algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
Wigner’s theorem asserts that a function $f : H \to H$ from a Hilbert space to itself (not assumed to be a linear functor)
is linear and in fact a (anti-)unitary operator (up to a phase) if only the function is
norm-preserving.
In quantum mechanics every symmetric operation needs to be a norm-preserving bijection from a Hilbert space of states to itself. Hence Wigner’s theorem asserts that in quantum mechanics symmetries are presented by unitary operators (or more rarely anti-unitary operator?s, as for example time reversal?).
Other theorems on the foundations and interpretation of quantum mechanics include: