Quantum field theory
we obtain the equations of motion for the field, which describe the way it varies in time and space:
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand.
Feynman rules can be used to directly evaluate tree-level diagrams. However, naïve computation of loop diagrams such as the one shown above will result in divergent momentum integrals, which seems to imply that almost all terms in the perturbative expansion are infinite. The renormalisation procedure is a systematic process for removing such infinities.
Shown above is an example of a tree-level Feynman diagram in QED. It describes an electron and a positron annihilating, creating an off-shell photon, and then decaying into a new pair of electron and positron. Time runs from left to right. Arrows pointing forward in time represent the propagation of positrons, while those pointing backward in time represent the propagation of electrons. A wavy line represents the propagation of a photon. Each vertex in QED Feynman diagrams must have an incoming and an outgoing fermion (positron/electron) leg as well as a photon leg.