Editing Dirac equation (section)

=== Dirac's coup ===
Dirac thus thought to try an equation that was ''first order'' in both space and time. He postulated an equation of the form
<math display="block">E\psi = (\vec{\alpha} \cdot \vec{p} + \beta m) \psi</math>
where the operators <math>(\vec{\alpha}, \beta)</math> must be independent of <math>(\vec{p}, t)</math> for linearity and independent of <math>(\vec{x}, t)</math> for space-time homogeneity. These constraints implied additional dynamical variables that the <math>(\vec{\alpha}, \beta)</math> operators will depend upon; from this requirement Dirac concluded that the operators would depend upon {{nowrap|4 × 4}} matrices, related to the Pauli matrices.<ref>{{Cite book |last1=Duck |first1=Ian |url=https://www.worldscientific.com/worldscibooks/10.1142/3457 |title=Pauli and the Spin-Statistics Theorem |last2=Sudarshan |first2=E C G |date=1998 |publisher=WORLD SCIENTIFIC |isbn=978-981-02-3114-9 |language=en |doi=10.1142/3457}}</ref>{{rp|205}}

One could, for example, formally (i.e. by [[abuse of notation]], since it is not straightforward to take a [[functional square root]] of the sum of two differential operators) take the [[Energy–momentum relation|relativistic expression for the energy]]
<math display="block">E = c \sqrt{p^2 + m^2c^2} ~,</math>
replace {{math|''p''}} by its operator equivalent, expand the square root in an [[infinite series]] of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.

As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator (see also [[half derivative]]) thus:
<math display="block">\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)~.</math>

On multiplying out the right side it is apparent that, in order to get all the cross-terms such as {{math|∂<sub>''x''</sub>∂<sub>''y''</sub>}} to vanish, one must assume
<math display="block">AB + BA = 0, ~ \ldots ~</math>
with
<math display="block">A^2 = B^2 = \dots = 1~.</math>

Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's [[matrix mechanics]], immediately understood that these conditions could be met if {{math|''A''}}, {{math|''B''}}, {{math|''C''}} and {{math|''D''}} are ''matrices'', with the implication that the wave function has ''multiple components''. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of [[Spin (physics)|spin]], something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least {{nowrap|4 × 4}} matrices to set up a system with the properties required – so the wave function had ''four'' components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.

Given the factorization in terms of these matrices, one can now write down immediately an equation
<math display="block">\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t\right)\psi = \kappa\psi </math>
with <math>\kappa</math> to be determined. Applying again the matrix operator on both sides yields
<math display="block">\left(\nabla^2 - \frac{1}{c^2}\partial_t^2\right)\psi = \kappa^2\psi ~.</math>

Taking <math>\kappa = \tfrac{mc}{\hbar}</math> shows that all the components of the wave function ''individually'' satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is
<math display="block">\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t - \frac{mc}{\hbar}\right)\psi = 0 ~.</math>

Setting
<math display="block">A = i \beta \alpha_1 \, , \, B = i \beta \alpha_2 \, , \, C = i \beta \alpha_3 \, , \, D = \beta ~, </math>
and because <math>D^2 = \beta^2 = I_4 </math>, the Dirac equation is produced as written above.