Relativistic Algebra over Finite Ring Continuum

Physics of Particles and Nuclei, 2019

We first give a rigorous mathematical proof that classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a special degenerate case of finite one in the formal limit when the characteristic p of the field or ring in finite mathematics goes to infinity. We consider a finite quantum theory (FQT) based on finite mathematics and prove that standard continuous quantum theory is a special case of FQT in the formal limit p → ∞. The description of states in standard quantum theory contains a big redundancy of elements: the theory is based on real numbers while with any desired accuracy the states can be described by using only integers, i.e. rational and real numbers play only auxiliary role. Therefore, in FQT infinities cannot exist in principle, FQT is based on a more fundamental mathematics than standard quantum theory and the description of states in FQT is much more thrifty than in standard quantum theory. Space and time are purely classical notions and are not present in FQT at all. In the present paper we discuss how classical equations of motions arise as a consequence of the fact that p changes, i.e. p is the evolution parameter. It is shown that there exist scenarios when classical equations of motion for cosmological acceleration and gravity can be formulated exclusively in terms of quantum quantities without using space, time and standard semiclassical approximation.

Bulletin of Symbolic Logic, 2012

In his monographOn Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including −ω, ω/2, 1/ω,and ω − π to name only a few. Indeed, this particular real-closed field, which Conway callsNo, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Gödel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect,Nobears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields.In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom),Nomay be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying frameworkNoprovides ...

2002

The present article is the first in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the field of complex numbers. We call this number system ``the Quantionic Algebra''. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but "almost" one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The first is a careful analysis of the abstract structure of quantum mechanics (the first part of the present work), the ...

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel's sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.

Journal of the Association For Computing Machinery, 2007

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0 -1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?

Notre Dame Journal of Formal Logic, 2012

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by and independently by Ehrlich can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).

Contemporary Mathematics, 2009

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k ≥ 2. Arithmetic and transformation properties of these states are given, both for basis states representing rational numbers and linear superpositions of these states. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of q k string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical representations as kit string states in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. In particular, states of qukit strings are elements of Hilbert spaces, which are vector spaces over the complex field. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Some approaches to integrating this work with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of states of qukit strings.

Mathematical Logic Quarterly - MLQ, 1988

2007

ThispaperpresentstheArithmeticofOrder(AoO),aframeworkforfoundationalphysics built onthe singleprinciple that realityemerges fromadiscrete, computable, andnon associativealgebraicsubstrateovertherationalnumbers.WeformalizeanOrigin–Observation Duality,whereasingletri-octonionstatevectorvgeneratesauniqueHermitianJordanop eratorAthat satisfiesbothanonlinear internal eigen-equation(Av=v[v])andadmitsa classical rational-valuedspectral decomposition(Aw=λw,λ∈Q). Thedual natureof thisspectrumprovidesadeductiveoriginforanemergent3+1Dspacetimewithanintrinsic particle/anti-particlesymmetry.Weapplythis frameworktoreformulateLHCjetproduc tionas the spectral footprintof aG2 automorphismcascade, anon-perturbativeprocess governedbyoctonionalgebra.Finally,weshowhowthecoreprinciplesofAoOaremirrored inthearchitecturesofmoderngenerativemodels,groundingthetheoryinexperimentaland computationalpractice.