This is my second try on the same question -- see Self-orthogonal states in Quantum Theory. Suppose one considers a hypothetical "non-classical Quantum Theory": we work in a Hilbert-like space $k^n$ ($k$ some field) with Hermitian form $\langle \cdot \vert\cdot \rangle$, which is not necessarily an inner product, etc. The nonzero vectors in $k^n$ correspond to states. What would be the physical meaning of states $\vert \psi \rangle$ for which $\langle \psi \vert \psi \rangle = 0$, that is, self-orthogonal states ?
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