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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?

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  • $\begingroup$ This post (v2) seems very broad. Non-linear representations/realizations are used allover physics. $\endgroup$
    – Qmechanic
    Nov 11, 2020 at 7:17

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So I would point out that, of course, you can always define whatever you want. But there is a good reason why linear representations are so important, so that's what I'll talk about. Suppose we are dealing with a Lagrangian theory and we have some symmetry thereof. Then we know by Noether's theorem that there will be a conserved current, and hence a conserved charge.

What's more, we also know that this conserved charge will generate the action of said symmetry via the Poisson bracket and if we started with some collection of symmetries obeying some Lie algebra, the Poisson algebra of the associated charges will be isomorphic (up to possible central extension).

By the nature of the Poisson bracket, these representations are all linear (on the space of functions over the phase space). And I would argue that Noether's theorem is really the best tool we have, so the fact that it leads to linear representations (even if the original symmetry transformation was non-linear!) immediately promotes linear representations to being the most common thing we can talk about and work with.

But this is not to say non-linear representations are not important. For example when studying spontaneous symmetry breaking in a QFT, such non-linear representations do pop up.

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  • $\begingroup$ This is exactly the sort of answer I was looking for! Somehow Poisson brackets did not occur to me. They are the key! $\endgroup$
    – timur
    Nov 11, 2020 at 16:04

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