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An
Ecosystem of δ-Potentials - III Blip: A Special Discrete Measurement Space?
22nd April 2022
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"... we must first base such words as “between” upon clear concepts, a thing which is quite feasible but which I have not seen done." - Carl Friedrich Gauss "An interesting outcome of this discussion is the conclusion that the measurement of extremely small distances is physically impossible. The mathematician defines the infinitely small, but the physicist is absolutely unable to measure it, and it represents a pure abstraction with no physical meaning. If we adopt the operational viewpoint, we should decide to eliminate the infinitely small from physical theories, but, unfortunately we have no idea how to achieve such a program." - Leon Brillouin, Science and Information Theory, Academic Press. "We must have a mathematical theory which, in some way, will represent a suitable mathematical model or idealization and enable us to predict in a coherent way- in much the same manner as physics has always done - what the outcome of experiments will be if we are given correctly all the conditions that fully characterize the nature of the experiment." - Julian Schwinger, Quantum Mechanics: Symbolism of Atomic Measurements, Springer. "In Euclid's time, as now, there was a conceptual gulf between geometry and number theory-between measuring and counting, or between the continuous and the discrete, The major reason for this gulf was the existence of irrationals,....." - John Stillwell, A Concise History of Mathematics for Philosophers, Cambridge Elements. ***
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So far we have discussed various structures present in the discrete measurement space, as measured by the macroscopic observer ObsM. We also discussed the e- interactions based tools available to ObsM, along with their fundamentals as we understand them at present. The idea has been to build a geometric space, time⊗space, based on the physical measurements. We shall not be using time and space as independent entities, unless we were in Galilean frame of reference1.
Before continuing with our discussion, we postulate the following:
![]() This assumption forms the core of the discrete measurement space. It is equivalent to saying that a measurement is not complete, until the observer ObsM has arrived at the point Q from P. 2. Every entity, simultaneously an observer and an object, will measure the curve of the least disorder.
We are going to discuss the actual mechanism, the observers will be using while making measurements in j-space. In other words, how a δ-potential measures another δ-potential in j-space fabric2? The concepts from the Special Theory of Relativity will form the core of this discussion. A brief review is provided here as a reference.
We first assume that one of these δ-potentials is the macroscopic observer ObsM itself. Each of these observer is equipped with its own Hamiltonian, in his/her own frame of reference.
Next, we assume that ObsM is in the frame of reference S and the observer Obs'M being measured is in the frame of reference S'. Both frame of references are inertial frames. Both ObsM and Obs'Mare measuring the path PQ, where P and Q now represent two events. The observer in S' moving with velocity u is assumed to have more resources available or its Hamiltonian is more powerful than that of the observer in S. Both the observers are measuring an infinite source, however to simplify our arguments we do not consider accelerated frames of reference. For the same reason we also assume to = t'.
Note that the parameter t from the reference frame S is not a part of the measurement scheme in special relativity. As mentioned earlier we consider t only when we consider Galilean frame of reference of motion. Same applies to the space intervals in S. ![]() ![]() In above equations, the
quantity ds is time like interval. lo
and to are the proper length and
proper time as measured in S, whereas l and
p are length and time as measured in the
moving frame S'.
- The concept of
simultaneity plays an important role in the
Special Theory of Relativity. If events
measured in two frame of references, S and S' are
simultaneous with the absolute precision3,
the observers in S and S' have equivalent
measurement capability. In j-space language
we say that both observers have identical entropy
contents in their respective Hamiltonians.
- The speed of light c,
remains constant in S and S' frame both. In
j-space language the measurements made by
observers in S and S' frame, must obey the rules
established by the measurements made by the
observer Obsc.
(i) It is equivalent to saying
that the ground state of the system representing
Blip, is defined by Obsc. And if the coordinates in S
and S' frames are represented by (x,y,z,t) and
(x',y,'z',t') respectively, then:
![]() (ii) The velocity of the light
remains constant independent of the reference
frame, also sets a limit on the capabilities of
the measurement equipment used by the
observers. Hence observers in S and S'
frames would be measuring identical
length-contractions and time-dilation, as they
measure each other.
