

An
Ecosystem of δPotentials  II Corporeal Structures in Blip 22^{nd} December 2021

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"Physics is becoming so unbelievably complex that it is taking longer and longer to train a physicist. It is taking so long, in fact, to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them."  Eugene Wigner "The essential point is that, although coordinates are a powerful, and sometimes essential, tool in many calculations, the fundamental laws of physics can be expressed without the aid of coordinates; and, indeed, their coordinatefree expressions are generally elegant and exceedingly powerful."  Kip Thorne "Can one not go a step beyond Palatini and base a theory on affine connexion alone, which is after all the first and only one needed to obtain a basis for mathematical analysis?"  Erwin Schrödinger, SpaceTime Structure, Cambridge University Press, 1950. ***

Our objective is to introduce various physical structures corresponding to different information contents, in the jspace fabric. Sometime ago we discussed how a macroscopic observer Obs_{M }from IS_{q=3} (q = 3 information space), would measure a shallowwell from a higher IS_{q=2}, as a δpotential in the current IS_{q=3} (also known as Aku's universe Blip). In this blog, we shall continue this discussion for the formation of corporeal structures ^{}due to delta potentials of different information contents within IS_{q=3} ^{1}. We note that the definition of a delta potential in jspace is correlative, based on the measurements made by an observer, as discussed in the previous blog. We first however need to understand the concepts behind the measurement of time and the measurement of space, as performed by Obs_{M}. We can not take it for granted that time⊗space exist empirically, or it has a unique structure as we perceive it. The arguments presented next, are independent of the qvalue of IS. We shall consider the physical measurements in the external Region1 of KruskalSzekeres coordinates. We also remind ourselves of the basic assumption behind the discrete measurement space or jspace: "Every entity, simultaneously an observer and an object, is in a state of
measurement and will continue to measure until the
measurement is complete."
Building a physical space from the information space The question we will try to answer here is whether the physical space, can exist without any information content, keeping in mind that in jspace the information can be obtained only by measurements? Furthermore does the physical space combined with time contains all the information, which is there to be measured? In Blip, these measurements use the electronphoton interaction defined by the finestructure constant α (~1/137). As an example, we will be discussing the very first measurement towards building the conventional time⊗space in Blip^{2}. (Note: These measurements are performed by Obs_{M}.) The simplest measurement in jspace is equivalent to completing a circuit. The observer obeys VTSymmetry and arrives back at the point of origin. Let us call the completed circuit, Path AB. Here A and B are identical points, i.e. the points A and B have same information contents. Each eγ interaction has a momentum associated with it, which is related to the physical space as a Fourier Transform, assuming that the "Completeness" property is satisfied. In jspace, the completeness property implies that with a given Hamiltonian, the path AB can be measured with a complete precision. We note the truly brilliant induction experiment
correlating
relative motions of a physical body (magnet) and the magnetic coil connected to a source
(charge/current). Simply
put, if we move magnet the current is induced in the
magnetic coil and
if we introduce a current in the magnetic coil the
magnet is
moved. The end result for Obs_{M} does not change in either case,
except for the fact that a "relative" physical motion ( ≡ Path PQ),
has taken place^{3}.
Our objective is here to introduce the concept of the vector potential A, and the role it plays in the measurement of the infinitesimal path PQ. The curl of A results in the magnetic field B, which by the definition of A, is a solenoidal field. The presence of A allows an infinitesimal physical motion to take place when a Hamiltonian is activated. In a sense, we are discussing the physical mechanism underlying the classical geometry using a discrete measurement space. In geometry what is considered a single point, is actually the infinitesimal path PQ in jspace. Therefore the time and the space are intricately connected to each other, and we are referring to this certitude symbolically as "time⊗space". The nature of infinitesimal path will be different for observers with different Hamiltonians. Thus the description of time⊗space is strictly observer dependent. Once we have the measurements of infinitesimal arcs, we can build various structures in the geometrical space as we discussed earlier. Some important features of the vector potential A, which are of interest to us in jspace, are: 1. The vector field A is not measurable, however its curl is the magnetic field B which can be measured. We can also find a scalar potential φ, which combined with A, gives us the electrical field E, which again is a measurable quantity. 2. The vector potential A, satisfies a wave equation under a certain condition known as Lorentz Gauge. The wave equation is given as: Therefore in principle, if we made the measurements in our lab in the presence of a charge distribution, by studying the nature of waves (incident and scattered), we can make predictions about the charge distribution's properties. This is probably our earliest introduction to the topology underlying the physical laws of nature. We note that photons represent the solution to the above equation in homogeneous form, i.e. 3. We will notice that we are moving in the direction of the theory of Quantum ElectroDynamics (QED), where we are interested in describing the interaction at a single point in time⊗space, as precisely as possible. In QED, we have a moving charged particle described by the Schrödinger equation (H_{Sch}), we have electromagnetic fields described by the motion of photons (H_{EM}), and we have the interaction between these two pictures (H_{Interaction}). The complete Hamiltonian, which represents the resources available to an observer performing the measurements in jspace, is written as: We also note that H_{Interaction }is invariant with respect to translations and rotations in time⊗space, as the nature of the eγ interaction is universal for a given value of α. In short if we can write a Hamiltonian, it means that the measurements based on this Hamiltonian provide a valid description of time⊗space. 4. Assuming that we have been able to come up with a description of single one location in time⊗space, we have to ensure this description is consistent across the whole time⊗space in Blip. We achieve this by the condition of "Locality". The concept of Locality in essence, allows the Hamiltonian to be additive. So the measurements at different locations on a given geometry in time⊗space add up and we can then write H_{Total} for Blip itself. In ideal jspace, no two geometries in time⊗space will result in same H_{Total}, unless a symmetry is assumed. The locality allows same measurement procedure to be true for all points in time⊗space, if the condition of locality does not hold it means that Hamiltonian is not additive, and the presented definition of the time⊗space is not valid. It does not mean that the informationspace does not exist, it just means that Obs_{M} can not measure the information as time⊗space in Blip. And without a valid description of time⊗space, Blip can not exist. 5. The Hamiltonian H_{Total}, is sum of all possible interactions present in a physical situation across different information spaces. Of which, the measurements made using eγ interactions form a very small subset. For the description based on eγ interactions, we consider the contributions from very low energies or very large wavelengths. When we move to higher energies or shorter wavelengths, we move towards atomic and nuclear information domains. These structures represent higher informations spaces, and hence greater resources are required to make predictions about the structures present in these domains.^{4} 6. Finally, we need to understand that the QED based description for time⊗space, is for an infinitesimal slice of time for which the ever increasing entropy of a discrete measurement space, is ignored. However in reality that is not the case. As the time progresses the information obtained by each eγ interaction based measurement, is decreasing, whether an observer can measure this deterioration in α or not. Subsequently at some point further down the timeaxis, no more information will be obtained from the measurements, even though the timeaxis for the observer itself may be infinite. Thus the universe Blip as measured by Aku, is bounded. It is also very evident that the information obtained in the beginning of the epoch ( t = 0^{+}) by the measurements, is massive (α ~1). Thus the entropy based limitations inherent in a measurement system, will restrict the time⊗space description obtained by this system. Only way to improve the situation is to bring in new information into the system by reducing the qvalue of IS. This is equivalent to saying that Obs_{M} needs to figure out how to improve the value of the finestructure constant in Blip, which is our challenge. We are discussing the measurements made by Obs_{M} using the eγ interactions, in IS_{q=3}. Thus only corporeal structures are introduced in jspace fabric this time. The ethereal structures based on higher information spaces exist as coherent states in jspace, as discussed previously. We now have some intuitive understanding of the underlying time⊗space mechanism built into the jspace fabric, which is in the rest frameS of Obs_{M}. Next we can start introducing information δfunctions, into this fabric. Few facts: 1. The measurements to establish a geometrical space, are being made in the Sframe of reference, in which the macroscopic observer Obs_{M}, is at rest. The field(s) are established by the measurements made by force carriers, such as photons, mesons, gravitons, Higgs etc. We assign a tangential entity, to the each point on the manifold formed by each of the force carriers. The normal components in the field are necessitated by the presence of the δfunctions. In the absence of δfunctions in the neighbourhood, the normal components are assumed to be infinitesimally small. (Note: A field in jspace cannot exist without the presence of at least one δfunction.) 2. The δfunctions, are either singular or composite information structures which can not be completely measured by an observer with given resources. Thus the concept of a δfunction in jspace is a correlative concept, based on an observer's Hamiltonian. For example the δfunction deemed by Obs_{M}, is not necessarily a δfunction for Obs_{c} who has a much powerful Hamiltonian. 3. The observers and objects, both are examples of δfunctions. The infinite information source is the ultimate δfunction in jspace, which can not be precisely measured by Obs_{i}, Obs_{c}, or Obs_{M}. 4. Finite barriers represent the lowenergy measurements in the regionI of KruskalSzekeres coordinates. Similarly δpotentials represent the highenergy measurements. We are mainly interested in the structures formed by the measurements of δpotentials made by the observer Obs_{M }in the reference frame S. Our discussion on the vector potential A will not be complete without a brief introduction to KaluzaKlein theory. In this theory we use a 4vector vector potential A^{μ} = (φ/c, A) combined with a scalar field Φ, to unify the field theories of gravitation and electromagnetism. In this case a five dimensional metric tensor is written as, At present, our objective right now is to identify "the measurement" all observers in Blip are trying to make in a discrete measurement space. In jspace, this agreement between observers defines the "unification" in the measurement space. This particular discussion is far from over. _______________________
1. The q = 3 information space corresponds to α ~ 1/137. 2. We are not assuming the existence of either time or space. Only assumption right now is that a macroscopic observer is measuring an infinite information source with α ~ 1/137. 3. If we could find a similar interaction mechanism in a different spin space, nobody said that we could not do the same with celestial bodies. But then, that spin space has to be fermionic in nature. (Crazy stuff, please do not pay attention. Too much Star Trek!) 4. Furthermore, the Hamiltonian in the relativistic limits is not an invariant. Which is not surprising as different information spaces overlap with each other under the relativistic limits. This creates a problem, which is resolved by noticing that the action is an invariant over a given path. Therefore we use the Lagrangian formulation, which is easier to work with. We estimate the Hamiltonian next by integrating over all the possible trajectories for a given path, also known as the path integration formulation. The Lagrangian formulation represents a dynamic measurement space, which is also the nature of jspace. Therefore we seem to be in a reasonably good shape. *** 
Previous Blogs: An Ecosystem of δPotentials  I 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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