In quantum field theory, the field itself is promoted to an operator-valued distribution $\hat\phi(x)$ acting on a Fock space. Particle number can change, so a single-particle quantum mechanics description is insufficient for relativistic interacting systems. QFT is the framework that combines quantum mechanics, special relativity, and locality.

The free scalar propagator is the inverse of the Klein-Gordon operator and in momentum space is proportional to $\frac{i}{p^2-m^2+i\epsilon}$. Similar structures exist for fermions and gauge bosons, with gauge fixing required to define the inverse kinetic operator for gauge fields.

Perturbation theory is organized by interaction terms such as $\lambda\phi^4$, Yukawa couplings, or gauge couplings. Feynman rules translate the Lagrangian into diagrammatic contributions, with external legs, propagators, and vertices systematically encoding amplitudes.

Scattering amplitudes are computed as expansions in the coupling constant. Each connected diagram contributes a specific momentum integral and symmetry factor. Renormalizable theories are those whose ultraviolet divergences can be absorbed into a finite number of parameters of the original Lagrangian.