프로그래밍과 경제

Searching for Quantum gravity. what is distance? (3)

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.)

In the previous post, I argued a radical, sounding absurd idea that applying the mathematical abstraction directly to getting the measurable or observable property in the physical world such as finding distance between two separate points in space could be dangerous in that it can distort the true reality so may yield a wrong prediction.

To support this view, I adopted the notion of cost which is time for the case of determining distance in space and it can help us to recognize the pitfall. In mathematics, subtracting two numbers what is normally used to get the distance between two points over number line is just an abstracted description human can think of in mind and such abstracted description can be distinguishable from the specific physical implementation of it. Why? Because the mathematical abstracted operation is time free in that it does not care about the time cost needed for the determination of it. Meanwhile, the notion of time cost arises in the actual implementation in physical world. For example, computer can calculate the subtraction of two large numbers much faster than human with only bare brain. We cannot tell which one is better than the other and both are just valid but “approximated” implementations by either human or computer showing different time cost. For C++ programmers, the point that this distinction between abstraction and physical implementation must be treated cautiously can be easily understood. The distance between two points is just a final result or objective we want to see and mathematics do not care about what a specific route to final goal will be chosen to get the distance. We human need to make a choice on the various routes which will take much or less time to get there. Knowing that the time cost does not matter in mathematical abstraction which is ideal case and time cost actually matters in the physical implementation is crucial to keep going to next discussion.

For those who are still not sure about my reasoning so far, let me take another thought experiment. Here let assume that there are two observers living their own planets A and B separated at a distance of 1 billion light years. Also there is a very large rigid ruler connecting two planet so both observers know their location relative to the ruler by reading it. Here the ruler plays the same role described in the Einstein’s special relativity paper. By reading the ruler, observer A’ at A can read a scalar value X for the exact location in space and other observer B’ at B can read a scalar value Y. Also, at initial time t0, assume that they know others location that the distance between two is |X-Y|. For now let’s suppose that a new planet C suddenly appeared near B at t0 and observer B’ soon know C is created nearby. It is fine because it is thought experiment. Now the question is how to determine the distance between A and C. For observer B’ located at B near C, determining the distance between A and C can be done almost instantaneously by doing mathematical operation because he is very close to C. First he can read the approximated value for the location of C by reading the ruler, let’s say it is Z, and determine the distance by mathematical operation by subtracting two scalar values X and Z in almost no time.

Meanwhile, the observer A’ at the planet A at t0 when C just appeared at a distance of 1 billion light years away has a no way to know that event happened at B. If he tries to observe and take a photo to the direction of planet B soon after t0, what he can see in the photo at t0 is just only B at r0 even though C actually exists near B until 1 billion light years flow since t0. According to Einstein’s special relativity, nothing can move faster than light so if A’ want to measure the distance between A and C, he need to wait 1 billion light years and it’s the fastest way to measure although B’ can determine the distance between A and C without waiting 1 billion light years with mathematical abstracted operation.

From the exampled thought experiment, we can see there are two ways in determining the distance in space. The observer A’ can use only the physically feasible method of which time cost needed to complete the determination is proportional to the magnitude of distance but B’ at B near C can use the mathematical abstracted operation so that he can completed the determination much faster than any physically feasible way. I argue this shows clearly that there is some sort of inequality between the mathematical abstracted operation and the physically possible operation yielding almost same result and the gab is impossible to be filled by anything relevant to physical world.

The main goal in physics I think is to make a perfect prediction repeatedly. That being said, the main question I want to raise here is what applying mathematical abstraction to physical world directly will affect the prediction we want to see. Will it show us what really exists as it is without any distortion so we can observe the exact thing as predicted? Someone who knows well about quantum weirdness such as measurement problem, wave function collapse which is detectable at physically measuring state of the system may guess the answer would be negative. The double slit experiment shows that calculating the probability to find the location of elementary particle, the only one available option we can take for now, does not provide the precise prediction on where the particle to be found in near future. Simply putting, people do not have a clear understanding on what applying mathematical abstraction to physical world really means and what side effect we could not expect will arise when it is done in physical world.

