# Superposition principle and Kirchhoff's integral theorem

## TL;DR

This paper examines the relationships between the superposition principle, Kirchhoff's integral theorem, and Green's functions in linear and nonlinear field theories, highlighting the need for modifications in certain contexts.

## Contribution

It analyzes various formulations of Kirchhoff's integral theorem and explores alternative superposition schemes based on completeness conditions.

## Key findings

- Kirchhoff's theorem variants are connected to superposition principles.
- Modifications are needed for superposition in nonlinear and massive particle fields.
- An alternative superposition scheme based on completeness is discussed.

## Abstract

The need for modification of the Huygens-Fresnel superposition principle arises even in the description of the free fields of massive particles and, more extensively, in nonlinear field theories. A wide range of formulations and superposition schemes for secondary waves are captured by Kirchhoff's integral theorem. We discuss various versions of this theorem as well as its connection with the superposition principle and the method of Green's functions. A superposition scheme inherent in linear field theories, which is not based on Kirchhoff's integral theorem but instead relies on the completeness condition, is also discussed.