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Quantum Observables as Real-valued Functions and Quantum Probability

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Quantum observables
are commonly described by self-adjoint operators on a Hilbert space H. I will
show that one can equivalently describe observables by real-valued functions on
the set P(H) of projections, which we call q-observable functions. If one regards
a quantum observable as a random variable, the corresponding q-observable
function can be understood as a quantum quantile function, generalising the
classical notion. I will briefly sketch how q-observable functions relate to
the topos approach to quantum theory and the process called daseinisation. The
topos approach provides a generalised state space for quantum systems that
serves as a joint sample space for all quantum observables. This is joint work
with Barry Dewitt.