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AIP conference proceedings

Quantum Theory from Observer’s Mathematics Point of View

Dmitriy Khots, Boris Khots, Andrei Yu. Khrennikov

Published Jan 1, 2010• 0 citations
Publisher

Abstract

This work considers the linear (time‐dependent) Schrodinger equation, quantum theory of two‐slit interference, wave‐particle duality for single photons, and the uncertainty principle in a setting of arithmetic, algebra, and topology provided by Observer’s Mathematics, see [1]. Certain theoretical results and communications pertaining to these theorems are also provided.

DOI
10.1063/1.3431504

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This work considers the linear (time‐dependent) Schrodinger equation, quantum theory of two‐slit interference, wave‐particle duality for single photons, and the uncertainty principle in a setting of arithmetic, algebra, and topology provided by Observer’s Mathematics, see [1]. Certain theoretical results and communications pertaining to these theorems are also provided.
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" 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 proved the following theorems: 1) Theorem A (Two-slit interference). Let Ψ1 be a wave from slit 1, Ψ2 - from slit 2, and Ψ = Ψ1 + Ψ2. Then the probability of that Ψ is a wave equals to 0.5; and 2) Theorem B (Wave-particle duality for single photons). If v is small enough, then λ is a random variable.
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 today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's 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 Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.
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 Newton's 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 “Observer's Mathematics—Mathematics of Relativity” (see references). 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. 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.
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 Newton's 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 Observer's 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 Observer's 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. Theorem 3. There are some values of light intensity where anticorrelation parameter A > 1.