When quantizing a theory of one particle, we are used to taking the classical dynamical variable $\gamma:\mathrm{time}\to\mathrm{space}$, the trajectory in time, and replacing it with another, probabilistic dynamical variable, the wave function $\psi:\mathrm{time}\times\mathrm{space}\to\mathbb{C}$ interpreted in such a way that if $S\subseteq\mathrm{space}$ then $\int_{x\in S} |\psi(t,x)|^2\mathrm{d}x$ is the probability of finding the particle within $S$ at time point $t$. The dynamics is generated by the postulate that $\psi$ obeys the Schrodinger equation $$ \mathrm{i}\dot{\psi} = H\psi $$ for some yet to be determined operator $H$ on functions $\mathrm{space}\to\mathbb{C}$.
If we have a non-relativistic field theory, why not view now the field dynamical variable $\phi:\mathrm{time}\times\mathrm{space}\to X$ ($X$ is the set of values of the field, for example if it is a real field then $X=\mathbb{R}$) instead also as a trajectory, whose values at each time point is not a space location, but rather functions on the whole of space, i.e., $$ \phi:\mathrm{time}\to(\mathrm{space}\to X)\,. $$
Now to quantize this field theory, we introduce a wave functional, i.e., $$\Psi:\mathrm{time}\times(\mathrm{space}\to X)\to\mathbb{C}$$ which, for every fixed time point, acts on functions of space (i.e. the set of classical field configurations is its domain) and spits out a complex number. We interpret its meaning probabilistically by saying that if $S\subseteq(\mathrm{space}\to X)$ then the probability of finding the field configuration within $S$ at time point $t$ is given by $$ \int_{\phi\in S}|\Psi(t,\phi)|^2\mathrm{d}\phi $$(this last integral should be interpreted as a path integral or perhaps a Bochner integral)
This immediately yields a square-integrability constraint on $\Psi$ at any time point so that a large class of wave functionals must be excluded and we have a functional $L^2$ Hilbert space.
The dynamics should then follow the same Schrodinger equation, now for the wave functional, as $$ \mathrm{i}\dot{\Psi} = H\Psi $$ where $H$ is some yet to be determined operator on this $L^2$ functional Hilbert space.
For example, if we want a free field theory, then $H$ should be proportional to the Laplacian. However, now the Laplacian is given by functional derivatives: $$ -\Delta = -\int_{j}(\frac{\delta}{\delta\phi_j})^2\mathrm{d}j\,. $$ Perhaps one can guarantee this makes more sense by choosing for field configurations $L^2(\mathrm{space}\to X)$ separable (i.e. $X$ a separable topological space), so that it has a countable basis, and then $j$ above ranges on a countable (but infinite) basis.
The main idea is essentially this: to get from particles to fields, one replaces space (a finite dimensional Euclidean space) with $L^2(\mathrm{space}\to X)$, a Hilbert space (an infinite dimensional Euclidean space). Otherwise the very same quantization procedure of non-relativistic quantum mechanics follows.
My question: I have read Peskin & Schroeder and the various other QFT books. Is this completely equivalent to the two approaches presented there (of path integrals and of considering fields as operators)? Has this approach been considered before?
