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| {{Quantum mechanics|cTopic=Equations}}{{quantum field theory}}
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| :''"[[Relativistic quantum field equations]]" redirects to here.''
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| In [[physics]], specifically [[relativistic quantum mechanics]] (RQM) and its applications to [[particle physics]], '''relativistic wave equations''' predict the behavior of [[particles]] at high [[energy|energies]] and [[velocity|velocities]] comparable to the [[speed of light]]. In the context of [[quantum field theory]] (QFT), the equations determine the dynamics of [[quantum field]]s.
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| The solutions to the equations, universally denoted as {{math|ψ}} or {{math|Ψ}} ([[Greek language|Greek]] [[Psi (letter)|psi]]), are referred to as "[[wavefunction]]s" in the context of RQM, and "[[field (physics)|field]]s" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a [[wave equation]] or are generated from a [[Lagrangian density]] and the field-theoretic [[Euler–Lagrange equation]]s (see [[classical field theory]] for background).
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| In the [[Schrödinger picture]], the wavefunction or field is the solution to the [[Schrödinger equation]];
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| :<math> i\hbar\frac{\partial}{\partial t}\psi = \hat{H} \psi</math>
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| one of the [[Mathematical formulation of quantum mechanics#Pictures of dynamics|postulates of quantum mechanics]]. All relativistic wave equations can be constructed by specifying various forms of the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]] ''Ĥ'' describing the [[Physical system|quantum system]]. Alternatively, [[Richard Feynman|Feynman]]'s [[path integral formulation]] uses a Lagrangian rather than a Hamiltonian operator.
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| More generally - the modern formalism behind relativistic wave equations is [[Lorentz group]] theory, wherein the spin of the particle has a correspondence with the [[representations of the Lorentz group]].<ref name="T Jaroszewicz, P.S Kurzepa">{{cite article
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| | author = T Jaroszewicz, P.S Kurzepa
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| | year = 1992
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| | location = California, USA
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| | publisher =
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| | title = Geometry of spacetime propagation of spinning particles
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| | journal = Annals of Physics
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| | arxiv =
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| | url = http://www.sciencedirect.com/science/article/pii/000349169290176M
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| }}</ref>
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| ==History==
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| ===Early 1920s: Classical and quantum mechanics===
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| The failure of [[classical mechanics]] applied to [[molecule|molecular]], [[Atom|atomic]], and [[Atomic nucleus|nuclear]] systems and smaller induced the need for a new mechanics: ''[[quantum mechanics]]''. The mathematical formulation was led by [[Louis de Broglie|De Broglie]], [[Niels Bohr|Bohr]], [[Erwin Schrödinger|Schrödinger]], [[Wolfgang Pauli|Pauli]], and [[Werner Heisenberg|Heisenberg]], and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The [[Schrödinger equation]] and the [[Heisenberg picture]] resemble the classical [[equations of motion]] in the limit of large [[quantum number]]s and as the reduced [[Planck constant]] {{math|''ħ''}}, the quantum of [[action (physics)|action]], tends to zero. This is the [[correspondence principle]]. At this point, [[special relativity]] was not fully combined with quantum mechanics, so the Schrödinger and Heisenberg formulations, as originally proposed, could not be used in situations where the particles travel near the [[speed of light]], or when the number of each type of particle changes (this happens in real [[fundamental interaction|particle interaction]]s; the numerous forms of [[particle decay]]s, [[annihilation]], [[matter creation]], [[pair production]], and so on).
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| ===Late 1920s: Relativistic quantum mechanics of spin-0 and spin-½ particles===
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| A description of quantum mechanical systems which could account for ''relativistic'' effects was sought for by many theoretical physicists; from the late 1920s to the mid-1940s.<ref name="Esposito">{{cite article | author = S. Esposito | year = 2011 | title = Searching for an equation: Dirac, Majorana and the others | arxiv = 1110.6878 | url = http://arxiv.org/pdf/1110.6878v1.pdf}}</ref> The first basis for [[relativistic quantum mechanics]], i.e. special relativity applied with quantum mechanics together, was found by all those who discovered what is frequently called the [[Klein–Gordon equation]]:
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| {{NumBlk|:|<math>
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| -\hbar^2\frac{\partial^2 \psi}{\partial t^2} +(\hbar c)^2\nabla^2\psi = (mc^2)^2\psi \,,
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| </math>|{{EquationRef|1}}}}
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| by inserting the [[energy operator]] and [[momentum operator]] into the relativistic [[energy–momentum relation]]:
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| {{NumBlk|:|<math>
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| E^2 - (pc)^2 = (mc^2)^2\,,
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| </math>|{{EquationRef|2}}}}
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| The solutions to ({{EquationNote|1}}) are [[scalar field]]s. The KG equation is undesirable due to its prediction of ''negative'' [[energy|energies]] and [[probability|probabilities]], as a result of the [[quadratic equation|quadratic]] nature of ({{EquationNote|2}}) - inevitable in a relativistic theory. This equation was initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the [[Schrödinger equation]]) was still of importance. Nevertheless - ({{EquationNote|1}}) is applicable to spin-0 [[boson]]s.<ref>{{cite book|title = Particle Physics | edition = 3rd | author = B. R. Martin, G.Shaw | series = Manchester Physics Series|publisher = John Wiley & Sons|year = 2008| page = 3|isbn = 978-0-470-03294-7}}</ref>
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| Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the [[hyperfine structure]] in the [[Hydrogen spectral series]]. The mysterious underlying property was ''spin''. The first two-dimensional ''spin matrices'' (better known as the [[Pauli matrices]]) were introduced by Pauli in the [[Pauli equation]]; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in [[magnetic field]]s, but this was ''phenomological''. [[Hermann Weyl|Weyl]] found a relativistic equation in terms of the Pauli matrices; the [[Weyl equation]], for ''massless'' spin-½ fermions. The problem was resolved by [[Paul Dirac|Dirac]] in the late 1920s, when he furthered the application of equation ({{EquationNote|2}}) to the [[electron]] - by various manipulations he factorized the equation into the form:
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| {{NumBlk|:|<math>
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| \left(\frac{E}{c} - \boldsymbol{\alpha}\cdot\mathbf{p} - \beta mc \right)\left(\frac{E}{c} + \boldsymbol{\alpha}\cdot\mathbf{p} + \beta mc \right)\psi=0 \,,
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| </math>|{{EquationRef|3A}}}}
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| and one of these factors is the [[Dirac equation]] (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices {{math|'''α'''}} and {{math|''β''}} in a relativistic wave equation, and explained the hyperfine structure of hydrogen. The solutions to ({{EquationNote|3A}}) are multi-component [[spinor field]]s, and each component satisfies ({{EquationNote|1}}). A remarkable result of spinor solutions is that half of the components describe a particle, while the other half describe an [[antiparticle]]; in this case the electron and [[positron]]. The Dirac equation is now known to apply for all massive [[spin-½]] [[fermion]]s. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.
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| Although a ''landmark'' in quantum theory, the Dirac equation is only true for spin-½ fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular - not all physicists were comfortable with the "[[Dirac sea]]" of negative energy states).
