Below I derive the mass – space equation from the microscopic angle. It concerns modeling, charge, particle physics, QM, etc. The course of derivation is long and complicated, but the result is very very simple. (content abstracted from X4 Theory 2020 Part 1)
X4 Theory of photon (Ⅲ)
X4 Model of Unit Electrical Charge
When we watch a spiral spring, we find that the chirality, left handed or right handed, is natural spacial property and will not change following movement / reference frames (Figure 5).
The concept of dimension should be continues and integral. Why this kind of natural spacial property seems to disappear down to microscopic scale? One solution might be that it reflects in the structure of basic particles.
Another question is that if a unified property existing in nature to judge “anti”, including charges?
A nonstandard model is initiated here in X4 Theory and have a try to solve these problems.
Let’s see the unit step function:
μ(t) = 0 (t＜0) μ(t) = 1 (t＞0)
It could be interpreted here as below:
Because t＜0 meaningless, so the signal or the state of matter does not exist.
When t＞0, the signal or the state of matter exist.
Next, play a mathematical game of “anti”…
With Fourier transformation (detailed calculation omitted), we got:
1 = (1/2) + (1/π)∫0→+∞（1/ω）sinωt dω
Then, replace variants, we got:
1 = (1/2) + (1/π)∫0→+∞（1/R）sinR P dR
1 = (1/2) + (1/2) = (1/π)∫0→+∞（1/R）sinR P dR + (1/π)∫0→+∞（1/R）sinR P dR ①
And so on, we got:
-1 = -[(1/2) + (1/π)∫0→+∞（1/R）sinR P dR]
= (-1/2) + (-1/2) = (1/π)∫0→+∞（1/-R）sin(-R) P d(-R）+ (1/π)∫0→+∞（1/-R）sin(-R) P d(-R)
If we take R’ = -R, then
-1 = (-1/2) + (-1/2) = [(1/π)∫0→-∞（1/R’）sinR’ P dR’] + [(1/π)∫0→-∞（1/R’）sinR’ P dR’]
= [-(1/π)∫0→+∞（1/R’）sinR’ P dR’] + [-(1/π)∫0→+∞（1/R’）sinR’ P dR’] ②
And so on, we got:
0 = (1/2) + (-1/2) = (1/π)∫0→+∞（1/R）sinR P dR +[- (1/π)∫0→+∞（1/R’）sinR’ P dR’ ] ③
Note: in the final result of equation②, the value of R’ is positive too because it has changed to be in the anti 4D space. The equation of R’ = -R just reflects the contrast relationship of two contrast 4D spaces.
Next, let’s analyze the characteristics of equation①②③
⑴ In macro, it’s a scalar constant.
⑵ In micro, it has something to do with space R.
⑶ The value of the constant inverts following the inversion of the four dimensional space. And there is a case of neutrality.
⑷ According to the replacement of variant, P ＞0,and could be regarded as the magnitude of three dimension spacial momentum.
According to the integration area, R ＞0, and could be regarded as the four dimension space for a particle R = X4 r, X4> 0, r > 0, r is the distance (3D space) from the origin of the coordinate system.
And（1/R）sinR P could be regarded as the space part of a position space wave function in triangular form in 4D space. If we apply wave function in momentum space in 4D space, it will beΨ(P ) =（1/r）sinr P , it represents a wave function distribution field adjacent to the origin of coordinate system. The property of this field is not even. It ∝1/r. When the distance trends to infinite, the field trends to zero. While the distance trends to zero, the field trends to infinite.
And (1/π)∫0→+∞（1/r）sinr PdX represents the comprehensive result of the field.
The situation of R’ could be regarded similar but for an anti particle.
We see that equation①②③ are very similar in nature to one physical quantity, it’s point electrical charge.
Next, we just use them as the mathematical model for unit electrical charge (+e or –e) and electrical neutrality and analyze them a further step. We got:
(A) Unit electrical charge has deeper cause in it. The deeper structure of matter could be called layer here. The electrical charge of such layer is + (1/2)e or - (1/2)e, simple equivalent to (1/π)∫0→+∞（1/R）sinR P dR or -(1/π)∫0→+∞（1/R’）sinR’ P dR’. Value inverted following the inversion of the four dimensional space R.
(B) Because the two wave function distribution fields：
（1/R）sinR P and（1/R）sinR P in case of equation①,
（1/R’）sinR’ P and（1/R’）sinR’ P in case of equation②,
（1/R）sinR P and （1/R’）sinR’ P in case of equation③,
Have the same origin of coordinate system in the respective case, and because only one direction is analyzed here, in fact, all direction should be the same situation,
So, the ideal Geometrical shape of layers in any case of equation①②③ should be concentric circle kind in any normal cutting plane of a sphere.
(C) Layers in any case of equation①②③ might be the two aspects of one thing naturally.
(D) We can’t exclude a very special case：there is a kind of layer which has no space inversion effect( namely 0 = 0 + 0). Of course, it will be electrical neutral.
