Searching for Quantum gravity. what is distance? (1)

Oh Youngjong

(dmqcka @ gmail.com) 

(This article contains my own personal view and not finished yet so be noted that it may contain wrong information. It can be modified at any time without notice.)

This series is written to share what I learnt from my almost 8 year effort to figure out what the theory of quantum gravity or theory of everything will look like. For now, I think I have my own not perfect but detailed picture on the theory enough to share. I bet it absolutely sounds crazy and sensational to readers if someone succeeded to figure out the most difficult problem which physicists face in marrying quantum mechanics and general relativity. If the person arguing such crazy thing is nameless and no doctoral degree in physics and cosmology like me, there is no doubt people treat him/her a liar and no pay attention at all. Even I pretty well know that, I will try to argue that because I pretty sure everyone will benefit from it regardless of whether it turn out to be true or false.

It all began when I happened to see a YouTube video with a title Dr. Quantum – double slit experiment. Especially I was curious on why observation change the behavior of light. In the video, light passing through double slit shows interference pattern with many bands at the end screen as much same as water wave. On the other hand, if an observer such as camera is placed to observe what slit light (a photon) pass through, interference pattern disappears and screen shows two bands of line where photon hit at. As Richard Feynman told, it is the most important weird one of many unsolvable mysteries in physics and no one knows what is going on. I also interested in the true nature hidden in this famous experiment. I just wanted to know a reasonable explanation but it seems weird and bizarre so no one ever can possibly think of any reasonable answer for it and so did I until I got caught of an idea that there could be a relation between string theory and chaos theory.

The shape of string (in string theory, everything is made of string and the differences of matter come from the difference vibrations of a common string) and strange attractor in chaos theory looked very similar to me because the trajectory of dynamical systems can have various different shapes such as strange attractor, periodic circulation and point like rotation, depending on the some changes of coefficients in their mathematical models. (To see this, refer to http://kevino.tistory.com/entry/Series-6-Chaos-theory-a-underlying-model-of-every-Quantum-particle). The similar pattern that a common base model can show different trajectory or vibration mode defined in their own state space by changing some parameters in its mathematical model can be found in both string theory and chaos theory and it looked so powerful and attractive to me. Since then, I extended my thought gradually to the much fundamental deepest level of our universe during last 8 years because I wanted to get more intuitive explanation without much mathematics enough to persuade myself to accept it. Finally I reached to believe that the most important key lies in the correct notion of distance in any dimension. In fact, there were lots of trials and failures I had to during my struggling but I will skip those all and begin with the notion of distance which we need to clarify necessary to get the Theory of Everything(TOE) because I think it is easier way to explain. Someday, there will be a chance to write it. So for now let’s dive into what is unclear in our understanding of the notion of distance of any dimension.

Why the notion of distance matters? Simply it is incompatible in two great principles in physics, quantum mechanics (QM) and general relativity (GR). QM usually deals with the light or massless, subatomic scale particles while GR on heavy cosmos scale objects such as stars and galaxies so they do not overlaps. In their own area, they work awesome precisely and it is believed there is no exception violating them. But the problem people keep failing to resolve arises when they need to be incorporated at some special cases such as black hole and big bang. At singularity, things get heavier and smaller so we need to use two principals at the same time to figure out the behaviors at the singularity but marrying them is not easy and nobody succeeds.

In QM, since Max Planck postulated the energy of electromagnetic wave was quantized, it is believed that everything including time and space is quantized. Plank time is the time required for light to travel a distance of 1 plank length in a vacuum and there not exist any value smaller than plank unit. But in classical mechanics and GR, length and time in cosmos scale are treated as smooth, continuous measurable which can be 1:1 mapped to real number. In Newton’s, law gravity works at a distance r which is real numbered. It can have any value, integer or irrational number. So if we need to unite QM and GR to study the behavior of singularity existing in our universe we need to pick one notion either continues or discrete medium for time and space. But whenever physicists tried to fix it, I read from many sources that all calculation always led to infinity which did not make sense. It is the most fundamental problem in marrying QM and GR which we need to solve. Without a clear picture on whether time and space are continuous or discrete, theory of everything won’t be available to us. Why the notion of time and space, specifically the notion of distance in time and space, are used incompatible to each other is the problem I would like to address here.

