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While studying a couple of concepts, I've understood the following premises to be true:

  1. Hermitian unitary matrices eigenvalues are unimodal (that's $\pm1$).

  2. In physics, operators/observables are represented by Hermitian matrices, its eigen(functions/vectors) forming a complete orthonormal base.

  3. In physics, each eigenvalue corresponds to a unique eigenfunction. If it happens to not be the case, it can be made so it is (according to Griffiths at least).

Therefore, it should follow that a Hermitian unitary matrix $\hat{Q}$ can only be 1 or 2-dimensional... right? If its eigenvalues can only be unimodal, and there can only be one of each, 2-dimensional self-adjoint unitary matrices are the only ones that can satisfy these constraints.

$$ \hat{Q}(\varphi + \psi) = \pm \varphi + \mp \psi $$

for $\varphi,\psi$ orthonormal would be the only way in which $\hat{Q}$ could act.

Am I missing something? Or is this fact that makes spin operators so beautifully unique?

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  • $\begingroup$ premise 3 seems to me to be not-true, as it precludes all degeneracies. This would be a great problem not only for unitary Hermitian operators but also for operators like angular momentum $L^2$... Can you bring the relevant part in Griffiths? $\endgroup$
    – user275556
    Commented Dec 7, 2021 at 11:34

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You are misinterpreting point 3. Indeed, the idea that all eigenspaces are one-dimensional is contradicted by the existence of the identity operator (on any space), which is both Hermitian and unitary and whose $+1$ eigenspace is the entire Hilbert space on which it acts.

Without having additional context or a direct quote it's not obvious to me where the misunderstanding is taking place.

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