This paper considers homomorphism of the Lie group SU(2) to the Lie group SO(3) of all rotations of 3- dimensional Euclidean space from Observers Mathematics point of view. In our work, we proved that in Observers Mathematics the probability of spin-j transformation is a homomorphism (representation) of SU(2 ) to the set of matrix transformations of a linear space of polynomial functions is less than 1, and got corresponding results for elementary fermions and bosons. As a continuation of these results we proved here the following theorems: Theorem 1. In Observers Mathematics the probability of two-to-one transformation of SU(2) to SO(3) is Lie groups homomorphism (representation) is less than 1. Theorem 2. The probability of two-to-one transformation and spin-j transformation (j = 1) are equivalent in Observers Mathematics is less than 1.
This paper considers spin - j representation of the Lie group SU(2) and its applications to contemporary physics from Observer's Mathematics point of view (see www.mathrelativity.com). This mathematics was introduced by authors based on denial of infinity idea. We proved here the following. Theorem. In Observers Mathematics the probability of spin - j transformation is a homomorphism (representation) of SU(2) to the set of matrix transformations of a linear space of polynomial functions is less than 1. Theorem. The probability of elementary fermions have half-integer spin in Observer's Mathematics is less than 1. The probability of elementary bosons have integer spin in Observer's Mathematics is less than 1.
KEYWORDS: Mathematics, Probability theory, Special relativity, Quantum physics, Quantum mechanics, Theory of relativity, Mathematical modeling, Relativity, Logic, Clocks
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.
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 = m2 v2 is less than 1: P(L = m2 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.
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.
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