What are some good resources to learn about Hilbert spaces corresponding to QFT? I want to deepen my understanding about Hilbert spaces corresponding to QFT. Could you please introduce some good resources?
 A: For say a real scalar Bosonic QFT in $d$ dimensional flat spacetime, I believe that regardless of interacting or not, the Hilbert space should be $\mathcal{H}=L^2(\mathscr{S}'(\mathbb{R}^{d-1}),d\nu)$. Here $\mathscr{S}'(\mathbb{R}^{d-1})$ is the space of real-valued Schwartz temperate distributions. The measure $\nu$ is a Borel probability measure on $\mathscr{S}'(\mathbb{R}^{d-1})$.
In case the Euclidean path integral can be realized as integration with respect to an honest probability measure $\mu$ on $\mathscr{S}'(\mathbb{R}^{d}$ and if one has the so-called Markov property, then one should be able to get $\nu$ as follows. One must show that for $\mu$-almost all $\phi\in\mathscr{S}'(\mathbb{R}^{d})$ one can define the restriction of $\phi(\tau,x_1,\ldots,x_{d-1})$ to the $\tau=0$ hyperplane. Here $\tau$ is the Euclidean or imaginary time, while $x_1,\ldots,x_{d-1}$ are the spatial coordinates. Then this restriction operation gives a Borel-measurable map $R: \mathscr{S}'(\mathbb{R}^{d})\rightarrow\mathscr{S}'(\mathbb{R}^{d-1})$ and $\nu$ is just the push-forward measure $R_{\ast}\mu$, namely the law of the time zero restriction.
This can be done rather explicitly for the free field while recovering the isomorphism of $\mathcal{H}$ with Fock space. See:

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*J. Glimm and A. Jaffe, Quantum Physics, 2nd ed., Springer-Verlag, New York, 1987

*B. Simon, The $P(\phi)_2$ Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics.

*A. Klein, "The semigroup characterization of Osterwalder-Schrader path spaces and the construction of Euclidean fields", J. Funct. Analysis, vol. 27, no. 3, pp. 277-291, 1978.

