# PlanetPhysics/Dirac Equation

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The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin ${\displaystyle 1/2}$ particles (fermions). It is given by:

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

The Einstein summation convention is used.

### Derivation

Mathematically, it is interesting as one of the first uses of the spinor calculus in mathematical physics. Dirac began with the relativistic equation of total energy:

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

As Schr\"odinger had done before him, Dirac then replaced ${\displaystyle p}$ with its quantum mechanical operator, ${\displaystyle {\hat {p}}\Rightarrow i\hbar \nabla }$. Since he was looking for a Lorentz-invariant equation, he replaced ${\displaystyle \nabla }$ with the D'Alembertian or wave operator

${\displaystyle \Box =\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}}$

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

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

Multiplying this out, we find that:

${\displaystyle A^{2}=B^{2}=C^{2}=D^{2}=1}$

And

${\displaystyle AB+BA=BC+CB=CD+DC=0}$

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

${\displaystyle A={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},B={\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{pmatrix}}}$

${\displaystyle C={\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}},D={\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}}}$

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

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

### Feynman slash notation

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

${\displaystyle \gamma ^{\mu }q_{\mu }={\cancel {q}}}$

Using this notation, the Dirac equation is simply

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