. Rev. D</i>. 26 (6): 1348–1367. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1982PhRvD..26.1348C">1982PhRvD..26.1348C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevD.26.1348">10.1103/PhysRevD.26.1348</a>.</li> <li>Mann, R.B.; Ralph, T.C. (2012). "Relativistic quantum information". <i>Class. Quantum Grav</i>. 29 (22): 220301. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2012CQGra..29v0301M">2012CQGra..29v0301M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0264-9381%2F29%2F22%2F220301">10.1088/0264-9381/29/22/220301</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123341332">123341332</a>.</li> <li>Low, S.G. (1997). "Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)". <i>J. Math. Phys</i>. 38 (22): 2197–2209. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/physics/9703008">physics/9703008</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2012CQGra..29v0301M">2012CQGra..29v0301M</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0264-9381%2F29%2F22%2F220301">10.1088/0264-9381/29/22/220301</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123341332">123341332</a>.</li> <li>Fronsdal, C.; Lundberg, L.E. (1997). "Relativistic Quantum Mechanics of Two Interacting Particles". <i>Phys. Rev. D</i>. 1 (12): 3247–3258. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/physics/9703008">physics/9703008</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1970PhRvD...1.3247F">1970PhRvD...1.3247F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevD.1.3247">10.1103/PhysRevD.1.3247</a>.</li> <li>Bordovitsyn, V.A.; Myagkii, A.N. (2004). <a href="https://cds.cern.ch/record/488497/files/0102107.pdf">"Spin–orbital motion and Thomas precession in the classical and quantum theories"</a> (PDF). <i>American Journal of Physics</i>. 72 (1): 51–52. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/physics/0310016">physics/0310016</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004AmJPh..72...51K">2004AmJPh..72...51K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.1615526">10.1119/1.1615526</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119533324">119533324</a>.</li> <li>Rȩbilas, K. (2013). "Comment on 'Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession'". <i>Eur. J. Phys</i>. 34 (3): L55 – L61. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2013EJPh...34L..55R">2013EJPh...34L..55R</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0143-0807%2F34%2F3%2FL55">10.1088/0143-0807/34/3/L55</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122527454">122527454</a>.</li> <li>Corben, H.C. (1993). "Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession". <i>Am. J. Phys</i>. 61 (6): 551. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1993AmJPh..61..551C">1993AmJPh..61..551C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.17207">10.1119/1.17207</a>.</li></ul> <h2 id="further-reading">Further reading</h2> <h3>Relativistic quantum mechanics and field theory</h3> <ul><li><a href="/facts/Tommy_Ohlsson/1ob5aRfc">Ohlsson, T.</a> (2011). <a href="https://books.google.com/books?id=hRavtAW5EFcC&q=pauli-lubanski+pseudovector&pg=PA11"><i>Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory</i></a>. Cambridge University Press. p. 10. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-139-50432-4.</li> <li>Aitchison, I.J.R.; Hey, A.J.G. (2002). <a href="https://books.google.com/books?id=n7k_QS4Hb0YC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA50"><i>Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED</i></a>. Vol. 1 (3rd ed.). CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8493-8775-3.</li> <li>Griffiths, D. (2008). <a href="https://books.google.com/books?id=Wb9DYrjcoKAC&q=hyperfine+structure+in+relativistic+quantum+mechanics"><i>Introduction to Elementary Particles</i></a>. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-527-61847-7.</li> <li>Capri, Anton Z. (2002). <a href="https://books.google.com/books?id=tTJHB5hepQUC&q=relativistic+quantum+mechanics"><i>Relativistic quantum mechanics and introduction to quantum field theory</i></a>. World Scientific. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2002rqmi.book.....C">2002rqmi.book.....C</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-238-137-8.</li> <li><a href="/facts/Wu_Ta-You/HIPrdLot">Wu, Ta-you</a>; Hwang, W.Y. Pauchy (1991). <a href="https://books.google.com/books?id=gJR_gVU52NkC&q=relativistic+quantum+mechanics"><i>Relativistic quantum mechanics and quantum fields</i></a>. World Scientific. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-02-0608-6.