Editing Relativistic wave equations (section)

== History ==

=== Early 1920s: Classical and quantum mechanics ===

The failure of [[classical mechanics]] applied to [[molecule|molecular]], [[atom]]ic, 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).

=== Late 1920s: Relativistic quantum mechanics of spin-0 and spin-<sup>1</sup>/<sub>2</sub> particles ===

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 journal | author = S. Esposito | year = 2011 | title = Searching for an equation: Dirac, Majorana and the others | arxiv = 1110.6878 | doi=10.1016/j.aop.2012.02.016 | volume=327 | journal=Annals of Physics | issue = 6 | pages=1617–1644| bibcode=2012AnPhy.327.1617E | s2cid = 119147261 }}</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]]:
{{NumBlk||<math display="block">
 -\hbar^2 \frac{\partial^2 \psi}{\partial t^2} + (\hbar c)^2 \nabla^2 \psi = (mc^2)^2 \psi,
</math>|{{EquationRef|1}}}}
by inserting the [[energy operator]] and [[momentum operator]] into the relativistic [[energy–momentum relation]]:
{{NumBlk||<math display="block">
 E^2 - (pc)^2 = (mc^2)^2.
</math>|{{EquationRef|2}}}}

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 |url = https://archive.org/details/particlephysics00mart |url-access = limited | edition = 3rd | author = B. R. Martin, G. Shaw | series =  Manchester Physics Series |publisher = John Wiley & Sons |year = 2008| page = [https://archive.org/details/particlephysics00mart/page/n24 3] |isbn = 978-0-470-03294-7}}</ref>

Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the [[fine 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 ''phenomenological''. [[Hermann Weyl|Weyl]] found a relativistic equation in terms of the Pauli matrices; the [[Weyl equation]], for ''massless'' spin-1/2 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
{{NumBlk||<math display="block">
 \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,
</math>|{{EquationRef|3A}}}}
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 fine 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-1/2]] [[fermion]]s. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.

Although a landmark in quantum theory, the Dirac equation is only true for spin-1/2 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).

=== 1930s–1960s: Relativistic quantum mechanics of higher-spin particles ===

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"/>

This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of ({{EquationNote|3A}}):
{{NumBlk||<math display="block">
 \left(\frac{E}{c} + \boldsymbol{\alpha} \cdot \mathbf{p} - \beta mc\right) \psi = 0,
</math>|{{EquationRef|3B}}}}
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 {{EquationNote|3B}} 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 journal | author = R. Casalbuoni | year = 2006 | title = Majorana and the Infinite Component Wave Equations | journal = Pos Emc | volume = 2006 | pages = 004 | arxiv = hep-th/0610252| bibcode = 2006hep.th...10252C }}</ref><ref name = "Bekaert, Traubenberg, Valenzuela">{{cite journal |author1=X. Bekaert |author2=M.R. Traubenberg |author3=M. Valenzuela | year = 2009 | title = An infinite supermultiplet of massive higher-spin fields | arxiv = 0904.2533 | doi=10.1088/1126-6708/2009/05/118 | volume=2009 | journal=Journal of High Energy Physics |issue=5 | page=118|bibcode=2009JHEP...05..118B |s2cid=16285006 }}</ref>

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"/>

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 {{nobr|{{math|''n'' + 1/2}}}} for integer {{math|''n''}} (see [[Van der Waerden notation]] for the meaning of the dotted indices):
{{NumBlk||<math display="block">
 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},
</math>|{{EquationRef|4A}}}}
{{NumBlk||<math display="block">
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},
</math>|{{EquationRef|4B}}}}
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"/> The direct derivation of the Dirac–Pauli–Fierz equations using the Bargmann–Wigner operators is given by Isaev and Podoinitsyn.<ref>
{{cite journal
 | last1 = A. P. Isaev
 | last2 = M. A. Podoinitsyn
 | year = 2018
 | title = Two-spinor description of massive particles and relativistic spin projection operators
 | url = https://www.sciencedirect.com/science/article/pii/S0550321318300580
 | journal = Nuclear Physics B
 | volume = 929
 | issue = 
 | pages = 452–484
 | doi = 10.1016/j.nuclphysb.2018.02.013
 | arxiv = 1712.00833
 | bibcode = 2018NuPhB.929..452I
 | s2cid = 59582838
 | access-date = 
}}</ref>

In 1941, Rarita and Schwinger focussed on spin-3/2 particles and derived the [[Rarita–Schwinger equation]], including a [[Lagrangian (field theory)|Lagrangian]] to generate it, and later generalized the equations analogous to spin {{math|''n'' + 1/2}} 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 wave functions must obey.<ref>{{cite journal |author1=R. K. Loide |author2=I. Ots |author3=R. Saar | year = 1997 | title = Bhabha relativistic wave equations  | doi=10.1088/0305-4470/30/11/027|bibcode = 1997JPhA...30.4005L | volume=30 | journal=Journal of Physics A: Mathematical and General |issue=11 | pages=4005–4017}}</ref>

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 name="Esposito"/><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=211–223 |issue=5 |bibcode = 1948PNAS...34..211B |doi = 10.1073/pnas.34.5.211 |pmid=16578292 |pmc=1079095 |doi-access=free}}</ref> In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by [[H.&nbsp;Joos]] and [[Steven Weinberg]], the [[Joos–Weinberg equation]]. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.<ref name="T Jaroszewicz, P.S Kurzepa"/><ref name="E.A. Jeffery 1978">
{{cite journal
 | author =E. A. Jeffery
 | year =1978
 | title =Component Minimization of the Bargman–Wigner wavefunction
 | journal =Australian Journal of Physics
 | volume=31 | issue =2
 | pages=137–149
 | bibcode = 1978AuJPh..31..137J
 | doi=10.1071/ph780137| doi-access =free
}}</ref><ref>
{{cite journal
 | author = R. F. Guertin
 | year = 1974
 | title = Relativistic hamiltonian equations for any spin
 | journal = Annals of Physics
 | doi=10.1016/0003-4916(74)90180-8
 | bibcode = 1974AnPhy..88..504G
 | volume=88
 | issue = 2
 | pages=504–553
}}</ref>

=== 1960s–present ===

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"/>