as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for and , the anticommutation relations for the coefficients can be computed. In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that creates a fermion of momentum p and spin s, and creates an antifermion of momentum q and spin r. The general field is now seen to be a weighted (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, , is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions. With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity . This makes the reason for the choice of clear. This is because the general Lorentz transform on is not unitary so the quantity would not be invariant under such transforms, so the inclusion of is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is . Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the system recover the Dirac equation. Such an expression has its indices suppressed. When reintroduced the full expression is The Hamiltonian (energy) density can also be constructed by first defining the momentum canonically conjugate to , called With that definition of , the Hamiltonian density is: where is the standard gradient of the space-like coordinates, and is a vector of the space-like matrices. It is surprising that the Hamiltonian density doesn't depend on the time derivative of , directly, but the expression is correct. Given the expression for we can construct the Feynman propagator for the fermion field: we define the time-ordered product for fermions with a minus sign due to their anticommuting nature Plugging our plane wave expansion for the fermion field into the above equation yields: where we have employed the Feynman slash notation. This result makes sense since the factor is just the inverse of the operator acting on in the Dirac equation. Note that the Feynman propagator for the Klein–Gordon field