Can observables with discrete and continous eigenvalues be commuting?

Question

In Dirac's The Principles of Quantum Mechanics he often uses the notion of "a complete set of commuting observables". This means a set of observables $\{\xi_1, \xi_2, \ldots, \xi_v, \xi_{v+1}, \ldots, \xi_u\}$ where $\{\xi_1, \xi_2, \ldots, \xi_v\}$ have discerete eigenvalues, and $\{\xi_{v+1}, \ldots, \xi_u\}$ have continous eigenvalues and each pair of observables commute, and to each set of eigenvalues only one simultanious eigenvector belongs.

The definiton of observables by Dirac is that each $\xi_i$ is a real linear operator, and the set of eigenvectors of the operator is a complete set. Now my question is: how is it even possible that obsevables with discrete and continous igenvalues are commuting? I mean if an opeator with discrete eigenvalues is an observable, that means that the set of basis vectors for the vector space is a countably infinite set. However if an operator with continous eigenvalues is also an observable (acting on the same vector space), that means that the set of basis vectors for the same vector space is an uncountably infinite set. For me, these two properties contradict with eachother.

Not to mention that Dirac also gives a proof of two commuting observables having a complete set of simultanious eigenvalues, the "side result" of his proof being that if two observables commute, their sets of eigenvectors mutually contain eachother, which means that the two sets are equal. To me this is also a contradiction.

Also can even two observables with discerete and continous eigenvlaues act on the same space?

Can somebody give me an explanation?

Answer 1

After some inquiry on math.stackexchange.com and writing a letter to one of my teachers, I've found the key: in infinite dimensional vector spaces the basis of the space is a dense subset of the vector space, and two dense subsets may have different cardinalities.

A very primitive example of this is the space of real numbers (over the field of real numbers). In this space the rational and irrational numbers both form dense subsets (that is any real number may be "synthesized" as the limit of a sequence of rational numbers, and the same is true for irrationals). This way both the rational and irrational numbers form a (Scahuder) basis of the vector space.

A more advanced example of the same space having bases of different cardinalities is the space of square integrable functions which equal their Fourier series. In bra-ket notation we may write that any such function $f(t)$ has an abstract $|f\rangle$ vector representing it, and there are two bases on the vector space, the time basis: $|t'\rangle, t' \in (-\infty, +\infty)$; and the angular frequency basis: $|k'*\omega_0\rangle, k' \in \mathbb{Z}$.

Since in the continuous basis: $|f\rangle=\int_{-\infty}^{+\infty}dt'|t'\rangle\langle t'|f\rangle=\int_{-\infty}^{+\infty}dt'|t'\rangle f(t')$, the function in the time domain may be looked upon as the coordinates of the vector $|f\rangle$ in the "time" basis.

At the same time $|f\rangle=\sum_{k'=-\infty}^{+\infty}|k'*\omega_0\rangle\langle k'*\omega_0|f\rangle=\sum_{k'=-\infty}^{+\infty}|k'*\omega_0\rangle c_{k'}$. Here the $c_{k'}$-s are the Fourier coefficients of the function, and may be seen as the coordinates of the vector $|f\rangle$ in the "frequency" basis.

From these one may calculate the transformation functions between the two bases.

$f(t')=\langle t'|f\rangle=\langle t'|\sum_{k'=-\infty}^{+\infty}|k'*\omega_0\rangle\langle k'*\omega_0|f\rangle=\sum_{k'=-\infty}^{+\infty}\langle t'|k'*\omega_0\rangle\langle k'*\omega_0|f\rangle=\sum_{k'=-\infty}^{+\infty}\sqrt{2\pi/\omega_0}e^{jk'\omega_0t}c_{k'}$.

Also $c_{k'}=\langle k'*\omega_0|f\rangle = \langle k'*\omega_0|\int_{-\infty}^{+\infty}dt'|t'\rangle\langle t'|f\rangle=\int_{-\infty}^{+\infty}dt'\langle k'*\omega_0|t'\rangle\langle t'|f\rangle=\int_{-\infty}^{+\infty}dt'\sqrt{2\pi/\omega_0}e^{-jk'\omega_0t}f(t')$

Here $\langle t'|k'*\omega_0\rangle=\sqrt{2\pi/\omega_0}e^{jk'\omega_0t}$ and $\langle k'*\omega_0|t'\rangle=\sqrt{2\pi/\omega_0}e^{-jk'\omega_0t}$ are the representatives of the frequency vectors in the time basis and of the time vectors in the frequency basis, respectively. They are also the transformation functions between the two bases.

Here is the link to the same question I've asked on math.stackexchange.com.