프로그래밍과 경제

시공간에 관련한 시리즈 3탄

시공간에 대한 개념은 인간의 일상생활에서 뗄레야 뗄수 없는 너무나도 친숙한 개념이어서 고대부터 기록에 다양하게 나와 있다. 역사기록이 남기 시작한 이래 주요 사건의 시대를 우리가 구체적으로 알수 있고, 주요 인물들의 나이를 알고 있을 수 있는 인간이 시간개념을 고대부터 인지하고 있음을 보여준다. 고대부터 형성된 이 시간과 공간 개념은 뉴튼에 와서 성립된 고전 역학 법칙에서 어떠한 다른 요인에 의해서도 간섭받지 않고 절대적으로 독립적인 지위를 가지는 변수로 정립될수 있었다.

갈릴레이(15 February 1564 – 8 January 1642) 상대성 원리

시간과 공간 개념을 얘기할때 빼놓을수 없는 존재가 갈릴레이로, 뉴튼이 "자신이 남보다 멀리 볼수있었던 것은 오직 거인의 어깨위에서 볼수 있었기 때문이다"이라고 표현했을때 그 거인중의 하나가 바로 그였다. 뉴튼의 역학을 만들어내는데 핵심적인 사상을 제공한 갈릴레이는 뉴튼에 이어 아인슈타인에 이르기까지 커다란 영향을 끼쳤는데 그 핵심사상이 갈릴레이의 상대성원리다.

오늘날 가장 최신의 관점에서 볼때 시간과 공간은 서로 독립해서 존재하는 것이 아닌 한데 어우러져 움직일수 있는 동적인 존재로 비춰지는데 이 시간과 공간의 관계를 가장 밑바닥에서 연결짓는 개념이 속도가 된다. 

where s is change in position for duration of t.

속도방정식을 보면 시간의 변화 대비 위치의 변화량임을 알수 있는데, 시간과 공간을 엮어서 생각할때 더 이상 간단한 관계를 생각해 낼수 없을 정도로의 기본적인 관계가 속도이고 속도의 개념을 정확하게 파악하는 것은 시공간개념을 더 잘 이해하는데 필수불가결한 것이다. 이 속도가 바로 갈릴레이의 상대성원리의 핵심 구성요소이다. 

이 핵심원리가 말하는 바는 운동의 법칙은 모든 관성계(Inertial frame)에서 동일하게 적용된다는 것이다. 여기서 나오는 관성계를 갈릴레이는 관심의 대상인 물체의 속도와 관련해서 시간에 따라 속도의 변화가 없이 유지되는 기준시스템으로 정의했다. 이 관성계는 그 자체가 일정한 속도를 유지하는데 계자체속도는 반드시 계외부와 비교될때 검출될수 있지 계내부에서는 알수 있는 방법이 없다. 예를 들어 일정한 속도로 강을 흘러가는 배의 밀폐된 실내에 움직이지 않고 앉아있는 사람은 자신이 움직이고 있는지 멈춰있는지 알아낼수 없다는 것을 의미한다.

여기 두개의 서로 다른 속도 v1, v2를 가진 배가 둘 사이에 공유된 한 공간에서 존재한다고 가정하고 각각의 배에서 상대적으로 정지해 있는 관측자가 동일한 힘으로 배의 진행방향으로 돌을 던지고 돌의 움직임을 관측할때 두명의 관측자는 구분할수 없는 동일한 돌의 움직임만을 검출할 것이라는 것이 갈릴레이의 상대성원리의 기본 가설(hypothesis)이다. 이 핵심가설로부터 물리학자들은 속도는 관측자시점과 무관하게 존재하는 절대적인 성격을 가지는 것이 아닌 상대적으로만 측정가능할수 밖에 없다는 결론을 도출하게 된다. 두개의 관성계가 존재할때 하나를 정지계(A)로 놓고 다른 하나를 정지계와 상대적으로 동일한 속도로 관성운동하는 운동계(B)라 보면 이 두 관성계에서 관측가능한 모든 운동의 법칙이 동일하다면, 또 상대적 정지계와 운동계의 관점만 달리해서 볼때도 역시 모든 운동의 법칙이 동일하다면 절대적인 속도는 알수 있는 방법이 없으며 오직 상대적으로 관측가능한 속도만이 유효한 것이다.

