Skalarfeld

Skalarfeld, or scalar field, is a field that assigns a single real-valued number to every point in space and time. Unlike vector or tensor fields, it is invariant under coordinate transformations. In classical contexts it is usually real-valued; in quantum theory it may be real or complex. Common examples include a temperature distribution in a solid, an electrostatic potential, and in high-energy physics the Higgs field.

Mathematically, a scalar field is a function φ: M → R (or C), defined on a spacetime or space domain, and often assumed smooth. In the Lagrangian formalism its dynamics are described by a Lagrangian density such as L = 1/2 ∂μφ ∂^μφ − V(φ). The Euler–Lagrange equation gives the equation of motion □φ + dV/dφ = 0; for V(φ) = 1/2 m^2 φ^2 this is (□ + m^2) φ = 0. In curved spacetime one uses the covariant d'Alembertian.

In quantum field theory the field is promoted to an operator φ̂(x); its excitations are particles. Interactions come from nonlinear terms in V(φ) and couplings to other fields. Scalar fields are central in the Standard Model (the Higgs field) and in cosmology (inflaton, quintessence).

In condensed matter physics a scalar order parameter describes the degree of order in a system near phase transitions. Mathematically, scalar fields also appear as potential functions in classical mechanics and differential geometry.