I thought QFT is using hilbert space still?

Hawking radiation and the Unruh effect both rely upon the fact that there is no absolute Hilbert space in which QFT unfolds, the Hilbert space used is observer dependent.

For example, in the Unruh effect an inertial observer sees the vacuum in their Hilbert space as having no particles. However, a uniformly accelerating observer sees that same vacuum as containing particles.

This state of affairs is addressed via algebraic quantum field theory (AQFT), which doesn't take a Hilbert space as the primary thing one starts with, but views Hilbert spaces as derived concepts.

google "Fock space"