We will discuss the equivalence between Lorentz transformation in Special Theory of Relativity and Möbius transformation in Complex Space C.  The Lorentz transformation ensures that two observers moving with respect to each other with a constant velocity, preserve the space-time interval between two events.  However a physical picture seems to be missing.   What about respective geometric shapes?  Would a sphere measured by an observer would also be measured as a sphere by another in a relative frame, or as a cube.  How about the relationships, temporal and spatial both, between geometric shapes?  In next few blogs we will be discussing the characterization of j-space in C.  Both Möbius and Lorentz transformations shall play crucial roles in forming various structures in j-space.

    But first let us set up the problem of measurement.  A finite-capacity observer (v/c << 1) Obs_M, is in the state of measurement of an infinite source.  We call this observer a macroscopic observer, for example human beings.  In the j-space, the infinite source is defined as "a point containing infinite information" within itself ¹.

    Since in reality, all the information within the-point can never be precisely measured by Obs_M, we also called the infinite information source as "indeterminate or i-space".  The observer Obs_M, can precisely measure only an infinitesimal portion of "indeterminate or i-space", called δ_i. The subscript i refers to the information measured from the i-space. 

    As the observer Obs_M  is in the state of measurement, δ_i essentially represents the universe (∞j) the macroscopic observer exists in, also called the j-space.  We represent this rather annoying revelation as,

                                                                     ∞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.

Thus if the observers are able to measure the unit-point sphere completely, albeit their descriptions of the events and how they 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 with 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.
 

To be continued...........

 ________________________
1. The exact nature of the information is not known.  The information could be related to biological, chemical, physical or mathematical structures or a combination of them.  Only conditions we place are, that the observer making measurements can report only what it could measure scientifically, and that the observer will remain in the state of measurement until the PE1 measurement is complete.
2. In space-time continuum Δs²= g_ik dxⁱ dx^k,  the invariance of  Δs² is also stated as the imposition of the metrical connexion up on the space-time continuum.  The metric g_ik represents scalar density.  In conventional (t, x, y, z) metric, the metrical connexion translates into Lorentz invariance.
3. For details please refer to Needham's Visual Complex Analysis.

 

As the legend goes, Schrödinger disappeared after his seminal work in quantum mechanics.  It was with the assistance of Heisenberg, that Möbius found him inside a black body.  The following exchange took place between two great minds: "Why are you running, Dr. Schrödinger?" "I am trying to dodge this photon with my name on it." "Um...What happened?" "I forgot to account for spin in my formulation." "Surely you can tunnel your way out.  After all Pixel did." "Pixel was dodging a bullet.  Photon has much higher capability.  It is bound to find me before I could tunnel." "Have you spoken to Dr. Einstein?  He does not believe in photons."                                                (Long pause) "Holy Sagittarius A*! " "Now where are you off to, Dr. Schrödinger?" "I am going to find myself the nearest black-hole.  That is the only place this blue devil would turn into red."                       (Here a moment of silence to acknowledge the sheer brilliance of the man.)§ "It may still sting, Dr. Schrödinger." "Perhaps, but there it will not be able to annihilate." "Good luck, Dr. Schrödinger! Thank you for your time and please remember." "Lord sure is subtle." ____________________ §  EBlue =  2.50-2.75eV,  ERed =  1.65-2.00eV.