σ_{z }and I |
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11^{th} October 2015 |
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We have developed a rather important geometrical structure with central-force nature, with associated 4π-symmetry. 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 measurement-framework would not change. The measured physical description must follow the constraints of the framework.The important characteristics of the measurement-framework in the case under consideration, were the shell-structure and the central-force laws. It should be clarified that the shell structure shown in Figure-1, was not a property of the the anharmonic coordinate representation. Instead it was a 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 z-axis measurements. The discrete nature of the structure was due to the constructibility constraints imposed upon the coefficients l, m, and n.
Figure-1: Anharmonic Coordinates + Spin Matrices -> Shell Structure. The σ_{z} matrix is given as follows,
Some of the properties of zero-trace matrices such as matrix
The combination of anharmonic coordinate system with spin matrices, describe a system based on the interaction between two pure-states, 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 is 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 time-axis on which we have no control. At the same time the spin-matrices 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 trace-zero 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 j-space 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 pure-states with no internal structures corresponding to the fundamental interaction in our measνrement space (q = 3 space) should not be that difficult to imagine. Our friend Zork even though from a higher information state, will also measure similar structures.1. For details please refer to Olga Taussky, "Matrices with Trace Zero", Mathematical Notes, pp. 40-42, Jan. 1960. 2. ^{ }Please note that the true j-space is essentially a Hausdorff space, with the volume of the neighbourhood for each point in Hausdorff space equal to zero. Hence a continuous path is not possible in j-space. A continuous path is possible only when a symmetry is assumed and 0_{j} is defined accordingly. We will discuss the topological aspects of the problem intermittently during our discussions. We will find it easier to do as in j-space we are not restricted by the requirements of the Euclidean space. 3. If a perfect straight line could be formed between two points which included the origin, we would not be in Mordor to begin with. |
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