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| ===1930s–1960s: Relativistic quantum mechanics of higher-spin particles===
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| The natural problem became clear: to generalize the Dirac equation to particles with ''any spin''; both fermions and bosons, and in the same equations their [[antiparticle]]s (possible because of the [[spinor]] formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by [[Bartel Leendert van der Waerden|van der Waerden]] in 1929), and ideally with positive energy solutions.<ref name="Esposito"/>
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| This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of ({{EquationNote|3A}}):
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| {{NumBlk|:|<math>
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| \left(\frac{E}{c} + \boldsymbol{\alpha}\cdot\mathbf{p} - \beta mc \right)\psi=0 \,,
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| </math>|{{EquationRef|3B}}}}
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| where {{math|ψ}} is a spinor field now with infinitely many components, irreducible to a finite number of [[tensor]]s or spinors, to remove the indeterminacy in sign. The [[matrix (mathematics)|matrices]] {{math|'''α'''}} and {{math|β}} are infinite-dimensional matrices, related to infinitesimal [[Lorentz transformation]]s. He did not demand that each component of to satisfy equation ({{EquationNote|2}}), instead he regenerated the equation using a [[Lorentz covariance|Lorentz-invariant]] [[action (physics)|action]], via the [[principle of least action]], and application of [[Lorentz group]] theory.<ref>{{cite article | author = R. Casalbuoni| location = Firenze, Italy | year = 2006 | title = Majorana and the Infinite Component Wave Equations | arxiv = hep-th/0610252v1 | url = http://arxiv.org/pdf/hep-th/0610252v1.pdf}}</ref><ref name = "Bekaert, Traubenberg, Valenzuela">{{cite article | author = X. Bekaert, M.R. Traubenberg, M. Valenzuela | year = 2009 | title = An infinite supermultiplet of massive higher-spin fields | arxiv = 0904.2533v4 | url = http://arxiv.org/pdf/0904.2533v4.pdf}}</ref>
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| Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939), see [[Duffin–Kemmer–Petiau algebra]]. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana’s, as spinors were new mathematical tools in the early twentieth century, although Majorana’s paper of 1932 was difficult to fully understand; it took Pauli and Wigner some time to understand it, around 1940.<ref name="Esposito"/>
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| Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors {{math|''A''}} and {{math|''B''}}, symmetric in all indices, for a massive particle of spin {{math|''n'' + ½}} for integer {{math|''n''}} (see [[Van der Waerden notation]] for the meaning of the dotted indices):
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| {{NumBlk|:|<math>
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| p_{\gamma\dot{\alpha}}A_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcB_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n}
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| </math>|{{EquationRef|4A}}}}
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| {{NumBlk|:|<math>
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| p^{\gamma\dot{\alpha}}B_{\gamma\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n} = mcA_{\epsilon_1\epsilon_2\cdots\epsilon_n}^{\dot{\alpha}\dot{\beta}_1\dot{\beta}_2\cdots\dot{\beta}_n}
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| </math>|{{EquationRef|4B}}}}
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| where {{math|''p''}} is the momentum as a covariant spinor operator. For {{math|''n'' {{=}} 0}}, the equations reduce to the coupled Dirac equations and {{math|''A''}} and {{math|''B''}} together transform as the original [[Dirac spinor]]. Eliminating either {{math|''A''}} or {{math|''B''}} shows that {{math|''A''}} and {{math|''B''}} each fulfill ({{EquationNote|1}}).<ref name="Esposito"/>
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| In 1941, Rarita and Schwinger focussed on spin-{{frac|3|2}} particles and derived the [[Rarita–Schwinger equation]], including a [[Lagrangian]] to generate it, and later generalized the equations analogous to spin {{math|''n'' + ½}} for integer {{math|''n''}}. In 1945, Pauli suggested Majorana's 1932 paper to [[Homi J. Bhabha|Bhabha]], who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in ({{EquationNote|3A}}) and ({{EquationNote|3B}}) by an arbitrary constant, subject to a set of conditions which the wavefunctions must obey.<ref>{{cite article | author = R.K. Loide, I. Ots, R. Saar | year = 1997 | title = Bhabha relativistic wave equations | arxiv = | url = http://iopscience.iop.org/0305-4470/30/11/027}}</ref>
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| Finally, in the year 1948 (the same year as [[Feynman]]'s [[path integral formulation]] was cast), [[Valentine Bargmann|Bargmann]] and [[Eugene Wigner|Wigner]] formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the [[Bargmann–Wigner equations]]. <ref>{{cite journal|author1=Bargmann, V.|author2=Wigner, E. P.|title=Group theoretical discussion of relativistic wave equations|year=1948|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=34|pages=211–23|url=http://www.pnas.org/cgi/content/citation/34/5/211|issue=5}}</ref><ref name="Esposito"/> In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by [[H. Joos]] and [[Steven Weinberg]]. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.<ref>{{cite journal | author =E.A. Jeffery
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| | year =1978
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| | title =Component Minimization of the Bargman–Wigner wavefunction
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| | journal =Australian Journal of Physics
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| | location = Melbourne
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| | publisher = CSIRO
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| | url =http://www.publish.csiro.au/?act=view_file&file_id=PH780137.pdf
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| }}</ref><ref name="T Jaroszewicz, P.S Kurzepa"/><ref>{{cite article
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| | author = R.F Guertin
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| | year = 1974
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| | location = Texas, USA
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| | publisher =
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| | title = Relativistic hamiltonian equations for any spin
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| | journal = Annals of Physics
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| | arxiv =
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| | url = http://www.sciencedirect.com/science/article/pii/0003491674901808
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| }}</ref>
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| ===1960s–Present===
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| The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of the present-day research, because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.<ref name = "Bekaert, Traubenberg, Valenzuela"/>
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| ==Linear equations==
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| {{further|Linear differential equation}}
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| The following equations have solutions which satisfy the [[superposition principle]], that is, the wavefunctions are [[Additive function|additive]].
