Schwartz space, Gaussian measures and integration over paths

Question

I'm studying the Wiener measure motivated by the path integral in quantum mechanics. For that I'm using the book by Glimm & Jaffe "Quantum Physics: a Functional Integral Point of View" that deals with it from that perspective.

Now, I'm having a problem in understanding a part of the book, that I think might be just a notation problem.

The notations I'm used to

For me $\mathscr{S}(\mathbb{R}^d)$, the space of Schwartz functions, is the space of smooth functions $f : \mathbb{R}^d\to \mathbb{R}$ with the property that:

$$\sup_{x\in \mathbb{R}^d}x^\alpha D^\beta f(x)<\infty,\quad \alpha,\beta \ \text{multi-indices}.$$

Similarly, for me $\mathscr{S}'(\mathbb{R}^d)$ means the space of continuous linear functionals on $\mathscr{S}(\mathbb{R}^d)$. I know then that we have:

$$\mathscr{S}(\mathbb{R}^d)\subset \mathscr{S}'(\mathbb{R}^d),$$ so that $\mathscr{S}'(\mathbb{R}^d)$ can be seen as an enlargement of $\mathscr{S}(\mathbb{R}^d)$.The key point is: in what I'm used to a Schwartz distribution is a linear functional acting on $f : \mathbb{R^d}\to \mathbb{R}$ maps which can be seen as a more general class of such maps.

What Glimm and Jaffe seem to do

My issue is that Glimm & Jaffe talks many times about constructing gaussian measures (and in particular the Wiener measure) on the space of Schwartz distributions.

In particular, since $\mathscr{S}(\mathbb{R}^d)\subset \mathscr{S}'(\mathbb{R}^d)$ this measure allows to integrate over functions $f : \mathbb{R}^d\to \mathbb{R}$. These are not paths by any means.

Let me make my issue crystal clear: my issue is not on working with the dual. This seems a standard thing to do to deal with singular objects. Furthermore, later on one could see how to restrict the measure to the original space.

My issue is that the original space here is a space of functions $f : \mathbb{R}^d\to \mathbb{R}$ while paths must be functions $f: [a,b]\to \mathbb{R}^d$.

So I ask: how the Schwartz space relates to paths in Glimm & Jaffe treatment? How one integration over paths (Wiener measure) can be related with an integration over real-valued functions?

Is it perhaps another Schwartz space of functions, whose elements are indeed paths, and hence another Schwartz distribution space, whose elements are linear functionals on paths?

Answer 1

If the distribution $\phi\in S'(\mathbb{R}^d)$ is not too singular then you may be able to restrict it to (affine) hyperplanes $x_d=t$ where the coordinates of a point $x\in\mathbb{R}^d$ are say $(x_1,\ldots,x_d)$. So this way $\phi$ can be seen as a path or map $\mathbb{R}\rightarrow S'(\mathbb{R}^{d-1})$, $t\rightarrow \phi(x_1,\ldots,x_{d-1},t)$ where the latter is seen as a (singular/generalized) function of the first $d-1$ variables. In QFT, $\phi$ is sampled according to some probability measure and this retriction can be done almost surely if the singularities of the moments on the diagonal are not too bad, i.e., if the scaling dimension of the field is not too high. Note that being able to do this restriction (defining sharp-time fields) is important physically because that's how you get the physical Hilbert space of the theory. Note that for quantum mechanics aka QFT on spacetime of dimension $d-1=0$, what I described really corresponds to your intuition of a path, i.e., a function from $\mathbb{R}$ to $\mathbb{R}$.