Can projective geometry find out its application in physics?

https://www.math.nagoya-u.ac.jp/~yamagami/teaching/symmetry2015.pdf

Thank you for your reply, fresco. It let me know that I am not alone in this respect of researching.
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Actually, symmetry might be the simplest and most direct starting point/or say "angle" to this new researching area.
Their method of representation seems a bit complicated...

You may find that attitudes to Projective Geometry are that it borders on 'esotericism' because it has been linked with Goethe's alternative (spiritual/phenomenological) views of science, which were taken up Rudolph Steiner.

Interesting "attitudes"...
I heard some guys said that QM is also a bit "esoteric". So, PG and QM seems to be perfect couple.
Cosmos is a bit "esoteric" too...
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When I was in university, I often wandered along the riverside and drunk on the beautiful scenary in spring and autumn. I wrote some poems too.
Seems that the right guy doing the right thing. haha...

Actually projective geometry is four dimension spacial geometry. The coordinate of the space point is called homogenous coordinate: (X1, X2,
X3, X4). It contains four spacial elements and X4 is a very special element. One step forward than the traditional xyz.

We have some discussions about how to apply PG in physics in physicshelpforum.com
Relevant threads:
topsquark's thread "X4" in the special and general relativity column;
htam9876's thread "tramp and electron" in the philosophy column;
neila9876's thread "wavicle, an old topic" in the quantum column
neila9876's thread "balloon vs SR" in the special and general relativity column.

The phsical meaning of the application of affine geometry in physics is the existence of mass.
What should be the phsical meaning of the application of projective geometry in physics?
I suggest that it's the existence of the matter states.
X4 Theory is to research the (matter) state dimension in real cosmos.

Li Qiang Chen
Jan 1, 2020

Prelude
In ancient times, people stood on the ground and asked a question: how big is the ground? They never got an explicit concept until they climbed high enough in the third dimension.
Now, people wander in cosmos and ask a similar question: how big is cosmos?
The question of the beginning of cosmos, antimatter, dark matter, Relativity, QM, background microwave, etc, are established things or things under exploration. How can they relate together? Cosmos is just that one thing.
Maybe, one more dimension in physics, one percent more clear in cosmos. And this dimension should not be a length dimension again but a state dimension.


The Appropriate Mathematical (Geometric) Method to Describe Cosmos (Physics)

Next, watch the homogeneous coordinate in projective geometry: (X1, X2, X3, X4). The space point contains four elements, so actually projective geometry is four dimensional space geometry, one step forward than the traditional xyz three dimensional space. The component X4 which represents proportion of space is just alike a special dimension. So, we might try to use this mathematical (geometric) method to represent the state dimension.
Let’s talk about the concept of a straight line. People have two concepts. First, in an open space, people say a straight line extends faraway in two directions, just go. Second, in a close space, a straight line extends in two directions and meets together at the infinite point at last, the so-called hyper line in projective geometry.
Next are some engagements in projective geometry:
A new real number “∞” called “infinite maximum” is added to the traditional real numbers. This is called the “hyper real numbers”. And the rules of calculation are:
⑴ x / ∞ = 0 ( 0 ≠ x ∈ R)
⑵ x / 0 = ∞ ( 0 ≠ x ∈ R)
⑶ ∞ / ∞ = 1


4D (physical) Space Definition

One important task for X4 Theory is to research the space and time in different matter state (world).
We put aside the topic of time temporarily
At this stage, some ordinary representations (such as “homogeneous coordinate”, etc) in previous articles should change.
We use specialized and explicit four dimension spacial method of representation and use capitalized letters to denote physical quantities in four dimensional space.
The equation X i = X4χi (i = 1,2,3) or simplified as X = X4χ means four dimensional space is three dimensional space controlled (or effected) by X4 state. It represents the affiliation of space with regard to matter.
X4 ∈“the hyper real numbers”.
I denote it as below to highlight its characteristic.
X4 ∈ (∞,…,-n,…, -3.222xxx,… -2, -1, 0, +1, + 2,… +3.222xxx, … +n, …∞)
In math, the element X4 represents contrary (“not anti” vs “anti”) as well as proportion of space which will not change in different reference frames/movement in 3D space.
In philosophy, the value area of (∞,…,-n,…, -3.222xxx,… -2, -1, 0, +1, + 2,… +3.222xxx, … +n, …∞) reflects zero point, contrary, similar, certain and uncertain, infinite, with border. It demonstrates a state dimension might exist in cosmos.
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More Sufficient Concept of Physical Space

Next, I try to analyze from another angle to explore why this equation X i = X4χi (i = 1,2,3) or simplified as X = X4χ can appear.
Put aside y and z, just use x to represent 3D space, for the most simplicity.
The question here is what does the symbol “ – x ” mean?
A. The negative part of the axis;
B. Just “parity”; But what does “parity” mean exactly?
C. Inversion of 3D space.
We do be able to draw two opposite 3D frames. (see the second row in Figure 2, z axis omitted for convenience of watching.)
1. It is just a math game.
2. Inversion of 3D space.
If the answers are C and 2, why not use +1x to represent the 3D space which is in consistent with the 3D space in the first row while use –x to represent the 3D space which is opposite? Then we use the symbol X4 to accommodate the specific values of +1 and -1. We got an equation X = X4x.
Ⅰ. It’s a more clear and explicit method of representation in form. Actually, it’s a representation of 4D space because there is an additional element X4 attached to the 3D space x. Of course, if X4 also represents proportion of space, it seems to be more perfect. Actually, we can zoom those two frames in the second row.
Ⅱ. In meaning, X = X4x is a more sufficient/integral concept of space. It contains two parts. The part of x is length element which will vary following movement/frames while the part X4 is contrary or proportion element which will not vary following movement/frames.
In the past, people emphasize on variation of space following movement. Now, in X4 theory, we emphasize on the part which will not vary following movement.




Figure 2


A strange question is “what cause physical space to flip/inflate or deflate”? Math or human?
We can assume it’s matter state.



Concept of Full Scene Four Dimensions

I like to use vivid analogy to talk something.
1. Watch the rabbit in the Figure 3. From the angle perpendicular to the paper, people see a rabbit running in space – time four dimension.
The method of description in math is: x = x0 + vt
2. Watching from the angle parallel to the paper, people see the back of the rabbit zooming in 4D space.
The method of description in math is: X = X4x
If we use (X, t) to describe it, the concept will be Full Scene Four Dimensions.

Figure 3
(pictures failed to upload)

The Inherent Property of the Fourth Dimension of Cosmos

The simplified equation in the definition of 4D (physical) space X = X4χ can be transformed to be χ = X / X4 . Next, let’s calculate the differential quotient against X4. Pay attention please, not against time t. Here X4 is a variant.
Go on: dχ / dX4 = (X4dX / dX4 －XdX4 /dX4) / X4²= -X / X4² = -χ / X4
So, χ = - X4dχ / dX4
Also, χ = ∫vdt
We arrive at ∫vdt = -X4vdt/dX4 assumes v a constant for simplicity
So t +c = -X4dt / dX4 Here c is an integration constant,
So (1/(t+c)) dt = - (1/X4)dX4
∫(1/(t + c))d(t+c) = -∫(1/ X4)d X4
ln(t+c) = -lnX4
t = exp (-lnX4)
For convenience, here we take c = 0 and only talk about the positive value of X4.
We can see X4 is a real fourth dimension of cosmos, and time t is not imagined by people. It has real number corresponding relationship with X4. So X4 and time t are both real fourth dimension of cosmos.
(The sufficient solution of the differential equation is t = (1/ X4) expC1 + C2, details will be in the article X4 Theory of Time.)


Some Thinking about Wavicle


Start the topic from the experiment of single slit diffraction of electrons: electrons emitted one by one, poop, poop, poop…passes through a slit…then, a tiny point appears in this place or that place randomly on the screen/film. Time long enough, the diffraction pattern of electron appears on the screen/film.
Conclusion: It is the presentation of wave character of electron. Is it sufficient?
The randomly shooting is surely the behavior of wave character. But what’s tiny point on the screen/film? If it’s not a particle, what’s it?
So, in all around consideration, the conclusion should be: It’s the presentation of wavicle character of electron.

Below is a short story.
A professor asked a student: why a (moving) glass seems to have no wavicle character?
The student answered: because the Duc de Brogiel wavelength is too short, people is not easy to perceive it. (λ = h / p, for a glass, p is big enough to render λ very small)
The professor kept silent.
Why? It's seems that the student deal with it as mechanic wave. The logic is wrong first
The appropriate analysis in QM should be: The glass is a neutral big free particle, so in the equation Ψ = A exp( i(p•x– Et )) (omit “ℏ” and vector marks for convenience of observation), “A” is a constant,︱A︱² is a constant. It has nothing to do with λ, even xyz or t. It means that the probability of presentation of the glass at any space point is the same. That’s to say you can see the glass everywhere.
Contradiction with fact.

Or, it demonstrates that cosmos has two aspects and classical physics describes the certain aspect while QM describes the uncertain aspect?
How to solve the “seems” contradiction? The fourth (special) spacial dimension is needed to accommodate the uncertainty (probability of appearance).



