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Fri 9 Oct, 2020 02:07 am

Anyone can go back in time to ask Einstein what does the mathematical symbol “-t” mean in physics?

The mathematical symbol “-t” can be seen in the CPT theory

Below is abstract from an authentic physicist’s post in another site:

“Under a parity inversion we take x → -x but t → t. In order to get t → -t we need an extra time conversion.

In general there are three main transforms to "turn a particle" into an anti-particle. They are parity, charge conjugation, and time inversion. (Charge conjugation turns a particle into an anti-particle.) This is known as the CPT theorem. Under CPT symmetry anti-particles look like they are following the reverse path that a particle would take. So if we have an electron traveling at speed v under the CPT symmetry we would have an anti-electron (a positron) traveling a speed -v with the coordinate system transforming as x → -x.

For this symmetry we have that the electron has a positive energy and so does the positron. There are no negative energy states in Physics.”

My question is that what does the mathematical symbol “-t” mean exactly in physics?

What does “going back in time” mean exactly in physics? To see Einstein? Anyone can do that?

“Anti” should be the inherent property of the “anti particle”. Describing it with its behavior is just shallow method.

A vivid analogy: We need no reference frame or movement or the element of time to identify what’s a male rabbit and what’s a female rabbit.

Moreover, what’s charge? Why charge inverted following the inversion of particle?

Is there a natural property existing in nature to indentify “anti”?

The “spirit of science” should be “exploration never stop its feet”.

In X4 Theory, one factor “X4” is employed to denote the matter state, or say “the inherent property” of particle. If the positive X4 value represents “particle”, then the negative X4 value represents “anti - particle”.

The inversion of space, time, and charge is decided by the inversion of X4 value.

(The primary concept of X4 Theory could be found in the thread “Can projective geometry find out its application in physics?” in the geometry column.)

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Liqiang Chen 陈力强

Oct 11, 2020

I point out in the chapter The Basic Mechanics for Anti Matter that the four dimensional space of an anti particle is inverted: X’ = - X = - x

People might imagine that time t could be inverted too, and the symbols – t or d (-t) appeared in some lecture books. They seem meaningful to describe anti matter in math, but they are meaningless to describe anti matter in physics (or this real cosmos). I will interpret why in X4 theory of time.

Now, let’s turn to the negative energy state about particle in the chapter Klein - Gordon equation in Relativistic QM. The wave function for such situation of free particle could be seen:

Ψ（x, t）= N exp( -i(Et – p•x)) E＜0

How it is reinterpreted with – t and d (-t), see those lecture books. Next, I reinterpret it in X4 theory.

Ψ（x, t）= N exp( -i( - |E|t – p•x))

= N exp( i( |E|t + p•x))

= N exp( i( |E|t – p•（- x))）

= N exp( i( |E|t – p•（-X))）

So:

Ψ（X’, t）= N exp( i( |E|t – p’•X’)), (Note: three dimensional space momentum p = p’)

And we notice that the above representation is equivalent to

Ψ（X’, t）= N exp( i(p’•X’– |E|t ))

= N exp(- i(|E|t– p’•X’))

The reinterpretation of the above wave function is: positive energy anti particle flows forward in time in its four dimensional space.

Next, make a further step transformation:

= N exp(- i(|E|t– (- p’ )•(-X’)))

= N exp(- i(|E|t+ p•X))

It means the probability wave for a particle flying backward is the same with the anti probability wave for its anti particle flying forward.

Regardless the topic of negative energy state, it’s important to see the four dimension spacial method of representation of probability wave.

For a free particle, it is:

Ψ（X, t）= N exp(- i(Et– p•X))

For the corresponding anti particle, it is:

Ψ（X’, t）= N exp(- i(Et– p’•X’))

E > 0

Pay attention please, p and p’ are three dimension spacial momentum while X and X’ are four dimension space. So they are both four dimension spacial wave functions.

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.

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Piggy now knows how to insert diagram for better understanding of the theory. Below is a picture of spiral spring which demostrates the natural spacial property.

............................................

Note: Actually the elementary particle model in X4 Theory should be a spherical surface. But for intuition sake, the “circle kind standing wave” can serve as a simplified model. And for convenience in calculation in chapters below, the simplified model is applied. And it’s considered that the energy on the spherical surface be converted entirely into the “circle kind standing wave”.

Liqiang Chen

Oct 20, 2020

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