Question

I have managed to prove that a finite set of compatible observables [i.e. pairwise commuting (possibly unbounded) selfadjoint operators] on an Hilbert space {H}

admits an orthonormal basis of simultaneous eigenkets by proceeding by induction on the number of observables and by constructing the required basis piecewise, starting from each eigenspace of the last operator in the set.

I would like to know if one can generalize this Theorem to an arbitrary family of compatible observables on {H}
. I tried doing it by extending my own proof: instead of using induction we may use transfinite induction, however in that case we have the problem of dealing with limit ordinals.

Answer 1

The generalization of the theorem for simultaneous eigenkets in an infinite set of compatible observables on a Hilbert space is complex, often requiring advanced functional analysis techniques, and closely relates to the principles of Lorentz invariance and predictive power in physics theories.

Step-by-step explanation:

The generalization of the theorem you've proven for a finite set of compatible observables to an arbitrary family of compatible observables on a Hilbert space is indeed challenging, particularly when you encounter limit ordinals while using transfinite induction. The compatibility, meaning that the observables commute with each other, is essential for the existence of a complete set of simultaneous eigenkets. However, for an infinite set, you may need to consider additional technical methods within functional analysis such as spectral theory or the theory of unbounded operators. The main complication arises in limit cases if the family of observables is truly uncountable.

Since quantum mechanics and observables are deeply rooted in the postulates of relativity, understanding how various quantities such as the space-time interval remain Lorentz invariant is crucial in this discussion. This invariance ensures that physical laws are consistent across different inertial frames.

When you construct a theoretical framework or hypothesis, much like the inventors of the quark hypothesis did, you must ensure that all possibilities are accounted for, anticipating results that may have not yet been observed. Such predictive power is a hallmark of robust physics theories.