Next, analyze what specific physical structure could realize the conditions mentioned above from (A) to (D).
We think about circle kind of electromagnetic standing wave in any normal cutting plane of a sphere.
It’s two aspects (two travelling waves go in opposite direction) of one thing (the standing wave).
And we got the important character of layer: No alone layer exists in nature. Layers which construct a basic particle could not be separated by means of collision. There seems to be a strong force constraining that two lays but in fact that strong force is just a false impression.
We check out the ordinary standing wave function:
It’s an even function and has no space inversion effect.
We consider the track of the standing wave:
If the track is a smooth circle, its shape is too simple. Look at a smooth circle in the XY plane. The parametric equation is:
X = r cosθ Y = r sinθ Z = 0
If space inverted θ= -θ’ then：
X = r cosθ’ Y = - r sinθ’ Z = 0
That’s another smooth circle in the XY plane which derived from reflection of the original circle against X axis and in fact is the copy. No space inversion effect too. We put it aside temporarily.
If the track is a helical line, it has chirality, left handed or right handed（called natural identification of space here）.
Look at the parametric equation of a helical line:
X = r cosθ Y = r sinθ Z = kθ
If space inverted θ= -θ’ then：
X = r cosθ’ Y = - r sinθ’ Z = - kθ’
The chirality inverted.
Then the specific physical forms of that circle kind standing wave could be:
(Ⅰ) Superposition of two right handed helical circle kind travelling waves go in opposite direction, namely, Superposition of two right handed layers.
(Ⅱ) Superposition of two left handed helical circle kind travelling waves go in opposite direction, namely, Superposition of two left handed layers.
(Ⅲ) Superposition of one right handed helical circle kind travelling wave and one left handed helical circle kind travelling wave go in opposite direction, namely, Superposition of one right handed and one left handed layers.
If we artificially define situation(Ⅰ) as basic particle with positive unit electrical charge, then, situation (Ⅱ) would be basic anti particle with negative unit electrical charge. Namely, “anti” is just relative. Situation(Ⅲ) would be basic neutral particle with electrical neutrality.
Now, consider the track of a smooth circle mentioned above again. It would be:
(Ⅳ) Superposition of two circular travelling waves go in opposite direction, namely, Superposition of two neutral layers. It would also be basic neutral particle with electrical neutrality.
At this moment, we can talk about the micro standard for the determination of some X4 states:
If we define basic particle made up of two right handed layers as in the state of X4 = +1, then, basic particle made up of two left handed layers will be in the state of X4 = -1, namely, the anti particle. Namely, “anti” is just relative. Basic particle made up of one right handed layer and one left handed layer will be in the superposition states of X4 = +1 and X4 = -1. Basic particle made up of two neutral layers is in the state of X4 = +1 or X4 = -1, which means its anti state is itself.
The micro standard is also applicable for the determination of the X4 states of the layer itself.
And so on, the micro standard principle is also applicable to the determination of the states of X4 = +2,+3,+4,+5……+n and their counter part X4 = - 2,-3,-4,-5……-n.
So, if we define proton “particle”, then electron actually is a kind of “anti particle”, while positron is a kind of “particle”.
Watch the context in counter way, it appears that the chirality of layer determines the contrary of four dimensional space. Inversion of the chirality of layer leads to the inversion of four dimensional space. Inversion of four dimensional space leads to the inversion of unit electrical charge.
Maybe, equation①②③ is not necessarily the accurate-enough equations to describe the real wave function distribution fields of basic particles, but it still can demonstrate some properties and has no contradiction to facts:
1. Under such a model, unit electrical (charge) interaction between basic particles will be just the determination of probability method of existence each other, and only has something to do with distance (∝1/x or say∝|1/x|²= 1/x²).
2.Since the value of three dimension spacial momentum P does not affect the result of integration in equation①②③, that means the relative motion of basic particles do not affect the volume of unit electrical charge.
Below are some predictions based on the above context:
1. When the circle kind standing wave mentioned in the above context broken, it will transform into a kind of released photon. So, the flying track of leased photon in flat space should have three types: straight line, straight right handed helical line, straight left handed helical line.
2. In micro, the anti photon of a released photon is itself in case of straight line track. But in case of straight helical track, the anti photon of a released photon is its counter part with inverted chirality. But in macro, namely, in respect of space and time effect, they are all the same and in the state of X4 =∞.
X4 Theory of Photon (Ⅳ)
Photon VS Wavicle
This chapter starts from a strange topic: Analysis of Probability Wave Speed
The speed of probability wave of a free particle could be calculated as below:
If v is the moving speed of the particle, then, the speed of the corresponding probability wave:
u = γλ = (E/h)( h/P) = E/P =（mc²－m0c²）/ mv
= c²/v – (c²/v)√1－(v²/c²) = c²/v – (c/ v)√c²－v²
Here c is light speed.
