Translational symmetry

In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.

Analogously, an operator A on functions is said to be translationally invariant with respect to a translation operator ${\displaystyle T_{\delta }}$ if the result after applying A doesn't change if the argument function is translated. More precisely it must hold that

$${\displaystyle \forall \delta \ Af=A(T_{\delta }f).}$$

Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law.

Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.