Wave Functions in 4D Space (Ⅰ)

Once upon a time, in the equation X = X4x and then the equation V = X4v, the reference in fact is 3D space.
In order to prevent losing the meaning of contrary or reference, in 4D space wave function, we date that space and momentum, one is 4D space physical quantity, then the other is 3D space physical quantity. This is called the 1/2 Rule or the Special Engagement.
1. The position space wave function for a free particle in 4D space is:
Ψ(X, t) = A exp( i(P •X– Et )) Amplitude A is a constant. (omit “ℏ” and vector marks for convenience of observation)
X = X4x
Then, the probability density｜A｜² is a constant. It means the probability of appearance of the particle in any point of 4D position space is the same. In turn, it means the probability of the X4 state the free particle be in is the same. The situation is the same in any position x in the 3D space.
Because it’s a free particle, no interaction is concerned, the above analysis reflects the property of the particle itself regardless it is moving or not (It has nothing to do with wavelength λ = h/ P ).
2. The momentum space wave function for a free particle in 4D space is:
Ψ(P, t) = A exp( i(P•x– Et )) Amplitude A is a constant. (omit “ℏ” and vector marks for convenience of observation)
P = X4 P
Then, the probability density｜A｜² is a constant. It means the probability of appearance of the particle in any point of 4D momentum space is the same. In turn, it means the probability of the X4 state the free particle be in is the same. The situation is the same in any momentum P in the 3D space.
Because it’s a free particle, no interaction is concerned, the above analysis reflects the property of the particle itself regardless it is moving or not (It has nothing to do with wavelength λ = h/P).

Look at the 3D space wave function for free particle in QM again.
Ψ = A exp( i(p•x– Et )) omit “ℏ” and vector marks for convenience of observation.
When position x and momentum p are both determined at the same time, it will no longer be a wave function, instead, an oscillation function. It seems that the “wave character” should disappear either.
What interesting is it could be considered as a special situation and the calculation of probability is still applicable. “A” is a constant. ｜A｜² is a constant. It means the probability of appearance of the particle in any space point is the same. In turn, it means the position of the free particle can not be determined.
Contradiction…
Next, resort to 4D space wave function. In the item P •X or P•x, among space and momentum, if one physical quantity is 3D space quantity, then the other is 4D space quantity. It reflects that they two can not be determined simultaneously and avoid the dispute.
Classical physics describes the certainty in 3D space while QM describes the uncertainty in 4D space.

Schrodinger Equation in 4D Space

When talking about Hydrogen atom in QM, we get the concept of electronic cloud (the probability distribution of electron) in some lectures and see a strange phenomenon: The shape of the electronic cloud (the probability distribution of electron) is symmetric about the Z axis, and so, when the Z axis changes its orientation, the shape of the electronic cloud (the probability distribution of electron) will follow.
Think about it: the shape of the electronic cloud (the probability distribution of electron) should be the own characteristic of the Hydrogen atom, while the Z axis (the coordinate system) is artificial. If the electronic cloud is the true natural characteristic of the Hydrogen atom, its shape should not follow the change of the artificial coordinate system. You establish a coordinate system while I turn the Z axis an angle to build another coordinate system, which probability distribution is right? That is to say the wave functionψ（r, θ，φ）= R(r)Y(θ，φ) might not be the natural solution but the artificial solution. The concept of electronic cloud might not be natural but artificial.
What’s the mistake?
We analyze it carefully and find that people might use 3D mathematics to represent alive real cosmos, just as using plane geometry to explain solid，and is not the exact way.

Next, expand the thought and try to explore if a better solution could be found with the concept of four dimensional space, the real physical space which the electron actually situates in. It could be denoted with the equation:
X = X4x , or X / X4 = x
We know that the Schrodinger equation was induced in the 3D space. And it is:
Eψ= [-(ℏ²/2μ)(d²/dx ²) + V(x)] ψ
Next, let’s try to transform it in four dimensional space and see what the situation will be.
dψ/ dx = (dψ/ dX)( dX / dx) = (dψ/ dX) [X4+x ( dX4 / dx) ]
= X4(dψ/ dX) +x (dψ/ dX) ( dX4 / dx)
d²ψ/ dx²= ( dX4/ dx) (dψ/ dX) + X4(d²ψ/ dX²)( dX / dx) + (dψ/ dX) ( dX4 / dx)
+x ( dX4 / dx) (d²ψ/ dX²)( dX/ dx) + x (dψ/ dX) ( d²X4 / dx²)
= ( dX4 / dx) (dψ/ dX) + X4(d²ψ/ dX²)[X4+x ( dX4 / dx) ] + (dψ/ dX) ( dX4 / dx)
+x ( dX4 / dx) (d²ψ/ dX²)[X4 +x ( dX4 / dx) ] +x (dψ/ dX) ( d²X4 / dx ²)
= ( dX4 / dx) (dψ/ dX) + X4²(d²ψ/ dX²) +x X4(d²ψ/ dX²)( dX4 / dx)
+ (dψ/ dX) ( dX4 / dx) +x X4(d²ψ/ dX²)( dX4 / dx)
+x ²( dX4 / dx)²(d²ψ/ dX²) +x (dψ/ dX) ( d²X4 / dx ²)
= [ X4²+ 2 X( dX4 / dx) +x ²( dX4 / dx)²] (d²ψ/ dX²)
+ [2( dX4 / dx) +x ( d²X4 / dx ²)](dψ/ dX)

Then: Eψ=﹛-( ℏ²/ 2μ) [ (X4²+ 2 X( dX4 / dx) +x ²( dX4 / dx)²)(d²/ dX²)
+ (2( dX4 / dx) +x ( d²X4 / dx ²))(d / dX) ] + V (x)﹜ψ

It’s the Schrodinger equation in four dimensional space (4D space -S equation). Let’s analyze it:
1. In case of free particle or V does not affect the X4 state of the space point, X4 value is an invariant. The four dimensional space is not elastic.
dX4 / dx ≡0, d²X4/ dx ²≡0, then:
Eψ= [-(ℏ²/ 2μ) X4²(d²/dX²) + V(x)] ψ
When X4 = +1, X =x, then:
Eψ= [-(ℏ²/ 2μ)(d²/dx ²) + V(x)] ψ
It’s the Schrodinger equation we used to be familiar with, in the 3D space.
2. In case of V affects the X4 state of the space point, (we guess the central Coulomb field is such a case), the traditional Schrodinger equation is no longer applicable to describe the exact situation. It’s the 4D space -S equation does.
How to solve the complicated 4D space -S equation is a question and we also have to determine the gradient of the X4 state field, dX4 / dx, of the electron of the Hydrogen atom elsewhere first.
Just to guess its meaning with philosophy here.
If the 4D space -S equation has a solutionψ(X, x , X4), it should be the sufficient solution ,and should have the same characteristic in any direction. So the analysis in one direction is enough to see the characteristic.
It might be a series of wave functions. It even might be a concept of field：wave function distribution field. We know that the shape of field often does not follow the change of artificial coordinate system and only have something to do with the own structure of the matter concerned.
And it seems reasonable for the electron to reflect the distribution situation of the central Coulomb field (or exactly say, to describe the electric interaction in wave character).

When X4 takes a certain value, dX4 / dx ≡0, d²X4 / dx ²≡0, then:
Eψ= [-(ℏ²/ 2μ) X4²(d²/ dX4²x ²) + V(x)] ψ

= [-(ℏ²/ 2μ)(d²/ dx ²) + V(x)] ψ
It’s the Schrodinger equation we used to be familiar with, in the 3D space. It demonstrates that X4 value does not affect the energy levels. It’s a very important point.



An appropriate method / angle can describe an appropriate thing

At first glimpse, my action of putting x = X / X4 into the traditional 3D space Schrodinger equation rather than replace 3D space x with 4D space X directly is up to man’s own will. Think it carefully, the presence of 3D space element x is necessary for a method to describe “field” (or say exactly “interaction”). So, actually, it’s a natural choice.
A vivid analogy is you can describe the long nose of a pig from the side view while you can describe two holes from the front view…

Full context of X4 Theory 2020 Part 1 has been deposited in Mendeley.

Below is some abstract from X4 Theory 2020. (This tramp in cosmos suggests that guys who try to read this long post should pay attention to the points at the end of this context first.)

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.
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
Namely:
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 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:
Y =（2Acos2πx/λ）cos2πγt
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.

Note:
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.
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 in the rabole 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 (hidden variables) represents the equal probability effect. It reflects the uncertain property (contrast of ability) of the free particle itself.
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The Explanation of Double slits Experiment of Electron in X4 Theory:
Because the wave function for free particle is just a mathematical form while the physical meaning of wave already lost, the application of it to explain the generation of interference pattern in the double slits experiment of electron is inappropriate.
Actually the electron moves in electric field in the apparatus.
The electron passes through either one of the slits. No matter which slit the electron passes through, the wave characteristic of electric interaction will cause interference information in the slits.
When electro- magnetic detector(s) fixed to observe, those interference information created in the slits will be disturbed by the physical interaction between electron and detector(s), and then the interference pattern will disappear.
The principle is also applicable to explain the diffraction pattern in the single slit experiment for electron.

Li Qiang Chen
April 10, 2020

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