Scalar field

A scalar field assigns one scalar to each point of spacetime. Under a change of coordinates, its value at a physical point is unchanged.

A scalar potential is the derivative-free energy density $V(\varphi )$ of a scalar field. Its derivative supplies the force term in the field equation, and its local minima describe candidate vacuum field values.

A scalar-field shift symmetry acts by $\varphi (x)\mapsto \varphi (x)+c$ for constant $c$. Its Noether charge is the spatial integral of the canonical momentum, up to the normalization of $c$, and interactions invariant under the symmetry contain derivatives of the scalar field.