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Theory of Analytical Space-Time (II)

Author:  Cui Silong    PhD

For thousands of years, it has long been considered that what we see or observe is objective reality, however the impersonal world is in existence without relying on our consciousness. In 1905, Einstein rose in revolt first. He pointed out that, according to the Principle of constancy of light velocity, the length of an object in the direction of its movement looked shorter when relative speed of the object was high enough. Subauditionally, if we make two rulers of same length on the earth and put either of them in an airship that leaves the earth at an ultra high speed, we will find that the ruler on the airship is shorter than the other left behind on the earth. A philosophic problem here occurs: Is the length of the ruler objective on the airship? What is the standard of objective reality, stationary state or movement state? Einstein did not give a sound philosophic explanation.

Theory of Analytical Space-time (TAST) states that all facts we have observed may not be their real existence or objective matter in physics. Real existence is absolute but the result of observation is relative. 'Objective existence' as we are used to thinking about is nothing but misunderstanding of objectivity and it depends upon the state of observers. The faster an object moves, the greater difference it has between real existence and result of observation. Deflection of space-time allows us to know the difference.

When the airship is stationary, we say the real length of the ruler is the same as its measured length, however as long as the airship leaves the earth with a high speed, we can find there is error between its real length and measured length. Furthermore, this error will become bigger with the increase of speed of the airship. Due to the deflection of space-time, the real length of the ruler can not be measured precisely. We therefore use mathematical language to explain what uncertainty effect is both in microcosm and macrocosm.

It is known that uncertainty effect takes place in measurement of microcosm. What is the cause?

When an object leaves the earth with a very high speed u, with formula (1-1):  l = l'cosq,  l , l' are measured length and real length respectively, the error between l and l' is shown as d, 

d = l'- l = l'- l'cosq = l'(1-cosq)                   (2-1)

If measured length and real length of the object are near or equal,

then    d ® 0 and (1-cosq) ® 0   or  (1-cosq) = 0,

since cosq = (1-u²/c²)^1/2 , we must have u ® 0 or u = 0.

Apparently, this result does not agree with the assumed condition at the beginning.

Therefore, we conclude that the real length or position of an object could never be made certain by observation when the object is in a high relative speed. There is a same situation to momentum. The higher the relative speed of an object is, the more it shows uncertainty. From Chapter 1, we know that the measuring results are relative in a high speed situation, but few of us are often aware that the units we use to measure are absolute! Since uncertainty results from the deflection or rotation of space-time, we naturally infer that quantum uncertainty is due to space-time rotation and we can further understand the cause of quantum uncertainty in the paragraphs below. It can be said that uncertainty both in microcosm and macrocosm results from the deflection or rotation of space-time! TAST has expanded the concept of uncertainty and uncertainty effect is not the 'patent' of microcosm. It resides in superposition, depending on the method of measurement. For example, we are never certain in what direction the deflection or rotation is in three-dimensional space until a measurable impact acts on the object. Then, what about observation? Philosophically, perceptive existence is hardly the same as real existence and we should not request 'two existences' to be unified. Only on admitting objectivity of 'two existences' can we stride a firm step in searching for so called 'Theory of Everything'.

Chapter 2

Quantization of Analytical Space-time & Function of Space-time Wave

2.1 Compton effect

We begin with the quantization of analytical space-time in the scope of 10^-10 m -- 10^-14 m.

With formula (2-1), d = l' (1-cosq), we introduce Compton wavelength lc (lc=h/mc=2.426 x 10^-12 m).

Let l' = lc and put into (2-1),

then

Formula (2-2) is the Compton math model of dispersed X-ray and the deflective angle q proves the same as Compton dispersion angle.

