Invariant differential operators

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Invariant differential operators appear often in mathematics and theoretical physics. There is no universal definition for them and the meaning of invariance may depend on the context.

Usually, an invariant differential operator $D$ is a map from some mathematical objects (typically, functions on $\mathbb{R}^n$ , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle) to object of similar type. The word differential indicates that the value $D f$ of the image depends only on $f (x)$ and the derivations of $f$ in $x$ . The word invariant indicates that the operator contains some symmetry. This means that there is a group $G$ that have action on the functions (or other objects in question) and this action commutes with the action of the operator:

$D(g\cdot f)=g\cdot (Df)$

Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.

[edit] Examples

(1) The usual gradient operator $\nabla$ acting on real valued functions on Euclidean space is invariant with respect to all Euclidean transformations.

(2) The differential acting on functions on a manifold with values in 1-forms (its expression is

$d=\sum_j \partial_j dx_j$

in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on differential forms is just the pullback).

(3) The Dirac operator in physics is invariant with respect to the Poincare group (if we choose the proper action of the Poincare group on spinor valued functions. This is, however, a subtle question and if we want to make this mathematically rigorous, we should say that it is invariant with respect to a group which is a double-cover (see covering map) of the Poincaré group)