Ecosystem of delta potentials 1

    So far we have developed the idea of a discrete measurement space or j-space, which is solely based on the physical measurements of an infinite source. Since physical measurements do not allow the precise determination of origin, j-space has to have lower bounds for the measurements of various physical quantities. Similarly upper bounds must also exist.

    In j-space 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 (v²/c² ~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 j-space consists of j-pixels, which we have discussed earlier. In next few blogs we will be discussing, how to introduce physical structures, cosmic and elementary both, in j-space fabric. We will be using basic quantum mechanics concepts, such as Levinson's theorem, bound states and δ-potentials, along with Kruskal-Szekeres coordinates, in j-space.

δ-potentials and bound states:

The presence of δ-potentials 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 assumption behind the discrete measurement space or j-space, that every object in j-space is in a state of measurement, thus performing a measurement as well as being measured itself.

Why do elementary particles exist in j-space, 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 Least-Energy 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 region-I of Kruskal-Szekeres coordinates, shown below:

The position of the measurement circuit in Kruskal-Szekeres coordinates is around the Event-Horizon 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 Region-1 only, or in other words a macroscopic observer Obs_M, exists in the external Region-I 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 Region-1, 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 Region-1.

In j-space, the measurement circuit is around LES and the intersection of an hyperbola and an arc of measurement circuit (shown in red) in the external Region-1, represents the measurement of an elementary particle, made by Obs_M¹. Thus we have elementary particles corresponding to each red-arc (or equivalently the part of a blue-measurement circuit in Region-I) 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 m₂ and m₃, but not m₁.

To measure the particle m₁, the observer making measurement needs to be provided additional resources in the external Region-1. This situation is shown by the dashed path in the following diagram.

This is what particle-colliders 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 region-1, 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, and predict, the results obtained by Astronomy and Cosmology.

Nature of bound states in j-space

Let us consider an observer Obs_M in j-space, with resources in the range (E_i, E_f). Providing a precise value of the energy to an observer in j-space is not possible, as it implies ΔE = 0. According to uncertainty principle, ΔEΔt > ħ, ΔE = 0 means a state with infinite life-time is measured by Obs_M. An infinite life-time 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_Mwill be measuring the potentials ranging from a shallow-well to a deep-well 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 j-space. This observer Obs_c will have values for energy ranging from 0_j to ∞_j, in j-space 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 shallow-wells and represent zero-entropy measurements. However in the case of q = 2 information space, the situation becomes rather interesting. A shallow-well in q = 2 space is equivalent to δ-potential in q = 3 space even for Obs_c. And δ-potential 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 δ-potential 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 δ-potential 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 region-I in K-S 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 × (c²)^-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 region-I 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 life-time t_coh in external region-I, as ΔE_coh × t_coh ~ ħ. Since ΔE_coh is very small, the coherent state life-time t_coh will be exceeding large (~∞_j).

    Next, in external Region-I, 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 energy-source in Blip

Therefore in the discrete measurement space, the shallow-well from a higher-information-space|_q=2 is equivalent to a δ-potential in the current-information-space|_q=3. The information corresponding to the shallow-well|_q=2, will be measured as a coherent-state|_q=3 by a macroscopic observer Obs_M, in Blip (equivalently q=3 information state or the external region-I of KS coordinates).

This coherent state will result in an extremely massive entity in Blip, with an infinite life-time and infinite temperature. The infinite-ness of the coherent state E_coh, disappears as E_coh slowly radiates away, in the external Region-I (in a few billion years probably).

Lastly, the "measured-mass" 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 life-time t_coh.


	An Ecosystem of δ-Potentials - I An ethereal energy source in Blip 19^th July 2020	HOME
"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 well-connected 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 j-space, which is solely based on the physical measurements of an infinite source. Since physical measurements do not allow the precise determination of origin, j-space has to have lower bounds for the measurements of various physical quantities. Similarly upper bounds must also exist. In j-space 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 (v²/c² ~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 j-space consists of j-pixels, which we have discussed earlier. In next few blogs we will be discussing, how to introduce physical structures, cosmic and elementary both, in j-space fabric. We will be using basic quantum mechanics concepts, such as Levinson's theorem, bound states and δ-potentials, along with Kruskal-Szekeres coordinates, in j-space. δ-potentials and bound states: The presence of δ-potentials 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 assumption behind the discrete measurement space or j-space, that every object in j-space is in a state of measurement, thus performing a measurement as well as being measured itself. Why do elementary particles exist in j-space, 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 Least-Energy 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 region-I of Kruskal-Szekeres coordinates, shown below: The position of the measurement circuit in Kruskal-Szekeres coordinates is around the Event-Horizon 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 Region-1 only, or in other words a macroscopic observer Obs_M, exists in the external Region-I 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 Region-1, 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 Region-1. In j-space, the measurement circuit is around LES and the intersection of an hyperbola and an arc of measurement circuit (shown in red) in the external Region-1, represents the measurement of an elementary particle, made by Obs_M¹. Thus we have elementary particles corresponding to each red-arc (or equivalently the part of a blue-measurement circuit in Region-I) 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 m₂ and m₃, but not m₁. To measure the particle m₁, the observer making measurement needs to be provided additional resources in the external Region-1. This situation is shown by the dashed path in the following diagram. This is what particle-colliders 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 region-1, 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, and predict, the results obtained by Astronomy and Cosmology. Nature of bound states in j-space Let us consider an observer Obs_M in j-space, with resources in the range (E_i, E_f). Providing a precise value of the energy to an observer in j-space is not possible, as it implies ΔE = 0. According to uncertainty principle, ΔEΔt > ħ, ΔE = 0 means a state with infinite life-time is measured by Obs_M. An infinite life-time 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_Mwill be measuring the potentials ranging from a shallow-well to a deep-well 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 j-space. This observer Obs_c will have values for energy ranging from 0_j to ∞_j, in j-space 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 shallow-wells and represent zero-entropy measurements. However in the case of q = 2 information space, the situation becomes rather interesting. A shallow-well in q = 2 space is equivalent to δ-potential in q = 3 space even for Obs_c. And δ-potential 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 δ-potential 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 δ-potential 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 region-I in K-S 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 × (c²)^-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 region-I 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 life-time t_coh in external region-I, as ΔE_coh × t_coh ~ ħ. Since ΔE_coh is very small, the coherent state life-time t_coh will be exceeding large (~∞_j). Next, in external Region-I, 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 energy-source in Blip Therefore in the discrete measurement space, the shallow-well from a higher-information-space\|_q=2 is equivalent to a δ-potential in the current-information-space\|_q=3. The information corresponding to the shallow-well\|_q=2, will be measured as a coherent-state\|_q=3 by a macroscopic observer Obs_M, in Blip (equivalently q=3 information state or the external region-I of KS coordinates). This coherent state will result in an extremely massive entity in Blip, with an infinite life-time and infinite temperature. The infinite-ness of the coherent state E_coh, disappears as E_coh slowly radiates away, in the external Region-I (in a few billion years probably). Lastly, the "measured-mass" 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 life-time t_coh. ____________________ 1. The physical world of Blip, exists at Λ_-plane. 2. Action in j-space 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 region-I 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: 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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