2.2 Basic property of analytical space-time

Let a plane of a moving coordinate (S') be set G' of complex number z'=x'+iy' and, at meantime, the other set G is complex number w, w=u+iv

according to formulas (1-20) and (1-21):

x = x'cosq - y'sinq

y = x'sinq + y'cosq

Let u = x and v = y, obviously there is a definite rule between z' and w, which makes every complex number z' in set G' correspond to the other complex number w = u + iv,   

w = (x'cosq - y'sinq) + i(x'sinq + y'cosq)

or   z = reⁱa                   (2-4)

r = (x² + y²)^1/2 ,   argw = a ,    tga = x/y

u = xcosq - ysinq           (2-6)

v = ycosq + xsinq           (2-7)

With mapping concept of complex function, in geometry, the function w =f(z) is the mapping of set G' of plane z to set G of plane w, we call z the space-time primary image and w the mirror image in physics. Space-time mirror image is determined by mapping

w = z_o z           (2-8)

z_o= cosq + isinq

z_o=r_oeⁱq      (r_o=1)

\ w = eⁱqreⁱa

w = reⁱ(a+q)            (2-9)

With the definition of complex function, we know each point z₁, z₂, ... z_n in the observing coordinate (S) is mapped to w₁, w₂, ... w_n by mapping   w = z_o z, so the plane image in the moving coordinate (S') has changed to the other geometry image of the observing coordinate(S). From (2-4) and (2-9), the difference between w and z is that w has deflected by an angle q which is the deflective angle in principle (II) of TAST.

• Mirror image composed of w₁, w₂, ... w_n differs from primary image.
• Space-time mirror image is determined by the mapping  w = z_o z or w = reⁱ(a+q) .
• Due to | z_o | =1, space-time mirror image does not flex , but it circumvolves a phase angle q which is the phasic difference between the observing coordinate (S) and the moving coordinate (S').
• The phasic angle q is an independent variable and has nothing to do with z.
• Because of argument Argw=(a + q)+2kp, the number of mirror images is infinite.

Now let us discuss the analyzation of deflected space-time according to Cauchy-Riemann function:

If a function f(z) is differentiable in field D and meets the condition above, w=f(z) is an analytical function in D.

With (2-6) and (2-7),         

u = xcosq - ysinq

v = ycosq + xsinq

Therefore, the function of space-time mirror image w=f(z) is an analytical space-time function.

2.3  Quantization of Analytical Space-time

We have discussed the characters of analytical space-time in complex function and got the function (2-9) w = reⁱ( a+q), where r and a are constant. The deflective angle q is concerned with a wave vibrated in x direction, then q = wt , t = x/u,    uT = l

Therefore

Formula (2-10) shows a wave-function under complex field. We make it twice differential:

Since

and

put into (2-11), then

and replacing w with y . Since E = E_c+U, we get

Formula (2-12) as the basic function of quantum mechanics has long been an assumptive or experiential equation, but under complex function of analytical space-time, Schrödinger wave function becomes the deduction from TAST. However it can not be regarded as a strict theoretical deduction because some basic concepts of quantum mechanics and Relativity such as E= hv, E=mc² and E=2E_c are used in above deduction. Therefore it is necessary for us to review the basic concepts of quantum mechanics and Relativity from the deflection notion of analytical space-time so as to get a profound math expression -- Space-time wave function (STWF) to link Relativity up to quantum mechanics.

Assuming that one plane cosine wave travels along X axis in a non-absorbed medium, wave-speed is u and displacement of a particle is y and swing y₀, the vibration function of the particle is:

y = y₀cosw t                   (2-13)

According to formula (1-1), l = l'cosq, replacing l and l' with y and y₀ respectively, then

y = y₀cosq                   (2-14)

On comparing ( 2-13 ) with ( 2-14 ), if q in formula (2-14) varies at a uniform angular-speed, i.e.  | q |= wt, cosq=cos(-q), ( 2-14) and ( 2-13 ) are identical. This shows that any free vibratory particle function can be expressed as the space-time wave function. In a broader view, a simple periodic motion is a form of space-time wave. Not only do we use formula (1-1) to express varieties of space for coordinates, but it also becomes the expression of transferring more information among different space-time systems by way of space-time wave. 

With space-time wave function y = y₀cosq, let us see what conclusion it would give.

Speed u in y direction:

According to principle (II): sinq =u/c, and putting (2-15) into it, then

Minus means that space-time circular frequency has a reverse direction to particle wave or the phasic difference between two circular frequencies is p while the absolute values are equal.

