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It's really necessary for an observable represented by an operator acting in a Hilbert space to be hermitian? It's known that not only hermitian operators have real eigenvalues and that also normal operators can have real eigenvalues, even if not always.

Reading the work of Bender https://arxiv.org/abs/hep-th/0703096 i saw that there's a particular class of operators that have real eigenvalues (Parity-Time invariant operators in the regime of unbroken PT simmetry). Why isn't common to study them? Which is the largest class of operators with real eigenvalues?

It's just a legacy or what?

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  • $\begingroup$ I'm not sure why it's not common to study. Perhaps it's not well-known even among physics faculty. Certainly one reason is that it's more complicated. My knowledge is limited, but my poor understanding is that these Hamiltonians beget mixed states. The eigenstates are not orthogonal. Mathematical complexities. They are useful to describe open systems and energy is not conserved so lossy systems and systems with gain are studied. "Exceptional points" in the parameter space exist where eigenvalues coalesce and "unusual" phenomena occur. Now we wait for a real answer. $\endgroup$
    – garyp
    Commented Sep 26, 2023 at 12:37
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    $\begingroup$ See physics.stackexchange.com/q/438719. Also, physics.stackexchange.com/q/631665. Is this something similar to what you are asking (at least initially)? $\endgroup$
    – VaibhavK
    Commented Sep 26, 2023 at 12:43

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