여기서 약간 논점에 벗어난 얘기를 하자면 이 핵심가설은 현대에 와서 이 우주가 빅뱅이라는 한점으로부터 시작해서 유한한 시간동안 연속적으로 크기를 확장해왔으니 유한한 우주크기를 긍정할수 밖에 없고 그러한 유한한 공간내에서 존재하는 인간이 관측가능한 영역이 우리가 살고있는 우주의 공간이내라는 점을 고려해볼때 우주자체의 속도를 알수 있는 방법이 없다는 점에서 인간의 근원적인 한계를 드러내놓고 있다고도 볼수 있다.

이 갈릴레이의 상대성원리의 핵심가설은 아인슈타인의 특수상대성이론을 만들어나가는 2가지 핵심 가설중 첫번째에 해당할 정도로 이후 물리학의 발전에서 의심의 여지없는 진리로 간주되어 왔고 어느 누구도 이를 부정하는 사람은 아직까지 필자가 알기로는 없어 보인다. 미리 이 시리즈의 결론을 말하자면 필자는 이 갈릴레이의 상대성원리를 버려야 할 필요성을 강력하게 느끼고 있으며, 향후 진행되는 전개에서 충분한 설명을 제시하도록 할 것이다.

뉴튼의 시간과 공간 개념

뉴튼은 고전역학의 기본운동법칙을 명확하게 제시함으로써 그 이전까지 설명할수 없었던 다양한 현상들을 정량적으로 명확하게 설명할 수 있었다. 그 한가지 예로 갈릴레이가 피사의 사탑에서 실행했던 서로 다른 무게를 가진 추를 동시에 같은 높이에서 자유낙하시켰을때 동시에 떨어지던 현상을 이해할수 있기 위해서는 뉴튼의 가속도의 법칙의 발견이 절대적으로 필요했다. 관성운동의 개념이나 진자의 운동에서부터 천체의 운동의 기술에 이르기까지 질량을 가진 물체의 다양한 운동들은 뉴튼이 정립한 고전역학 법칙에 의해 명확하게 이해될수 있었던 것이다. 

이러한 뉴튼의 운동법칙이 명확하게 기술될수 있었던 것은 그의 시간과 공간에 대한 정의의 바탕위에서 가능했다. 그가 이해했던 시간과 공간에 대한 개념은 절대성을 가진것으로 다음의 표현에서 구체적으로 확인가능하다.

Absolute space in its own nature, without relation to anything external, remains always similar and immovable.

...absolute and mathematical time, of itself, and from its own nature, flows equally without relation to anything external, and by another name is called duration.

뉴튼은 또한 그의 저작 Principia의 도입부분에 존재하는 주해(Scholium)부분에서 상당분량을 시간과 공간의 참으로 존재하는 절대성과 측정가능한 상대성의 명확한 구분이 필요하다는 주장에 할당했다.(참고 1)

• Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year. 
• Newton, Principia; Definitions

뉴튼의 이러한 시공간에 대한 가정은 그의 기본운동법칙의 기반이 되었고 이 법칙은 그 이전까지 생각할수 없었던 예측들을 가능하게 만들어 줌으로써 이후 수백년동안 자연운동법칙의 기술의 진리로 받아들여질 수 있었다.

뉴튼운동 법칙 제1은 관성의 법칙으로 물체는 외부의 힘에 의해 간섭받지 않는 이상 자신의 원래 운동인 등속으로 공간상에서의 위치를 시간에 따라 균일한 차이로 정기적으로 바꾸거나 또는 절대적인 공간에서 고정된 위치에 항상 남아있게 된다고 설명하고 있다. 이 때의 관점은 갈릴레이의 상대성원리의 관점인 관성기준계가 정지해 있는지 아니면 움직이고 있는지를 알수 없다라는 관점을 벗어나서 제2의 관측자 시점을 완전히 배제하고 관심의 대상인 물체(A)이 절대적으로 정지하는 상태가 존재가능한 절대적인 공간과 시간 개념을 지지한다. 외부의 어떤 존재 여부와 상관없이 변화하지 않는 공간 개념과 균일한 비율로 일정하게 흐르는 시간 개념이 뉴튼이 자연을 보는 핵심 가정이 된다. 