Let  | q | = wt, then y₀w = c 

y₀ = cT/2p                 (2-16)

and putting (2-16) into (2-14), then

When discussing energy of space-time wave function (STWF), we have to add a restricted condition: for observers, STWF should meet the condition y³0, because it seems that space can not be minus. See Fig 2-1, in periods T, domain [0, T/4] and [3T/4, T] meets y³0, correspondingly for q, the domain is [0, p/2] and [3p/2, 2p].

Note: t = 0 is a factitious or relative situation.

With (2-15) and (2-16),

Energy E in formula (2-18) is determined at time t, we need to know the sum of the energy in a period T, which is defined as quantum energy E_Q. Because of incontinuity of STWF in the domain [0, +2p], quantum energy E_Q can not be figured out by average value of sin²wt in domain [0, T]. Since space-time wave is elastic, we calculate its wave-power: energy at time t₁ corresponds to E₁, t₂ to E_2, then DE=E₂-E_1. Average rate of wave energy in Dt is DE/Dt, so wave-power W at time t should be, as Dt®0, the limit of DE/Dt.

The result above shows that the sum of energy SE in domain [0, p/2] is E_C=mc²/2 (E_C means classical energy). Quantum mechanics deals with energy in domain [0, +¥) while quantum energy E_Q is the positive sum of energy in domain [2kp, 2(k+1)p] or unit positive energy in any period T. The algebraic sum of energy in any period T is null, so we can only measure positive energy anyway. The relation of E_Q and E_C is as follows:

E_Q=2E_C=mc²                       (2-20)

The general solution of SE or E_Q should be:

With formula (2-19),

then W(t) = mc²v, put into (2-21), then quantum energy E_Q = (mc²T)v.

Let h = mc²T, so the result is E_Q= hv         (2-22)

Apparently, h is a constant called Planck's constant (h = 6.63 x 10^-34 Js) and commonly used in quantum mechanics.

So far, we have got the basic energy expressions of quantum mechanics and Relativity from TAST:

E_Q= mc², E_Q=hv and E_Q=2E_C

Furthermore, according to STWF, we will demonstrate Schrödinger equation strictly, and make it, a 'hypothesis' in quantum mechanics, a theoretical outcome of TAST.

2.4    Space-time wave function

According to the description of  space-time wave function in 2.3, supposing space-time wave is simple periodic vibration at speed u in X direction where all matters vibrate freely. Circular frequency is w, period T, wavelength and swing l and A, so the vibration is expressed:

y = Acoswt     

Also, the complex function should be:

R_e(y) = Aeⁱwt

where R_e(y) takes the real part of the complex only. We do not discuss imaginary space-time in this Chapter.

with w = 2p/T, t = x/u and uT = l

we have

According to formula (2-20), E_Q=2E_C, E_Q = mc²

l =cT,   c² = l²/T²,   E_Q= ml²/T²,

l² = E_QT²/m = E_Q/v²m                     (2-24)

and with formula (2-22) E_Q=hv, v = E_Q /h put into (2-24), then

putting (2-25) into (2-23), we have

and E= E_C +U, where E is overall energy of mass in the space and U is potential energy.

Also, the function is the same in directions of y and z.

If we analyze y(t) of STWF from the meaning of math, the Fourier transform of y(t) is as follows:

According to energy integral of Parseval equation of Fourier transform, we get

where

S(w) is called energy density function, on which the distribution of energy y(t) depends. The sum of energy can be obtained by integral to all frequency. But as for incontinuity of y(t) in the domain, we have to use the method of statistics, which is called probability wave function in Quantum-mechanics, to determine the distribution of energy in general space.

(Here, we will not discuss energy density function of STWF further, since Fourier transform is a math issue only.)

As formula (2-27) is a complete expression of three-dimensional space, Schrödinger equation is no longer a hypothesis and becomes a theoretical result of STWF hereon. Space-time wave has a far greater and wider significance than particle wave (vibration). No matter what mechanical vibration, electromagnetic surge or vocal wave, they are all manifestation of space-time vibration or a certain wave in special scope, all of which are general phenomena in nature of both macrocosm and microcosm, and can be expressed by STWF. Furthermore, with overlapping of space-time waves, we know that quantum phenomena including wave-particle duality result from the impact of measurement and discover what hides behind the Schrödinger equation.

