Editing Relativistic wave equations (section)

=== 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Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16).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>