144 research outputs found
Does Newtonian space provide identity to quantum systems?
Physics is not just mathematics. This seems trivial, but poses difficult and
interesting questions. In this paper we analyse a particular discrepancy
between non-relativistic quantum mechanics (QM) and `classical' (Newtonian)
space and time (NST). We also suggest, but not discuss, the case of the
relativistic QM. In this work, we are more concerned with the notion of space
and its mathematical representation. The mathematics entails that any two
spatially separated objects are necessarily \ita{different}, which implies that
they are \ita{discernible} (in classical logic, identity is defined by means of
indiscernibility) --- we say that the space is , or "Hausdorff". But when
enters QM, sometimes the systems need to be taken as \ita{completely
indistinguishable}, 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 \ita{identity} to them by means of their individuation,
which seems to be contra the physical situation, where individuation
(isolation) does not entail identity (as we argue in this paper). Here we
discuss this topic by considering a case study (that of two potentially
infinite wells) and conclude 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 simple way
apparently never realized before.Comment: Preprint, 21 pages, 1 figur
Linear superposition as a core theorem of quantum empiricism
Clarifying the nature of the quantum state is at the root of
the problems with insight into (counterintuitive) quantum postulates. We
provide a direct-and math-axiom free-empirical derivation of this object as an
element of a vector space. Establishing the linearity of this structure-quantum
superposition-is based on a set-theoretic creation of ensemble formations and
invokes the following three principia: quantum statics,
doctrine of a number in the physical theory, and
mathematization of matching the two observations with each
other; quantum invariance.
All of the constructs rest upon a formalization of the minimal experimental
entity: observed micro-event, detector click. This is sufficient for producing
the -numbers, axioms of linear vector space (superposition
principle), statistical mixtures of states, eigenstates and their spectra, and
non-commutativity of observables. No use is required of the concept of time. As
a result, the foundations of theory are liberated to a significant extent from
the issues associated with physical interpretations, philosophical exegeses,
and mathematical reconstruction of the entire quantum edifice.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements;
70 pages(+2), further improvement
Quantum Mechanics, Ontology, and Non-Reflexive Logics
This is a general philosophical paper where I overview some ideas concerning the non-reflexive foundations of quan- tum mechanics (NRFQM). By NRFQM I mean formalism and an interpretation of QM that considers an involved on- tology of non-individuals as explained in the text. Thus, I do not endorse a purely instrumentalist view of QM, but believe that it speaks of something, and then I try to show that one of the plausible views of this ‘something’ is as en- tities devoid of identity conditions
Quasi-set theory for a quantum ontology of properties
In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for treating
aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure
On the Axiom of Canonicity
The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox (without reference to the L\"{o}wenheim-Skolem theorem) and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications
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