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| Throughout, the standard conventions of [[tensor index notation]] and [[Feynman slash notation]] are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the indexed quantities. The wavefunctions are denoted ''{{math|ψ}}'', and {{math|∂<sub>''μ''</sub>}} are the components of the [[four-gradient]] operator.
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| In [[matrix (mathematics)|matrix]] equations, the [[Pauli matrices]] are denoted by ''{{math|σ<sup>μ</sup>}}'' in which {{math|''μ'' {{=}} 0, 1, 2, 3}}, where {{math|''σ''<sup>0</sup>}} is the {{math|2 × 2}} [[identity matrix]]:
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| :<math>\sigma^0 = \begin{pmatrix} 1&0 \\ 0&1 \\ \end{pmatrix} </math>
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| and the other matrices have their usual representations. The expression
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| :<math>\sigma^\mu \partial_\mu \equiv \sigma^0 \partial_0 + \sigma^1 \partial_1 + \sigma^2 \partial_2 + \sigma^3 \partial_3 </math>
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| is a {{math|2 × 2}} [[matrix (mathematics)|matrix]] [[Operator (mathematics)|operator]] which acts on 2-component [[spinor field]]s.
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| The [[gamma matrices]] are denoted by ''{{math|γ<sup>μ</sup>}}'', in which again {{math|''μ'' {{=}} 0, 1, 2, 3}}, and there are a number of representations to select from. The matrix {{math|''γ''<sup>0</sup>}} is ''not'' necessarily the {{math|4 × 4}} [[identity matrix]]. The expression
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| :<math>i\hbar \gamma^\mu \partial_\mu + mc \equiv i\hbar(\gamma^0 \partial_0 + \gamma^1 \partial_1 + \gamma^2 \partial_2 + \gamma^3 \partial_3) + mc \begin{pmatrix}1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{pmatrix} </math> | |
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| is a {{math|4 × 4}} [[matrix (mathematics)|matrix]] [[Operator (mathematics)|operator]] which acts on 4-component [[spinor field]]s.
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| Note that terms such as "{{math|''mc''}}" [[scalar multiplication|scalar multiply]] an [[identity matrix]] of the relevant [[Dimension (vector space)|dimension]], the common sizes are {{math|2 × 2}} or {{math|4 × 4}}, and are ''conventionally'' not written for simplicity.
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| :{| class="wikitable" | |
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| ! scope="col" width="100px" | Particle [[spin quantum number]] ''s''
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| ! scope="col" width="200px" | Name
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| ! scope="col" width="300px" | Equation
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| ! scope="col" width="200px" | Typical particles the equation describes
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| |-valign="top"
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| | 0
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| | [[Klein–Gordon equation]]
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| | <math>(\hbar \partial_{\mu} + imc)(\hbar \partial^{\mu} -imc)\psi = 0</math>
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| | Massless or massive spin-0 particle (such as [[Higgs boson]]s).
