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The Ubiquitous zAxis 15^{th} July 2015 The problem we have faced in developing a physical description in the discrete measurement space or jspace, is the precise determination of the origin due to the infinitesimal capacity of the macroscopic observer Obs_{M}. It is reflected in the value of the finestructure constant. Since the absolute values are not available, we need to find comparative methods to analyze a given system. The twostate system is one of the consequences. An example of a twostate system is the concept of "spin" in Quantum Mechanics, described by the Pauli matrices. (We will discuss them within the context of Hamilton's Anharmonic Coordinates a little later.) The prerequisites to establish a twostate system, is the definition of the 0_{j} using VTsymmetry, which is dependent up on Obs_{M}'s capacity. As we have discussed earlier this definition is equivalent to ¬∈ = ∈. ,We are allowed to measure continuum and space, represented by OX and OY respectively as shown in Fig1. Both OX and OY can extend up to infinity. We define a unit point U, (1,1), whose projections on OX and OY, represent the unit measurements in continuum and space respectively. We keep in mind that OX and OY do not represent axes in Euclidean space. The objective is to define an arbitrary point P in terms of U in XY plane.
Figure1: The definition of unitpoint U in jspace. Figure2: The description of the measurement space for Obs_{M}. The unitpoint U is now assigned a value (1,1,1). Therefore
the origin O in XY space is replaced by the unit point
U. The vertex represented by O is now taken as one of the vertices of the ΔOXY. Note that the progress along OX and OY is limited by XY representing
observer's upper limit and as a consequence the measured infinite for Obs_{M} is projected on to O. Thus UO
represents the traditional zaxis used to describe the physical measurements of a twostate system. The vertices X, Y and O are represented by
(100), (010), and (001) respectively.
Figure3: AnharmonicCoordinate representation of the measurement space with respect to unit point U. Let us now consider the ΔOXY, and define UX, UY and UO as vectors having length α, β, γ and following the relationship that UX = XU, UY = YU, UO = OU as shown in Figure4. Please note that we have provided the points connected to U directional properties of vectors.
Figure4: Directional properties of the net associated with the unit point U.
Given vectors UX, UY, and UO ( α, β, γ ), it is possible to find scalars l, m, n such that (Figure3), if OB/BX = m/n; XC/CY = l/m; YA/AO = n/l, and OB/BX × XC/CY × YA/AO = 1, then lα + mβ + nγ = 0. The values of the scalars l, m, and n must be non zero. Further they must be either integers or rational numbers. The ΔOXY with OUC, XUA, and YUB can be considered the primary net of point U (1, 1, 1). It has the following properties: (i) if l + m+ n = 0 then the net drops
to a line, (iii) The line [l, m, n] = [1, 1, 1] is called the unit line for an arbitrary point (α, β, γ). For unit point (1, 1, 1) the unit line [1, 1, 1] is denoted by Λ. The relationship between a unit line corresponding to the unit point is that of a polar and its pole such that the condition, α + β + γ = 0, is satisfied. We will discuss the description of an arbitrary point in ΔOXY next. We should note that the construction being described here is analogous to the definition of "constructibility" from a unit length using compass and straightedge in field theory. Similar constraints of extensions of a field being rational numbers, are used to form the group of extensions which eventually lead to Galois theory to find solutions of higher order polynomials. The irrational numbers are not part of such groups as they are not constructible. In jspace we are forming similar structures in twostate system by constructing geometric nets around unitpoint U, with projections of U on OX and OY axes as unit lengths. For details on Geometric Nets and Anharmonic Coordinates, please refer to the works by Hamilton and Hime.
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