In this paper we discuss and outline a version of non-relativistic quantum mechanics based on a new non-reflexive logic, where the basic entities (elementary particles) lack identity conditions. Some relationships with quantum field theories are also sketched.
In the non-reflexive formulation of quantum physics, attention is given to a metaphysical view of quantum objects (q-objects) as non-individuals, characterised as entities to which the standard notion of identity lacks sense. The reference to the nature of q-objects is usually omitted in other interpretations and the most we can find is some vague reference to “to each [quantum] system we associate a Hilbert space”, and nothing more is said about them; we need to infer from the theory their characteristics, and it gives us several alternatives. One of them, to be justified here in broad aspects, is the non-individual view. It is important to enlighten that we are not trying to just find a way to deal with quantum entities but to defend a possible ontology of non-individuals, which can be applied also to ‘particles’ arising from quantum fields. Our account is based on two assumptions: (i) the ‘classical theory of identity’ does not apply to q-objects, and (ii) their non-individuality must be attributed from the start, that is, as a primitive notion and not ‘made a posteriori’ say by assuming symmetry principles. The underlying logic is called non-reflexive because the reflexive law of identity ∀x(x=x) is supposed not to hold for q-objects. Since all the details demand a lot of space, here we only sketch and justify our view. The references provide a more detailed account.
This is a general philosophical paper where I overview some ideas concerning the non-reflexive foundations of quantum mechanics (NRFQM). By NRFQM I mean formalism and an interpretation of QM that considers an involved ontology 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 entities devoid of identity conditions. Warning note: This is a revised version of a paper with the same name that was written by invitation to be published in a book titled \textit{The Mammoth Book on Quantum Mechanics Interpretations}, edited by Open Academic Press, Berlin, and having as editor a certain Ulf Edvinsson, who has invited me. The book was announced in the page of OAP and should appear by 2016. This never happened. Later I discovered that OAP is in a list of predatory editorial houses and that "Ulf Edvinsson" is (apparently) a fake name. Furthermore, I couldn't contact anyone responding by OAP to retire my name from the announcement of the book and for impeding them to publish the paper. I strongly apologize for such a fault, which is completely mine. Since the subject presented here has been among my preoccupations ever since I met Franscicso Antonio Doria for the first time (in 1987), it is a pleasure to dedicate the stuff to him. And of course I thank the editors for accepting this version of the paper for this book.
In this article we try to justify a way of looking for an alternative approach to quantum mechanics based on a non-classical logic. The main motivation is both to accommodate a metaphysical view according to which quantum objects are non-individuals and to introduce, so to say, these objects within the formalism, for as we know, the usual formalism treats of states of quantum systems and not (directly) of quantum systems properly speaking. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. Our way of looking to these concepts motivate us to construct a metaphysical view according to which quantum objects do not present identity criterion (they are ‘non-individuals’), and then we try to construct a quantum theory based on such a view, which is grounded on a non-classical logic, termed non-reflexive. From the formal point of view, the main motivation comes from the fact that in using classical logic and classical set theory (such as ZFC), we are necessarily committed to individuals, and non-individuals can be considered only by assuming ‘strong’ symmetry principles, which mask the basic idea involving non-individuality. In the core of the article, due to questions of space, only the main philosophical arguments are presented, but in two appendixes we outline, firstly, quasi-set theory, the non-reflexive mathematical framework that sustain our view, and a bottom-top construction of a quantum mechanics based on quasi-set theory.
Nonreflexive quantum mechanics is a formulation of quantum theory based on a non-classical logic termed \ita{nonreflexive logic} (a.k.a. `non-reflexive'). In these logics, the standard notion of identity, as encapsulated in classical logic and set theories, does not hold in full. The basic aim of this kind of approach to quantum mechanics is to take seriously the claim made by some authors according to whom quantum particles are \ita{non-individuals} in some sense, and also to take into account the fact that they may be absolutely indistinguishable (or indiscernible). The nonreflexive formulation of quantum theory assumes these features of the objects already at the level of the underlying logic, so that no use is required of symmetrization postulates or other mathematical devices that serve to pretend that the objects are indiscernible (when they are not: all objects that obey classical logic are \ita{individuals} in a sense). Here, we present the ideas of the development of nonreflexive quantum mechanics and discuss some philosophical (mainly metaphysical) motivations and consequences of it.
Abstract Traditionally by ‘quantum logic’, one understands the study of the lattice of the subspace of Hilbert space. Here, ‘quantum logic’ is taken to be the underlying logic of quantum physics. Following Schrödinger’s idea that the standard concept of identity cannot be applied to quantum objects, a kind of ‘non-reflexive’ logic called Schrödinger logic is presented, in which the reflexive law of identity is taken not to apply. First order Schrödinger logics are presented, and extended to higher order logics and also to an intensional logic. Classical semantics is presented, and a weak completeness theorem is sketched. The last part of the chapter covers a logic of sortal predicates for which a quasi-set semantics is delineated. This completes the formal framework capable of accommodating quantum non-individuality and thus the initial understanding of Born, Heisenberg, and Schrödinger, as well as the second horn of the above metaphysical underdetermination, can be formally represented.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.