Dirac equationDiracEquation2008-03-16 15:18:172008-03-16 15:18:17Definition% this is the default PlanetPhysics preamble. as your knowledge
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% define commands hereThe Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin $1/2$ particles (fermions). It is given by:
\[
(\gamma^\mu \partial_\mu - im)\psi = 0
\]
The Einstein summation convention is used.
\subsection{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:
\[
E = \sqrt{p^2c^2 + m^2c^4}
\]
As Schr\"odinger had done before him, Dirac then replaced $p$ with its quantum mechanical operator, $\hat{p} \Rightarrow i\hbar \nabla$. Since he was looking for a Lorentz-invariant equation, he replaced $\nabla$ with the D'Alembertian or wave operator
\[
\Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
\]
Note that some authors use $\Box^2$ 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:
\[
\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + c \frac{\partial}{\partial z} + D\frac{i}{c} \frac{\partial}{\partial t})^2
\]
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 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:
\[
A =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{pmatrix},
B = \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 \end{pmatrix}
\]
\[
C = \begin{pmatrix}
0 & 0 & 0 & -i \\
0 & 0 & i & 0 \\
0 & i & 0 & 0 \\
-i & 0 & 0 & 0 \end{pmatrix},
D = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \end{pmatrix}
\]
These matrices are usually given the symbols $\gamma^0$, $\gamma^1$, etc. They are also known as the generators of the special unitary group of order 4, i.e. the group of $n \times n$ matrices with unit determinant.
Using these matrices, and switching to natural units ($\hbar = c = 1$) we can now obtain the Dirac equation:
\[
(\gamma^\mu \partial_\mu - im)\psi = 0
\]
\subsection{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
\[
(\cancel{\partial} - im)\psi = 0
\]