Möbius & Lorentz Transformations
Equivalence in j-space - II
1st October 2018

"We often forget  the wonder that we felt as children,  when the cares of the activities of the "real world" have begun to settle upon our shoulders."

- Roger Penrose

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

- Hermann Minkowski


    We continue the discussion started earlier on the equivalence between Möbius and Lorentz transformations.  We would like to develop an intuitive understanding of what it means when we say that the descriptions of the universe are equivalent, as measured by two different macroscopic observers.  Let us say that these observers are earthlings (ObsEarth) and martians (ObsMars).  The very first condition that these observers must agree upon is, the Lorentz invariance in their measurements.  The Lorentz invariance implies that the space-time interval between two events must be same, as measured by either of these observers.  The condition of Lorentz invariance ensures that both observers stay on the same Λ-plane.

    We know that the discreet measurement space is a two-state system represented in the complex space C.  The measurements made by the observers are represented by the points on the complex plane, also known as the Argand plane.  If  all the information available in the unit circle on a complex plane, can be precisely measured by the observer, we can safely assume that Argand plane is complete.  The phrase "If", is loaded with implications.  We will discuss in detail some other time, the assumption that the Argand plane is complete for a macroscopic observer's measurements.  The Argand plane is equivalent to the Λ-plane in anharmonic coordinates.

    We can map the information available on Argand plane, onto the surface of a sphere known as the Riemann sphere.  Essentially we have placed the observer inside the Riemann sphere and the information available in the observer's universe is contained on the surface of the Riemann sphere.  At a given instant 1, this information can be transformed over to another Riemann sphere using Möbius transformation.  The situation is shown in the following figure.  

    The measurements of Obsearth are mapped on to Λ-plane on the left.  They are then stereographically projected upon the corresponding Riemann sphere Σ, which is next mapped on to another Riemann sphere Σ' using Möbius transformation.  Now the information from Σ', can be extracted and mapped on to Λ-plane on the right, for Obsmars.  The Lorentz invariance makes sure that Λ-planes are identical for both observers.  However the interpretation of the information extracted by Obsmars, will not be necessarily identical to that by Obsearth.  The difference may be of the nature that you say "tomato", I say "tomaato", but it will be appreciable.  This process can be reversed to transform the description of the universe per Obsmars, to the description per Obsearth, using inverse Möbius transformation.  We can generalize above process to n observers, where n is a large number.  It will require a minimum of n-1 Möbius transformations, if each of these observers is distinct from others. 

The set of these Möbius transformations and any combination of them will form a group.  In j-space, an important feature of these groups is that the required identity element e is the null element {}.  In discrete measurement space the null measurement is not possible.  The null element essentially is 0j, which has a finite value.  We start counting with our abacus using VT Symmetry in j-space, only after determining 0j or {}, (i.e. {} + {} + {} + {} +...........).  Thus the problem of the transformation of Riemann spheres, is reduced to the transformation of the null element {} or the identity element e of the group formed by Möbius transformations.  Therefore rather than collecting all the information from Riemann spheres and then transforming it, we can simply apply Möbius transformations to e or {}, and then extrapolate the result to the whole landscape in complex plane i.e. just add up2.

    Now let us up the ante a bit.  Rather than designing a mere interstellar GPS, we consider the scenario where life-forms other than those carbon based, are also in communication. The situation is as following:

The nature of the information is extremely complex in this case.  While the values of the fundamental physical constants such as the fine-structure constant 'α', Planck's constant 'h' and the speed of light 'c' may change, the underlying principles such as Lorentz invariance and Möbius transformation at a given instant, will remain unchanged.  The measurement space will remain discrete and VT symmetry  will be required to define the origin or {}.  The values of the physical constants will depend on the definition of the identity element e or {}, as determined by VT symmetry.  The observers making measurements here, will have to agree on Lorentz invariance to ensure that the Λ'-planes are identical, then apply respective Möbius transformations to exchange information. 

    An important point to note, is the definition of the observer making measurement as shown above.  In j-space, every entity is simultaneously an observer, as well as an object being measured.  The observer and the objects will stay in the state of measurement, until the PE1 measurement is completed.  We can not simply lock up the lab and call it a night until the measurements are complete.  An observer could be a planet, a galaxy, an atom, or an unknown.  Nevertheless the equivalent observers, have to follow the principles behind Lorentz invariance and Möbius transformations.  The conventional or the deterministic description of the human observers and their instruments, as independent of the objects being measured, is true only for the inertial frame of reference.

    Finally we have a rare treat for ourselves.  The Master himself is being interviewed by F. Hund in 1982 at Göttingen, Germany.  We are truly not worthy!  There is also a set of four lectures on Quantum Mechanics by Prof. Dirac given in 1975 at Christchurch, New Zealand, available on YouTube.  An incredible wealth of knowledge.  A must view for non-experts.

1.  Again a word of caution here, the definition of "instant" is very important.  For example when synchronizing geostationary satellites with the signal from earth in a GPS system, the time resolution of 10-9 seconds may be adequate.  However in higher information spaces, resolutions required for the precise definition of an "instant" may not be possible with QED instruments based on the electron-photon interaction only.  The limitations of the mechanism behind the measurement apparatus is super critical in j-space.  
2. The definition of {} or 0j, is not possible with the current digital computer technology.  A quantum computer or at least a quantum annealer would be required, where all the relevant information for {} for an observer in its j-space, is programmed as its internal configuration.  Next apply the appropriate Möbius transformation to obtain a new configuration within the quantum annealer.  Now the system can be allowed to relax adiabatically into the minimum energy state, which is based on the new configuration and hence it represents the description of {} in other observer's j-space.  Once the new configuration for {} is programmed in the quantum annealer, the communication channels can be established between both observers.  This procedure can be repeated for multiple observers, provided their respective Möbius transformations are available and the condition of Lorentz invariance is satisfied.   Maybe we can design an "universal translator" after all. 

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