In the end of this article, I will show that the distance, which is a measurement of how far two different positions or locations are separated in a dimension, can be defined only after the dimension can be quantized. If a dimension cannot be quantized, it is impossible to define the notion of distance. The quantization here means that there is an indivisible minimum unit length in the dimension and all observable distance is an integer multiple of this unit length. Here I pretty sure that almost everyone will laugh at me for arguing it is impossible to define the distance in continuous space and time because mathematics we learned from school taught us the space is continuous and differentiable at any point in the dimension and uses the distance without any problem so the notion of distance is self-explanatory enough no requiring additional description or condition. But it is the very important trap or subtlety which people have not recognized so far. I am not telling simply it is impossible to define the distance of any continuous differentiable dimension. Instead, if there is a way to quantize a dimension, we can define its distance as we do it with space and time. This will lead a conclusion that GR must be modified to encapsulate any form of quantized time and space in its theory in order to describe unambiguously the notion of distance in its 4 dimensional space.

This is not entirely my own view. In fact, according to the [1] http://plato.stanford.edu/entries/paradox-zeno/, it seems that the problem in defining the notion of distance was known to ancient Greek philosophers. Below are quotes from it:

“In response to this criticism Zeno did something that may sound obvious, but which had a profound impact on Greek philosophy that is felt to this day: he attempted to show that equal absurdities followed logically from the denial of Parmenides' views. You think that there are many things? Then you must conclude that everything is both infinitely small and infinitely big! You think that motion is infinitely divisible? Then it follows that nothing moves! (This is what a ‘paradox’ is: a demonstration that a contradiction or absurd consequence follows from apparently reasonable assumptions.) “

“But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. (Simplicius(a) On Aristotle's Physics, 141.2)”

I interpret this as what follows: If a dimension is continuous and there is infinite number of points between two “finite” points or two points are separated from each other with a finite distance r, they must be both small and large. As Zeno mentioned, it is a contradiction and it leads that it is meaningless to define the notion of length or distance between two points. In other words, this conclusion that the distance cannot be defined in a continuous differentiable dimension is logically valid but paradoxical.

Let me restate this paradox in a different way. We know that a distance can be a real number including irrational numbers such as pi. The circumference of circle with radius r is 2*PI*r and we can map this length onto the 1 dimensional line. Actually real number R can fill the entire region of a single dimension completely and it means there is infinite number of points between any two points A and B separated from each other with a finite distance d. For simplicity, assume there is a point like creature living on the 1 dimensional continuous line and it is allowed to move only along the line without jumping. Initially it is located at point A and it will measure the distance to the point B by moving itself. Since the path from A to B is continuous and jumping is not allowed, it needs to visit the third position P3 at the middle of A and B. Again, before it can move to P3, it must move to P4 at the middle of A and P3 and P5 at the middle of A and P4. Because there is infinite number of intermediate points between any two points, this finding new intermediate point repeats endlessly so it cannot move at all.

By intuition with the number line, the two distances D1 of between 1 and 2 and D2 between 3 and 4 have an equal length 2-1=4-3=1 but we all learned from school that the distance D3 between 1 and 100 is bigger than D1 and D2. But if we use the strict definition for distance which is how many intermediate points must be visited to arrive at the destination from start, then D1,D2 and D3 all have infinitely many so all are equally large as Aristotle mentioned already 2000 years ago. This logical deduction based on the things which is infinitely many divisible is correct but unacceptable to us because we all know well that light travels with velocity c=300,000,000m/s in vacuum along straight line. Light also need to face the aforementioned situation that it need to visit infinitely many intermediate points between any two points so it will need infinitely many times if the path light travel is continuous and light do not skip or jump in the course. But assuming the distance is 300,000,000 meter, 1 second suffice for light to travel such distance in real life as we can see. It is the discrepancy between idealism and realism which we need to resolve. So does it mean that we need to abandon the continuity or differentiability of a dimension including time and space? Absolutely not because any physical object including light can move in a finite time in reality.

Here comes a crazy assumption familiar to who knows QM that the distance exists only in a quantized form as Max Planck postulated that energy is quantized. It is the quantization that there is an indivisible minimum length for the distance in space. With this assumption, we can easily define the notion of distance unambiguously by counting the all valid intermediate points of the distance between any two separate points A and B.

My definition for the distance explained so far may sound weird and crazy to readers. In mathematics, the distance between two numbers is usually expressed by getting an absolute value of the subtraction and people use it freely without adopting the existence of indivisible finite unit length. Mathematics supports that every dimension is continuous and there is no such thing for indivisible finite unit length in it. Backed by this mathematical view, classical mechanics treats every physically measureable dimension such as space, time, mass and energy as continuum which turns out to false because we believe QM is correct than classical mechanics and believe in Planck time, Planck length. It is problematic that the continuum in principle is transformed into a quantized form which is an integer multiples of an indivisible finite unit in reality. Here I strongly believe that this problem can be solved by understanding where the quantization is engaged in nature.

To be continued…

차원과 실수에서 발견할 수 있는 절대성과 객관성

우리가 현재 알고 있는 시공간의 가장 진보된 개념은 아인슈타인의 상대성 이론에서 보여주고 있는 시간과 공간이 서로 별개로 떨어져서 독립되어 존재하는 것이 아닌 시공간이라는 한데 뭉쳐서 마치 옷감처럼 공간에 존재하는 물체의 분포여부에 따라 구부러지고 휘어질 수 있는 역동적인 객체라는 것이다. 이렇듯 배경에 의존적인 시공간의 개념(fabric like space-time which is background dependent)은 양자역학에서 필요로 하는 뒷배경에 독립적인 시간과 공간이라는 개념과 완전히 상반되고 있고 이러한 하나의 객체에 대해 완전히 상반된 해석은 물리학자들의 수십 년간 지속된 양자역학과 상대성이론을 통합하려는 노력을 번번히 물거품으로 만들고 있다.[1]. 수많은 뛰어난 물리학자들이 수십년간 같은 목표를 가지고 연구에 연구를 거듭했지만 실패했었던 이유가 무엇일까? 필자가 그 이유를 판단해 볼 때 그것은 가장 기초적인 뿌리에 해당하는 시간과 공간의 개념에 대해 완벽한 이해를 가지지 못했기 때문이다. 다수의 물리학자들 또한 이러한 견해를 가지고 있다. 그들 중 한 명인 Lee Smolin은 그의 저서 Three roads to Quantum Gravity, page 4에서 다음과 같이 언급하기도 했다.

“The problem is that while quantum theory changed radically the assumptions about the relationship between the observer and the observed, it accepted without alteration Newton's old answer to the question of what space and time are. Just the opposite happened with Einstein's general relativity theory, in which the concept of space and time was radically changed, while Newton's view of the relationship between observer and observed was retained. Each theory seems to be at least partly true, yet each retains assumptions from the old physics that the other contradicts. The problem is that while quantum theory changed radically the assumptions about the relationship between the observer and the observed, it accepted without alteration Newton's old answer to the question of what space and time are. Just the opposite happened with Einstein's general relativity theory, in which the concept of space and time was radically changed, while Newton's view of the relationship between observer and observed was retained. Each theory seems to be at least partly true, yet each retains assumptions from the old physics that the other contradicts.”

현재 진행되고 있는 연재물은 바로 많은 물리학자들이 알고 싶어하는 시간과 공간의 실체에 대한 하나의 개인적인 생각에 대한 설명이 주를 이루고 있다. 이에 대한 필자의 견해의 핵심은 이전 게시물에서 이미 언급했듯이 시간과 공간이 뉴튼의 절대적인 시간과 공간 개념과 아인슈타인의 시공간 개념이 동시에 존재한다라는 것이다. 그리고 시간 및 공간이라는 차원들의 절대값과 상대값의 관계는 결론을 미리 얘기하자면 질량의 역수가 된다.

$$T_r=\frac{1}{m}T_a$$

$$D_r=\frac{1}{m}D_a$$

위의 식은 시간, 공간, 그리고 질량이 서로 긴밀히 엮여 있음을 수식으로 보여주는데, 이를 선형대수(linear algebra)의 용어를 사용하여 표현하면 어느 특정 에너지 공간(Energy field)에서 존재하는 시간(temporal distance) 또는 공간(spatial distance)은 각각 절대값과 상대값 이렇게 두가지의 표현들을 가지고 있는데, 이때 절대값을 eigenvector로 간주하면 이 eigenvector v를 취해서 에너지 공간에서 측정 가능한 스칼라 값 λ(eigenvalue)를 곱하는 선형 변환 T를 생각할 수 있는데, v->T(v) 이렇게 해서 나온 결과값이 시간 또는 공간의 상대값이 된다는 의미이다. 여기서 eigenvalue는 우리가 지금 관심을 가지는 시스템(독립 입자)이 경험하는 질량(total mass)의 역수가 된다.

이를 양자역학에서 친숙한 용어를 가지고 표현한다면 아인슈타인의 (관측가능한 상대적인) 시간과 공간은 observable or measurable state이고 뉴튼의 절대적인 시간과 공간은 관측 이전의 상태(state before measurement)이며 질량의 역수가 operator가 된다. 필자의 이러한 견해가 지극히 개인적이고 또 대단히 중요한 몇 가지의 개념(질량이 무엇인가 등)에 대한 충분한 설명이 뒷받침되어 있지 않아 독자들에게 굉장하게 불편하게 읽힐 수 있음을 잘 이해한다. 미안하지만 아직 질량에 대한 구체적인 설명이전에 우선적으로 이루어져야 하는 보다 근본적인 개념들에 대한 이해들이 아직 남아 있다. 지금 필자는 구체적인 실증을 들기 이전에 필자가 나누고자 하는 전반적인 견해에 대해 기본적인 줄기를 포괄적으로 우선 제시해서, 추후에 이어질 구체적인 설명들이 좀더 독자들에게 잘 이해될 수 있도록 청사진을 보여주고자 의도하고 있다. 바라건데 이번 글이 독자들의 외면을 불러일으키지 않았으면 한다. 필자가 약속할 수 있는 것은 이러한 개인적인 견해가 완전 헛소리로 드러날 가능성이 존재하겠지만 적어도 이제껏 보지 못했던 가장 흥미로운 소설일 것이라는 점이다. 그리고 현재까지 나와 있는 모든 생각 중 가장 직관적이고 아름다운 환상을 느낄 수 있을 것이라 확신한다.

