Relation to tree diagramms

Perturbative solutions of classical field equations

Amputation of the last leg

Perturbative solutions of classical field equations

Let us take to be the space of perturbative solutions of nonlinear equations of the form:

where is some linear differential operator, and is a nonlinear function
describing the interaction. We assume that is a polynomial starting with
quadratic or higher order terms.

The point will be the zero solution .
Then can be identified with the space of solutions of
the linearized equation:

Tree level perturbation theory can be thought of as a 1-parameter map

where satisfies:

The definitions of the operator has an ambiguity
(because one can add a solution of the free equation).
Suppose that we made some choice of .

Amputation of the last leg

Let us define as follows: