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Given a single-particle Hilbert space, it's not hard to construct a Fock space using tensor products and symmetrization/anti-symmetrization projection operators, from which we can define creation/annihilation operators. However, is the converse true? More specifically, suppose that we have an algebra of operators acting on some Hilbert space which satisfy the CAR/CCR, e.g., $\{c_k,c_l\}=\delta_{kl}$, is the Hilbert space (on which such an algebra acts) unique up to a isomorphism?

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The answer is yes for finitely many operators - this is the content of the Stone-von Neumann theorem: Up to unitary equivalence, there is only a single unitary representation of the CCR $[x_i,p_j] = \delta_{ij}$ (or rather their more well-behaved cousins, the Weyl relations), and it is given by $L^2(\mathbb{R}^n)$ with $n$ the number of position (or momentum) operators and position as multiplication and momentum as differentiation (or vice versa via Fourier transform).

The answer is no for infinitely many operators - this is the content of Haag's theorem: There are many different unitary representations of an infinite collection of CCRs, which is a notoriously annoying issue for rigorous attempts at quantum field theory.

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