여기서 밝힌 필자의 견해가 독자들에게 만의 하나라도 일리가 있다고 느껴진다면 그 다음은 지금까지 수많은 물리학자들이 찾아 헤맸던 시간과 공간의 양자화를 가능케 해 줄 개념적인 프레임워크(Conceptual framework for the quantization of time and space)가 준비되어 있다. 시간과 공간이 어떻게 양자화 될수 있을까에 대한 필자의 기본 생각은 대략적으로 다음과 같다.

우선 아인슈타인의 에너지는 질량과 같다라는 수식을 떠 올려보자(##E=mc^2##). 그리고 Max Plack의 양자가설 (E=hν)은 에너지는 언제나 양자화 되어 존재함을 얘기한다. 이 두가지 식을 조합하면 에너지가 항상 양자화 되어 존재하면 에너지가 바로 질량이므로 질량 또한 양자화 되어 존재한다는 논리를 도출할 수 있다. 여기에서 필자의 견해인 시간의 절대값과 상대값의 관계를 대입하게 되면 인간이 관측가능한 시간값(즉 아인슈타인의 특수상대성이론에 나오는 입자의 속도에 따르는 상대적 시간값 – 이건 필자의 추후 자세히 설명하게 될 견해와 다르며 필자의 견해가 보다 완전한 이해를 독자들에게 가져다 줄 것이다)이 연속적인 시간의 절대값이 양자화된 질량 상수의 역수와 곱의 결과로 양자화 될 수 있다! 여기서 독자는 한가지 근본적인 의문을 분명히 가질 것이다. 질량이 0인 입자는 이것의 역수를 취하면 무한대가 될 것이고 이를 위의 식에 넣으면 시간의 상대값이 무한대가 되는데 시간의 상대값의 무한값이 어떻게 양자화 될 수 있을 것인가 하는 의문이 바로 그것일 것이다. 이를 설명하자면 길어지므로 여기에서 깊이 다루지는 않겠지만 짧은 답을 한다면 이때는 진공에서 이동하는 빛의 유한하고 불변의 속도와 관련된 시간은 무한대의 값이 아니라는 점을 상기하면 된다고 하겠다. 진공에서 움직이는 질량이 없는 빛이 자체적으로 느끼는 시간은 상대값이고 무한값을 가지긴 하나 우리가 모든 실험을 통해 얻어낸 진실인 빛이 진공에서 유한하고 불변인 속도로 움직일 때 걸리는 시간은 무한대가 아닌 것을 인간은 명확하게 알고 있다.

이뿐만 아니라 필자의 견해가 가져다 줄 이익들은 무궁무진하다. 아인슈타인이 특수상대성 이론에서 얻어낸 시간수축(time dilation)보다 더 정확하고 완전한 수식이 존재할 수 있음을 보여줄 수 있다. 이 단계에 이르면 GPS에서 정확한 시간을 얻기 위해 속도와 중력장 두가지의 상태를 각각 고려하게 되는데 실은 질량 하나만 고려해도 충분함을 이해할 수 있게 될 것임을 장담한다. 이 모든 얘기가 헛소리로 들릴 독자들이 대부분일 것이지만 또 실제 헛소리임이 증명된다 하더라도 적어도 필자의 견해가 독자들에게 신선한 생각거리를 줄 수 있을 것으로 믿는다. 또 아는가? 필자가 헛소리하는 게 아닐지도 모른다.

이 다음 연재글은 이번 글에서 제시한 전반적이고 개념적인 생각 수준을 보다 엄밀하게 뒷받침하기 위해 보다 근본적인 개념 즉 시간과 공간이 속하는 일반적인 차원의 성질에 대해 좀더 자세히 설명하고자 한다. 1차원은 다중차원의 근본으로서 1차원의 속성을 알게 된다는 얘기는 일반적인 차원을 양자화할 수 있는 근본적인 원리를 이해할 수 있는 토대가 될 것이기 때문이다.