Graduate Stone's theorem on one-parameter unitary groups and new observables?

[victorvmotti]

TL;DR

Using unitary groups looking for new observables

I have been following the proof of the Stone's theorem on one-parameter unitary groups.

The question is if the current list of self-adjoint operators used in quantum mechanics, including position and momentum operators, is exhaustive or not?

Put it another way, can we say that there is no other one-parameter unitary groups, that can give us yet new self-adjoint operators, in addition to position and momentum ones, and therefore new observables?

 

[vanhees71]

Of course there's one more important, which is spin. In non-relativistic QM it's just another angular-momentum operator in addition to the orbital-angular momentum operator $\hat{\vec{L}}=\hat{\vec{x}} \times \hat{\vec{p}}$. While the orbital-angular momentum realizes only integer-representations of the rotation group, $\ell \in \{0,1,2,\ldots \}$, the spin realizes also half-integer representations (which are representations of the covering group SU(2) of the rotation group SO(3)).

In the representation with wave functions you get spinor-valued fields, transforming under rotations like
$$\psi_{\sigma}'(\vec{x}')=D_{\sigma,\sigma'}(R) \psi_{\sigma'} (R^{-1} \vec{x}),$$
here $D_{\sigma,\sigma'}$ with $\sigma,\sigma' \in \{-s,-s+1,\ldots,s-1,s \}$ (and using the Einstein summation convention) is the representation of the rotation $R$ for particles with spin $s \in \{0,1/2,1,\ldots \}$.

 

[victorvmotti]

Thanks so much for the reply.

But my point wasn't about spin.

I actually meant if there is a classification theorem on one parameter unitary groups.

Do we know now that there is no other such group that if exist at all then its generator will give us some yet unknown observable in quantum theory?