What is the general form of the duality transform for the Fock space?

Question

I am interested in properties of the symmetric Fock space, looked at via the associated Wiener space. It is well known that for a Hilbert space $k$, the symmetric Fock space $$\mathcal{F}(L^2(\mathbb{R}_+; k))\cong W,$$ where $W$ is the Wiener space, i.e. the space associated to a Wiener process taking its values in $k$. This isomorphism is sometimes called the Wiener-Ito-Segal isomorphism or the duality transform (for reference cf. e.g. D. Nualart, The Malliavin Calculus and Related Topics). However, usually the transform is only described explicitly for exponential vectors of real valued functions via $$U\varepsilon(f) \mapsto \exp(\int_0^t f(s) {\rm d}W_s -\frac12 \|f\|^2).$$

From this one can derive that the sum of the annihilation and creation operator on the Fock space corresponds to multiplication by Brownian motion on the Wiener space.

My questions are:

1. How does this transform work for vectors other than (real) exponentials? For example, for vectors only supported on a particular number of particles?
2. How can we see other multiplication operators on the Wiener space as operators on Fock space?

I come from a more mathematical background - I wouldn't be surprised if the answers to these questions were known in the world of physics and I just could not find them.

Thank you for your help.

Answer 1

The linear space generated by the exponential vectors contains any orthonormal basis of the Fock space as a subspace. This is basically its overcompleteness property. The transform of nonexponential types of vectors can be computed as follows:

Since $\epsilon(f) = \sum_{n \geqslant 0} \frac{f^{\otimes^n}}{\sqrt{n!}}$. The transform of vectors of the type $f^{\otimes^m}$, can obtained by scaling f by a constant $f\rightarrow cf$ and equating the coefficient of $c^m$ from both sides.

Now substituting $f = \sum_{i=1}^m c_i f_i$ in the expression of $Uf^{\otimes^m}$, and taking the coefficient of $\prod_{i=1}^m c_i$ from both sides, we obtain the transform of a general vector of the form: $\otimes_{i=1}^m f_i$

Other commuting operators can be obtained for example from powers of sums of creation and annihilation operators of the form $A(f)+A^{+}(f)$.