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Changed dimension to $d$ to avoid confusion with particle number $n$ subsequently introduced
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gmvh
gmvh
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Let $\mathcal{S}'(\mathbb{R}^n)$$\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^n$$\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^n)$$\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the Hilbert space $H = L^2(\mathcal{S}'(\mathbb{R}^n), d\phi_C)$$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.

Let $\mathcal{S}'(\mathbb{R}^n)$ be the set of tempered distributions over $\mathbb{R}^n$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^n)$ with covariance operator $C$. Consider the Hilbert space $H = L^2(\mathcal{S}'(\mathbb{R}^n), d\phi_C)$. 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.

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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CBBAM
CBBAM
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Identification of Fock space and the $L^2$ space of tempered distributions

Let $\mathcal{S}'(\mathbb{R}^n)$ be the set of tempered distributions over $\mathbb{R}^n$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^n)$ with covariance operator $C$. Consider the Hilbert space $H = L^2(\mathcal{S}'(\mathbb{R}^n), d\phi_C)$. 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.