Nullfeld

Nullfeld is a term encountered in physics and mathematics to denote a field that vanishes identically on a given domain. The word is formed from the German words null (“zero”) and Feld (“field”), and in English-language texts it is often translated as “null field.” The term appears in specialized German-language literature on differential equations, electromagnetism, and gauge theories, where it serves to describe the trivial or vacuum solution of a field equation under certain boundary conditions.

Definition and formalism

Let F be a field defined on a domain Ω with values in a vector space V. F is called a Nullfeld on Ω if F(x) = 0 for every x in Ω. In differential geometry, the concept aligns with the zero section of a vector bundle or with the trivial solution to a system of partial differential equations. In this sense, a Nullfeld represents the simplest possible solution within a given model or boundary-value problem.

Context and interpretation

In electromagnetism, for example, a region devoid of sources may admit the Nullfeld as a possible solution where the electric and magnetic fields are zero throughout the region. More generally, the Nullfeld serves as a reference state against which nonzero field configurations are compared. It is also used in the study of linear PDEs to describe the zero element of the solution space.

Relation to boundary data

Whether a solution is a Nullfeld can depend on boundary or initial data. Imposing nontrivial boundary conditions can enforce nonzero fields inside the domain, yielding non-null solutions, while homogeneous or trivial boundary data may admit the Nullfeld as the unique solution.