I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged Hilbert spaces in this context. The topic caught my attention one more time after my discussion on physics stack exchange about the Dirac formalism.
Repeating a little bit what I've said in the linked question, physicists use $|x\rangle$ as eigenvectors of the position, and write every element $|\psi\rangle$ of the underlying Hilbert space as:
\begin{eqnarray}
|\psi\rangle = \int dx \,\psi(x) |x\rangle \tag{1}\label{1}
\end{eqnarray}
This is analogous to what is usually found in the theory of Hilbert spaces, in which for every $x\in \mathscr{H}$ and $\{e_{n}\}_{n\in \mathbb{N}}$ Hilbert basis:
\begin{eqnarray}
x = \sum_{n\in \mathbb{N}}\langle e_{n},x\rangle e_{n} \tag{2}\label{2}
\end{eqnarray}
As it turns out, the space of vectors $|\psi\rangle$ are usually taken to be $L^{2}(\mathbb{R}^{d})$, so that the position operator $\hat{x}$ makes sense as a multiplication operator on a dense subspace. However, it is a self-adjoint operator with continuous spectrum, so it has no eigenvectors at all in $L^{2}(\mathbb{R}^{d})$ and, thus, $|x\rangle$ does not make mathematical sense in this context. In any case, the use of (\ref{1}) demand that the coefficients $\psi(x)$ to be elements of $L^{2}(\mathbb{R}^{d})$ themselves, just as the coefficients $\alpha_{n} = \langle e_{n},x\rangle$ are elements of $\ell^{2}(\mathbb{N})$ in (\ref{2}), so we end up using the components $\psi(x)$ to do actual calculations most of the time (at least, explicit calculations such as solving the Schrödinger equation).
This is what intrigues me. The above scenario is present at the very beginning of every discussion of quantum mechanics and, at least to me, these very few words are more than enough to justify the need of "some bigger space", which seems to be a rigged Hilbert space. However, a quick search on the internet and you find only a few things about rigged Hilbert spaces, in general papers discussing its connections with quantum mechanics. I've checked also my books on rigorous quantum mechanics and none of them seem to address the problem. However, all of them discuss spectral theory, which I've heard is another way to approach all this.
Question: Why does it seem that rigged Hilbert spaces are still a paper subject, not fully incorporated into the usual expositions of rigorous quantum mechanics, while spectral theory is deeply rooted? Is the spectral theory formalism preferable to treat the problems mentioned above instead of rigged Hilbert spaces by any reason? And why is this?
Best Answer
Why are rigged Hilbert spaces a paper subject, not usually treated in rigorous textbooks: this is a good question. If you learn QM the way I did, you start by understanding that the physicists' Dirac delta is fictional (i.e., doesn't live in $L^2(\mathbb{R}^d)$), then you learn von Neumann's approach via spectral theory, which makes everything rigorous, and then sometime later you hear about rigged Hilbert spaces as a way to make the Dirac delta function rigorous. At that point your reaction may be that this just seems like extra work for the purpose of giving literal meaning to the things physicists say, and why bother doing this?
The answer one is looking for is some clearly important, basic topic in QM which can be treated more easily with rigged Hilbert spaces. I don't know of such a thing, but that is likely because I don't know enough physics. Hopefully someone more knowledgeable can give a good example.
The thesis Quantum Mechanics in Rigged Hilbert Space Language by de la Madrid looks like a good resource. Similar questions on mathoverflow and the physics stack exchange are here and here.