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Möbius and Lorentz
Transformations Equivalence in j-space - II 1st October 2018
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"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 discrete measurement space is
a two-state system represented in the complex
space C
(Pauli operators). 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
at the center of 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 for communication
between these observers, 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 e, 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 true 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.
*** |
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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