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Spintronics

The spin degree of freedom from observer’s mathematics point of view

Boris Khots, Dmitriy Khots

Published Aug 20, 2020• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.2567105

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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.
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.
Abstract We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics, and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection.
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 Newtonʼs definition of a limit in analysis. We believe that is where the main problem lies in the contemporary study of nature. This work considers physical aspects in a setting of arithmetic, algebra, geometry, analysis, and topology provided by Observerʼs Mathematics (see www.mathrelativity.com). In this paper, we consider Dirac equations for free electrons. Certain results and communications pertaining to solutions of these problems are provided.
Spin is what makes the world go 'round. The Groups SO (3) and SU (2) How does SU(2) act on its Lie algebra? For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ 1 , …, ψ N ) T . He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates ( x 1 , x 2 , x 3 ) of space, the N -tuples must transform under some representation ρ: SO (3) → U ( N ) of the rotation group SO (3) of space. This is not quite accurate; since e i γ Ψ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ( g )ρ( h ) = e i γ( g,h ) ρ( gh ) for a function γ( g, h ). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU (2) onto SO (3), thus yielding a (perhaps double valued) representation of SU (2) into U ( N ).
The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group SU(2). The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-1/2 system is obtained and expressed in terms of matrix elements of the group SU(2). The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities.