Editing Dirac equation (section)

== Comparison with related theories ==

=== Pauli theory ===
{{See also|Pauli equation|Lévy-Leblond equation}}
The necessity of introducing half-integer [[Spin (physics)|spin]] goes back experimentally to the results of the [[Stern–Gerlach experiment]]. A beam of atoms is run through a strong [[Homogeneity and heterogeneity|inhomogeneous]] [[magnetic field]], which then splits into {{math|''N''}} parts depending on the [[Spin (physics)|intrinsic angular momentum]] of the atoms. It was found that for [[silver]] atoms, the beam was split in two; the [[ground state]] therefore could not be [[integer]], because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with {{math|''L<sub>z</sub>'' {{=}} −1, 0, +1}}. The conclusion is that silver atoms have net intrinsic angular momentum of {{sfrac|1|2}}. [[Wolfgang Pauli|Pauli]] set up a theory that explained this splitting by introducing a two-component wave function and a corresponding correction term in the [[Hamilton's Principle|Hamiltonian]], representing a semi-classical coupling of this wave function to an applied magnetic field, as so in [[SI units]]: (Note that bold faced characters imply [[Euclidean vectors]] in 3&nbsp;[[dimensions]], whereas the [[Minkowski space|Minkowski]] [[four-vector]] {{math|''A''<sub>''μ''</sub>}} can be defined as {{tmath|1= A_\mu = \left( \phi/c, -\mathbf A \right) }}.)
<math display="block"> H = \frac{ 1 }{\ 2\ m\ }\ \Bigl( \boldsymbol{\sigma}\cdot\bigl(\mathbf{p} - e\ \mathbf{A} \bigr)\Bigr)^2 + e\ \phi .</math>

Here {{math|'''A'''}} and <math>\phi</math> represent the components of the [[electromagnetic four-potential]] in their standard SI units, and the three sigmas are the [[Pauli matrices]]. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual [[Hamiltonian mechanics#Hamiltonian of a charged particle in an electromagnetic field|classical Hamiltonian of a charged particle]] interacting with an applied field in [[SI units]]:
<math display="block"> H = \frac{ 1 }{\ 2\ m\ }\ \bigl(\mathbf{p} - e\ \mathbf{A}\bigr)^2 + e\ \phi - \frac{ e\ \hbar }{\ 2\ m\ }\ \boldsymbol{\sigma} \cdot \mathbf{B} ~.</math>

This Hamiltonian is now a {{nowrap|2 × 2}} matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as [[minimal coupling]], it takes the form:
<math display="block">\Bigl(\gamma^\mu\ \bigl( i\ \hbar\ \partial_\mu - e\ A_\mu \bigr) - m\ c\Bigr)\ \psi = 0 ~.</math>

A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by {{math|''i''}}, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the [[gyromagnetic ratio]] of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:
<math display="block">
  \begin{pmatrix}
     mc^2 - E + e\phi\quad &
    +c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e\mathbf{A}\right) \\
    -c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e\mathbf{A}\right) &
     mc^2 + E - e\phi
  \end{pmatrix} \begin{pmatrix} \psi_{+} \\ \psi_{-} \end{pmatrix} =
  \begin{pmatrix} 0 \\ 0 \end{pmatrix} ~.
</math>
so
<math display="block">\begin{align}
   (E - e\phi)\ \psi_{+} - c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e\mathbf{A}\right)\ \psi_{-} &= mc^2\ \psi_{+} \\
    c\boldsymbol{\sigma}\cdot \left(\mathbf{p} - e\mathbf{A}\right)\ \psi_{+} - \left(E - e\phi\right)\ \psi_{-}  &= mc^2\ \psi_{-} \end{align} ~.</math>

Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its [[rest energy]], and the momentum going over to the classical value,
<math display="block">\begin{align}
   E - e\phi &\approx mc^2 \\
  \mathbf{p} &\approx m\mathbf{v}
\end{align}</math>
and so the second equation may be written
<math display="block"> \psi_- \approx \frac{ 1 }{\ 2\ mc\ }\ \boldsymbol{\sigma}\cdot \Bigl(\mathbf{p} - e\ \mathbf{A} \Bigr)\ \psi_{+} ,</math>
which is of order <math>\ \tfrac{\ v\ }{ c } ~.</math> Thus, at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement
<math display="block"> \bigl(E - mc^2\bigr)\ \psi_{+} = \frac{ 1 }{\ 2m\ }\ \Bigl[ \boldsymbol{\sigma} \cdot \bigl( \mathbf{p} - e\mathbf{A}\bigr) \Bigr]^2\ \psi_{+} + e\ \phi\ \psi_{+}\ </math>

The operator on the left represents the particle's total energy reduced by its rest energy, which is just its classical [[kinetic energy]], so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious {{math|''i''}} that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although ostensibly in the form of a [[diffusion equation]], actually represents wave propagation.

It should be strongly emphasized that the entire Dirac spinor represents an ''irreducible'' whole. The separation, done here, of the Dirac spinor into large and small components depends on the low-energy approximation being valid. The components that were neglected above, to show that the Pauli theory can be recovered by a low-velocity approximation of Dirac's equation, are necessary to produce new phenomena observed in the relativistic regime – among them [[antimatter]], and the [[matter creation|creation]] and [[annihilation]] of particles.

=== Weyl theory ===
In the massless case <math>m = 0</math>, the Dirac equation reduces to the [[Weyl equation]], which describes relativistic massless spin-1/2 particles.<ref name="Ohlsson2011">{{cite book |first=Tommy |last=Ohlsson |author-link=Tommy Ohlsson |date=22 September 2011 |title=Relativistic Quantum Physics: From advanced quantum mechanics to introductory quantum field theory |page=86 |publisher=Cambridge University Press |isbn=978-1-139-50432-4 |url=https://books.google.com/books?id=hRavtAW5EFcC&pg=PA86}}</ref>

The theory acquires a second <math>\text{U}(1)</math> symmetry: see below.