The

This

again indicating that this is the

To compute the **Hamiltonian density**, we start by finding the momenta conjugate to the fields
.

There is no time derivative of so those momenta are zero. The Hamiltonian can then be computed.

We may expand the field
in the **complete set of plane waves**
either using the four spinors
for
or using the electron and positron spinors
and
for
.
For economy of notation, we choose the former with a plan to change to the later once the quantization is completed.

The conjugate can also be written out.

Writing the

where previous results from the Hamiltonian form of the Dirac equation and the normalization of the Dirac spinors have been used to simplify the formula greatly.

**Compare this Hamiltonian** to the one used to quantize the Electromagnetic field

for which the Fourier coefficients were replaced by operators as follows.

The Hamiltonian written in terms of the creation and annihilation operators is.

By analogy, we can skip the steps of making coordinates and momenta for the individual oscillators,
and just **replace the Fourier coefficients for the Dirac plane waves by operators**.

(Since the Fermi-Dirac operators will anti-commute, the analogy is imperfect.)

The **creation an annihilation operators**
and
satisfy **anticommutation relations**.

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

A **state of the electrons in a system** can be described by the occupation numbers (0 or 1 for each plane wave).
The state can be generated by operation on the vacuum state with the appropriate set of creation operators.

Jim Branson 2013-04-22