

A Timeless Constant
9^{th} May 2019

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"An essential feature of measurement is the difference between the “determination” of an object by individual specification and the determination of the same object by some conceptual means."  Herman Weyl, Space Time Matter "One could also ask a slightly different question. What are the principles that govern the interaction of a microscopic system with a classical system? One would like to know whether it is possible to define classical physical systems that cannot interact with quantum systems." "Is colour metabolic efficient or coding efficient?"  James V Stone, Principles of Neural Information Theory. "And so these men of Indostan  Elephant and the blind men, a childhood story. ..And it took a Witten to put the Melephant together. 
In jspace the perception of space and time, are based on the measurements made by an observer, and therefore they are specific to the observer. Different observers, unless they are following Lorentz Invariance condition, may have measurements which do not agree with each other. Hence they may view universe differently. For example an observer named Taku may not be able to measure certain features of the universe, features which can be measured by Baku, another observer with higher measurement capacity. Further the existence of time is required, before the measurement of space itself can be thought of. This requirement is crucial as the conventional space can not exist without the existence of an observer to make measurements. For a give topology or equivalently a given path, each observer^{1} comes with its own timeaxis based on the observer's measurement capacity. Each observer is measuring the infinite source or the exact source. The observers obeying the Lorentz invariance, make measurements along a congruent timeaxis. In the previous blog, we highlighted this problem of time itself not being a universal entity. You see, each measurement is supposed to be a zeroentropy measurement, and hence no timeaxis^{2}. But entropy inherent in the system, a system represented by the car named Bubble Aku is travelling in, does not allow a zeroentropy measurement, and subsequently the timeaxis starts stretching out towards infinity. Our objective or at least one of them, is to come up the description of a topological space which is being measured by Aku, and this topological space must be time independent. Can we figure out based on Aku's measurements which are assisted by precision observers ^{133}Cs and HeNe, a constant which is time independent? The universal constants available to Aku are h, c, and G. We can also include k_{B}, but it does not represents relativistic limits.^{3}
j_{ML} represents a constant in discretemeasurement space or jspace. The subscripts ML remind us that the constant j_{ML, }has the dimensions of mass multiplied by length, and it is independent of time. Next question to ask, is what does j_{ML }really represent? Turns out that we do not have to go far. We are already familiar with Compton wavelength, a property associated with a particle in its relativistic limits. Compton wavelength is given as:
In this formula, h is the Planck constant, m is the mass associated with a particle or a structure , and c is the velocity of light. An interesting topic of discussion would be the nature of the mass m, whether it is inertial or gravitational. For the time being, we assume that the inertial and the gravitational mass are equal in topological space. Therefore we can write j_{ML} in terms of Compton wavelength λ_{Compton }as:
The dimensions of j_{ML }remain mass × length and hence timeless. At this point it will be prudent to discuss the significance of Compton wavelength. The physics behind Compton scattering can be found in the Wikipedia article here. We want to understand how λ_{Compton} provides a better understanding of structures with different information contents when they are measured in jspace, and if j_{ML} can truly be considered a topological constant in a discrete measurement space or jspace. In terms of photonscattering, λ_{Compton }represents the wavelength which combined with the angle of scattering provides with the shift in the wavelength of an photon after it is scattered by a target. Important fact to note is, that this shift in wavelength is independent of the energy of the incoming photon, and it solely depends on the target and angle by which the photon is scattered. In essence we are talking about relativistic region in measurement space, and indeed it were the relativistic considerations which led to the derivation of Compton wavelength. One of the basic assumption in the classical theory of fields is that the source of a field (for example, charge or mass), is an ideal pointparticle. λ_{Compton }represents the distance below which a particle can not be considered a single pointlike particle. Below this value the particle must be treated as a system with an inner mechanism, which means that spin must be accounted for while calculating total angular momentum of what used to be a point particle in classical limits (> λ_{Compton}). The interaction between constituents becomes important and the energy values are in relativistic regime (E^{2} = p^{2} + m_{o}^{2}; c =1). Subsequently the photons with energy greater than λ^{1}_{Compton}, will be able to generate electronhole pairs, rather than just knocking electrons off from atomic orbits. λ_{Compton }also represents the distance below which virtual particles are allowed to exist. Hence in terms of nuclear forces, the region below λ_{Compton} represents the range of the nuclear forces. And finally, in the region below λ_{Compton}, the Euclidean space does not exist. The measurement space corresponding to Euclidean geometry, will exist only in the region, whose dimensions are greater than λ_{Compton}. We keep in mind that we arrived at Compton wavelength, by using a simple argument that zeroentropy measurements represent the nonexistence of the timeaxis. Subsequently we should be able to derive a constant out of the known universal constants, such as h, c, G, which are measured in jspace, and this derived constant we called j_{ML}, would represent the topological space for the observer making measurements. For sanity check, we have to verify that the value of j_{ML} does not change for different known particles. Furthermore since we want to identify the topological space underlying the discrete measurement space with j_{ML}, we must associate j_{ML }with all known structures in jspace, from blackholes to elementary particles. We will perform the sanity check next. ___________________
1. An observer here, may mean a group of billions of amoebae inside Bubble, bound by Lorentz Invariance and Möbius Transformations. 2. Zero entropy measurement is equivalent to the condition ds = 0, where ds is the spacetime interval given as ds^{2} = g_{μν }ds^{μ}ds^{ν}. The zeroentropy measurement is performed by Obs_{c} (v/c ~1) or equivalently photons. In jspace we write zero entropy measurement as ds = 0_{j}, i.e. ds is not exactly zero, but well below the observer's measurement threshold, hence considered zero for the particular measurement space in classical limits. 3. A discussion on the correlation between Boltzmann's constant k_{B} and Planck's constant h and their significance in jspace, can be found here. ***

Previous Blogs: 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 Discrete Space, 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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