I understand this question is rather marginal, but still think I might get some help here. I previously asked a question regarding the so-called computable Universe hypothesis which, roughly speaking, states that a universe, such as ours, may be (JUST IN PRINCIPLE) simulated on a large enough computer, and the question was resolved quite successfully.(adsbygoogle = window.adsbygoogle || []).push({});

This is to say that everything, that has a meaning in terms of observation, might be in principle simulated (up to a finite precision).

Now, the question. (Forgive me my mediocre knowledge) Letbe a Hermitian operator acting on anAn-dimensional Hilbert spaceH. By the spectral theorem, we can decomposeinto the sum ΣA_{i=1}^{m}λ_{i}Pwhere λ_{i}_{i}-s aremmutually distinct eigenvalues ofandAP-s are the corresponding orthogonal projections. Then,_{i}Hcan be rewritten as a direct sum of the corresponding subspaces. Now, if we were to simulate all (observable in real world) physical systems, we would need to know whether the eigenvalues of all Hermitian operators that correspond to the real physical systems are distinguishable. Otherwise, our "supercomputer" would be unable to determine, which eigenstate the system falls into after measurement. In particular, it is true when all the operators are represented by non-degenerate matrices.

Are there (or have there been observed) real-world physical systems known to have indistinguishable eigenvalues?

My question is motivated by the following work:

Computable Spectral Theorem

Another discussion on the topic (quite old though)

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# I Eigenvalue degeneracy in real physical systems

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