| Version 8 |
Version 7 |
| 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: |
| \[ |
\[ |
| \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + c \frac{\partial}{\partial z} + D\frac{i}{c} \frac{\partial}{\partial t})^2 |
\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + c \frac{\partial}{\partial z} + D\frac{i}{c} \frac{\partial}{\partial t})^2 |
| \] |
\] |
| Multiplying this out, we find that: |
Multiplying this out, we find that: |
| \[ |
\[ |
| A^2 = B^2 = C^2 = D^2 = 1 |
A^2 = B^2 = C^2 = D^2 = 1 |
| \] |
\] |
| And |
And |
| \[ |
\[ |
| AB + BA = BC + CB = CD + DC = 0 |
AB + BA = BC + CB = CD + DC = 0 |
| \] |
\] |
| 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: |
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: |
| \[ |
\[ |
| A = |
A = |
| \begin{pmatrix} |
\begin{pmatrix} |
| 1 & 0 & 0 & 0 \\ |
1 & 0 & 0 & 0 \\ |
| 0 & 1 & 0 & 0 \\ |
0 & 1 & 0 & 0 \\ |
| 0 & 0 & -1 & 0 \\ |
0 & 0 & -1 & 0 \\ |
| 0 & 0 & 0 & -1 \end{pmatrix}, |
0 & 0 & 0 & -1 \end{pmatrix}, |
| B = \begin{pmatrix} |
B = \begin{pmatrix} |
| 0 & 0 & 0 & 1 \\ |
0 & 0 & 0 & 1 \\ |
| 0 & 0 & 1 & 0 \\ |
0 & 0 & 1 & 0 \\ |
| 0 & -1 & 0 & 0 \\ |
0 & -1 & 0 & 0 \\ |
| -1 & 0 & 0 & 0 \end{pmatrix} |
-1 & 0 & 0 & 0 \end{pmatrix} |
| \] |
\] |
| \[ |
\[ |
| C = \begin{pmatrix} |
C = \begin{pmatrix} |
| 0 & 0 & 0 & -i \\ |
0 & 0 & 0 & -i \\ |
| 0 & 0 & i & 0 \\ |
0 & 0 & i & 0 \\ |
| 0 & i & 0 & 0 \\ |
0 & i & 0 & 0 \\ |
| -i & 0 & 0 & 0 \end{pmatrix}, |
-i & 0 & 0 & 0 \end{pmatrix}, |
| D = \begin{pmatrix} |
D = \begin{pmatrix} |
| 0 & 0 & 1 & 0 \\ |
0 & 0 & 1 & 0 \\ |
| 0 & 0 & 0 & -1 \\ |
0 & 0 & 0 & -1 \\ |
| -1 & 0 & 0 & 0 \\ |
-1 & 0 & 0 & 0 \\ |
| 0 & 1 & 0 & 0 \end{pmatrix} |
0 & 1 & 0 & 0 \end{pmatrix} |
| \] |
\] |
| These matrices are 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. |
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: |
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 |
| \[ |
\[ |
| (\cancel{\partial} - im)\psi = 0 |
(\cancel{\partial} - im)\psi = 0 |
| \] |
\] |
