PlanetPhysics/Dirac Equation

The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic [[../QuarkAntiquarkPair/|spin]] $1/2$ [[../Particle/|particles]] ([[../AntiCommutationRelations/|fermions]]). It is given by:

$({\gamma }^{\mu }{\partial }_{\mu }-im)\psi =0$

The [[../EinsteinSummationNotation/|Einstein summation convention]] is used.

Derivation

Mathematically, it is interesting as one of the first uses of the [[../ECartan/|spinor]] calculus in [[../PhysicalMathematics2/|mathematical physics]]. Dirac began with the relativistic equation of total [[../CosmologicalConstant2/|energy]]:

$E=\sqrt{p^2{c}^2+m^2{c}^4}$

As Schr\"odinger had done before him, Dirac then replaced

$p$

with its quantum mechanical [[../QuantumOperatorAlgebra4/|operator]],

$\hat{p}\Rightarrow i\hslash \mathrm{\nabla }$

. Since he was looking for a Lorentz-invariant equation, he replaced

$\mathrm{\nabla }$

with the D'Alembertian or [[../DAlembertOperator/|wave operator]]

$◻={\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}$

Note that some authors use ${◻}^2$ for the D'alembertian. Dirac was now faced with the problem of how to take the [[../PiecewiseLinear/|square]] root of an expression containing a differential operator. He proceeded to factorise the d'Alembertian as follows:

${\mathrm{\nabla }}^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}=(A\frac{\partial }{\partial x}+B\frac{\partial }{\partial y}+c\frac{\partial }{\partial z}+D\frac{i}{\frac{\partial }{\partial t}}{)}^2$

Multiplying this out, we find that:

$A^2=B^2=C^2=D^2=1$

And

$AB+BA=BC+CB=CD+DC=0$

Clearly these [[../Bijective/|relations]] cannot be satisfied by [[../Vectors/|scalars]], so Dirac sought a set of four [[../Matrix/|matrices]] which satisfy these relations. These are now known as the Dirac matrices, and are given as follows:

$A=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),B=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ -1& 0& 0& 0\end{array}\right)$

$C=\left(\begin{array}{cccc}0& 0& 0& -i\\ 0& 0& i& 0\\ 0& i& 0& 0\\ -i& 0& 0& 0\end{array}\right),D=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& -1\\ -1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right)$

These matrices are usually given the symbols ${\gamma }^0$, ${\gamma }^1$, etc. They are also known as the [[../Generator/|generators]] of the special unitary [[../TrivialGroupoid/|group]] of order 4, i.e. the group of $n\times n$ matrices with unit [[../Determinant/|determinant]]. Using these matrices, and switching to natural units ($\hslash =c=1$) we can now obtain the Dirac equation:

$({\gamma }^{\mu }{\partial }_{\mu }-im)\psi =0$

Feynman slash notation

Richard Feynman developed the following convenient notation for terms involving Dirac matrices:

${\gamma }^{\mu }{q}_{\mu }=\cancel{q}$

Using this notation, the Dirac equation is simply

$(\cancel{\partial }-im)\psi =0$