Is there an alternative to Fock Space and Hilbert Space in quantum field theory?

Question

Why were Fock Space and Hilbert Space used in quantum field theory? What was the motivation for choosing them over other mathematical techniques?

Answer 1

Hilbert spaces are used throughout all Quantum Mechanical theories. In fact all Hilbert spaces are isomorphic, so we might as well just call this structure "the abstract Hilbert space". That observables are represented by operators acting on the Hilbert space is one of the axioms of Quantum Mechanics.

The Fock space is a Hilbert space, so it is mathematically isomorphic to the abstract Hilbert space. What makes it special is that on the Fock space, the Canonical (Anti-)Commutation Relations of the field theory are represented by field operator-valued distributions, defined by picking a very special basis that consists of $n$-particle subspaces; then defining the creation and annihilation operators and making quantum field operator-valued distributions from them.

For interacting QFT, the Fock space is not the correct representation of the CCR/CAR, due to Haag's theorem. So contrary to the widespread belief, interacting QFT does not use the Fock space at all – instead it uses a different (read, unitarily inequivalent) representation of the CCR/CAR on the abstract Hilbert space.

However, it is still possible to use the Fock space to label approximately the states of the interacting QFT for which the particles are far enough apart for the interaction to be neglected. We can then make these particles come close to each other, interact, and fly far apart again. This process can be approximately described by the scattering operator that acts on the Fock space. Usually the scattering operator is evaluated perturbatively by summing up expressions that correspond to Feynman diagrams.