Skalarfelder

Skalarfelder are mathematical objects that assign a single scalar value to every point in a space or spacetime region. Formally, a scalar field on a domain M is a function φ: M → R (or C). In Euclidean space, φ: R^n → R. In physics, scalar fields often depend on position and time, written as φ(x, t), representing quantities such as temperature or energy density.

A scalar field is distinct from vector and tensor fields, which assign vectors or tensors to each point. Properties of scalar fields include regularity classes such as continuity, differentiability, or smoothness. Differentiation yields other fields: the gradient ∇φ is a vector field that points in the direction of greatest increase, and the Laplacian ∇^2φ is a scalar that describes diffusion-like behavior.

Prominent examples include a temperature distribution in a solid, the gravitational potential φ = -GM/r in classical mechanics, and the electric potential in electrostatics. In differential geometry, a scalar field is simply a smooth function on a manifold; in physics, it is common to consider time-dependent fields and to study their evolution via field equations.

In physics, scalar fields are interpreted as spin-0 quantum fields in quantum field theory; they can be real or complex. They are used to model phenomena ranging from classical temperature distributions to fundamental particles in field theories. Common equations involve the gradient and Laplacian, such as Poisson’s equation ∇^2φ = −ρ/ε0 in electrostatics, or the homogeneous form ∇^2φ = 0 in charge-free regions.