Editing Dirac equation (section)

=== Covariant form and relativistic invariance ===
To demonstrate the [[Lorentz covariance|relativistic invariance]] of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:
<math display="block">\begin{align}
  D &=   \gamma^0, \\
  A &= i \gamma^1,\quad B = i \gamma^2,\quad C = i \gamma^3,
\end{align}</math>
and the equation takes the form (remembering the definition of the covariant components of the [[4-gradient]] and especially that {{math|1=∂<sub>0</sub> = {{sfrac|1|''c''}}∂<sub>''t''</sub>}})

{{Equation box 1
|title='''Dirac equation'''
|indent=:
|equation = <math>(i \hbar \gamma^\mu \partial_\mu - m c) \psi = 0</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

where there is an [[Einstein notation|implied summation]] over the values of the twice-repeated index {{math|''μ'' {{=}} 0, 1, 2, 3}}, and {{math|∂<sub>''μ''</sub>}} is the [[4-gradient]]. In practice one often writes the [[gamma matrices]] in terms of 2 × 2 sub-matrices taken from the [[Pauli matrices]] and the 2 × 2 [[identity matrix]]. Explicitly the [[gamma matrices#Dirac basis|standard representation]] is
<math display="block">
\gamma^0 = \begin{pmatrix} I_2 &        0 \\         0 & -I_2 \end{pmatrix},\quad
\gamma^1 = \begin{pmatrix}   0 & \sigma_x \\ -\sigma_x &    0 \end{pmatrix},\quad
\gamma^2 = \begin{pmatrix}   0 & \sigma_y \\ -\sigma_y &    0 \end{pmatrix},\quad
\gamma^3 = \begin{pmatrix}   0 & \sigma_z \\ -\sigma_z &    0 \end{pmatrix}.
</math>

The complete system is summarized using the [[Minkowski metric]] on spacetime in the form
<math display="block">\left\{\gamma^\mu, \gamma^\nu\right\} = 2 \eta^{\mu\nu} I_4</math>
where the bracket expression
<math display="block">\{a, b\} = ab + ba</math>
denotes the [[anticommutator]]. These are the defining relations of a [[Clifford algebra]] over a pseudo-orthogonal 4-dimensional space with [[metric signature]] {{math|(+ − − −)}}. The specific Clifford algebra employed in the Dirac equation is known today as the [[Dirac algebra]]. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this ''[[geometric algebra]]'' represents an enormous stride forward in the development of quantum theory.

The Dirac equation may now be interpreted as an [[eigenvalue]] equation, where the rest mass is proportional to an eigenvalue of the [[4-momentum operator]], the [[proportionality constant]] being the speed of light:
<math display="block"> \operatorname{P}_\mathsf{op} \psi = m c \psi .</math>

Using <math>{\partial\!\!\!/} \mathrel{\stackrel{\mathrm{def}}{=}} \gamma^\mu \partial_\mu</math> (<math>{\partial\!\!\!\big /}</math> is pronounced "d-slash"),<ref>{{cite book |last=Pendleton |first=Brian |url=http://www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf |archive-date=2022-10-09 |url-status=live |title=Quantum Theory |year=2012–2013 |at=section&nbsp;4.3 "The Dirac Equation"}}</ref> according to Feynman slash notation, the Dirac equation becomes:
<math display="block">i \hbar {\partial\!\!\!\big /} \psi - m c \psi = 0 .</math>

In practice, physicists often use units of measure such that {{math|''ħ'' {{=}} ''c'' {{=}} 1}}, known as [[natural units]]. The equation then takes the simple form

{{Equation box 1
|title='''Dirac equation''' '' (natural units)''
|indent=:
|equation = <math>(i{\partial\!\!\!\big /} - m) \psi = 0</math>
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

A foundational theorem{{which|date=September 2024}} states that if two distinct sets of matrices are given that both satisfy the [[Clifford algebra|Clifford relations]], then they are connected to each other by a [[Matrix similarity|similarity transform]]:
<math display="block">\gamma^{\mu\prime} = S^{-1} \gamma^\mu S ~.</math>

If in addition the matrices are all [[unitary transformation|unitary]], as are the Dirac set, then {{math|''S''}} itself is [[unitary matrix|unitary]];
<math display="block">\gamma^{\mu\prime} = U^\dagger \gamma^\mu U ~.</math>

The transformation {{math|''U''}} is unique up to a multiplicative factor of absolute value 1. Let us now imagine a [[Lorentz transformation]] to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator {{math|''γ''<sup>''μ''</sup>∂<sub>''μ''</sub>}} to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the previously mentioned foundational theorem,{{which|date=September 2024}} one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form
<math display="block">\begin{align}
  \left(iU^\dagger \gamma^\mu U\partial_\mu^\prime - m\right)\psi\left(x^\prime, t^\prime\right) &= 0 \\
              U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi\left(x^\prime, t^\prime\right) &= 0 ~.
\end{align}</math>

If the transformed spinor is defined as
<math display="block">\psi^\prime = U\psi</math>
then the transformed Dirac equation is produced in a way that demonstrates [[Manifest covariance|manifest relativistic invariance]]:
<math display="block">\left(i\gamma^\mu\partial_\mu^\prime - m\right)\psi^\prime\left(x^\prime, t^\prime\right) = 0 ~.</math>

Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation.

The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the ''standard'' representation – in it, the wave function's upper two components go over into Pauli's 2&nbsp;spinor wave function in the limit of low energies and small velocities in comparison to light.

The considerations above reveal the origin of the gammas in ''geometry'', hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as {{math|''γ''<sub>''μ''</sub>''γ''<sub>''ν''</sub>}} represent ''[[oriented surface]] elements'', and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is
<math display="block">V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta .</math>

For this to be an invariant, the [[Levi-Civita symbol|epsilon symbol]] must be a [[tensor]], and so must contain a factor of {{math|{{sqrt|''g''}}}}, where {{math|''g''}} is the [[determinant]] of the [[metric tensor]]. Since this is negative, that factor is ''imaginary''. Thus
<math display="block">V = i \gamma^0\gamma^1\gamma^2\gamma^3 .</math>

This matrix is given the special symbol {{math|''γ''<sup>5</sup>}}, owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is
<math display="block">\gamma_5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}.</math>

This matrix will also be found to anticommute with the other four Dirac matrices:
<math display="block">\gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0</math>

It takes a leading role when questions of ''[[parity (physics)|parity]]'' arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.