# Uniqueness of Fock space

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?

• Jul 28 at 22:11

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).