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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-γγ interactions based tools available to ObsMalong with their fundamentals as we understand them at present.  The idea has been to build a geometric space, timespace,  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:


1.  Every entity, simultaneously an observer and an object, is in a state of measurement and will continue to measure until the measurement is complete.

 

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.

  •     This is our definition of the unification in a discrete measurement space.
  •     This particular assumption is based on a straightforward principle, which states that resources available to an observer to make a measurement must be optimized, or equivalently the entropy inherent in the travel from P to Q must be minimized. 

  •    Our eventual goal, or one of them, is to determine if phenomena observed in nature derive from this simple requirement.  And if that is the case, can we explain structures and their interactions as measured in the ObsM's rest frame S?
  •   We have discussed the measurement of the curve of least disorder in j-space earlier. We recall that the curve of least disorder is formed by combining infinite-many infinitesimal curves, each with its own curvature, by matching their respective tangents.

  •   Each of these infinitesimal curves, in essence represents a path "PQ".  The curvature of a curve, represents the difficulty an observer faces while making a measurement.  Or in other words, the curvature represents the entropy in the measurement.  If we can measure the curves of least disorder, we can generalize the argument to the surface and the volume, using Stokes and Gauss theorems.   

     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.


 

 
     We can write the Lorentz factor in four-coordinates (t, x, y, z) as:
 
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.
 
  • The proper volume Vo contracts, but the proper hydro-static pressure Po remains unchanged. 
    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).

 
  • Energy (E), Work (W), Heat (Q), and Temperature (T) transformations are given as:
Very very cold, isn't it?  We also note that from the relations stated above:
 
 
  • Finally we discuss two very important concepts in j-space, Entropy and Mass.  Entropy relates to the internal property of a given system in an inertial frame.  We can simply write:


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.
 
  • In Special Relativity, the conservation of momentum allows the association of the kinetic energy of a particle to its mass.  Therefore if the rest mass of a particle is mo and it is moving with a velocity u, then the mass of the moving particle is given as:
m = γ(u) mo .

The kinetic energy E of a particle of rest mass mo, which is moving with a velocity u, is given as

E = 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?

    The discrete-measurement space is based on the measurements made by an observer with a finite capability.  The possibility to have a system with capability to be at complete rest, viz.,
j potential energy, is a part of the picture in j-space.  Therefore the contribution to the mass in S-frame comes from kinetic energy and potential energy together.  This mass was measured by Obsi and we had called it  Zero-Entropy mass MZE

    The problem for us, is that we do not know how to incorporate MZE in the description presented so far? 
We simply can not go on assuming mass values based on the contributions from the kinetic energies only.  It is fairly certain that MZE can not be measured in the exterior region-1 of KS-Coordinates or the statistical-thermodynamic world as we know it. 

    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 sum up:

1.   Unification in j-space means the agreement between all frames of references on measuring the curve of least disorder.  The physical structures and laws governing the interactions between these structures, should emerge from this requirement.


2.    Special Relativity provides us with an extremely sophisticated machinery to develop the concepts of j-space.  Each δ-potential becomes an observer and is assigned a frame of reference.  While Special Relativity refers to inertial frames, ideas developed should provide us with a much better understanding of phenomena occurring in the non-inertial frames or the accelerated frames.


3.   The δ-potential being measured, can be single or composite in nature.  The first postulate ensures that both cases are possible.


4.   We must incorporate the effect of the potential energy along with the kinetic energy on calculating the mass of a given body.  We do know about it in a way,  in the relationship between the binding energy and the mass defect.

.....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.

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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

A Timeless Constant

Space Time and Entropy

Nutshell-2018

Curve of Least Disorder

Möbius & Lorentz Transformation - II

Möbius & Lorentz Transformation - I

Knots, DNA & Enzymes

Quantum Comp - III

Nutshell-2017

Quantum Comp - II

Quantum Comp - I

Insincere Symm - II

Insincere Symm - I

Existence in 3-D

Infinite Source

Nutshell-2016

Quanta-II

Quanta-I

EPR Paradox-II
 
EPR Paradox-I 

De Broglie Equation

Duality in j-space

A Paradox

The Observers
 
Nutshell-2015
 
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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