The question suggests several implicit answers: 1) the particle is one-dimensional, 2) the particle line is closed, 3) a vacuum "particle" of unit length has zero mass. However, this topic contains many unanswered questions: about the shape of the particle line, about the dimension and shape of the space in which the particle lines lie, about the formation of chains of particles, and so on.
Preston Guynn ,
Okay, let it be a rotational motion in Minkowski space. And if we compactify the Minkowski space into a product of spheres S^1xS^3, then what will be the rest mass of the particle?
Igor Bayak Igor, while I cannot state that (yet) for the mass of all particles, the mass of the electron (and with this positron) the both most fundamental particles of the (quantized!) atomic world and are for sure governed by the most simple rules of iSpace geometry, able to derive most but all constants of nature from simple integer geometric first order multiplicative base assumptions - so for sure not on mathematically continuum style approximating by design logarithm (of whatever entity):
Conference Paper iSpace - Exact Symbolic Equations for Important Physical Con...
Preprint iSpace - Quantization of Time in iSpace-IQ Unit-System by 1/...
Christian G. Wolf ,
And what is the essence (in words, not formulas) of your geometric approach?
mass and length have different units, and logarithm of only numbers (not units) make sense. So, is this even a valid question?
one may consider mass of a particle divided by some normalizing mass m0(planck mass, electron mass...) and length (or perhaps de-broglie wavelength or wavelength of photon whose mass-energy is equal to that particle) divided by some normalizing length L0(classical electron radius, planck length etc), then one can set that mass ratio equal to (10 or e-based) logarithm of length ratio. For e-based logarithm
m/m0 =ln(l/L0). Now, if mc2=ch/l; l=h/mc; or as per de-broglie formula, l=h/mv. Now putting the numbers m/m0 =ln(h/mvL0), or M=ln(C/M), where M=m/m0 , C=h/m0vL0 ; M*exp(M)=C You need to numerically solve this equation, the solution depending on the h/m0vL0 . You may use relativistic corrections like E2=(pc)2+(mrestc2)2 =(hc/l)2 for better results as well
Igor Bayak That is self evident - obvious - from what I wrote before already: If time is discrete and speed of light c is constant, then length is necessarily discrete as well. If length is discrete and angular momentum of hBar is discrete (which it is, accepted by all) then any momentum is discrete. As any force is minimal momentum over a length, force is discrete as well (at the exact same level of quantized granularity). So if length is discrete obviously area and volume are discrete and hence everything is discrete.
And in a primary discretized world consisting of quanta only, of course any model of spacetime - including relativity - is based on exact same quantized base geometry automatically, inherently whatever one calculates with pseudo 4D continuum geometry, there is no such thing as a continuum when looking at the core of things from a perspective of constants of nature.
Hence each and every math using any continuous aspect whatsoever like Euklidean-Pi, exponential function and logarithm is just an approximation as integration requires a continuum to operate (functions based on R or C), while as shown clearly above, only summation based approaches can ever be exact. The summation is not the approximation of integration, it’s exactly the other way round.
Nevertheless one can do very good approximations with e or log if objects are large enough - but not if they are quantum at nature.
Indeed, the length in our Euclidean space has a metric dimension, and for example, the length in an 8-dimensional space can have the dimension of action.
Discreteness and continuity do not always contradict each other. Take, for example, the winding of a circle (into which the spiral degenerates), with an integer number of turns. Isn't winding a continuous process and isn't the result of winding discrete.
Igor Bayak Igor, very wise insight indeed with "Discreteness and continuity do not always contradict each other" - but - on the other hand sometimes they do - and then with massive consequences (for theories, values, etc.).
Best example I can only recommend to study in detail, is starting with "unquantized" h or hBar (action) and getting extremely wrong ultraviolet catastrophe, but when/with (Planck) properly quantising h or hBar fully correct blackbody radiation curve arises from "seemingly out of nowhere".
The action is discrete only for an observer who crosses the particle line. Therefore, there is really no contradiction here.
Igor Bayak Igor, I cannot unfortunately be so optimistic like you are with respect to all the nasty details, especially given that I recently read the following paper of @ Lev Verkhovsky about a (very plausible!) *omission* in theory of relativity inherited already from day one but also kept by all over to our present time over decades, which is basically an *additionally* required term multiplying in required correction for the doppler effect (simple as its sounds), but imho such changes a lot in relativity, if not everything:
Preprint Memoir on the Theory of Relativity and Unified Field Theory
That said, I am way outside my own well known domain of constants of nature, so assuming you know relativity as is accepted mainstream by now way better than I do (or likely better than I ever will know, for that matter). Nevertheless, you should have a deep look on above link before doing too many premature assumptions about relativity in its present form (based on).
And - as usual - a particles "length" is something tricky to define when all current measurement results tell us such should be "(invalid!) "zero-sized" so called "point particles" in most but all other theories (except iSpace theory and few reasonable others, where the size, length and mass of everything is most clearly simple yet effective metrically defined in integer geometric multiples of well know and well defined base values like GoldenRatio).
In the article
Preprint Chaotic dynamics of an electron
there is a conclusion of the heuristic formula for the masses of leptons.
Perhaps I will explain why, when deriving the heuristic formula for the masses of leptons, a golden spiral is taken, in which the polar radius of each quarter turn is equal to the Fibonacci number. The fact is that in one revolution of the (1,1)-toric knot, which is drawn on a torus stretched over a sphere with the poles gouged out, we pass from one pole to the other four times.
Просто бред. Масса частицы определяется ее объемов в 3-х мерном пространстве.
http://msm.omsu.ru/jrns/jrn63/boldov.pdf
Не хватает воображения. Не могу представить многогранник в виде границы частицы.
Все просто до неприличия. Наше пространство дискретно-непрерывно. Об это хорошо писали Н.Вяльцев и В.Корухов с О. Шарыповым.https://studylib.ru/doc/680403/model._-diskretno-nepreryvnogo-prostranstva?ysclid=lt361klk6731359935
Отрицание дискретности - привет апории Зенона. Единственный способ их избежать - признать, что размерность 3-х мерного пространства органичена снизу мимнимальным размером (Планковская длина 10 в мину 33 степнни.)Т.е пространство состоит из 0-размерных дискрет. Логично, что кроме них могут быть 1- и 2-х размерные образования, А 3-х мерные - это элементарные частицы.
У меня другой подход. Частицы - топологические особенности векторного поля, в которых линии тока замыкаются. Поэтому, имея в виду длину замкнутой линии тока, я говорю о длине частицы. Однако надо понимать, что хотя линия тока и одномерна, сама топологическая особенность многомерна.
I have a different approach. Particles are topological features of a vector field in which the current lines are closed. Therefore, bearing in mind the length of the closed current line, I am talking about the length of the particle. However, it must be understood that although the current line is one-dimensional, the topological feature itself is multidimensional.
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Yes the mass of a particle can be a function of its Compton lentgh. See my article:
Preprint On the unification of the quantum mechanics and the theory o...
Alaya Kouki,
You write in French. Maybe you can say a few words in Russian or English here. I will note, however, that by particle length I meant the length of a closed curved line of a certain mass flow.