σ_{z }and I  
11^{th} October 2015 

" When you observe an interesting property of numbers, ask if perhaps you are not seeing, in the 1 x 1 case, an interesting property of matrices. "  Olga Taussky " I wrote this book for myself. I wanted to piece together carefully my own path through Galois Theory, a subject whose mathematical centrality and beauty I had often glimpsed, but one which I had never properly organized in my own mind."  Charles R. Hadlock, “Field Theory and its Classical Problems”.

We have developed a rather important geometrical structure with centralforce nature, with associated 4πsymmetry. The physical description corresponding to S_{U} was measured on Λ_{∞ }plane by what we call the macroscopic observer Obs_{M}. This physical description as measured by Obs_{M}, could have been of any dimensions except 0 or 1, but the underlying measurementframework would not change. The measured physical description must follow the constraints of the framework. The important characteristics of the measurementframework in the case under consideration, were the shellstructure and the centralforce laws. It should be clarified that the shell structure shown in Figure1, was not a property of the the anharmonic coordinate representation. Instead it was the consequence of how we applied a measurement scheme, in this case spin matrices σ_{x}_{ }, σ_{y }, and σ_{z}, to the anharmonic coordinates and then developed the physical description based on zaxis measurements. The discrete nature of the structure was due to the constructibility constraints imposed upon the coefficients l, m, and n, i.e. theses coefficients must be rational. Figure1: Anharmonic Coordinates + Spin Matrices > Shell Structure. The σ_{z} matrix is given as follows, Some of the properties of zerotrace matrices such as the matrix σ_{z} , are as follows ^{1}: (GellMann matrices are another example of zerotrace matrices besides Pauli Spin Matrices.)
The combination of anharmonic coordinate system with spin matrices, describe a system based on the interaction between two purestates, represented by the measurements along the axes OX and OY of the triangle OXY.
Please note that the trace is not zero for I, hence there is no convex surface in this case. If we think of a measurement space where the measurements are represented by the values between 0 and 1, the Identity Matrix I represents a measurement matrix with maximum possible value for the trace (1 + 1 = 2). In such systems the probability of each measurement is alike or the results from each measurement are independent of others. This is the case of maximum entropy represented by the Uniform Distribution in statistics, similar to white noise. An example is the progression of the timeaxis on which we have no control. At the same time the spinmatrices σ_{x}_{ }, σ_{y }, and σ_{z}, represent a system which is likely to follow Gaussian or Normal Distribution. The above discussion can be generalized for n > 2 tracezero matrices. Please note that we have discussed the framework describing the existence of interaction between two pure states with no internal structure, using anharmonic coordinates and the Basis of the Hilbert Space. We are not concerned with the values of physical measurements of the states at Λ_{∞ }plane. The point is that all the observers with varying capacities in the discrete measurement space or jspace must observe such pure states and interaction between them, although the values obtained from physical measurements characterizing these states, will be different for each observer. The existence of purestates with no internal structures corresponding to the fundamental interaction in our measνrement space (q = 3 space) should not be that difficult to imagine (eg. electronphoton). Our friend Zörn even though from a higher information state, will also measure similar structures. __________________________________________ 
Previous Blogs: Spin Matrices
Rational behind Irrational Numbers The Ubiquitous zAxis Majorana ZFC Axioms Set Theory Nutshell2014 Knots in jSpace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, bField & Lower Mass Bound Incompleteness II The Supersymmetry The Cat in Box The Initial State and Symmetries Incompleteness I Discrete Measurement Space The Frog in Well Visual Complex Analysis The Einstein Theory of Relativity 
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