(iii) This fact reflects in the values of the physical constants, k, h and c, which are same for observers in S and S' frames. - We remind ourselves that the statistical world of Blip exists in the exterior region-1 of Kruskal-Szekeres (KS) Coordinates. This is where the thermodynamics of the system measured in the proper coordinates, becomes important 4. The following results refer to the measurements made in the proper coordinates. The proper coordinates are, the coordinates referring to the path PQ being measured by ObsM, in S frame of reference.
![]() It is an important
result as it ensures the stability
of the j-pixel. If Po
changed relativistically, then we could have
found a frame of reference S' for which the
j-pixel would have literally exploded in
S.
Vo and Po are the volume and the pressure being measured in proper coordinates respectively. V and P are the volume and the pressure as measured in the moving frame S' respectively. It is straightforward that the volume of j-pixel is infinitesimally small, as it is determined by the measurements of Obsc (u ~ c).
![]() Very very cold, isn't
it? We also note that from the
relations stated above:
![]()
S = So
.
Above relation again ensures that the speed of light c is constant in all reference frames. However if we consider the entropy density φ of an infinitesimal volume δV, such that S = φ.δV. Then the proper entropy density φo and the entropy density in S'-frame φ, are correlated as: ![]() Above
provides an important insight, as
our j-pixels by definition have
finite volume even though their
volume is in Planck's domain.
More on it later.
m =
γ(u)
mo
.
The kinetic
energy E of a particle of rest mass mo, which is
moving with a velocity u, is given asE = c2 (m -
mo)
,
dE = c2 dm . We note that we have considered only the elastic collisions, while calculating above relations. But in j-space and the life in general, perfect elastic collisions are not possible. There has to be some elastic deformation and potential energy associated with it, when a collision between particles takes place, even though these particles are measured as identical by ObsM. We can set a lower bound on the measured value of the combined mass M as: ![]() The assumptions made in deriving above equation are that both particles are at rest relative to each other in S'-frame which is moving with a velocity u, and both particles have the rest mass mo. The lower bound symbolizes the fact that the collision is assumed to be of an infinitesimal duration in S-frame.
But then what if this deformation
is not of infinitesimal duration,
and instead it extends across the
epoch time instead? How
would an observer in S-frame would
measure its effect on the mass?
MZE is
likely to behave like a structure
with a massively high value.
Which in essence is saying that
the structures with very high
information contents will behave
as massive bodies when
interacting with light. We
have to include the effect of some
sort of microscopic elastic
deformation in Planck's domain and
the impact of the associated
potential energy in to the value
of mass in j-space. We will
continue this discussion further.
.....To
be continued.
![]() 1. So in essence we develop geometry from measurements rather than assuming it to preexist by default. Further if we remove irrational numbers from our argument, we can not use the relationship ds2 = dx2 + dy2, as Pythagorean theorem is no longer in the picture (yet). 2. We note that the vacuum-state specific to an observer, is formed by the measurements made by the observer in j-space fabric. For example the conventional ground state in Quantum Mechanics, is based on the measurements made by Obsc. 3. The absolute precision in Blip, is determined by the observer Obsc. In time-space coordinates, ds = 0 or 0j, per measurements made by Obsc. However ds will be measured as finite by a macroscopic observer. (ds2 = -dx2-dy2-dz2+c2dt2) 4. A rather superb treatment of the thermodynamics of a relativistic system is provided in "Relativity Thermodynamics and Cosmology" by Richard C. Tolman. *** |
Previous Blogs: An Ecosystem of δ-Potentials - II An Ecosystem of δ-Potentials - I Nutshell-2019 Stitching the Measurement Space - III Stitching the Measurement Space - II Stitching the Measurement Space - I Mass Length & Topology 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 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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