Dmitriy Khots

Physical Aspects of Observer’s Mathematics

This work considers Lorentz transform, Newton, Schrodinger and geodesic equations in a setting of arithmetic provided by Observer’s Mathematics. Certain results and communications pertaining to solution of these problems are provided.

Quantum Theory and Observer's Mathematics

When we consider and analyze physical events with the purpose of creating corresponding mathematical models we often assume that the mathematical apparatus used in modeling, at least the simplest mathematical apparatus, is infallible. In particular, this relates to the use of infinitely small and infinitely large numbers in arithmetic and the use of Newtons definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature (physics, genetics). We have introduced a new concept of “Observers Mathematics—Mathematics of Relativity” (see references). Observers mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. The concept of a quantum arises automatically from the human thought process, rather than from physics. From this point of view classical mechanics is brought closer to quantum mechanics.

Observer’s mathematics – mathematics of relativity

Abstract Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The basic problems of Algebra, Geometry, Topology, Logic, and Functional Analysis are considered for this chain. Definition of Dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian Geometries become particular cases of the developed Geometry, although many others are possible. For example, we proved that two lines in a plane may intersect in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. Two of the classical Geometries depend on a particular neighborhood of a given line. For example, Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, Lobachevsky Geometry takes over. Developed Topology gives birth to Time, and Time becomes a function of Space. Also, the Axiom of Choice becomes invalid in the new model of Mathematics. The application of the new model to Einstein’s special physical theory of relativity is considered. The existence of Time and Space quantums is proved. We also construct a new system of coordinate transformations that substitute Lorenz transformations. We also consider the application of the new model to data-mining.

Special relativity from observer's mathematics point of view

When we create mathematical models for quantum theory of light we assume that the mathematical apparatus used in modeling, at least the simplest mathematical apparatus, is infallible. In particular, this relates to the use of ”infinitely small” and ”infinitely large” quantities in arithmetic and the use of Newton - Cauchy definitions of a limit and derivative in analysis. We believe that is where the main problem lies in contemporary study of nature. We have introduced a new concept of Observer’s Mathematics (see www.mathrelativity.com). Observer’s Mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. We use Einstein special relativity principles and get the analogue of classical Lorentz transformation. This work considers this transformation from Observer’s Mathematics point of view.

Photoelectric effect from observer's mathematics point of view

When we consider and analyze physical events with the purpose of creating corresponding models we often assume that the mathematical apparatus used in modeling is infallible. In particular, this relates to the use of infinity in various aspects and the use of Newtons definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature. This work considers Physical aspects in a setting of arithmetic, algebra, geometry, analysis, topology provided by Observers Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided. In particular, we prove the following Theorems, which give Observers Mathematics point of view on Einstein photoelectric effect theory and Lamb-Scully and Hanbury-Brown-Twiss experiments: Theorem 1. There are some values of light intensity where anticorrelation parameter A ∈ [0,1). Theorem 2. There are some values of light intensity where anticorrelation parameter A = 1. The...

What is behind small deviations of quantum mechanics theory from experiments? Observer's mathematics point of view

Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within todays physics is a possible reason. In particular, we consider the concept of infinity that exists in todays mathematics as the root cause of this problem. We have created Observers Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observers Mathematics may explain some of the contradictions in QM theory results. We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observers Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to soluti...

Geometrical and quantum mechanical aspects in observers' mathematics

When we create mathematical models for Quantum Mechanics we assume that the mathematical apparatus used in modeling, at least the simplest mathematical apparatus, is infallible. In particular, this relates to the use of ”infinitely small” and ”infinitely large” quantities in arithmetic and the use of Newton Cauchy definitions of a limit and derivative in analysis. We believe that is where the main problem lies in contemporary study of nature. We have introduced a new concept of Observer’s Mathematics (see www.mathrelativity.com). Observer’s Mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. We prove that Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, non-euclidean Geometry takes over. We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. We proved the following theorems: Theorem A (Lagrangian). Let L be a Lagrange function of free material point with mass m and speed v. Then the probability P of L = m 2 v2 is less than 1: P(L = m 2 v2) < 1. Theorem B (Nadezhda effect). On the plane (x, y) on every line y = kx there is a point (x0, y0) with no existing Euclidean distance between origin (0, 0) and this point. Conjecture (Black Hole). Our space-time nature is a black hole: light cannot go out infinitely far from origin.

The foundations of quantum mechanics in Observer's Mathematics

This work considers the Lagrange function in classical mechanics and Hamiltonian equations of general relativity and quantum theory, in a setting of arithmetic, algebra, and topology provided by Observers Mathematics. Certain results and communications pertaining to solutions of these problems are provided.

Probability in quantum theory from observer's mathematics point of view

This work considers the Hamilton equations of general relativity and quantum theory in a setting of arithmetic, algebra, and topology provided by Observers Mathematics, see [1], [2], [3]. The probability appears automatically, without apriori assumptions. Certain results and communications pertaining to solutions of these problems are also provided.

Quantum mechanics problems in observer's mathematics

This work considers the ontology, guiding equation, Schrodingers equation, relation to the Born Rule, the conditional wave function of a subsystem in a setting of arithmetic, algebra and topology provided by Observers Mathematics (see www.mathrelativity.com). Observers Mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. Certain results and communications pertaining to solutions of these problems are provided. In particular, we prove the following theorems: Theorem I (Two-slit interference). Let Ψ1 be a wave from slit 1, Ψ2 - from slit 2, and Ψ = Ψ1+Ψ2. Then the probability of Ψ being a wave equals to 0.5. Theorem II (k-bodies solution). For Wn from m-observer point of view with m>log10((2×102n−1)2k+1), the probability of standard expression of Hamiltonian variation is less than 1 and depends on n,m,k.