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 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.
Abstract We present an operational reconstruction of the well-known two-to-one homomorphism between the groups SU(2) and SO(3), grounded in the physical description of quantum state preparation and evolution. Starting from the connection between vectors in three-dimensional physical space and quantum states of two-level systems, we investigate how reversible transformations–modeled as completely positive and trace-preserving maps–give rise to a correspondence between spatial rotations and unitary operations. Our approach reveals how this group-theoretic structure naturally emerges from physical constraints, particularly the preservation of purity and reversibility in quantum processes. Beyond its theoretical relevance, the construction offers a pedagogically accessible framework for introducing core ideas in quantum mechanics and symmetry groups, making the abstract correspondence between SU(2) and SO(3) tangible through experimentally meaningful procedures.
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 homogeneous canonical transformation of boson operators is studied from the standpoint of the SU(1, 1) quasi-spin formalism. It is shown that this transformation can be regarded as a rotation in the SU(1, 1) quasi-spin space. This situation is just analogous to the SU(2) quasi-spin formalism of many-fermion problem, in which fermion Bogolyubov transformation can be regarded as a rotation in the SU(2) quasi-spin space. The transformation law of state vectors under this boson Bogolyubov transformation is then written down explicitly in terms of the SU(1, 1) d-functions obtained in the preceding paper. For clarity, we treat the simplest nontrivial example of one-dimensional harmonic oscillator or, equivalently, a system consisting of one kind of bosons. Generalization to many-boson problem and its possible applications to Bose fluid and nuclear structure are briefly discussed.
Spinors ψj of SU(2) represent group elements, i.e. three-dimensional rotations Rj , because they are shorthands of SU(2) representation matrices Rj obtained by taking their first columns. We explain that making linear combinations of spinors of SU(2) is feasible algebraically but geometrically meaningless. E.g. ψ1 + ψ2 ∈ C 2 would have to correspond to R1 + R2. But in SO(3), R1 + R2 would be a function that transforms r ∈ R 3 into R1(r) + R2(r), while in SU(2) it would correspond to a different function that is up to a multiplication constant equal to S : r → S(r) = R1(r) + R2(r) + 2 [ cos(ϕ1/2) cos(ϕ2/2) − s1•s2 ] r + 2 sin(ϕ1/2) sin(ϕ2/2)[ (s2•r) s1 + (s1•r) s2 ] + 2 sin(ϕ1/2) cos(ϕ2/2)(s1 ∧ r) + 2 cos(ϕ1/2) sin(ϕ2/2)(s2 ∧ r). The multiplication constant must be chosen such that |r| is preserved. Such extensions by linear combinations for spinors from a manifold to an embedding vector space are therefore mindless algebra with a spurious, conceptually impenetrable geometrical counterpart. This should not surprise anybody because the group axioms only define products of group elements, not linear combinations of them. Redefining spinors as vectors of a Hilbert space is therefore a purely formal, would-be scholar generalization.
Abstract A group is an algebraic structure equipped with binary associative multiplication, identity element and inverse operation with respect to identity. Lie group is a smooth manifold with group multiplication and the inverse being smooth functions. Therefore as one chooses manifold coordinate charts, various parametrizations of multiplication and inverse operations arise. Recently, the simple probability representation of qubit states and observables and an associated triangle geometrical picture called ‘quantum suprematism’ were introduced. At its core lies the representation of the Bloch sphere by correlated dichotomic probability distribution. The aim of the present paper is the introduction of probability parametrization of two classical compact and non-compact Lie groups, SU(2) and SU(1,1) respectively.