When v → c, Lim u = c,
When v → 0 , Lim u = 0,
When 0＜ v ＜ c, v²＜ c², v ^4 ＜ v²c²,
So, v ^4 － 2 v²c²＜ －v²c²
So, v ^4 － 2 v²c²+ c ^4 ＜ c ^4 － v²c²
So, (c²－ v²) ^2 ＜ c²(c²－ v²)
So, c²－ v²＜ c√c²－v²
So, v²＞ c²－ c√c²－v²
So, v ＞ c²/v – (c / v)√c²－ v²
So, v ＞ u
Since the frequencyγand wavelengthλof probability wave are meaningful in physics, their product, the wave speed u should be meaningful in physics too. And we got one strange phenomenon: the moving speed of a free particle (except photon) is always not in conformity with the wave speed of the associated probability wave. How to understand it?
Below is a try to answer this question in logic.
1. A “particle” (except photon) is actually a “wavicle” which has two aspects: the particle characteristic / or say the certain aspect as well as probability (wave) characteristic / or say the uncertain aspect.
2. For a released photon, seems that those two aspects merged. The question here is that which aspect disappears? The wave property of light wave is obviously not that kind of probability wave. So we can estimate primarily that it’s the probability (wave) characteristic / or say the uncertain aspect disappears.
Next we turn to X4 Theory. In X4 Theory we expect that X4 = +2, +3, +4, +5…+n… represent the states similar to X4 = +1. We can imagine that they are certain matter states (worlds) similar to but different from X4 = +1 matter states（world）. And 0 < X4 < ∞ is the relevant probability (wave) state of that matter world. A particle is actually a “wavicle”. While light / released photon has only one state X4 = ∞. It demonstrates that it’s the probability (wave) characteristic / or say the uncertain aspect disappears as well as that the wave property of light wave is not that kind of probability wave.
So, what’s a photon? An appropriate solution is that photon is a section of electromagnetic wave.
Something interesting is: considering a photon appears in a 3D space point, and it also propagates ahead in the form of (electromagnetic) wave. That means light speed in photon frame is also c.
X4 Theory of Photon (Ⅴ)
The Mass Space
First, there is a question, why the energy equation for a released photon is E = hγ in Quantum and need not to take the number of crests(n of λ, here n is a natural number) into account?
But a released photon is just that of a section of electromagnetic wave, its energy E should ∝the number of crests(n of λ) no matter what method is used to calculate.
It imply that all released photons have the same number of crests(n of λ)? Temporarily suppose it is and see what will happen.
Next, we refer to the chapter X4 Model of Unit Electrical Charge and play a game of Geometry.
We unfold the standing wave (suppose |X4| = 1), namely make its radius r →∞, and it turns into two released photons.(Pay attention please, it’s only a game of Geometry and might not be the real physical course of transformation)
We watch the game in counter way, it’s two released photons turn into a standing wave.
And we got: standing wave which constitute all kinds of basic particles have the same number of crests (nt of λt when referring to travelling waves, and ns of λs when referring to standing waves. The lower index t means travelling wave and s means standing wave in this context)
We turn to SR and talk about space at this moment temporarily.
The equation for the traditional Lorentz transformation of length contraction is: L’ = L / γ
The mass – speed equation is M = γM0
Then, ML’ = M0L so, L’ = M0L / M
Because M0 and L are constants, then: L’ ∝ 1 / M. This might be called the “mass – space equation”.
Mass M is a scalar, so the equation reflects the contraction of size in all directions due to the increase of mass.
It seems that the “mass – space equation” is a derived one. But its components “mass” and “space” are most basic physical elements. Actually, it should be a more basic equation than the equation for the traditional Lorentz transformation of length contraction and the mass – speed equation. It describes more basic natural rule.
During the course of exploration, people perhaps touch the leaves of a tree first, then the sticks, and so on the stock, at last the root. But for a tree, the root is the most basic.
From the angle of the physical substance of the object itself, perhaps the research might be more in depth and sufficient.
Moreover, when we replace the 3D space Gama factor γ with 4D space Gama factor Ý in the course of derivation above, we also can get the “mass – space equation”. That’s to say, the “mass – space equation” is also applicable to those matter worlds of |X4| = 2, 3, …, n.
It demonstrates that the three dimension physical space concept is induced by mass. Give it a nickname here: the Mass Space (Mspace). And Mspace ∝ 1/M.
Next, we try to use the basic particle model in the chapter X4 Model of Unit Electrical Charge and the temporary conclusion of same number of crests in this chapter to calculate the radius of basic particle.
Below, we calculate the central line of the helical track of layer instead of the helical track itself because the speed of light should be calculated on the central line. And the wavelengthλt should be the projective wavelength on the central line.
E = MC² = 2hγt = 2hC /λt then:
λt = 2h / MC then:
The radius of basic particle is:
r = ntλt / 2π = nth /πCM
Because nt, h, π, C are all constants, we use a constant “a” for simplicity and a = nth/πC, then:
r = a/M
Namely, r ∝1/M. In fact, it’s the same meaning as Mspace ∝ 1/M in SR above.
(note: picture and some symbol miss during cyber transition)
Li Qiang Chen
Jan 27, 2020