Why Only Three Dimensions? |
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12^{th} May 2017 |
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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 various 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. However 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 like to show that as a macroscopic observer Obs _{M} 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 are gravitational and electrostatic forces. The physical reality i.e. the universe exists in Λ_{∞ }plane.
In the diagram shown above, various structures being measured by Obs_{M }exist in Λ_{∞ }plane. The observer itself exists on each of these structures. An example is earth with organic mechanisms behaving as macroscopic observers. The positions of normal vectors as measured by the observer in the Euclidean space is shown. 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 physical space 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}, the smallest value which can be possibly measured by a measurement-system whose design is based on the electron-photon interaction. Thus the observer in Λ_{∞}-plane will always measure location of S_{U} as 'zero', since the precise measurement of S_{U }is beyond the observer's measurement capacity (S_{U }is probably somewhere in Planck's domain).Our description in j-space can be simplified to assume that 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 irrespective of the distance between them or their respective existence on different instances on the time scale, are measured with respect to S_{U}.
Essentially by introducing S _{U} as origin or the point of reference, we have transformed the physical space as we know it into Λ_{∞}-sphere around S_{U}. 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 which can measure only in Λ_{∞}-plane.
Whatever classical structure formation takes place, 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 growth for organics), must be able to measure the information space precisely to proceed, which 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}.The unit-point sphere essentially is a topological structure ^{5}, impenetrable and unmeasurable by the macroscopic observer. Whatever information Obs_{M} receives based on its measurement is a small subset of the information available at the surface of S_{U}. Only Obs_{C} with (v/c =1 ) capacity can measure all the information available on the surface of S_{U}. The fact that S _{U} is spherical, forces the 3-D constraint on the physical space as the growth or the progress in physical space, is allowed moving away from S_{U} on Λ_{∞}-sphere. It does not matter if the value of q is 3, 2, or 1. In each information space there will be three dimensions accompanied by an entropy driven irreversible time axis. The macroscopic observers in q = 1 and q = 2 states will also be restricted to three physical dimensions with a time axis, except that the information contained in single unit of length and time may be vastly different between q = 1, q = 2 and q = 3 states.We have to differentiate between the actual reality and the description provided by the mathematics. The use of probability in quantum mechanics does not make physical reality random. It merely provides us with better estimates of physical measurements. We can explain some of the facts known to us using quantum mechanics but not all with required accuracy and clarity. In fact we do not know anything at all about what we do not know. Similarly a higher dimension space is a mathematical construct to provide better understanding of an indeterminate situation. The physical dimensions of more than three, will not exist in q = 1 and q = 2 information states either. However an observer from q = 3 information state may use higher dimension algebra such as tensors to describe the q = 1 and q = 2 information states. Similarly an observer from q = 2 information state may use mathematical constructs with dimensions higher than three, to describe the q = 1 information space ^{6}. In determining the requirements for physical dimensions, we have not used any argument based on physical constants except for the phenomenological facts which came out while deriving the value for the fine structure constant. ------------------------------------------------------- 1. "How Many Dimensions Does the Universe Really Have?" by Paul Halpern, http://www.pbs.org/wgbh/nova/blogs/physics/2014/04/how-many-dimensions-does-the-universe-really-have/ 2. In discrete measurement space we can not take the existence of complex numbers for granted. We have to justify using them by providing appropriate context. 3. The fundamental form of a vector X in n-dimension is defined as Φ _{x} = x_{1}^{2}+ x_{2}^{2}+..............+ x_{n}^{2}. For an isotropic vector in j-space, the individual component may have negative values and combined together the value of Φ_{x} comes out to be below 0_{j}^{2}, the threshold of the measurement capacity in j-space. Other possibility is that the contribution of each component even though positive when summed up, is still below 0_{j}^{2}.4. The motion towards S _{U} is allowed within an infinitesimal limit iff the observer has quantum capacity.5. The spherical shape of this topological structure is the result of the constraints imposed by the complex-space where i-axis and j-axis are perpendicular to each other. This orthogonality between real and imaginary axes, is an assumption which is not supported by the discrete measurement space. Conceptually S _{U} can be of any form or shape unaffected by time. 6. We keep in mind what we stated before, a basic principle without proof as: "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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