세상에서는 수많은 변화가 지속적으로 일어나고 있다. 날씨가 흐렸다가 개인다던지 물체가 한장소에서 다른 위치로 움직인다던지 하는 변화가 인간에 의해 관측될수 있는데 이러한 경우 인간은 시간이 흐른다고 인지할수 있다. 뉴튼은 여기에서 더 나아가 물체가 공간상에서 움직이지 않고 어떠한 변화가 검출되지 않다고 하더라도 시간은 지속적으로 정기적으로 동일한 비율로 흐른다고 보았다. 아니 그의 운동법칙이 성립되기 위해서는 이러한 절대적인 시간의 흐름은 필수적이었다.

뉴튼은 다만 절대적인 시간과 공간이 존재함을 가정했음에도 불구하고 관성운동을 유지하는 물체가 절대적인 공간에서 움직이는 상태에 있는지 또는 정지상태있는지를 증명할수 있는 방법은 알지 못했고, 존재론(Ontology)관점에서 참된 운동 상태를 알수 없고 다만 관측자시점에서 바라볼수 있는 상대적인 운동 상태만을 관측가능하다고 보았다. 

뉴튼은 또한 그의 역학에서 무한대의 속도가 가능한 것으로 보았다. 그의 중력법칙은 두개의 질량을 가진 물체 사이에서 서로 끌어당기는 힘이 존재하고 이를 다음과 같이 정량적으로 계산할수 있는 방법을 제시했다.

그림 3-1. 뉴튼 중력 법칙

뉴튼 중력에서 특이한 점은 물체사이에 일견 아무것도 없는 진공상태임에도 불구하고 마치 보이지 않는 용수철이 둘사이를 연결해서 서로 끌어당기는 것처럼 끌어당기는 힘이 존재하고 이 힘은 "동시"에 양쪽에서 작용하는 것으로 간주되었다는 점이다. 두 물체가 우주에 존재하기 시작한 바로 그 순간에 존재한다는 정보가 각각 상대방에게 무한대의 속도로 전달되어 영향력을 미치는 것처럼 동작하는 듯 보였는데 뉴튼은 자신이 비록 이러한 가설을 제시하고 이를 통해 그 이전과는 비교불가할 정도의 천체운동들을 훌륭하게 설명했지만 빈공간에서 어떠한 매개체의 개입없이도 힘이 중계되는 것 처럼 보이는 현상에 대한 본질적인 설명을 할수는 없었다. 

또다른 유의할 점은 뉴튼 역학에서의 속도에는 제한이 없이 무한대값까지 가능했다는 것이다. 두 물체가 아무리 멀리 떨어져 있더라도 양쪽에 있어 항상 동일한 힘이 동시에 발생하고 적용되어야 했다. 그의 법칙어디에 있어서도 상호 동작하는 힘의 동시성은 필수 불가결했는데 절대적으로 또 선험적으로 존재하는 절대우주공간에 임의의 거리로 떨어져 존재하는 서로 다른 물체사이에 공유될수 있는 동시성의 개념은 당시 알려져 있는 모든 운동들을 이해하는데 절대적으로 유용해서 젊은 아인슈타인에 의해 반박되기까지 수백년동안 의심의 여지없는 진리로 군림했다.

정리하면 뉴튼 역학에서는 뉴튼 자신조차 설명하지 못하는 두가지 불만족스러운 부분이 있는데 첫번째가 시간과 공간 개념에서의 절대적인 면과 상대적인 면의 구분이 필요하지만 이를 어떻게 구분해야하는지는 알수 없었다는 점이고, 두번째가 동시성의 필수불가결함이다. 이 두가지 흠결은 아인슈타인에 의해 새롭게 정리되고 그 결과로 우리의 시간과 공간에 대한 개념은 혁명적으로 바뀌게 되었으며 새로운 분야의 미스테리들을 좀 더 잘 이해할수 있게 되었다.

