| Version 14 |
Version 13 |
| The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin $1/2$ particles (fermions). It is given by: |
The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin $1/2$ particles (fermions). It is given by: |
| \[ |
\[ |
| (\gamma^\mu \partial_\mu - im)\psi = 0 |
(\gamma^\mu \partial_\mu - im)\psi = 0 |
| \] |
\] |
| The Einstein summation convention is used. |
The Einstein summation convention is used. |
| \subsection{Derivation} |
\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: |
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: |
| \[ |
\[ |
| E = \sqrt{p^2c^2 + m^2c^4} |
E = \sqrt{p^2c^2 + m^2c^4} |
| \] |
\] |
| 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 |
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 |
| \[ |
\[ |
| \Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} |
\Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} |
| \] |
\] |
| 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: |
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: |
| \[ |
\[ |
| \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (\gamma^0 \frac{\partial}{\partial x} + \gamma^1 \frac{\partial}{\partial y} + \gamma^2 \frac{\partial}{\partial z} + \gamma^3\frac{i}{c} \frac{\partial}{\partial t})^2 |
\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (\gamma^0 \frac{\partial}{\partial x} + \gamma^1 \frac{\partial}{\partial y} + \gamma^2 \frac{\partial}{\partial z} + \gamma^3\frac{i}{c} \frac{\partial}{\partial t})^2 |
| \] |
\] |
| Multiplying this out, we find that: |
Multiplying this out, we find that: |
| \[ |
\[ |
| (\gamma^0)^2 = (\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = 1 |
(\gamma^0)^2 = (\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = 1 |
| \] |
\] |
| And |
And |
| \[ |
\[ |
| \gamma^0\gamma^1 + \gamma^1\gamma^0 = \gamma^1\gamma^2 + \gamma^2\gamma^1 = \gamma^2\gamma^3 + \gamma^3\gamma^2 = 0 |
\gamma^0\gamma^1 + \gamma^1\gamma^0 = \gamma^1\gamma^2 + \gamma^2\gamma^1 = \gamma^2\gamma^3 + \gamma^3\gamma^2 = 0 |
| \] |
\] |
| 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: |
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: |
| \[ |
\[ |
| \gamma^0 = |
\gamma^0 = |
| \begin{pmatrix} |
\begin{pmatrix} |
| 1 & 0 & 0 & 0 \\ |
1 & 0 & 0 & 0 \\ |
| 0 & 1 & 0 & 0 \\ |
0 & 1 & 0 & 0 \\ |
| 0 & 0 & -1 & 0 \\ |
0 & 0 & -1 & 0 \\ |
| 0 & 0 & 0 & -1 \end{pmatrix}, |
0 & 0 & 0 & -1 \end{pmatrix}, |
| \gamma^1 = \begin{pmatrix} |
\gamma^1 = \begin{pmatrix} |
| 0 & 0 & 0 & 1 \\ |
0 & 0 & 0 & 1 \\ |
| 0 & 0 & 1 & 0 \\ |
0 & 0 & 1 & 0 \\ |
| 0 & -1 & 0 & 0 \\ |
0 & -1 & 0 & 0 \\ |
| -1 & 0 & 0 & 0 \end{pmatrix} |
-1 & 0 & 0 & 0 \end{pmatrix} |
| \] |
\] |
| \[ |
\[ |
| \gamma^2 = \begin{pmatrix} |
\gamma^2 = \begin{pmatrix} |
| 0 & 0 & 0 & -i \\ |
0 & 0 & 0 & -i \\ |
| 0 & 0 & i & 0 \\ |
0 & 0 & i & 0 \\ |
| 0 & i & 0 & 0 \\ |
0 & i & 0 & 0 \\ |
| -i & 0 & 0 & 0 \end{pmatrix}, |
-i & 0 & 0 & 0 \end{pmatrix}, |
| \gamma^3 = \begin{pmatrix} |
\gamma^3 = \begin{pmatrix} |
| 0 & 0 & 1 & 0 \\ |
0 & 0 & 1 & 0 \\ |
| 0 & 0 & 0 & -1 \\ |
0 & 0 & 0 & -1 \\ |
| -1 & 0 & 0 & 0 \\ |
-1 & 0 & 0 & 0 \\ |
| 0 & 1 & 0 & 0 \end{pmatrix} |
0 & 1 & 0 & 0 \end{pmatrix} |
| \] |
\] |
|
These matrices are also known as the generators of the special unitary group of order 4, i.e. the group of $4 \times 4$ matrices with unit determinant.
|
These matrices 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: |
Using these matrices, and switching to natural units ($\hbar = c = 1$) we can now obtain the Dirac equation: |
| \[ |
\[ |
| (\gamma^\mu \partial_\mu - im)\psi = 0 |
(\gamma^\mu \partial_\mu - im)\psi = 0 |
| \] |
\] |
| \subsection{Feynman slash notation} |
\subsection{Feynman slash notation} |
| Richard Feynman developed the following convenient notation for terms involving Dirac matrices: |
Richard Feynman developed the following convenient notation for terms involving Dirac matrices: |
| \[ |
\[ |
| \gamma^\mu q_\mu = \cancel{q} |
\gamma^\mu q_\mu = \cancel{q} |
| \] |
\] |
| Using this notation, the Dirac equation is simply |
Using this notation, the Dirac equation is simply |
| \[ |
\[ |
| (\cancel{\partial} - im)\psi = 0 |
(\cancel{\partial} - im)\psi = 0 |
| \] |
\] |
| \subsection{Relationship with Pauli matrices} |
\subsection{Relationship with Pauli matrices} |
| The Dirac matrices can be written more concisely as matrices of Pauli matrices, as follows: |
The Dirac matrices can be written more concisely as matrices of Pauli matrices, as follows: |
| \begin{align*} |
\begin{align*} |
| \gamma_0 &= \begin{pmatrix} \sigma_0 & 0\\ |
\gamma_0 &= \begin{pmatrix} \sigma_0 & 0\\ |
| 0 & -\sigma_0 |
0 & -\sigma_0 |
| \end{pmatrix}\\ |
\end{pmatrix}\\ |
| \gamma_1 &= \begin{pmatrix} 0 & \sigma_1\\ |
\gamma_1 &= \begin{pmatrix} 0 & \sigma_1\\ |
| -\sigma_1 & 0 |
-\sigma_1 & 0 |
| \end{pmatrix}\\ |
\end{pmatrix}\\ |
| \gamma_2 &= \begin{pmatrix} 0 & \sigma_2\\ |
\gamma_2 &= \begin{pmatrix} 0 & \sigma_2\\ |
| -\sigma_2 & 0 |
-\sigma_2 & 0 |
| \end{pmatrix}\\ |
\end{pmatrix}\\ |
| \gamma_3 &= \begin{pmatrix} 0 & \sigma_3\\ |
\gamma_3 &= \begin{pmatrix} 0 & \sigma_3\\ |
| -\sigma_3 & 0 |
-\sigma_3 & 0 |
| \end{pmatrix} |
\end{pmatrix} |
| \end{align*} |
\end{align*} |
