short. The free two-point function, also known as the Feynman propagator, can be found for the real scalar field by either canonical quantization or path integrals to be In an interacting theory, where the Lagrangian or Hamiltonian contains terms or that describe interactions, the two-point function is more difficult to define. However, through both the canonical quantization formulation and the path integral formulation, it is possible to express it through an infinite perturbation series of the free two-point function. In canonical quantization, the two-point correlation function can be written as: where is an infinitesimal number and is the field operator under the free theory. Here, the exponential should be understood as its power series expansion. For example, in -theory, the interacting term of the Hamiltonian is , and the expansion of the two-point correlator in terms of becomesThis perturbation expansion expresses the interacting two-point function in terms of quantities that are evaluated in the free theory. In the path integral formulation, the two-point correlation function can be written where is the Lagrangian density. As in the previous paragraph, the exponential can be expanded as a series in , reducing the interacting two-point function to quantities in the free theory. Wick's theorem further reduce any -point correlation function in the free theory to a sum of products of two-point correlation functions. For example, Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order to calculate all physical quantities in the (perturbative) interacting theory. This makes the Feynman propagator one of the most important quantities in quantum field theory. Feynman diagram Correlation functions in the interacting theory can be written as a perturbation series. Each term in the series is a product of Feynman propagators in the free theory and can be represented visually by a Feynman diagram. For example, the term in the two-point correlation function in the theory is After applying Wick's theorem, one of the terms is This term can instead be obtained from the Feynman diagram . The diagram consists of external vertices connected with one edge and represented by dots (here labeled and ). internal vertices connected with four edges and represented by dots (here labeled ). edges connecting the vertices and represented by lines. Every vertex corresponds to a single field factor at the corresponding point in spacetime, while the edges correspond to the