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| |-valign="top"
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| |scope="row" rowspan="5"| 1/2
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| | [[Weyl equation]]
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| | <math> \sigma^\mu\partial_\mu \psi=0</math>
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| | Massless spin-1/2 particles.
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| |-valign="top"
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| | [[Dirac equation]]
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| | <math>\left( i \hbar \partial\!\!\!/ - m c \right) \psi = 0 </math>
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| | Massive spin-1/2 particles (such as [[electron]]s).
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| |-valign="top"
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| | [[Two-body Dirac equations]]
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| | <math>[(\gamma_1)_\mu (p_1-\tilde{A}_1)^\mu+m_1 + \tilde{S}_1]\Psi=0,</math>
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| <math>[(\gamma_2)_\mu (p_2-\tilde{A}_2)^\mu+m_2 + \tilde{S}_2]\Psi=0.</math>
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| | Massive spin-1/2 particles (such as [[electron]]s).
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| |-valign="top"
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| |[[Majorana equation]]
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| | <math> i \hbar \partial\!\!\!/ \psi - m c \psi_c = 0</math>
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| | Massive [[Majorana particle]]s.
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| |-valign="top"
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| |[[Breit equation]]
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| |<math> i\hbar\frac{\partial \Psi}{\partial t} = \left(\sum_{i}\hat{H}_{D}(i) + \sum_{i>j}\frac{1}{r_{ij}} - \sum_{i>j}\hat{B}_{ij} \right) \Psi </math>
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| | Two massive spin-1/2 particles (such as [[electron]]s) interacting electromagnetically to first order in perturbation theory.
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| |-valign="top"
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| |scope="row" rowspan="2"| 1
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| | [[Maxwell equations]] (in [[Quantum electrodynamics#Equations of motion|QED]] using the [[Lorenz gauge]])
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| |<math>\partial_\mu\partial^\mu A^\nu = e \overline{\psi} \gamma^\nu \psi </math>
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| | [[Photon]]s, massless spin-1 particles.
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| |-valign="top"
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| |[[Proca equation]]
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| |<math>\partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+\left(\frac{mc}{\hbar}\right)^2 A^\nu=0</math>
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| | Massive spin-1 particle (such as [[W and Z bosons]]).
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| |-valign="top"
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| |3/2
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| |[[Rarita–Schwinger equation]]
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| |<math> \epsilon^{\mu \nu \rho \sigma} \gamma^5 \gamma_\nu \partial_\rho \psi_\sigma + m\psi^\mu = 0</math>
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| | Massive spin-3/2 particles.
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| |-valign="top"
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| |''s''
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| |[[Bargmann–Wigner equations]]
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| |<math>\begin{align}
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| & (-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_1 \alpha_1'}\psi_{\alpha'_1 \alpha_2 \alpha_3 \cdots \alpha_{2s}} = 0 \\
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| & (-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_2 \alpha_2'}\psi_{\alpha_1 \alpha'_2 \alpha_3 \cdots \alpha_{2s}} = 0 \\
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| & \qquad \vdots \\
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| & (-i\hbar \gamma^\mu \partial_\mu + mc)_{\alpha_{2s} \alpha'_{2s}}\psi_{\alpha_1 \alpha_2 \alpha_3 \cdots \alpha'_{2s}} = 0 \\
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| \end{align}</math>
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| where ''{{math|ψ}}'' is a rank-2''s'' 4-component [[spinor]].
