X4 Model of Unit Electrical Charge vs the Wave Characteristic of Electric Interaction
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.
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 be Ψ(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 Pdr 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 of electric charge and has no contradiction to facts:
1. Under such a model, electric field or say unit electric (charge) interaction between basic particles will be just the determination of probability method of existence each other. According to the wave function Ψ(P ) =（1/r）sinr P , it only has something to do with distance (∝1/r or say∝|1/r|² = 1/r²).
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.
3. Charge conservation, because 4D space is naturally symmetric.
Moreover, the X4 unit charge model demonstrates that electric interaction has an aspect of wave characteristic.
A special situation in GR (without gravity) is SR. How about electric interaction? When electric interaction disappears, it’s a free particle. An ideal situation of this is the infinite distance. We can see that the amplitude 1 / r of the wave function Ψ(P ) =（1/r）sinr P in the X4 unit charge model trends to zero when distance r trends to infinite. Actually the wave function will no longer meaningful. How to maintain the uniform of representation in math? We can assume the amplitude of the wave function is a constant A and it means the situation of without electric interaction. That’s the wave function for free particle.
Of course, in X4 Theory, when the situation of without electric interaction happens, no need resorting to wave function. The factor X4 itself represents the equal probability effect. It reflects the uncertain property (contrast of ability) of the free particle itself.
Nov 3, 2020