Fock space vs. wavefunctionals

Question

There are at least two representations of the Hilbert spaces of quantum field theory. For a scalar field, we have

The Fock space representation, such that every state is represented as the Fock space of 'one particle' representations,

$$\mathcal H = \bigoplus_{n = 0}^\infty \text{Sym}(\mathcal H_1^{\otimes n})$$

with $\mathcal H_1^0 = \mathbb C$ and $\mathcal H_1 = \mathcal L^2(\Bbb R^3)$.

The wavefunctional representation, such that every state is represented by a functional on the space of configurations of the field on $\Bbb R^3$ which is square integrable with respect to some measure

$$\mathcal H = \mathcal L^2(\mathcal F[C(\Bbb R^3)], d\mu[\phi])$$

Since those two representations describe the same theory, I am guessing that there is some isomorphism between the two, but what would it be?

Answer 1

The two representations are not equivalent, except for free fields, and hence in perturbation theory.

Nonperturbatively, the Fock space representation is inappropriate (because of Haag's theorem, which asserts that the Hilbert space of an interacting relativistic quantum field theory has no natural Fock space structure; see the discussions at Haag's theorem and practical QFT computations and State space of interacting theories) whereas the wave functional representation (i.e., the functional Schroedinger picture) may still work - it can capture a lot of nonperturbative information such as instantons.

In the functional Schroedinger picture, states of a QFT are treated as functionals of the field coordinates in the same way as states in QM are treated as functions of the position coordinates. A thorough discussion of the functional Schroedinger picture is in the article by Jackiw, Analysis on infinite dimensional manifolds: Schrodinger representation for quantized fields (p.78-143 of the linked document). A problem is that the infinite-dimensional measure is not really well-defined, it is given formally (i.e., nonrigorously) by a functional integral.