아인슈타인의 상대적인 시간과 공간 개념

참고

1. http://plato.stanford.edu/entries/newton-stm/

2. http://physics.ucr.edu/~wudka/Physics7/Notes_www/node47.html

Challenging to solve the double slit experiment puzzle

Oh YeungJong (dmqcka _at_ gmail _dot_ com)

Note: This paper is intended to introduce my private view in a hope helping others to build the theory of everything. Please be noted that this has some radical or crazy ideas and may have wrong statements. I want to celebrate the 100 anniversary Einstein's general theory of relativity before the end of 2015. Welcome any comment and further discussion.

One of the weirdest things in Quantum Mechanics (QM) would be the subatomic particle’s behavior which can be observable in the double slit experiment. The microscopic scale particle such as photon and electron appears as a particle whenever we measure them but if single subatomic particle is not observed, then it seems to exist at multiple places at the same time. This observation confused many scientists. One major figure in modern physics, Richard Feynman mentioned in his physics lecture series [1] that (this) “phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot explain the mystery in the sense of “explaining” how it works.” For now, it is common belief that there is no one who can explain “how it works” in the double slit experiment similar to the Figure 1 introduced in his lecture.

With Schrödinger's Cat thought experiment, they are showing one famous counter-intuitive feature about quantum superposition and collapse. In quantum mechanics, superposition is a fundamental principle stating that a point like particle such as photon or electron exists in several different positions at the same time and collapse means that measurement gives a result corresponding to only one of the possible state. This observation has been so weird to many physicists including Feynman but it is a fact no one can deny.

How can a point like particle exist several places at the same time? This question based on the reality has confused many physicists since the discovery of quantum mechanics and still remains unanswered. Problem is that even though many physicists tried to understand the weird behavior during last few decades, there is no working model which can give us a reasonable explanation of the quantum superposition yet. Is it possible to think about such a physical model explaining quantum superposition and collapse using two great principles of quantum mechanics and theory of relativity?

Figure 1: Double slit experiment with electrons. [1]

Here I present a bold argument that there is another essential principle missing in building a theory of everything which will unify quantum mechanics and relativity. Without serious consideration for it, I argue no one will have clear picture on the fundamental structure beneath Quantum mechanics. I am pretty sure that if many scientists seek for something beyond the standard model to build the theory of everything, the first candidate must be the chaos theory and I hope the following discussion will make it clear. To show the usefulness of chaos theory, I begin by introducing a classical example which could be a hint for particle superposition similar to the quantum superposition and collapse.

Before delving into the discussion of a new model concerning quantum superposition, it would be helpful for understanding it to consider an mechanism to display Arabic number with a 7 segment display as a classical example which can show the same pattern with the superposed particle.

Figure 2: An example for particle superposition effect in 7 segment display utilizing Persistence of Vision

In the left of Figure 2, a typical numeric display device known as 7 segment display is shown and in it there are 7 lights which are tagged as a, b, c, d, e, f and g. It is used to display any single Arabic number among 0-9. In order to display number 5 as in the figure 2, we can choose one of two possible options. First is to turn on all 5 lights at a, c, d, f and g at the same time. It obviously consumes 5 times energy than turning on single light. To reduce the required energy, with help of persistence of vision, we can take the second option to turn on only one light at any instance time. The right of the Figure 2 shows the timing sheet showing how to schedule to turn a light in 7 segment display. The used algorithm is something like as follows.

Let suppose there is a system which can display a number 5 with 7 segment display and it has 7 states named from a to g. It is assumed to be in one state at certain instance time and change its state at a discrete time interval or tick t. With these limitations, it requires 7 ticks in minimum to visit all states so let assume its transition is always periodic with period 7. Basically it is scheduled to turn on ‘a’ during first tick out of period 7 and ‘b’ during second tick and so on. In order to display 5, it can visit ‘a’, ‘c’, ‘d’, ‘f’, and ‘g’ sequentially except ‘b’ and ‘e’. As described in the previous assumption, it is scheduled to keep the previous state during interval for ‘b’ and ‘e’.