It is known that a math equation in physics is indispensable to certain special physical notions, but we find that Schrödinger equation, as a principle in Quantum mechanics, is lack of a 'should have' physics meaning. It might as well be a conclusion as a hypothesis or principle. Though Schrödinger equation is useful in application, it does have a weak foundation as the principle of Quantum mechanics before TAST shows up. Also, General Relativity defines non-inertia space-time as a space of Riemann. For Riemann space has positive curvature, we have to doubt about where the minus curvature space is. Could it be said that God favors positive curvature? We can not talk about the 'Theory of Everything' convincingly before giving a right answer to above 'simple' questions.

2.5    Space-time panorama

Not only do we give a math show of Schrödinger equation in this chapter but we also set it up on substantial and more representational principles, therefore we have reasons to scan all physical theories with deflection of space-time:

2.5.1  Absolute space-time

It is the space-time where we live. We have absolute time and absolute space here, which has ruled humans' views for thousands of years. So far it has had deep influence on our thinking and philosophical viewpoints because the world we experience is at a low speed. No one can change low-speed reality any further in this space-time. Facing this 'reality', what physicists could do is to shorten the distance between subject and object at best.

2.5.2  Relativistic space-time

About a century ago, Einstein discovered the relativistic space-time and thought that space-time should not be absolute anymore and that absoluteness of space and time was no longer applicable in this space-time. After years elaborate design, he described the space-time as a curved, multi-dimensional and forth raised space, an end of which is absolute space-time and the other end is a black hole (q = p/2) where all classical physical theories are invalid. It is a pity that he did not find that the relativistic space-time is only a small part of entire space-time wave band.

2.5.3  Quantum space-time

Quantum space-time deals with a farther range than  relativistic space-time does. As it expands wave band from [0, p/2] to [0, +¥), it should be said that both absolute space-time and relativistic space-time are special cases of quantum space-time. Scientists find that it is difficult to determine both position and momentum of particles at the same time and, furthermore, energy distribution is not continuous. Why is energy distribution discrete and where is the 'lost space'? We have known from this article that there are minus space in [0, +¥) according to STWF and that is the 'lost space' we look for. Generally, space and energy are symmetrical along time axis, the only problem is that we can not feel minus energy and minus space directly. It is understandable for us to introduce negative space.

2.5.4 Negative space

It needs considerable courage to accept minus space that is a forbidden zone in classical theory. We should have a stably theoretical base to complete the span of space-time. With a very simple math model, TAST depicts a panorama of space-time and provides an essential tool for us to understand the entire analytical space-time system. If we expect to do something in space-time field, we have to cast away our intrinsic conception -- God is always in favor of humans. Positive space is wholly the same as negative space in symmetry and just depends on which side humans believe we stand by! 

So far we have unified the principles of Relativity (special and general) and quantum mechanics on the foundation of TAST, thus the principles of existing theories of physics become the results of TAST. The two principles of TAST seem beyond our experiences, simply because our experiences do have defects in experimenting the essence of the world. We are sure that the principles of the unified theory or the universal theory must be related to the field beyond our physical experiences.

From the development of pure science, we see that we take some things as phenomena while taking other things as essence to describe our experiences. When we reach a deeper level in science, the essence or principles at the upper level become the manifestation and we use more comprehensive principles as essence. We believe that all explanations of the world will converge upon simpler and simpler principles and that the laws that govern the behaviors of cosmos are already connotative in the theories such as Relativity, Quantum mechanics, Natural selection, etc. We will find that TAST is related to matter & energy, origin and evolution of cosmos, order of masterdom over lives, mechanism of mind or consciousness as well as to the old, primal and essential questions in philosophy, i.e. subjective & objective, mind & matter and consciousness & existence.

We understand the world by learning and discovering the regularity in our experiences. There are two kinds of regularity, mechanical laws and statistical order. It is TAST that may combine the both theoretically!

Now that TAST, as commented in a special review on TAST, has had us see the aureole of God through the general effect of deflection or rotation of space-time, would it lead us to see the true face of God or comprehend the mind of God?

Turn the key, you open the door.

November 3, 1999

Please also refer to the article at The American Physical Society Publish-Eprint .

(To be continued)

Following chapters of TAST will not be published at Internet for the time being.

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