Why Only Three Dimensions? |
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12^{th} May 2017 Updated on 19^{th} November 2018 |
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“ In what way does it become manifest in the fundamental laws of physics that space has
three dimensions?" - Paul Ehrenfest, KNAW, Proceedings, 20 I, 1918, Amsterdam, 1918, pp. 200-209, Reference: www.pbs.org "I shall therefore content myself with the statement that if the student, in solving these equations, does not fail to make use of division wherever possible, he will surely reach the simplest terms to which the problem can be reduced. "
- René Descartes in La Géometrié |
A basic fact of life which seems to puzzle us, is that we are restricted in a three dimensional world. It would be nice to be able to become higher dimension beings and to be able to do things no mortal can do. It is actually more of a fantasy when we allow matter over mind. Mind is known to have no dimensional constraints as long as the knowledge exists. Without this knowledge we are no better than a dead log with multiple physical dimensions, as useless in one dimension as in many. In fact the inner workings of mind can not be precisely described by theories and equations. Various explanations for the existence in a physical three dimensional universe, are based on string theory, the requirements of thermodynamics, the experiments performed at CERN and so on^{1}. Our objective here is not to summarize these theories. However we will like to point out that all these ideas are trying to explain a reality which is measurement based, measurements which themselves are limited by the observer's capacity to make them. The observer's capacity is infinitesimal when it comes to measuring an infinite source. We will like to show that the existence in three dimension should have a phenomenological explanation. Let us reintroduce the following diagram used in the previous blog describing an infinite source. We will be discussing q = 3 information-state which essentially is our universe, whose measurement is based on the electron-photon interaction. We will argue that as macroscopic observers Obs_{M}'s we have no option but to exist in 3-D space. More importantly the same is true for the macroscopic observers in q = 2 and q = 1 information-states too. While developing the physical description in discrete measurement space or j-space, we ran into the problem of the precise measurement of the origin. We used anharmonic coordinates combined with Pauli's Spin matrices to develop a geometrical structure with central-force nature, which had to followed by the any measurements made by Obs_{M}. The examples were gravitational and electrostatic forces. In this description, the physical reality i.e. the universe existed in Λ_{∞ }plane. In the diagram shown above, various structures being measured by Obs_{M }exist in Λ_{∞ }plane. The measurements by Obs_{M} on these structures, will follow the central-force nature. An example is our own earth with the organic mechanisms behaving as macroscopic observers. The positions of normal vectors as measured by the observer in the Euclidean space on one of the observers, is shown as reference. The normal vectors are drawn with respect to the physical origin of the structure in Euclidean space. We need to understand the correlation between the normal vectors in the Euclidean space of Obs_{M} to the unit-point sphere S_{U}. We note that S_{U} was the result of the application of Spin Matrices and therefore we were allowed to develop the description in the complex-space^{2}. We can assume that S_{U} is represented by an isotropic vector in the discrete measurement space. Therefore its fundamental form^{3} is equal to zero or in other words its value is less than 0_{j}^{2}, where 0_{j}^{2 }is the smallest value which can be possibly measured by a measurement-system, whose design is based on the electron-photon interaction or QED space. Thus the observer Obs_{M} in Λ_{∞}-plane will always measure location of S_{U} as 'zero' irrespective of the structure Obs_{M} is on and the location at which Obs_{M }is on, on a given structure. Consequently our description in j-space can be simplified to assume that respective normals for all the structures in Euclidean space, are drawn with respect to S_{U}. Or all the structures in Λ_{∞}-plane have the unit-point sphere S_{U }as their origin as shown below. Thus we have translated the limited capacity of the observer to measure a source into a geometrical context in Euclidean space, where all the structures in Λ_{∞}-plane irrespective of the distance between them or their respective existence on different instances on the time scale, are measured with respect to the unit-point sphere S_{U}.
Essentially by introducing S_{U} as origin or the point of reference, we have transformed the physical space or universe as we know it, into Λ_{∞}-sphere around S_{U}, as shown in the diagram below. Next we have to worry about time variable. We note that time will exist only in Λ_{∞}-plane or outside Λ_{∞}-sphere, as the time is an entropy driven characteristics associated with observer Obs_{M }which can measure structures only in Λ_{∞}-plane.
Whatever classical structure formation takes place as a consequence of measurements, it must be time-dependent and hence it is restricted to the region outside Λ_{∞}-sphere. We also note that the information content decreases as we move away from S_{U}^{4}. An structure which is allowed to progress (equivalently the growth of organics), must be able to measure the information space precisely to proceed. This is possible, only if the observer associated with the growing structure under consideration, measures the information-space in the region outside Λ_{∞}-sphere, moving outwardly away from the unit-point sphere S_{U}. (For example if the observer was to measure physical structures inside Λ_{∞}-sphere, while moving inwards in the direction of S_{U}, the observer's capacity must increase with respect to time. This is a contradiction since the entropy inherent in j-space dictates that the observer's measurement capacity must always go down, with the progression along the time axis.) The existence does not automatically translate into the stability. The stability of physical structures in j-space is based on the formation of least-energy surfaces, which are more likely to be ellipsoids than spheres. The formation of least-energy surfaces for the respective stability of the physical structures, applies to all states in j-space irrespective of their q-values (q = 1, q = 2, q = 3 and so on ). It represents a very important fundamental requirement in discrete measurement space, which we shall discuss some other time. "Every observer moves towards the state with maximum information, the observer can measure."
Please note the emphasis on "can measure". If a state with infinite information is available but the observer can not measure it then the observer can not move towards it, however the observer will stay in orbit. Gravity is not necessarily the only force we may have to worry about. |
Previous Blogs:
Chiral Symmetry
Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis Majorana ZFC Axioms Set Theory Nutshell-2014 Knots in j-Space Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions Discrete Space, b-Field & 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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