With these preparations, when we exploit the persistence of vision, one characteristic of the human eyes, we can experience the same effect as in the first method. If the unit period time T is decreased below a certain threshold value, fairly less than 1/60 second, human eye feels difficult to distinguish the observed behavior of it from one of turning on all lights and its appearance will be almost same as the interval of unit time tick is decreased to zero. Even though human eye can see the same effect as all lights are turned on with extremely short interval of unit time tick, whenever we try to stop the system running and see what state it show, we always observe one state instead of many states. Here the probability we can calculate and measure for all possible states are shown in Table 1. The example model used here gives us a hint in building a model deceiving us that a single particle can show up multiple places at the same time with the predictable probabilities but is detected at only one place.

Table 1 Probability to be in one state out of 7 states of 7 segment display system showing number 5.

A physical model explaining quantum superposition and collapse

My purpose here is to introduce a model describing properties of the most fundamental building block in our universe probably scale down to the plank scale distance using three theoretical frameworks; quantum gravity, relativity and chaos theory. It is expected to explain many weird behaviors what many physicists want to understand and the nature of time, space and mass. As like other great theories require certain number of postulates, my model needs many postulates but I think the number need to be decreased significantly as study goes.

Postulates

1.     Revival of Newton’s absolute time. Newton’s view on absolute time that absolute time exists independently of any perceiver and progresses at a consistent pace throughout the universe is a basis for further reasoning. Full range of absolute time can be mapped into continuous real number and absolute time flows always to the future at a uniform rate independently of any inertial frame of reference.

2.     Revival of Newton’s absolute space. According to Newton, “absolute space, in its own nature, without regard to anything external, remains always similar and immovable.” Full range of absolute space can be mapped into continuous real number.

3.     Similar to what string theory suggests, the time evolution of every elementary entity in 3 spatial dimensional space defined in the standard model such as electron, photon, dark matter and dark energy can be described as mathematical language and all share a common mathematical form and the general form can be rearranged as a set of first order differential equations defined in 11 dimensional space whose Eigen vectors include:

A.     3 absolute spatial vectors x, y, z

B.      1 absolute time vector t

C.      3 spinning forces f_r, f_g, f_b at position p= (x, y, z).

D.     3 directional forces f_x, f_y, f_z at position p= (x, y, z).

E.      1 energy vector E as a power source driving the time evolution.

4.     The general form in 3 for any elementary entity in can be simplified as follows:

, where h is plank constant, ν is light frequency, c is the speed of light in empty space and m is mass. This equation reflects the wave-particle duality. For a pure photon with energy E=hν, if it is got absorbed by a balckhole, it's wave like energy will turn into a pure mass form with $$E=mc^2$$. So m and ν are related as $$E=hν=mc^2$$.

5.     The p and q in equation (2) has the relation of

6.     Dark matter which is well known term in cosmology is assumed to be a point like particle with q=1 in the equation (2) and the pure photon with constant light velocity is considered as a point like particle with p=1 in the equation (2).

7.     Of any elementary particle system, Einstein’s relative time t_r and relative space x_r are related with Newton’s absolute time t and x as in what follows:

, where m is the mass term and m^-1 is called as a world evolving rate. For a photon, m=0.

8.     Of any system of elementary particle, the mass term m in equation (3) and (2) is defined as a sum of all interaction with dark matters included within its bounded area over a unit interval of absolute time domain. The full set of possible states of the system governed by the rule of equation (1) is known to be finite or bounded in the state space. [Refer to 2].

9.     The definition of measurement is to synchronize two different world evolving rates of two different inertial reference frames. For example, If a detector system with the world evolving rate r1 try to measure or detect a photon with the evolving rate r2=∞, then the measurement can be completed only after setting r2 to r1.

[2016-206. Added]

10.     The term "dark matter" mentioned in the postulate 8 can be thought as a particle having a tendency to stand still to a inertial reference frame and its role is generally to put some amount of mass to an encompassing particle. It is defined as the linear sum of 3 spinning forces f_r, f_g, f_b at a spatial position p= (x, y, z). 

Explanation for the postulate 10.

