|σz and I|
|11th 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 central-force nature, with associated 4π-symmetry. The physical description corresponding to SU was measured on Λ∞ plane by what we call the macroscopic observer ObsM. This physical description as measured by ObsM, 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 the consequence of how we applied a measurement scheme, in this case spin matrices σx , σy , and σz, 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, i.e. theses coefficients must be rational.
Figure-1: Anharmonic Coordinates + Spin Matrices -> Shell Structure.
The σz matrix is given as follows,
Some of the properties of zero-trace matrices such as the matrix σz , are as follows 1:
(Gell-Mann matrices are another example of zero-trace matrices besides Pauli Spin Matrices.)
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 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 time-axis on which we have no control. At the same time the spin-matrices σx , σy , and σz, 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.
Rational behind Irrational Numbers
The Ubiquitous z-Axis
Knots in j-Space
Cauchy and Gaussian Distributions
Discrete Space, b-Field & Lower Mass Bound
The Cat in Box
The Initial State and Symmetries
Discrete Measurement Space
The Frog in Well
Visual Complex Analysis
The Einstein Theory of Relativity
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