Do `classical' space and time provide identity to quantum particles?

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Krause, Décio
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Non-relativistic quantum mechanics is grounded on ‘classical’ (Newtonian) space and time (NST). The mathematical description of these concepts entails that any two spatially separated objects are necessarily different, which implies that they are discernible (in classical logic, identity is defined by means of indiscernibility) — we say that the space is T2, or "Hausdorff". But quantum systems, in the most interesting cases, some- times need to be taken as indiscernible, so that there is no way to tell which system is which, and this holds even in the case of fermions. But in the NST setting, it seems that we can always give an identity to them, which seems to be contra the physical situation. In this paper we discuss this topic for a case study (that of two potentially infinite wells) and con- clude that, taking into account the quantum case, that is, when physics enter the discussion, even NST cannot be used to say that the systems do have identity. This case study seems to be relevant for a more detailed discussion on the interplay between physical theories (such as quantum theory) and their underlying mathematics (and logic), in a way never considered before.