The phase space of a classical free field theory is a

symplectic vector space V. Choosing a complex structure (polarization)

turns V into a complex Hilbert space H, and the Fock space built on H

provides a quantization of V. When V is finite-dimensional, the Fock

representations built from different polarizations are equivalent (as a

consequence of the Stone-von Neumann theorem), but this is no longer

true in the infinite-dimensional case.

By constructing the state space of a quantum field theory as a

projective limit of simpler building blocs (truncations), it is possible

to lift the Stone-von Neumann theorem to the infinite-dimensional case.

Carefully selecting a "dense" collection of truncations yields a quantum

state space which is at the same time universal (it encompasses at once

all "reasonable" Fock spaces, and supports arbitrarily good

implementations of all bounded linear symplectomorphisms of V) and

constructive (all states can be constructed in a systematic way, in

contrast eg. to generic algebraic states).