RELATIONAL LOGIC
(with applications to Quantum Mechanics, String Theory, Cosmology, Statistical Mechanics)
ABSTRACT: Quantum mechanics defies classical logic, defined by Aristotle and extended into mathematics by George Boole. We suggest that the inner syntax of quantum mechanics is relational logic, a form of logic developed by C. S. Peirce in the years 1870-1880. Within relational logic, relation is the fundamental, primary constituent and everything else is expressed in terms of relations. The composition of relations leads naturally to the fundamental quantum laws. A double line representation for relations generates patterns similar to string world-sheets. A relation may be represented by a spinor and a single spinor gives rise to the Bloch sphere, which is topologically equivalent to the light cone of Minkowski spacetime. We examined the geometry emerging out of the quantum entanglement of two spinors. We found that quantum entanglement offers us a cosmology involving an extra dimension (its size is determined by the amount of quantum entanglement), with two “mirror” branes coexisting in the extra dimension. An interesting phenomenology for the neutrino mixing follows. We studied also a network composed of relations-links, using methods of statistical mechanics. Our model may serve as a prototype model to address different systems (internet network, virus propagation etc).
References
1) A. Nicolaidis, Categorical foundation of quantum mechanics and string theory, Int. J. Mod. Phys. A24, 1175-1183 (2009)
2) A. Nicolaidis and V. Kiosses, Spinor Geometry, Int. J. Mod. Phys. A27, ID: 1250126.
3) A. Nicolaidis and V. Kiosses, Quantum entanglement on cosmological scale, J. Mod. Phys. 4, 153-159 (2013)
4) A. Nicolaidis, Relational Quantum Mechanics, Chapter 16 in “Advances in Quantum Mechanics”, edited by Paul Bracken, InTech Publisher, 2013
5) A. Nicolaidis, Brane Cosmology and Neutrino Mixing, to be published
6) A. Nicolaidis, K. Kosmidis and V. Kiosses, Grand Canonical Approach to an Interacting Network, J. Mod. Phys. 6, 472-482 (2015) ((http://dx.doi.org/10.4236/jmp.2015.64051