In my view, the dark matter seems to play the same role what the "Higgs particle" is expected to show. The most important thing is this dark matter is also the source of gravitational lensing effect such as bending space-time as most physicists think. My understanding is that the major view point shared by most physicists are that Higgs particle and dark matter are two different things and not the same one but I argue they are the same thing. Then the question is how it, whatever is called as dark matter or Higgs particle, can play both roles; putting a some amount of mass to a particle or bending space-time.

I think if the dark matter particle is included within the bounded area of a particle or object, then the object will get the mass equal to the sum of the interaction with the dark matter particle. So the mass of our body is believed to be the sum of interaction with the dark matter enclosed by our body. In this view, any physical shape including us are the total sum of lights, massless particle, interacting with the dark matter particles within a bounded area in 3 dimensional space. If a massive rigid body occupying a finite area in space is in a consistent motion with the same velocity, we say it is associated with a inertial reference of frame IRF and the dark matter particles contributing the mass of it tend to standing still to IRF. This view shows the role of Higgs particle is expected to play. I think the mass created from the interaction with dark matter particles bounded within a thing(i.e electron, proton and etc) is the rest mass or invariant mass of the thing most physicists mean.

Another role which dark matter plays is to change the motion of a particle or bend the space-time. Let me describe what I have in mind. Dark matter can reside any point in 3D space and shows a tendency to stay still relative to a inertial reference of frame. It makes me to think the eye of typhoon, a floating force trying to pull everything nearby into its center. The dark matter distributed not only within the bounded area of a rigid body but also near out side of the rigid body constituting a field. The distribution of dark matter near a massive body of mass M is believed to meet the condition described in the postulate 5 and suggested by general relativity. 

For better understanding, let me give you a classical analogy. Suppose, in the center of ocean, that a ship is sailing with constant velocity v and will pass nearby a whirlpool strong enough to change the sailing direction of a ship. In this case the future life of the ship will be determined by the closet distance between the ship and the center of whirlpool with the assumption that the the whirlpool power is constant and no driver controlling the ship on board. As it gets to the center of the whirlpool, it will need more energy to escape from it. Here we need to change our classical view that the eye of typhoon is just one. Instead, we can think there are finite but infinitely many number of eyes within the bounded area of typhoon and each eye having different magnitude is orderly distributed so that its pulling power is inversely proportional to the distance to the center of typhoon. This view resolves the problem of acting with object at a distance which Newton want to know. I believe this classical example helps readers understanding intuitively the space-time curvature what Einstein recognized firstly 100 years ago.

So it is the source to give rise in the rest mass of a rigid body that the dark matters enclosed by it and standing still relative to the center of mass of it. The minimum distance between two dark matters is estimated to be the Planck scale distance 10^-35m. No matter where the rigid body moves, its paired dark matter will follow standing still to the IRF of it so there will be no change in the rest mass. Of course, near out side of the rigid body, there will be dark matters distributed consisting a gravitational field due to the rigid body. Returning to the classical example of the ship sailing near by a whirlpool, the ship can be thought as the rigid body and whirlpool as a black hole exerting humongous gravitational field. In such case, the space-time curvature what Einstein described in his general relativity can be described as the distribution of two groups of dark matters which are originated from the rigid body and black hole in space. This distribution which is continuous in space can provide definitely the guiding passage for the rigid body in its world line. Unfortunately, I don't have enough skills to describe it with a strict mathematical form for now but I believe some experts can derive it easily based on the description above.

It would be worth to note one additional philosophical view that nature hates sudden discontinuity and I agree on that. On the boundary of a massive rigid body, I think it would be nice and beautiful that the distribution of dark matter is decreased gradually as the distance from the boundary increases.

The dark matter seems to be a singularity attractor pulling any particle moving near by and black hole is believed to be a dark matter particle with huge energy. I borrowed the concept of attractor from the chaos theory. Some characteristics found in attractor such as the new concept of frequency of a strange attractor are especially important to understand the true nature of our universe but I will leave it here and discuss in another thread. 