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| |Free particles of arbitrary spin (bosons and fermions).<ref>{{cite journal | author =E.A. Jeffery
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| | year =1978
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| | title =Component Minimization of the Bargman–Wigner wavefunction
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| | journal =Australian Journal of Physics
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| | location = Melbourne
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| | publisher = CSIRO
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| | url =http://www.publish.csiro.au/?act=view_file&file_id=PH780137.pdf
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| }}</ref><ref>{{cite article
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| | author = R.Clarkson, D.G.C. McKeon
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| | year =2003
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| | title = Quantum Field Theory
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| | pages = 61–69
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| | url =http://www.apmaths.uwo.ca/people/QFTNotesAll.pdf
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| }}</ref>
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| |-
| |
| |}
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| ===Gauge fields===
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| The [[Duffin–Kemmer–Petiau algebra#Duffin–Kemmer–Petiau equation|Duffin–Kemmer–Petiau equation]] is an alternative equation for spin-0 and spin-1 particles:
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| :<math>(i \hbar \beta^{a} \partial_a - m c) \psi = 0</math>
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| ==Non-linear equations==
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| {{further|non-linear differential equation}}
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| There are equations which have solutions that do not satisfy the superposition principle.
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| ===Gauge fields===
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| * [[Yang–Mills theory|Yang–Mills equation]]: describes a non-abelian gauge field
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| * [[Yang–Mills–Higgs equations]]: describes a non-abelian gauge field coupled with a massive spin-0 particle
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| ===Spin 2===
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| * [[Einstein field equations]]: describe interaction of matter with the [[gravitational field]] (massless spin-2 field):
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| ::<math>R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}</math>
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| :The solution is a [[metric tensor|metric]] [[tensor field]], rather than a wavefunction.
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| ==See also==
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| * [[Scalar field theory]]
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| * [[Status of special relativity]]
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| * [[Mathematical descriptions of the electromagnetic field]]
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| * [[Minimal coupling]]
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| * [[Lorentz transformations]]
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| * [[List of equations in quantum mechanics]]
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| * [[List of equations in nuclear and particle physics]]
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| ==References==
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| ===Notes===
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| {{reflist}}
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| ===Further reading===
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| * {{citebook|title=Encyclopaedia of Physics|edition=2nd|author=R.G. Lerner, G.L. Trigg|publisher=VHC publishers|year=1991|isbn=0-89573-752-3}}
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| * {{citebook|title=McGraw Hill Encyclopaedia of Physics|edition=2nd|author=C.B. Parker|year=1994|isbn=0-07-051400-3}}
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| * {{citebook|title=The Cambridge Handbook of Physics Formulas|author=G. Woan, Cambridge University Press|year=2010|isbn=978-0-521-57507-2}}
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| * {{citebook|title=Relativity DeMystified|author=D. McMahon|publisher=Mc Graw Hill (USA)|year=2006|isbn=0-07-145545-0}}
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| * {{citebook|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman|year=1973|isbn=0-7167-0344-0}}
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| * {{citebook|title=Particle Physics (Manchester series)|author=2nd|author=B.R. Martin, G. Shaw|publisher=John Wiley & Sons|year=2008|isbn=978-0-470-03294-7}}
| |
| * {{citebook|title=Supersymmetry|author=P. Labelle, Demystified|publisher=McGraw-Hill (USA)|year=2010|isbn=978-0-07-163641-4}}
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| * {{citebook|title=Physics of Atoms and Molecules|author=B.H. Bransden, C.J.Joachain|publisher=Longman|year=1983|isbn=0-582-44401-2}}
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| * {{citebook|title=Quantum Mechanics|author=E. Abers|publisher=Addison Wesley|year=2004|isbn=978-0-13-146100-0}}
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| * {{citebook|title=Quantum Field Theory|author=D. McMahon|publisher=Mc Graw Hill (USA)|year=2008|isbn=978-0-07-154382-8}}
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| *{{cite article
| |
| | author = M. Pillin
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| | year = 1993
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| | location = Max-Planck-Institut fuer Physik, Werner-Heisenberg-Institut, Foehringer Ring 6, D-80805 Muenchen, Germany
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| | publisher =
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| | title = q-Deformed Relativistic Wave Equations
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| | arxiv = hep-th/9310097v1
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| | url = http://arxiv.org/pdf/hep-th/9310097v1.pdf
| |
| }}
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| | |
| {{DEFAULTSORT:Relativistic Wave Equations}}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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| [[Category:Equations of physics]]
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| [[Category:Special relativity]]
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| [[Category:Waves]]
| |