Pauli's Spin Matrices 

7^{th} September 2015 

The spacetime position (c^{2}t^{2}  x^{2} y^{2} z^{2}), an invariant under Lorentz transformation, is given by the (ct, x, y, z) metric in Clifford Algebra as, If the observer has the infinite capacity we can reduce the problem to the single measurement or a pure twostate system. The single measurement capacity implies relativistic limits i.e. zero entropy ⇒ t ≈ 0*. In that case matrix X reduces to zeroentropy spacetime matrix X_{o} ,
Writing X_{o} into its components, we get the following matrices:
X_{1}, X_{2}, and X_{3} are known as Pauli Spin Matrices. The matrix X_{3} corresponds to the actual physical measurement of a two state system with eigenvalues (EVs) [1, 1] or the measurements along zaxis. The offdiagonal elements equal to '0' represent zero probability of tunneling between the states represented by EVs 1 and 1. Pauli matrices are conventionally represented as σ_{x}_{ }, σ_{y }, and σ_{z}. An important property to note is that Pauli Spin Matrices are their own inverse(s) which means that the definition of the origin is mandatory within the structure described by these matrices. (It is not necessarily true for a simple reflection of a structure in which case the reference origin is in the plane of the mirror not in that of the structure itself). As we recall that the VT symmetry is required to determine the origin hence the quanta for the measurement space is predefined for the structure described by these matrices. Please also note that Pauli spin matrices do not represent Majorana states. When we see a structure like the measurement triangle ΔOXY in anharmonic coordinates, Pauli's spin matrices come to mind immediately. Let us try to evaluate the measurement triangle ΔOXY by placing σ_{x}_{ }, σ_{y }, and σ_{z} at X, Y, and O respectively and measure the unit point U. Effectively we have replaced the α, β, and γ by spin matrices. We should be able to find scalars l, m and n such that, lσ_{x} + mσ_{y} + nσ_{z} = 0 . 1
We plug in the values of matrices and solve to find, l^{2} + m^{2} + n^{2} = 0 . 2
The equation2 represents an sphere in measurement space with radius zero as measured by the macroscopic observer Obs_{M}. Please note that l, m, n, are the points lying on the measurement net. Each of them is an integer or a ratio of integers hence rational or "constructible"**. This is how the most elementary geometrical symmetry i.e. spherical symmetry is formed from a twostate system.
The unitpoint U will be measured as an sphere of zero radius by Obs_{M}. Let us call it unitpoint sphere S_{U}^{!}. This situation is rather fantastic. The quantities on LHS of equation2 are positive whereas RHS is equal to zero. However we must not forget Obs_{i}, the ispace observer who has higher capacity than Obs_{M}. Obs_{i} will measure 0 as 0_{j}, a finite value. Hence equation2 can be rewritten as, l^{2} + m^{2} + n^{2} = 0_{j} . 3
Above equation represents a sphere of finite radius √0_{j}, a radius whose measurement is not possible for Obs_{M}. We should be using the quantity i√0_{j} or an imaginary value for radius, but we are not. Why? Because in measurement space only PE1 measurable quantities are reported, hence 0_{j} which actually is the radius squared is of value. In other words the information on the internal structure of this sphere is not available to the macroscopic observer Obs_{M}. In a later section, this argument will result in shellstructure we are so familiar in classical picture.
The unitpoint sphere S_{U} has a symmetry of 4π or 720^{o}, centered around the unit point U. The 4πsymmetry is an essential feature of the systems with interacting components or the systems whose components are not completely independent of each other. We can later build a coordinate system with independent axes with S_{U} with 4π symmetry, as the origin. We can visualize the symmetry of 4π as a fundamental internal symmetry of a twostate system measured as S_{U} by Obs_{M} at Λ_{∞}. One may want to think of the symmetry with respect to the integral multiple or half integral multiple rotation of 360^{o}, but then not all the information contained within S_{U} would be measured. Had we considered only a classical singlestate system, the 360^{o} rotation would have been sufficient to describe the corresponding symmetry. Obviously 2 × 360^{o} rotations do not represent 4πsymmetry which is symmetry of the origin itself. In fact the information measured by ∞_{j} × 360^{o} rotations, will always be less than the information contained within the 4πsymmetry. In other words the resources available to a classical observer are not enough to redefine the origin and change the information structure being measured. The unitline Λ_{∞} has the property that it is parallel to every straight line in the anharmonic plane. We have used this property for drawing an important geometric feature in Figure1 related to Obs_{M}. Can you figure out this feature? Think shell theorems.
Please note that S_{U} will form the initial state for the measurements performed by Obs_{M}, the measurements represented by the polar Λ_{∞} in anharmonic coordinates (Figure1). The central force nature of the problem is quite obvious. We have so far refrained from using physical constants or known forces in nature while developing this description. One of the conditions we have discussed is related to values of scalars l, m, and n, which have to be either integer or rational. Consequently a continuous spectrum of measurable structures is prohibited in jspace.
__________________________________________________________ *The complete Hamiltonian of a twostate system is given as,
We have merely eliminated the timedependent phase which is a function of g_{0} to derive a description with the unit point U as reference. Please note that we have introduced the relativistic description by making time insignificant. For the unit point U the time has no significance, as time is the property associated with the observer making measurement, not with the source which is being measured. The conventional (t, x, y, z) description is valid only for Λ_{∞}, the measurement plane of the macroscopic observer with intrinsic entropy.
** What happens if (l, m, n) are complex numbers? ^{! }The unitpoint sphere S_{U} is not the same as traditional unit sphere S^{ 2}.

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