∞j  =  δ_i.

A bummer isn't it?  Here we are, fantasizing about the supremacy over "others" by any means possible, yet what we know including life, may not even be an infinitesimal of what we really do not know.  An amoeba in q=2 space is probably more capable than the whole universe combined in q=3 space.

    For macroscopic observers, the information ∞_j  or  δ_i, can be represented by the unit-point sphere S_U.  In the following figure S_U is being being measured by two observers who are in relative motion with constant velocity, with respect to each other.  The observers would be in disagreement with respect to the time and the space coordinates. 

    Assume that the time and the space coordinates for Obs₁ and Obs₂ are (t₁, x₁, y₁, z₁) and (t₂, x₂, y₂, z₂) respectively.  However the observers are restricted by the requirement that maximum velocity possible is c, which is the speed of light.

The measurements performed by both observers can be linearly mapped into each other using Lorentz transformations.  The consequence, is that they will have to agree that the space-time interval Δs² between two events as measured by either of them, will be the same, i.e.

s² = (c²Δt₁² -  Δx₁² - Δy₁² - Δz₁²)  =  (c²Δt₂² -  Δx₂² - Δy₂² - Δz₂²).

Thus if the observers are able to measure the unit-point sphere completely, albeit their descriptions of the events and how they occur occur may be different,  their measurements must be Lorentz Invariant i.e. Δs²does not change for either of them². 

    The Lorentz Invariance is required due to the fact that time and space are not linearly independent of each other.  The information gathered from the same space point at different time instants is not independent of each other, unless assumed otherwise as in Newtonian Mechanics.  Similarly the information gathered at two different space points at the same time instant is not independent of each other,  unless assumed otherwise as in Newtonian Mechanics.  (In j-space the terms "events" and "information gathered from measurements" are analogous.)

  It is also worthwhile to note that the independence of the space and time coordinates occurs only when there is no memory of the initial state <t = 0_j>.  In this case the structures are represented by the uniform distribution and the macroscopic observer is allowed to start and stop his or her measurement clock as he or she wills.  The uniform distribution is not a stable distribution. 

    As the memory of the initial state is brought in to the picture, we move towards stable distributions such as Gaussian, Cauchy and Levy.  In this case, the structures are not governed by the measurement clock of Obs_M but that of Obs_C instead.  (Obs_M does not have the memory of the initial state, Obs_C does.)

    In j-space the condition of Lorentz Invariance is enforced by the measurements made by the observer Obs_c (v/c =1), the maximum efficiency observer in q=3 state.  We can also say that the details of the universe as measured by Obs₁, will be linearly correlated to the details of the universe as measured by Obs₂, by Lorentz transformations.  Please note that the linear correlation between the details of the universe in complex plane,  does not imply linear independence of time and space coordinates as in Newtonian mechanics.

Now imagine that both observers are comparing observations made at "the same instant" in the j-space.  What will be the relationship between their measurements?  One thing for sure, is that the measurements must be linearly correlated as the Lorentz Invariance must still hold.  Thus at a given instant if two observers either stationary or moving with a constant velocity with respect to each other, make measurements then,

i)  The Lorentz Invariance must be satisfied, and

ii) The measurements at that instant are correlated by the Möbius Transformation in C.

    Why Möbius Transformation?  Because Möbius Transformation preserves forms in Complex space C.  For example if an observer measures a circle in the interior, then it is mapped using Möbius Transformation, as a circle in Λ_∞ region.  Furthermore Möbius Transformation is the unique transformation in C, which preserves the structure shape³. 

    This is a very important property as it uniquely allows the translation of the information without loss, from inside the unit-point sphere where it is measured, into the Λ_∞ region where the macroscopic observers can visualize it.  (Note: We will have to show how structures in C, transform into physical structures we observe and measure in real-life.)

    One but can not help notice the similarities between what we have discussed and wave-theory and matrix formulations of quantum mechanics.  The matrix formulation is based on the description of a system at a given instant which also includes the spin effect.  The wave theory does not include spin in its formulation and hence the concept of the electron-positron pair, does not exist.  The time in quantum wave theory is more of classical nature as it is used in conventional wave theory.

    This is turning out to be a rather fascinating discussion.  We will continue to explore this extraordinary result further.