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Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the Hilbert space $H = L^2(\mathcal{S}'(\mathbb{R}^d), d\phi_C)$. In Glimm & Jaffe's textbook on quantum field theory they give the identification $H = \mathcal{F}$ where $\mathcal{F}$ denotes a Fock space representation. I am having some trouble understanding this identification. Usually Fock space is given as a (symmetric) direct sum $$\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{F}_n$$ but what are the spaces $\mathcal{F}_n$ when working with this particular choice of $H$? How can one understand the Fock space representation of $H$ here?

I am interested in this because Glimm & Jaffe define $\mathcal{F}_n$ as the $n$ particle subspace of $\mathcal{F}$, but more importantly they define Wick monomials as an orthogonal projection onto the $\mathcal{F}_n$. That is, they define $$:\phi(f_1)\cdots\phi(f_n): = E_n \phi(f_1) \cdots \phi(f_n)$$ where $\phi \in \mathcal{F}$ (and is thus a tempered distribution) and $E_n: \mathcal{F} \rightarrow \mathcal{F}_n$ is an orthogonal projection.

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  • $\begingroup$ Fock........... $\endgroup$
    – no upstairs
    Commented Aug 8 at 10:46
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    $\begingroup$ First you define $\mathcal{G}_n$ as the space of polynomials in $\phi$ of degree at most $n$. Namely this is the linear span of functions $\phi\mapsto \phi(f_1)\cdots\phi(f_k)$ with $k\le n$. Then you do a kind of Gram-Schmidt orthogonalization, and define $\mathcal{F}_n$ as the orthogonal complement of $\mathcal{G}_{n-1}$ inside $\mathcal{G}_n$. If GJ is not clear, you can look this up in the probability literature. You will need to switch terminology to "Wiener chaos" instead of "Fock space". Standard refs are books by Svante Janson and David Nualart. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Aug 8 at 14:54
  • $\begingroup$ @AbdelmalekAbdesselam Thank you. Weiner chaos is new to me but seems to be exactly what I am looking for! $\endgroup$
    – CBBAM
    Commented Aug 8 at 15:24
  • $\begingroup$ @AbdelmalekAbdesselam After doing some more reading on Weiner chaos the general construction is clear, but how does it ensure that the spaces $\mathcal{F}_n$ are symmetric? Also, is the Weiner chaos the same thing as Fock space but with a different name in the probability literature or is there a more subtle connection between the two? $\endgroup$
    – CBBAM
    Commented Aug 8 at 17:07

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