(We could also add a tensor term but it is not needed to get the Dirac equation.) The

This

The Hamiltonian density may be derived from the Lagrangian in the standard way and the total Hamiltonian
computed by integrating over space.
Note that the Hamiltonian density is the same as the Hamiltonian derived from the Dirac equation directly.

We may expand in plane waves to understand the Hamiltonian as a sum of oscillators.

Writing the

By analogy with electromagnetism, we can replace the Fourier coefficients for the Dirac plane waves by operators.

The

is the occupation number operator. The anti-commutation relations constrain the

The Dirac field and Hamiltonian can now be **rewritten** in terms of electron and positron fields for
which the energy is always positive by replacing the operator to annihilate a ``negative energy state'' with
an operator to create a positron state with the right momentum and spin.

These

Now rewrite the fields and Hamiltonian.

All the

There is an (infinite) constant energy, similar but of opposite sign to the one for the quantized EM field,
which we must add to make the vacuum state have zero energy.
Note that, had we used commuting operators (Bose-Einstein) instead of anti-commuting, there would have been no
lowest energy ground state so this Energy subtraction would not have been possible.
**Fermi-Dirac statistics are required for particles satisfying the Dirac equation**.

Since the **operators creating fermion states anti-commute**,
fermion states must be antisymmetric under interchange.
Assume
and
are the creation and annihilation operators for fermions and that they anti-commute.

The

Its not hard to show that the

Note that the **spinors satisfy the following slightly different equations**.

Jim Branson 2013-04-22