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We introduce a formulation of quantum theory (QT) as a general probabilistictheory but expressed via quasi-expectation operators (QEOs). This formulationprovides a direct interpretation of density matrices as quasi-moment matrices.Using QEOs, we will provide a series of representation theorems, a' la deFinetti, relating a classical probability mass function (satisfying certainsymmetries) to a quasi-expectation operator. We will show that QT for bothdistinguishable and indistinguishable particles can be formulated in this way.Although particles indistinguishability is considered a truly "weird" quantumphenomenon, it is not special. We will show that finitely exchangeableprobabilities for a classical dice are as weird as QT. Using this connection,we will rederive the first and second quantisation in QT for bosons through theclassical statistical concept of exchangeable random variables. Using thisapproach, we will show how classical reality emerges in QT as the number ofidentical bosons increases (similar to what happens for finitely exchangeablesequences of rolls of a classical dice).

Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality