X4

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X4 idea

“I have been doing the SR thing because one of your basic equations seems to be a new definition of the Lorentz factor in SR. If the difference between what you are deriving and Lorentz's factor then I ask that you please show the derivation.”
Yes, the 4D space γ factor which is shown in the attached picture in my first post in this thread “seems” to be a new one. (Note again, it is not definition but the result of calculation. The course of derivation is shown in my first post in this thread too.) In fact, it has no much difference with the traditional 3D space γ factor we see in Einstein’s SR.
When X4 ≠ ∞, and is a certain value, V / X4 = v. Put it into the 4D space γ factor, and you can get back the traditional 3D space γ factor. Lorentz transformation is the same principle, but pay attention, please.They are carried out in different matter worlds.
When X4 = ∞, it’s very special situation, the 4D space γ factor = 1. (The detailed course of calculation can be found in one of rabbit htam9876’s threads. It’s not yet the time to talk about it here, so I put aside it temporarily). Then Lorentz transformation is L’ = L, △t’ = △t, so, L’/△t’ = L/△t, namely c’ = c, It means light speed is a constant in different inertial frames.(When X4 = ∞, it means light or electromagnetic wave. This can be found in one of rabbit htam9876’s threads. It’s not yet the time to talk about it here, so I put aside it temporarily.)
“The main focus of this thread I ask you to please write down the basics of your idea and how it applies to Physics, be it SR or otherwise.”
The idea of X4 theory is to initiate 4D space era in physics, not just in SR, but just as SR did one century ago. It tries to disclose the in depth secret in cosmos and also establish a universal/generalized model. So, X4 Theory belongs to none specific discipline but seems to be a philosophy of cosmos. One of its basic taskes is to research the space and time in different matter worlds. That’s why the concept of X4 first appeared in the philosophy column.
 

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More sufficient concept of space

“What does X = X4x mean? i runs from 1 to 3, but your symbol X4 has not been defined?”
What does X = X4x mean? It can be found in the 4D space definition in my first post in this thread.
But now, I try to analyze from another angle to explore why this equation 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 the attached picture, 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 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.
……….
How to fully define the symbol X4 is the key step in X4 Theory. Next post.
 

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From geometry/math to physics

Fully defining X4 is the key step of X4 Theory.
Again set out from the angle of space.
In projection geometry, X4 is merely the proportion of space; in the derivation of the more sufficient concept of space, X4 is also the contrary or proportion of space. It’s geometry/math concept.
But we are researching physics/real matter cosmos. The space we talking are physical space.
I have to make an analogy here and refer to SR temporarily. What render 3D space contracted? It’s movement or in my own way, the increase of mass.
So on, what render 3D space inverted/proportionate? We imagine “anti matter world” / “similar matter worlds”.
If a guy considers that the physical space in “anti matter world” / “similar matter worlds” are the same, then his concept of space is a new type concept of absolute space.
This direction of thought ultimately lead to the fully definition of X4 is the (matter) state dimension of cosmos.
 

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time

"...which then we can talk about a space-time"
When I got the X4 concept in projection geometry one and a half years ago, I wondered quite a while that what should X4 exactly be in physics/cosmos?
Actually, the dramatic step in development of X4 Theory was the calculation of differential quotient againt the element of X4. It's shown in rabbit htam9876's first thread. I just move it here, still in old style. What funny in history of X4 is the breakthrough point actually is from the angle of time.

The Fourth Dimension of Cosmos

I suppose that a straight line in the real cosmos is a so-called extension line and we use homogeneous coordinate(X1, X2) to represent a point on it in the article The Appropriate Mathematics to Explain Cosmos (Physics). Next, let’s calculate the differential quotient against X2. Pay attention please, not against time t. Here X2 is a variant.
X=X1/X2 So, dX/dX2=(X2dX1/dX2-X1dX2 /dX2)/X2²= -X1/X2² = -X/X2
So, X = - X2dX/dX2
Also, X = ∫Vdt
We arrive at ∫Vdt = -X2Vdt/dX2 suppose V a constant for simplicity
So t +c = -X2dt/dX2 Here c is an integration constant,
So (1/(t+c)) dt = - (1/X2)dX2
∫(1/(t + c))d(t+c) = -∫(1/ X2)d X2
ln(t+c) = -lnX2
t = exp (-lnX2)
For convenience, here we take c = 0 and only talk about the positive value of X2.
When representing a point in the three dimension space, the homogenous coordinate is (X1,X2,X3,X4) , the symbol of the real fourth dimension change from X2 to X4. And the equation above change to be t = exp (-lnX4)
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, this is the "time-state equation". Details will be in the X4 Theory of Time. It will be a successive theory to X4 Theory.)
 
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time vs state

"As to the GR comment you made GR talks about the metric of space-time, not just the metric for 3D space..."
Yes, ...space-time is integral.it seems that I even can't put aside time for simplicity, but I occationally try to do that.
In X4 Theory, space - state is integral. And it 's more explicit and and convenient to use state (X4).
Time and state are the common fourth dimension of cosmos. So,"space-time integral" and "space-state integral" are actually talking the same kind of ****
The differential equation:
(1/t) dt = - (1/X4) dX4
might be called the differential form of time state equation. It seems to show that the experience of a period of time is equivalent to the experience of a series of state changes.
 
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The (matter) state dimension of cosmos

Watch the characteristic of X4 value area below:
X4 ∈ (∞,…,-n,…, -3.222xxx,… -2, -1, 0, +1, + 2,… +3.222xxx, … +n, …∞)
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.
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.
It demonstrates a state dimension might exist in cosmos. (Why not expect / imagine or even assume that there is such a state dimension and then try to verify it?)
…….
Next, turn to SR.
Here we use the symbol γ4 to represent the 4D space γ element while the symbol γ to represent the traditional 3D space γ element.
Here, we use the capitalized letter M to represent 4D space mass while the small letter m to represent 3D space mass.
Here we use the symbol EK to represent the 4D space kinematic energy while the symbol Ek to represent the traditional 3D space kinematic energy.
Here, we use the capitalized letter P to represent 4D space momentum while the small letter p to represent 3D space momentum.
When X4 ≠ ∞ , V = X4v or v = V / X4. Put it into the 4D space γ element shown in the attached picture of my first post in this thread. We got γ4 = γ.
The 4D space mass – speed equation is:
M = γ4m0 = γm0 = m
That means 4D space mass equals to 3D space mass.
The 4D space kinematic energy equation is:
EK = Mc²﹣ m0c² = mc²﹣ m0c² = Ek
That means 4D space kinematic energy equals to 3D space kinematic energy.
The 4D space momentum equation is:
P = MV = m* X4v = X4* m v = X4 p
That means 4D space momentum equals to 3D space momentum multiply X4.
Next, compare them to QM.
We can see that, one X4 value one 4D space momentum P value. This is somewhat the same as the state of a free particle in Quantum theory, which is represented by wave function Ψnp(r, t)
The value of X4 represents a state. So, X4, could really represent the state dimension.
It also hints that the issue of QM might be the issue of 4D space.


Li Qiang Chen
September 10th, 2019
 

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Wave function for free particle

Look at the simplest wave function for free particle in QM below.
Ψ = A exp( i(p•x– Et )) “h-” and vector marks omitted for convenience of watching. Amplitude A is a constant.
So, according to authentic QM, the probability density︱A︱² is a constant. It has nothing to do with wavelength λ (= h/p), even 3D space position xyz or time t. It means that the probability of presentation of the free particle at any 3D space point is the same. That’s to say people can see the free particle everywhere no matter it moves or not. It seems not so understandable.
……………
Another case, Of course, if position x and momentum p are both determined at the same time, it will be classical physics.
Look at the simplest wave function for free particle in QM below again.
Ψ = A exp( i(p•x– Et )) “h-” and vector marks omitted for convenience of watching.
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 density is still applicable. “A” is a constant. |A|² is a constant. It means the probability of appearance of the particle in any 3D space point is the same. In turn, it means the position of the free particle can not be determined.
Contradiction…
Classical physics vs QM, who is wrong?
Or, it demonstrates that a coin has two aspects and classical physics describes the certain aspect while QM describes the uncertain aspect?
……………………
How to make QM more understandable or to solve the “seems” contradiction?
If we use 4D space in the wave function for free particle,
Ψ = A exp( i(p•X– Et )) “h-” and vector marks omitted for convenience of watching.
Then,|A|² is a constant. It means the probability of appearance of the particle in any point of 4D 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.

In traditional QM, value area for 3D space x is (-∞, +∞). In X4 Theory, it’s a bit different, positive values of X4 represent “not anti” while negative values of X4 represent “anti”. So, 4D space has something to do with matter state. The value area can be either positive or negative. Put it aside temporarily.
The purpose of this step is to use a special (space) dimension to accommodate the uncertainty.


Li Qiang Chen
September 13th, 2019
 

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X4 Physical Meaning of Momentum Space

We start from the wave function in QM shown in the attached picture. 0< p < ∞
Here the form of sum rather than the form of integration is used for the sake of intuition and only talk about positive value for simplicity.
Watch the left side, it’s the wave function determined by a free particle (assume it’s an electron) with certain value of momentum p.
Watch the right side, it’s the sum of a series of wave functions C1Ψ1p(r, t), C2Ψ2p(r, t), …
…CnΨnp(r, t)…
Each CnΨnp(r, t)is the wave function determined by that electron with corresponding certain value of momentum np .
Certain value of momentum is a prerequisite condition for wave function.
Think about it carefully and a question emerges: Which one the hell is the momentum of the electron moving with at this moment?
Feel logic a bit confused?
Situation seems more serious when talking about the case of electrons reflected from crystal lattice.
There is the saying in some materials that the above equation could be regarded as the superposition of the reflected electronic waves.
Regardless the discussion about superposition of the reflected electronic waves, the focus here is the value of momentum of the reflected electron.
When a beam of electron with certain value of momentum p is injected into the crystal, there will really be a reflected electron moving with so big value of momentum → ∞ ? So, the concept of momentum space in QM is just for fun in math while without true physical meaning?
How to address these questions?
We refer to the concept of four dimension spacial momentum space.
(Four dimension spacial momentum) P = X4p (three dimension spacial momentum)
Three dimension spacial momentum describes a particle moving with the certain value of momentum p. It describes the particle character aspect of matter.
Four dimension spacial momentum P = X4p describes the aspect of probability (wave) character. It means the particle also probably exists in all X4 states from 0 to ∞ (0<X4<∞). The result is 0< P = X4p <∞.
That’s the true physical meaning of momentum space in QM. And superposition is the superposition of X4 states.
To the very end, QM is just to describe the aspect of probability (wave) character.

Li Qiang Chen
September 13th, 2019
 

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Schrodinger Equation in Four Dimensional Space

(This article was post by rabbit htam9876 last year. I just rewrite it in formal 4D space math and made minor amendment at the end. In this content, we don't replace 3D space x with 4D space X directly because that method seems to get no interesting result here. So change method.)
........
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 4D alive real cosmos, just as using plane geometry to explain solid,and is not the exact way. The ordinary spherical or rectangular coordinate system is established in 3D space.
Even if people find a way to solve the Schrodinger equation in rectangular coordinate system, the probability distribution of electron which is represented with the wave functionψ(X, Y,Z)will also follow the coordinate system.
Next, turn to X4 Theory, and have a try to solve this problem with the concept of four dimensional space. The alive real 4D space which the electron actually situates in could be denoted with the equation:
X = X4χ or X / X4 =χ

We know that the Schrodinger equation was induced in the 3D space. And it is:
Eψ= [-( h²/2μ)(d²/dχ²) + V(χ)] ψ
Next, let’s try to transform it in four dimensional space and see what the situation will be.
dψ/ dχ= (dψ/ dX)( dX/ dχ) = (dψ/ dX) [X4+χ( dX4/ dχ) ]
= X4(dψ/ dX) +χ(dψ/ dX) ( dX4/ dχ)
d²ψ/ dχ²=( dX4/ dχ) (dψ/ dX)+ X4(d²ψ/ dX²)( dX/ dχ)+ (dψ/ dX) ( dX4/ dχ)
+χ( dX4/ dχ) (d²ψ/ dX²)( dX/ dχ)+χ(dψ/ dX) ( d²X4/ dχ²)
=( dX4/ dχ) (dψ/ dX) + X4(d²ψ/ dX²)[X4+χ( dX4/ dχ) ] + (dψ/ dX) ( dX4/ dχ)
+χ( dX4/ dχ) (d²ψ/ dX²)[X4+χ( dX4/ dχ) ] +χ(dψ/ dX) ( d²X4/ dχ²)
=( dX4/ dχ) (dψ/ dX) + X4²(d²ψ/ dX²)+χX4(d²ψ/ dX²)( dX4/ dχ)
+ (dψ/ dX) ( dX4/ dχ)+χX4(d²ψ/ dX²)( dX4/ dχ)
+χ²( dX4/ dχ)²(d²ψ/ dX²)+χ(dψ/ dX) ( d²X4/ dχ²)
=[ X4²+2 X( dX4/ dχ) +χ²( dX4/ dχ)²] (d²ψ/ dX²)
+[2( dX4/ dχ)+ χ( d²X4/ dχ²)](dψ/ dX)

Then: Eψ=﹛-( h ²/2μ) [ (X4²+2 X( dX4/ dχ) +χ²( dX4/ dχ)²)(d²/ dX²)
+(2( dX4/ dχ)+χ( d²X4/ dχ²))(d/ dX) ]+V (χ)﹜ψ

It’s the Schrodinger equation in four dimensional space (4D -S equation). Let’s analyze it:
1. In case of free particle or V does not affect the X4 state of the particle (space point), X4 value is an invariant. The four dimensional space of the particle’s own world is not elastic.
dX4/ dχ≡0, d²X4/ dχ²≡0, then:
Eψ= [-(h ²/2μ) X4²(d²/dX²) + V(χ)] ψ
When X4 =1, X =χ, then:
Eψ= [-(h ²/2μ)(d²/dχ²) + V(χ)] ψ
It’s the Schrodinger equation we used to be familiar with, in the 3D space. The solutionψ(X) means the probability distribution in the particle’s four dimensional space is the same as that in the 3D space .
2. In case of V affects the X4 state of the particle (space point, and 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-S equation does.
How to solve the complicated 4D -S equation is a question and we also have to determine the gradient of the X4 state field, dX4/ dχ, of the electron of the Hydrogen atom elsewhere first.
Just to guess its meaning with philosophy here.
If the 4D -S equation has a solutionψ(X,χ,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 4D 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/ dχ≡0, d²X4/ dχ²≡0, then:
Eψ= [-(h ²/2μ) X4²(d²/dX4²χ²) + V(χ)] ψ

= [-(h ²/2μ)(d²/dχ²) + V(χ)] ψ
It’s the Schrodinger equation we used to be familiar with, in the 3D space. It means that X4 value does not affect the energy levels. It’s a very important point.
(Cyber transition errors such as "h" and inexplicit of some letters disclaimed. )

Li Qiang Chen
Semptember 14th, 2019
 

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Appropriate method

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 my 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.
This semi – tramp often says “pig vs pig” / “two hole vs a long nose”…An appropriate method / angle can describe an appropriate thing.
 

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