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"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space anymore." - John von Neumann

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Surprising that he could have that intuition in 1935, before the Unruh effect was known of, before AQFT was invented, and really before most of QFT was formulated.

But then again, he was John von Neumann.

I thought QFT is using hilbert space still?

google "Fock space"

Aren't Fock spaces still hilbert spaces?

Yes, they are complete inner product spaces.

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.

reading about AQFT, found this: "no examples are known of AQFT-compatible interacting field theories, and in particular the standard model cannot at present be made AQFT compatible". Should I continue to read?

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