In my view, all Higgs particle and dark matter and black hole are actually the same thing with magnitude variations: a energy force trying to pull any object passing near by into its center. I think most physicists will agree on that a physical model with fewer assumptions is better than one with more assumptions. From my intuition, it seems to prevent the time evolution of a particle and shrink its perception on the spacial space similar to what special relativity proposes.  

[/2016-206. Added]

How can quantum superposition and collapse be interpreted using the above postulates?

For simplicity, I will begin with the most basic element particle, a massless photon, other than electron with rest mass. From postulate 3 and 8, a photon evolves in time in a bounded volume of 3 spatial dimensional space deterministically ruled by a set of ordinary differential equation. Bounded area means that the photon oscillate within a bounded area or has a finite size instead of a point independently of any perceiver.

As quantum mechanics already discovered that the probability to find a photon at certain volume area can be predicted, the previous work with chaos theory [3] showed that the probability distribution of a particle moving along the trajectory of chaos system such as Lorentz system can be determined by means of numerical analysis before actual measurement.

The size of a photon is appeared as almost zero to any perceiver close to human scale but in absolute time and space, the bounded area of a photon is non-zero volume and remains unchanged if there is no interaction with any external energy form.

What is the frequency of light in free space and is it quantizable?

This is next topic.

[Added 2016-01-03]

See the recent comments in the additional part in 2016-2-06.

I missed one important postulated so here I add more postulate about the distribution of dark matter in absolute space. Dark matter which is known as a hypothetical kind of matter causing gravitational effect in cosmology is, in my model, a sort of frictional like energy particle instead of matter. It is made up of the linear combination of f_r, f_g, f_b. It is same with the inertial mass in classical mechanics.

Postulate 10.

Given a point like matter with mass m, assuming 1 directional space, dark matter is distributed around it in decreasing way as the distance r from the mass position: Unfortunately, I don't have the exact equation for the dark matter distribution but my guess is that it could be a equation whose integrals from 0 to infinity distance is matched with the rate of percentage of dark matter over visible matter: 

Figure 3. Conceptual image for dark matter distribution. The integral of the distribution between 0 to infinity is expected to be matched with the rate between the percentage of dark matter and visible matter. 

Understanding quantum superposition in double slit experiment with the new model.

From postulate 3, it is known that any particle system of which time evolution is governed by the set of ordinary differential equations is oscillating in a bounded area in absolute time space in similar way to the strange attractor. It means it is possible that the bounded area of the particle is bigger than the distance between two slits. Figure 5 shows such a case. 

Figure 4. A conceptual image. An oscillating particle in the bounded area is passing through double slits with light velocity c. The trajectory of the particle is big enough to contain the wall between two slits.

As can be seen in the previous example of 7 segment display system, for a system which can have only one state at any instance time, it can produce an illusionary effect as if it can have multiple states at the same time by reducing the duration of a unit tick to zero. Same principle can be applied with the double slit experiments with photon. What important information is that a photon has no mass and a world evolving rate $$ \alpha = \frac{t}{t_r} = \infty $$

What the world evolving rate α of a system is infinity means that it takes no time to complete the evolution which is analogous with increasing the clock speed of CPU to infinity or decreasing the duration of a unit tick. It should be noted that any system experience only relativistic time and space and there is no way to measure or observe the absolute time and space. It is similar to what we are unable to measure π. Instead of using ideal symbol, we can only use the approximated value of π in a calculation and the approximated value is normally relativistic to who make a decision of the level of approximation. 

Back to the double slit experiment, due to the mass of wall included in the bounded area of photon particle, the world evolving rate will be decreased but negligible because the mass is small and still it's moving from one slit to other along its trajectory will take almost zero time. 

I think the model described so far using chaos theory and special relativity is the only available non-mathematical explanation about the quantum superposition and easy to understand.

[/Added 2016-01-03]

1.      [↑] Feynman, R.P., Leighton, R.B., and Sands, M. (1965). The Feynman Lectures in Physics Volume 3, Section 1–1, Addison–Wesley.

2.     Chaos. http://plato.stanford.edu/entries/chaos/

3.      Jos Leys, É. G. (n.d.). Chaos-viii-statistics. Retrieved from CHAOS: http://www.chaos-math.org/en/chaos-viii-statistics