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:

As Schrödinger 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

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:

Multiplying this out, we find that:

And

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:

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. Using these matrices, and switching to natural units ($\hbar=c=1$) we can now obtain the Dirac equation:

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

Using this notation, the Dirac equation is simply

The Dirac matrices can be written more concisely as matrices of Pauli matrices, as follows: