

An Ecosystem of δPotentials  I An ethereal energy source in Blip
19^{th}^{} July 2020

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"Why, for example, do we still not have an international center for climate predictions, which by current estimates would cost “only” $1 billion spread over 10 years? That’s peanuts compared to what particle physics sucks up, yet vastly more important. Or why, you may have wondered recently, do we not have a center for epidemic modeling? It’s because too much science funding is handed out on the basis of inertia. In the past century, particle physics has grown into a large, very influential and wellconnected community. They will keep on building bigger particle colliders as long as they can, simply because that’s what particle physicists do, whether that makes sense or not. It’s about time society takes a more enlightened approach to funding large science projects than continuing to give money to those they have previously given money to. We have bigger problems than measuring the next digit on the mass of the Higgs boson."  Sabine Hossenfelder in The World Doesn’t Need a New Gigantic Particle Collider, Scientific American, June 19, 2020. ***

So
far we have developed the idea of a discrete measurement space or
jspace, which is solely based on the physical measurements of an infinite source. Since physical measurements do not allow the precise determination of
origin, jspace has to have lower bounds for the measurements of
various physical quantities. Similarly upper bounds must also
exist.
In jspace the resource management is very important and different observers have different resources and thus different capacities to perform precise measurements. The observer Obs_{c}, with measurement capacity (v/c ~1), determines the lower and the upper bounds, based on which the guidelines for the measurements in Aku's universe Blip, are established. The fabric of jspace consists of jpixels, which we have discussed earlier. In next few blogs we will be discussing, how to introduce physical structures, cosmic and elementary both, in jspace fabric. We will be using basic quantum mechanics concepts, such as Levinson's theorem, bound states and δpotentials, along with KruskalSzekeres coordinates, in jspace. δpotentials and bound states: The presence of δpotentials and subsequent bound states for a particle, is a standard problem in Quantum Mechanics. The case of a single delta potential and the corresponding bound state for a particle of mass m, is shown below: The bound state E_{b}
is same as the ground state if the potential is symmetric.
If there are more than one delta functions, there can be more than one
bound states depending upon the magnitude of αδ(x). Our objective in next few blogs, is to discuss the basic conjecture
behind the discrete measurement space or jspace, that every
object in jspace is in a state of measurement, thus performing a
measurement as well as being measured itself.
Why do elementary particles exist in jspace, to begin with?
Let us review the expression for the bound state energy in the presence of a delta function: The important quantities here are m, α,
and ħ. But first we need to understand what do we mean when we
make the statement that "a particle of mass m exists"?
In a measurement space, a measurement is completed when the corresponding circuit is completed i.e. the observer has arrived back at its point of origin. In the discrete measurement space the objective is to measure the LeastEnergy Surface (LES) with the absolute precision, but clearly not all the observers can perform this measurement. We also note that the physical world exists in regionI of KruskalSzekeres coordinates, shown below: The position of the measurement circuit in KruskalSzekeres coordinates is around the EventHorizon Point (EHP), R_{sch}
= 2MG. In the following diagram various measurement circuits
around EHP are shown. We note that the conventional space and
time exist in the external Region1 only, or in other words a macroscopic observer Obs_{M}, exists in the external RegionI only.
The
measurement circuits are shown in blue. The macroscopic observer's capability
to make measurements is decreasing as we move from the left to right in the external Region1,
and it is represented by the hyperbolic paths in black.
Sharper the curvature at the vertex, more difficult the
measurement, i.e. more and more resources are needed to make a
measurement as we move from the right to the left in the external Region1.
In jspace, the measurement circuit is around LES
and the intersection of an hyperbola and an arc of measurement circuit
(shown in red) in the external Region1, represents the measurement of an elementary
particle, made by Obs_{M}^{1}.
Thus we have elementary particles corresponding to each redarc (or
equivalently the part of a bluemeasurement circuit in RegionI) which
potentially can be measured by a macroscopic observer Obs_{M}, given Obs_{M} has
sufficient resources to measure it^{}^{2}. In the following diagram, an observer can measure m_{2} and m_{3}, but not m_{1}.
To measure the particle m_{1},
the observer making measurement needs to be provided additional
resources in the external Region1. This situation is shown by
the dashed path in the following diagram.
This is what particlecolliders are trying to do in a rather rudimentary way. However
it must be clearly understood that no matter how much resources are
provided to an observer making measurements in the external region1,
LES can never be measured with an absolute precision. At best we
will have an enormous catalog of particles, afflicted with la maladie exotique,
but no new fundamental principle is likely to be discovered. For
that we will have to analyze the results obtained by Astronomy and Cosmology.
Nature of bound states in jspace Let us consider an observer Obs_{M} in jspace, with resources in the range (E_{i}, E_{f}). Providing a precise value of the energy to an observer in jspace is not possible, as it implies ΔE = 0. According to uncertainty principle, ΔEΔt > ħ, ΔE = 0 means a state with infinite lifetime is measured by Obs_{M}. An infinite lifetime measured by Obs_{M}, is finite (∞_{j}) per the measurements of Obs_{c} or Obs_{i}, and hence Δt is always finite and subsequently ΔE is always greater than 0.
The observer Obs_{M }will be measuring the potentials ranging from a shallowwell to a deepwell as shown below: We note that as the depth of the potential well increases the observer Obs_{M} has less and less freedom in the free space.
Next we consider an observer Obs_{c} who has infinite, ∞_{j}, resources in jspace. This observer Obs_{c} will have values for energy ranging from 0_{j} to ∞_{j}, in jspace corresponding to the current information space (q = 3). We want Obs_{c} to measure a shallow potential well in a higher information space (q = 2).^{3} In other words we want Obs_{c}, if possible, to perform a zero entropy measurement in q = 2 information space. For Obs_{c} since it has ∞_{j} resources, all the potential wells in q = 3 information space, are equivalent to shallowwells and represent zeroentropy measurements. However in the case of q = 2 information space, the situation becomes rather interesting. A shallowwell in q = 2 space is equivalent to δpotential in q = 3 space even for Obs_{c}. And δpotential is, what Obs_{c} is measuring even though Obs_{c} has infinite, ∞_{j}, resources in q = 3 space, as shown below: In this case, all the states are inside the δpotential and they form in essence a single bound state or a "coherent" state of the width ΔE ~ 0.
Thus the nature of the bound state in q = 3 measurement space, while
measuring a shallow well from q = 2 space, is that of a coherent state E_{coh}.
The measurement of δpotential in Blip Our interest is in understanding how the measurements of the coherent state E_{coh}, manifest themselves into physical parameters measured in Blip. Blip is represented by the measurements made by the macroscopic observer Obs_{M}, in the external regionI in KS coordinates. Since the energy of the coherent state E_{coh} in q = 3 information space, has a minimum limit ∞_{j}, the corresponding rest mass m_{o} = E × (c^{2})^{1}, will be infinite or extremely heavy, even though the physical dimensions may be in Planck's domain, in the q = 3 space or equivalently in the external regionI of KS Coordinates. The coherent state will have a minimum spread ΔE_{coh} ~ 0j, per Obs_{M} measurements. The uncertainty principle correlates ΔE_{coh} with the coherent state lifetime t_{coh} in external regionI, as ΔE_{coh} × t_{coh} ~ ħ. Since ΔE_{coh} is very small, the coherent state lifetime t_{coh} will be exceeding large (~∞_{j}). Next, in external RegionI, an infinitesimal change in entropy of the coherent state dS_{coh} and the change in the energy to bring about the infinitesimal change dS_{coh} in the entropy dE_{coh}, are correlated to the temperature T_{coh} of the coherent state as: It will take infinite amount of resources (dE_{coh} ~ ∞), to bring about infinitesimal change in the entropy (dS_{coh}
~ 0), of the coherent state under discussion. Hence the
temperature of the coherent state as measured by the macroscopic
observer Obs_{M}, will be exceedingly high.
Finally an infinite energysource in Blip Therefore in the discrete measurement space, the shallowwell from a higherinformationspace_{q=2} is equivalent to a δpotential in the currentinformationspace_{q=3}. The information corresponding to the shallowwell_{q=2}, will be measured as a coherentstate_{q=3} by a macroscopic observer Obs_{M}, in Blip (equivalently q=3 information state or the external regionI of KS coordinates). Lastly the restmass and the temperature of the coherent state, as they are determined in Blip, are statistical quantities inversely related to the spread in the energy ΔE_{coh}, of the coherent state_{q=3}, similar to the lifetime t_{coh}_{}. ____________________
1. The physical world of Blip, exists at Λ_{}plane. 2. Action in jspace is defined as the resources provided to an observer traveling the path PQ, at the point P, such that the observer can arrive at the point Q unassisted. In external regionI of KS coordinates, action translates into the physical variables, energy E or the momentum k. 3. Different information spaces do not agree upon the definitions of Lorentz and Möbius invariances. *** To be continued.. 
Previous Blogs: Nutshell2019 Stitching the Measurement Space  III Stitching the Measurement Space  II Stitching the Measurement Space  I Mass Length & Topology Chiral Symmetry
Sigmaz and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous zAxis Majorana ZFC Axioms Set Theory Nutshell2014 Knots in jSpace Supercolliders Force Riemann Hypothesis Andromeda Nebula Infinite Fulcrum Cauchy and Gaussian Distributions bField & 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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