# Are non-linear representations important?

In quantum theory, physical states are elements of a Hilbert space, and the transformations must be unitary, implying that the states must transform under a representation of the symmetry group.

However, in classical physics, the states do not necessarily form a linear space, and even if they form a linear space, there is nothing that says that any transformation between states must be a linear operator. So it seems that linear representation theory is not as important in classical physics.

My questions is, do there exist quantities that transform under "nonlinear representations" of a group? Or can one argue that classical quantities must always transform under linear representation?

• This post (v2) seems very broad. Non-linear representations/realizations are used allover physics. Nov 11 '20 at 7:17