Editing Relativistic wave equations (section)

=== 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).