ABSTRACT
The neutral scalar field with https://www.w3.org/1998/Math/MathML" display="inline"> λ ϕ 4 interaction is the simplest example of an interacting field theory, widely used to introduce the more complex cases of QED or QCD interactions. Today, the interest of the interaction is not only for illustrative purposes, however, since this is the form of the self-interaction of the scalar Higgs field of the Standard Theory, see Chapter 21.
In this Chapter, the perturbative expansion of the generating functional in powers of https://www.w3.org/1998/Math/MathML" display="inline"> λ is considered and the rules to construct the amplitudes corresponding to propagation and mutual interaction of the field quanta are derived. The final outcome is a set of simple and elegant prescriptions, Feynman Diagrams and Feynman Rules, to obtain the 2k-point Green's function to the n-th order of perturbation theory. Feynman diagrams are obtained starting from the external, initial or final, particles, which are generated or absorbed in given points with coordinates https://www.w3.org/1998/Math/MathML" display="inline"> x 1 , x 2 , … , x 2 k , and interact in points of coordinates https://www.w3.org/1998/Math/MathML" display="inline"> x , y , z … (vertices). Four lines stem out from each vertex (a consequence of https://www.w3.org/1998/Math/MathML" display="inline"> λ ϕ 4 ) to be connected, in all possible ways, to the other vertices, possibly including the vertex they started from, or to the external particles. Each independent diagram corresponds to a possible “path” from initial to final state, the corresponding amplitude being the product of the amplitudes of the different components of the path: propagation, i∆F (x − y) and interactions, iλ. The analysis of Green's functions in terms of Feynman diagrams corresponds fully to the idea of the sum over paths introduced at the start of this book, Chapter 2.
