Quantum Mechanics as a Space-time Theory

We show how quantum mechanics can be understood as a space-time theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare this new description of space-time with the Bohmian picture of quantum mechanics. 1. What is quantum space-time? Both modern mathematics and modern physics underwent serious foundational crises during the 20th century. The crisis in mathematics occured at the beginning of the century and the main problem was to deal with certain infinities that are directly related to the concept of real number. Poincaré [31] explained this crisis in terms of different attitudes to infinity, related to Aristotle’s actual infinity and the potential infinity (the first attitude believes that the actual infinity exists, we begin with the collection in which we find the pre-existing objects, the second holds that a collection is formed by successively adding new members, it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, non-standard definitions of real numbers. The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of non-locality. In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number [10].Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of non-standard real numbers that we call qrumbers. It is our goal in the present paper to analyze non-locality and the concept of space-time at the light of the new conceptual tools that we developed in the past. Mathematics Department, Macquarie University, Sydney, NSW 2109, Australia, email: [email protected] TENA-TONA Free University of Brussels, Pleinlaan 2, B-1050 Brussels, Belgium. email: [email protected]