What makes discrepancy between continuum and discontinuum?

In previous post, I argued that there is no physically feasible way to define the distance in continuum. Instead, we usually use a mathematical abstracted operation, subtracting two scalars which is the only one available option we know so far. And such mathematical operation does not limited by Einstein’s famous second postulate in his special relativity saying nothing moves faster than light as we can see previous thought experiment. Here what I want to do is to find the physical intuitive meaning of that mathematical operation can be completed much faster than speed of light which is the absolute limiting factor residing in every event happening in our universe.

To be continued …

Searching for Quantum gravity. what is distance? (2)

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.)

In the previous writing, there proposed a radical idea that the notion of distance in any continuous dimension can be defined only after there is a way to generate the indivisible finite length unit out of it. The problem in defining the notion of distance can be traced back to ancient Greek philosophers. Specifically Dichotomy paradox which is one of Zeno’s paradoxes shows the paradoxical conclusion that traveling over any finite distance can neither be completed nor begun in any continuous dimension or space and so leading to another conclusion that distinguishing distance is meaningless.

Many scalars interested in it tried to provide reasonable explanation to resolve it. From the wiki page https://en.wikipedia.org/wiki/Zeno%27s_paradoxes, two suggestions for possible resolution would be worth to be mentioned for further discussion. First resolution is Hermann Weyl’s one that the paradox can be resolved if it is not true that between any two different points in space including time, there is always another point. I strongly support his view with the confidence that his assumption is true, which will be explained later with a new working model. Secondly, Pat Corvini claims that the paradox can be resolved by distinguishing the physical world from the abstract mathematics used to describe it. According to the wiki, it seems that Corvini distinguishes the abstract mathematical notion that between any two different points in space, there is always another point from the physical world of a moving object to traverse an infinite number of divisions. So I totally agree with the quote “The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution”.

Regarding Corvini’s view, I found a new concept would be quite useful to understand the necessity for distinction between the mathematical abstracted notion and the physical world and it is the cost which is needed to complete the determination of distance. In mathematics, i.e. over real number line which is a manifestation for any continuous dimension, the needed process to measuring distance is just a subtraction of two numbers which requires a constant cost over any distance. It means that no matter how far two numbers are separated from each other, the cost for the subtraction is all the same. For example, in the computer, the required time to complete subtracting two numbers in CPU is constant so it is independent on distance. On the other hand, in the physical world, our daily experiences show that the cost for measuring distance must increase as distance is getting larger. So regarding in determining distance, from this observation of time independence in mathematics and time dependence in physics, it is obvious that it would be too dangerous to apply a mathematical abstraction directly to the physical world.

To get the better understanding on the notion of distance in any dimension, especially space and time, I focus three things in this article. First is the conflict of quantum mechanics and theory of general relativity at singularity due to mainly (I think) the different notion of space and time. Since Max Planck opened a door leading to quantum mechanics by postulating the existence of quantized energy which was considered as continuous measurable property in classical physics, QM assume that space and time also must be quantized while GR dealing with cosmos scale objects treats space and time as continuous. In order to predict the behaviors of heavy matter such as stars and galaxies being absorbed by black hole, I believe that we must get a clear understanding about how heavy matters ruled by general relativity at larger continuous space moves at the scale of quantum mechanics assuming discrete Planck length inside of Schwarzschild radius of black hole.

Similar to what Corvini suggested that mathematical abstraction cannot be applied directly to the physical world, I think that there is a crucial boundary between the mathematical abstracted concepts and the physical tools used to describe the physical world. The key element making the separation is infinity. Actually to spoil my conclusion in advance, it is the mass. We human are all massive and have limited time, meanwhile light is massless so it has unlimited time. The mass plays a core role in assigning time and space to every physical object residing in the universe. The reason for this will be explained in details later.

Putting it with two suggested solutions for Zeno’s paradoxes mentioned above together, I argue that in physical world, there should be a way or model to generate an indivisible finite unit length in space and time which can be observable or measureable out of the absolute continuous space and time.

In next time, the common underlying model for every elementary particle will be introduced.

 References

[1] https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

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…