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An observable is a self-adjoint operator $\mathcal{O}$ on the Hilbert space of states $\mathcal{H}$. The spectral theorem tells us that such an operator has an orthonormal basis of eigenvectors in $\mathcal{H}$, if it is compact. If it is not compact, we have to "enlarge" the Hilbert space to something called rigged Hilbert space or Gelfand tripel. A ...

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Unfortunately not every operator, which is used to get an observable gives you enough elements to make a base of the Hilbert space. As far as I know every eigenstate is orthogonal to each other. But the “length” could be arbitrary. Mostly used eigenstates in quantum mechanics are normalized for better usage.

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[...] concepts of Space and Time arising [...] An important guideline in this direction is surely Einstein's foundational principle that: All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points. The foundational "privileged ...

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