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A
Timeless Constant 9th
May 2019
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"An essential feature of measurement is the difference between the “determination” of an object by individual specification and the determination of the same object by some conceptual means." - Herman Weyl, Space Time Matter "One could
also ask a slightly different question. What are the
principles that govern the interaction of a microscopic
system with a classical system? One would like to know
whether it is possible to define classical physical
systems that cannot interact with quantum systems." "Is colour metabolic efficient or coding efficient?" - James V Stone, Principles of Neural Information Theory. "And so these men of
Indostan - Elephant and the blind men, a childhood story. ..And it took a Witten to put the M-elephant together. |
In j-space the perception of space and time, are based on the measurements made by an observer, and therefore they are specific to the observer. Different observers, unless they are following Lorentz Invariance condition, may have measurements which do not agree with each other. Hence they may view universe differently. For example an observer named Taku may not be able to measure certain features of the universe, features which can be measured by Baku, another observer with higher measurement capacity.
Further the existence of time is required,
before the measurement of space itself can be
thought of. This requirement is crucial
as the conventional space can not exist
without the existence of an observer to make
measurements. For a give topology or
equivalently a given path, each observer1
comes with its own time-axis based on the
observer's measurement capacity. Each
observer is measuring the infinite source or the
exact source. The observers
obeying the Lorentz invariance, make
measurements along a congruent time-axis.
In the previous blog, we highlighted this
problem of time itself not being a universal
entity. You see, each measurement is
supposed to be a zero-entropy measurement, and
hence no time-axis2.
But entropy inherent in the system, a system
represented by the
car named Bubble Aku is travelling in,
does not allow a zero-entropy measurement, and
subsequently the time-axis starts stretching
out towards infinity. Our objectives or at least one of them, is to come up the description of a topological space which is being measured by Aku, and this topological space must be time independent. Can we figure out based on Aku's measurements which are assisted by precision observers 133Cs and HeNe, a constant which is time independent? The universal constants available to Aku are h, c, and G. We can also include kB, but it does not represents relativistic limits.3
![]() jML
represents a constant in discrete-measurement
space or j-space. The subscripts ML
remind us that the constant jML,
has
the dimensions of mass multiplied by length,
and it is independent of time.
Next question to ask,
is what does jML really
represent? Turns out that we do not have
to go far. We are already familiar with
Compton wavelength, a property associated
with a particle in its relativistic
limits. Compton wavelength is given
as:
In
this formula, h is the Planck
constant, m
is the mass associated with a particle or a
structure , and c is the velocity of
light. An
interesting topic of discussion would be the
nature of the mass m, whether it is inertial
or gravitational. For the time being, we
assume that the inertial and the gravitational
mass are equal in topological space.
Therefore we can write jML
in terms of Compton wavelength λCompton as:
![]() The
dimensions of jML
remain
mass ×
length and hence timeless.
At
this point it will be prudent to discuss the
significance of Compton wavelength.
The physics behind Compton scattering can be
found in the Wikipedia article here.
We want to
understand how λCompton
provides a better understanding of
structures with different information
contents when they are measured in j-space,
and if jML
can truly be considered a topological
constant in a discrete measurement space or
j-space.
In terms of
photon-scattering, λCompton
represents
the wavelength which combined with the angle
of scattering provides with the shift in the
wavelength of an photon after it is
scattered by a target. Important fact
to note is, that this shift in wavelength is
independent of the energy of the incoming
photon, and it solely depends on the target
and angle by which the photon is
scattered. In
essence we are talking about relativistic
region in measurement space, and indeed it
were the relativistic considerations
which led to the derivation of Compton
wavelength.
One of the basic
assumption in the classical theory of
fields is that the source of a field
(for example, charge or mass), is an
ideal point-particle. λCompton
represents
the distance below which a particle can not
be considered a single point-like
particle. Below this value the
particle must be treated as a system with an
inner mechanism, which means that spin must
be accounted for while calculating total
angular momentum of what used to be a point
particle in classical limits (> λCompton). The
interaction between constituents becomes
important and the energy values are in
relativistic regime (E2 = p2
+ mo2; c =1).
Subsequently the photons with energy greater
than λ-1Compton,
will be able to generate electron-hole
pairs, rather than just knocking electrons
off from atomic orbits.
λCompton
also
represents the distance below which virtual
particles are allowed to exist. Hence
in terms of nuclear forces, the region below
λCompton
represents the range of the nuclear
forces. And finally, in the region
below λCompton,
the Euclidean space does not exist.
The measurement space corresponding to
Euclidean geometry, will exist only in the
region, whose dimensions are greater than λCompton.
We keep in mind that we arrived at Compton
wavelength, by using a simple argument that
zero-entropy measurements represent the
non-existence of the time-axis.
Subsequently we should be able to
derive a constant out of the known universal
constants, such as h, c, G, which
are measured in j-space, and this derived
constant we called jML,
would represent the topological space for
the observer making measurements. For sanity check, we have to verify that the value of jML does not change for different known particles. Furthermore since we want to identify the topological space underlying the discrete measurement space with jML, we must associate jML with all known structures in j-space, from black-holes to elementary particles. We will perform the sanity check next. ___________________
1. An observer here, may mean a
group of billions of amoebae inside Bubble,
bound by Lorentz Invariance and Möbius
Transformations. 2. Zero entropy measurement is
equivalent to the condition ds = 0,
where ds is the space-time interval given as ds2 = gμν
dsμdsν.
The zero-entropy measurement is performed
by Obsc (v/c ~1) or
equivalently photons. In j-space we write
zero entropy measurement as ds = 0j,
i.e. ds
is not exactly zero, but well below the
observer's measurement threshold, hence
considered zero for the particular measurement
space in classical limits. 3. A discussion on the
correlation between Boltzmann's constant kB
and Planck's constant h and their significance
in j-space, can be found here. *** |
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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