</li> <li>Nagashima, Y. (2010). <i>Elementary particle physics, Quantum Field Theory</i>. Vol. 1. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-527-40962-4.</li> <li>Bjorken, J.D.; Drell, S.D. (1965). <i>Relativistic Quantum Fields (Pure & Applied Physics)</i>. McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-005494-3.</li> <li><a href="/facts/Steven_Weinberg/G61D6kJw">Weinberg, S.</a> (1996). <a href="https://archive.org/details/quantumtheoryoff00stev"><i>The Quantum Theory of Fields</i></a>. Vol. 2. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-55002-4.</li> <li><a href="/facts/Steven_Weinberg/G61D6kJw">Weinberg, S.</a> (2000). <a href="https://archive.org/details/quantumtheoryoff00stev"><i>The Quantum Theory of Fields</i></a>. Vol. 3. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-66000-6.</li> <li>Gross, F. (2008). <a href="https://books.google.com/books?id=o9dXoK_2s6MC&q=fractional+quantum+mechanics+books&pg=PP2"><i>Relativistic Quantum Mechanics and Field Theory</i></a>. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-527-61734-0.</li> <li>Nazarov, Y.V.; Danon, J. (2013). <a href="https://books.google.com/books?id=BrTP8hEWRGUC&q=Relativistic_quantum_mechanics"><i>Advanced Quantum Mechanics: A Practical Guide</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-76150-5.</li> <li>Bogolubov, N.N. (1989). <a href="https://books.google.com/books?id=Ef4zDW1V2LkC&q=lamb+shift&pg=PA187"><i>General Principles of Quantum Field Theory</i></a> (2nd ed.). Springer. p. 272. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7923-0540-8.</li> <li>Mandl, F.; Shaw, G. (2010). <a href="https://books.google.com/books?id=7VLMj4AvvicC&q=pauli-lubanski+pseudovector&pg=PA273"><i>Quantum Field Theory</i></a> (2nd ed.). John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-49683-0.</li> <li>Lindgren, I. (2011). <a href="https://books.google.com/books?id=noUF83SAl2wC&q=hyperfine+structure+in+relativistic+quantum+mechanics&pg=PA9"><i>Relativistic Many-body Theory: A New Field-theoretical Approach</i></a>. Springer series on atomic, optical, and plasma physics. Vol. 63. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-8309-1.</li> <li>Grant, I.P. (2007). <a href="https://books.google.com/books?id=yYFCAAAAQBAJ"><i>Relativistic Quantum theory of atoms and molecules</i></a>. Atomic, optical, and plasma physics. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-34671-7.</li></ul> <h3>Quantum theory and applications in general</h3> <ul><li>Aruldhas, G.; Rajagopal, P. (2005). <a href="https://books.google.com/books?id=GnCWJF6TQUwC&q=electromagnetic+moments+for+high-spin+particles&pg=PA395"><i>Modern Physics</i></a>. PHI Learning Pvt. Ltd. p. 395. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-81-203-2597-5.</li> <li>Hummel, R.E. (2011). <a href="https://books.google.com/books?id=TsHFou6RftkC&q=electromagnetic+moments+for+high-spin+particles&pg=PA395"><i>Electronic properties of materials</i></a>. Springer. p. 395. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4419-8164-6.</li> <li>Pavia, D.L. (2005). <a href="https://books.google.com/books?id=FkaNOdwk0FQC&q=magnetic+moments+allowed+by+spin+quantum+number&pg=PA105"><i>Introduction to Spectroscopy</i></a> (4th ed.). Cengage Learning. p. 105. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-495-11478-9.</li> <li>Mizutani, U. (2001). <a href="https://books.google.com/books?id=zY5z_UGqAcwC&q=magnetic+moments+allowed+by+spin+quantum+number&pg=PA387"><i>Introduction to the Electron Theory of Metals</i></a>. Cambridge University Press. p. 387. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-58709-9.</li> <li>Choppin, G.R. (2002). <a href="https://books.google.com/books?id=IsAEjPpvyrkC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA308"><i>Radiochemistry and nuclear chemistry</i></a> (3 ed.). Butterworth-Heinemann. p. 308. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7506-7463-8.</li> <li>Sitenko, A.G. (1990). <a href="https://books.google.com/books?id=IcF5GV7ftT0C&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA443"><i>Theory of nuclear reactions</i></a>. World Scientific. p. 443. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-9971-5-0482-3.</li> <li>Nolting, W.; Ramakanth, A. (2008). <a href="https://books.google.com/books?id=vrcHC9XoHbsC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA748"><i>Quantum theory of magnetism</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-85416-6.</li> <li>Luth, H. (2013). <a href="https://books.google.com/books?id=9zrbFSHWcCAC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA149"><i>Quantum Physics in the Nanoworld</i></a>. Graduate texts in physics. Springer. p. 149. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-31238-0.</li> <li>Sattler, K.D. (2010). <a href="https://books.google.com/books?id=08BWNlciXx4C&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=SA40-PA3"><i>Handbook of Nanophysics: Functional Nanomaterials</i></a>. CRC Press. pp. 40–43. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4200-7553-3.</li> <li>Kuzmany, H. (2009). <a href="https://books.google.com/books?id=Jg89d0h9SIoC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA256"><i>Solid-State Spectroscopy</i></a>. Springer. p. 256. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-01480-2.</li> <li>Reid, J.M. (1984). <a href="https://books.google.com/books?id=rcdRAQAAIAAJ&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA277"><i>The Atomic Nucleus</i></a> (2nd ed.). Manchester University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7190-0978-5.</li> <li>Schwerdtfeger, P. (2002). <a href="https://books.google.com/books?id=9tO_9Tf6dZgC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA208"><i>Relativistic Electronic Structure Theory - Fundamentals</i></a>. Theoretical and Computational Chemistry. Vol. 11. Elsevier. p. 208. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-054046-7.</li> <li>Piela, L. (2006). <a href="https://books.google.com/books?id=nbdITbfsP6oC&q=magnetic+multipole+moments+allowed+by+spin+quantum+number&pg=PA676"><i>Ideas of Quantum Chemistry</i></a>. Elsevier. p. 676. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-08-046676-7.</li> <li>Kumar, M. (2009). <a href="/facts/Quantum_(book)/cpeMNWXr"><i>Quantum (book)</i></a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-84831-035-3.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Pfeifer, W. (2008) [2004]. <a href="http://www.walterpfeifer.ch/index.html"><i>Relativistic Quantum Mechanics, an Introduction</i></a>.</li> <li>Lukačević, Igor (2013). <a href="https://web.archive.org/web/20140826114453/http://www.fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_11_Relativistic_quantum_mechanics.pdf">"Relativistic Quantum Mechanics (Lecture Notes)"</a> (PDF). Archived from <a href="http://www.fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_11_Relativistic_quantum_mechanics.pdf">the original</a> (PDF) on 2014-08-26.</li> <li><a href="http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_relqu.pdf">"Relativistic Quantum Mechanics"</a> (PDF). <a href="/facts/Cavendish_Laboratory/nh4qJsWs">Cavendish Laboratory</a>. <a href="/facts/University_of_Cambridge/0la7aX9o">University of Cambridge</a>.</li> <li>Miller, David J. (2008). <a href="https://web.archive.org/web/20201219112349/http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf">"Relativistic Quantum Mechanics"</a> (PDF). <a href="/facts/University_of_Glasgow/D0XAYMtq">University of Glasgow</a>. Archived from <a href="http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf">the original</a> (PDF) on 2020-12-19. Retrieved 2020-11-17.</li> <li>Swanson, D.G. (2007). <a href="https://archive.org/details/quantummechanics00swan">"Quantum Mechanics Foundations and Applications"</a>. Alabama, USA: Taylor & Francis. p. <a href="https://archive.org/details/quantummechanics00swan/page/n173">160</a>.</li> <li>Calvert, J.B. (2003). <a href="http://mysite.du.edu/~jcalvert/phys/partelec.htm">"The Particle Electron and Thomas Precession"</a>.</li> <li>Arteha, S.N. <a href="http://www.antidogma.ru/english/node58.html">"Spin and the Thomas precession"</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Perkins, D.H. (2000). Introduction to High Energy Physics. Cambridge University Press. ISBN 978-0-521-62196-0. <a href="978-0-521-62196-0" target="_blank">978-0-521-62196-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. p. 3. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Reiher, M.; Wolf, A. (2009). Relativistic Quantum Chemistry. John Wiley & Sons. ISBN 978-3-527-62749-3. <a href="978-3-527-62749-3" target="_blank">978-3-527-62749-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Strange, P. (1998). Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics. Cambridge University Press. ISBN 978-0-521-56583-7. <a href="978-0-521-56583-7" target="_blank">978-0-521-56583-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Mohn, P. (2003). Magnetism in the Solid State: An Introduction. Springer Series in Solid-State Sciences Series. Vol. 134. Springer. p. 6. ISBN 978-3-540-43183-1. <a href="978-3-540-43183-1" target="_blank">978-3-540-43183-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. pp. 5–6. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Messiah, A. (1981). Quantum Mechanics. Vol. 2. North-Holland Publishing Company. p. 875. ISBN 978-0-7204-0045-8. <a href="978-0-7204-0045-8" target="_blank">978-0-7204-0045-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Schweber, Silvan S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton, N.J: Princeton University Press. p. 573. ISBN 978-0-691-21328-6. <a href="978-0-691-21328-6" target="_blank">978-0-691-21328-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Bhaumik, Mani L. (2022). "How Dirac's Seminal Contributions Pave the Way for Comprehending Nature's Deeper Designs". Quanta. 8 (1): 88–100. arXiv:2209.03937. doi:10.12743/quanta.v8i1.96. S2CID 212835814. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. p. 3. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Other common notations include ms and sz etc., but this would clutter expressions with unnecessary subscripts. The subscripts σ labeling spin values are not to be confused for tensor indices nor the Pauli matrices. <a href="/wiki/Tensor_index_notation" target="_blank">/wiki/Tensor_index_notation</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Forshaw, J.R.; Smith, A.G. (2009). Dynamics and Relativity. Manchester Physics Series. John Wiley & Sons. pp. 258–259. ISBN 978-0-470-01460-8. <a href="978-0-470-01460-8" target="_blank">978-0-470-01460-8</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Greiner, W. (2000). Relativistic Quantum Mechanics. Wave Equations (3rd ed.). Springer. p. 70. ISBN 978-3-540-67457-3. <a href="978-3-540-67457-3" target="_blank">978-3-540-67457-3</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Wachter, A. (2011). "Relativistic quantum mechanics". Springer. p. 34. ISBN 978-90-481-3645-2. <a href="978-90-481-3645-2" target="_blank">978-90-481-3645-2</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Weinberg, S. (1964). "Feynman Rules for Any spin" (PDF). Phys. Rev. 133 (5B): B1318 – B1332. Bibcode:1964PhRv..133.1318W. doi:10.1103/PhysRev.133.B1318. Archived from the original (PDF) on 2020-12-04. Retrieved 2014-08-24.; Weinberg, S. (1964). "Feynman Rules for Any spin. II. Massless Particles" (PDF). Phys. Rev. 134 (4B): B882 – B896. Bibcode:1964PhRv..134..882W. doi:10.1103/PhysRev.134.B882. Archived from the original (PDF) on 2022-03-09. Retrieved 2013-04-14.; Weinberg, S. (1969). "Feynman Rules for Any spin. III" (PDF). Phys. Rev. 181 (5): 1893–1899. Bibcode:1969PhRv..181.1893W. doi:10.1103/PhysRev.181.1893. Archived from the original (PDF) on 2022-03-25. Retrieved 2013-04-14. <a href="https://web.archive.org/web/20201204053451/http://theory.fi.infn.it/becattini/files/weinberg1.pdf" target="_blank">https://web.archive.org/web/20201204053451/http://theory.fi.infn.it/becattini/files/weinberg1.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Masakatsu, K. (2012). "Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation". arXiv:1208.0644 [gr-qc]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. pp. 1193–1194. ISBN 978-0-07-051400-3. <a href="978-0-07-051400-3" target="_blank">978-0-07-051400-3</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Resnick, R.; Eisberg, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. p. 274. ISBN 978-0-471-87373-0. <a href="978-0-471-87373-0" target="_blank">978-0-471-87373-0</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Landau, L.D.; Lifshitz, E.M. (1981). Quantum Mechanics Non-Relativistic Theory. Vol. 3. Elsevier. p. 455. ISBN 978-0-08-050348-6. <a href="978-0-08-050348-6" target="_blank">978-0-08-050348-6</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics. Shaum's outlines (2nd ed.). McGraw–Hill. p. 181. ISBN 978-0-07-162358-2. <a href="978-0-07-162358-2" target="_blank">978-0-07-162358-2</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Abers, E. (2004). Quantum Mechanics. Addison Wesley. p. 425. ISBN 978-0-13-146100-0. <a href="978-0-13-146100-0" target="_blank">978-0-13-146100-0</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Wachter, A. (2011). "Relativistic quantum mechanics". Springer. p. 5. ISBN 978-90-481-3645-2. <a href="978-90-481-3645-2" target="_blank">978-90-481-3645-2</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. p. 3. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Abers, E. (2004). Quantum Mechanics. Addison Wesley. p. 415. ISBN 978-0-13-146100-0. <a href="978-0-13-146100-0" target="_blank">978-0-13-146100-0</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Penrose, R. (2005). The Road to Reality. Vintage Books. pp. 620–621. ISBN 978-0-09-944068-0. <a href="978-0-09-944068-0" target="_blank">978-0-09-944068-0</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Bransden, B.H.; Joachain, C.J. (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 634. ISBN 978-0-582-44401-0. <a href="978-0-582-44401-0" target="_blank">978-0-582-44401-0</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Penrose, R. (2005). The Road to Reality. Vintage Books. pp. 620–621. ISBN 978-0-09-944068-0. <a href="978-0-09-944068-0" target="_blank">978-0-09-944068-0</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. pp. 5–6. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Grandy, W.T. (1991). Relativistic quantum mechanics of leptons and fields. Springer. p. 54. ISBN 978-0-7923-1049-5. <a href="978-0-7923-1049-5" target="_blank">978-0-7923-1049-5</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Abers, E. (2004). Quantum Mechanics. Addison Wesley. p. 423. ISBN 978-0-13-146100-0. <a href="978-0-13-146100-0" target="_blank">978-0-13-146100-0</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>McMahon, D. (2008). Quantum Field Theory. Demystified. McGraw Hill. p. 114. ISBN 978-0-07-154382-8. <a href="978-0-07-154382-8" target="_blank">978-0-07-154382-8</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. pp. 1193–1194. ISBN 978-0-07-051400-3. <a href="978-0-07-051400-3" target="_blank">978-0-07-051400-3</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Bransden, B.H.; Joachain, C.J. (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. pp. 632–635. ISBN 978-0-582-44401-0. <a href="978-0-582-44401-0" target="_blank">978-0-582-44401-0</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Martin, B.R.; Shaw, G. (2008-12-03). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. p. 3. ISBN 978-0-470-03294-7. <a href="978-0-470-03294-7" target="_blank">978-0-470-03294-7</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics. Shaum's outlines (2nd ed.). McGraw–Hill. p. 181. ISBN 978-0-07-162358-2. <a href="978-0-07-162358-2" target="_blank">978-0-07-162358-2</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>This spinor notation is not necessarily standard; the literature usually writes ψ = ( u 1 u 2 ) {\displaystyle \psi ={\begin{pmatrix}u^{1}\\u^{2}\end{pmatrix}}} or ψ = ( χ η ) {\displaystyle \psi ={\begin{pmatrix}\chi \\\eta \end{pmatrix}}} etc., but in the context of spin 1/2, this informal identification is commonly made. <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Again this notation is not necessarily standard, the more advanced literature usually writes ψ = ( u v ) = ( u 1 u 2 v 1 v 2 ) {\displaystyle \psi ={\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}u^{1}\\u^{2}\\v^{1}\\v^{2}\end{pmatrix}}} etc., but here we show informally the correspondence of energy, helicity, and spin states. <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Parker, C.B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 1194. ISBN 978-0-07-051400-3.. <a href="978-0-07-051400-3" target="_blank">978-0-07-051400-3</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>Labelle, P. (2010). Supersymmetry. Demystified. McGraw-Hill. ISBN 978-0-07-163641-4. <a href="978-0-07-163641-4" target="_blank">978-0-07-163641-4</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Esposito, S. (2011). "Searching for an equation: Dirac, Majorana and the others". Annals of Physics. 327 (6): 1617–1644. arXiv:1110.6878. Bibcode:2012AnPhy.327.1617E. doi:10.1016/j.aop.2012.02.016. S2CID 119147261. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Bargmann, V.; Wigner, E.P. (1948). "Group theoretical discussion of relativistic wave equations". Proc. Natl. Acad. Sci. U.S.A. 34 (5): 211–23. Bibcode:1948PNAS...34..211B. doi:10.1073/pnas.34.5.211. PMC 1079095. PMID 16578292. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079095" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079095</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Wigner, E. (1937). "On Unitary Representations Of The Inhomogeneous Lorentz Group" (PDF). Annals of Mathematics. 40 (1): 149–204. Bibcode:1939AnMat..40..149W. doi:10.2307/1968551. JSTOR 1968551. S2CID 121773411. Archived from the original (PDF) on 2015-10-04. Retrieved 2013-04-14. <a href="https://web.archive.org/web/20151004025027/http://courses.theophys.kth.se/SI2390/wigner_1939.pdf" target="_blank">https://web.archive.org/web/20151004025027/http://courses.theophys.kth.se/SI2390/wigner_1939.pdf</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Jaroszewicz, T.; Kurzepa, P.S (1992). "Geometry of spacetime propagation of spinning particles". 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