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Observables, Disassembled

This paper argues that non-self-adjoint operators can be observables. There are only four ways for this to occur: non-self-adjoint observables can either be normal operators, or be symmetric, or have a real spectrum, or have none of these three properties. I explore each of these four classes of observables, arguing that the class of normal operators provides an equivalent formulation of quantum theory, whereas the other classes considerably extend it.