It is well known that any subatomic entity including light and electron has both wave and particle like properties but no one can see both of them simultaneously. Wave-particle duality is quite different property with everything experienced in our real life, simply unbelievable thing.

 

But I will argue that the weirdness can be easily understood in much intuitive way if you can find similar pattern found in the Realization uncertainty introduced in my previous post here.

 

Similar patterns

 

1. Measurement of subatomic entity and realization of irrational number are same in that ideal entity which reside only in human mind is realized in physical form which can be compared with other same kind of form.

 

For a subatomic entity, before measurement, it can have multiple possibilities and after measurement, all possibilities disappear suddenly and instantaneously but except one candidate appear.

 

As for irrational number, to find out similar behavior, let me take an example related to number quantization issue in digital computer. No computer existing in the world can compute any irrational number without error due to limit in number resolution. All computer can output is finite number only and impossible to represent true value of any irrational number. It means that for any irrational number there is always quantization error for to be it expressed with finite memory or bits. 

 

Picture 1 shows that given number representing resolution, there is a relative decision problem to make a much suitable position somewhere between mr and (m+1)r due to limitation of finite memories or bits so called resolution limit. It is obvious that it is impossible to express a number with infinite number of digits with finite digits. It is possible to increase number precision by adding more bits but even with increased precision, it is impossible to express infinite number of digits including floating points. Same difficulty prevent human to get exact same value in real life and true form of all irrational number can be imagined and handled only in human mind. Without a realized form for the irrational number, there is no way to compare with other irrational number. Comparing two numbers is only possible when they are represented in realized form.

 

 

 


 

Picture 1. number realization is like picking a position within a range of quantization error due to limit in finite bits to store number. For a given resolution r, Any point on the line between mr and (m+1)r such as (mr +r/2) and (mr+r/4) can be selected as an approximated value for the target number n. Once decision is made, then only single point will be used and effective as like we can see only one particle once measured.

 

 

Everyone will agree that any irrational number is single value and is impossible to have multiple values. On the other hand, to realize an irrational number, it is possible for someone to pick a position somewhere between mr and (m+1)r and there are infinite number of possibilities.

 

 

Below table shows summary of similarity between particle and number based on the above description. Especially it is very important to know that the whole range of many possibilities is actually dependent to the observer or measure. It is relative metric which can be varied over different person who measure it or pick. If two persons having different number resolution without common rule for choice of realizing form for the irrational number want to realize it, than the quantization error is also different.

 

 

 before

 triggering event

 After

 Wave-particle

 Many possibilities 

 measurement or observe

 singe particle

 Irrational number

 

Many possibilities , 

 pick any finite number

 Finite single value. Ex)3.1415926


The possibility density function for the number realization would be varied upon persons who realize it with his own decision. If normal computer will quantize it, one possible candidate for the density function would be delta function peaky at a low end, mr as seen in Picture 1. In normal case it would be normal distribution function with peak at the center of quantization error range. But again, it all depend on user's own decision. Synchronization or advance promise on what is the common rule for choices is a must for getting all agreed single output for object as the same way computer is constructed.

 

This topic also related to the sensitivity to the initial condition for a Chaos system. Chaos system is a deterministic system but extremely sensitive to the change in initial condition. If there is a tiny little error in the initial value, the final evolving result will be differ greatly and that is why prediction is impossible for a chaos system. 



Conclusion


For a subatomic entity such as light, electron etc, the particle form appears only when it is observed and measured, in other word realized. There is no wave-like form we can sense but it disguise in wave-like form if it is not realized. If we think about the process for number realization and apply it to wave-particle duality, then we can find that both shared the vary same mechanism: before realization many possibility given to us and after realization only one value is defined.



Definition

Realization: Being defined as a physical form comparable with other same type out of something unmeasured or unobserved by any means yet.


Posted by kevino
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Uncertainty in number realization


Realization Uncertainty


There is a much more stronger fundamental limit to the precision of measurement for any analog value overpowering the effect of the Heisenberg's uncertainty principle. The uncertainty principle was derived from the fact that the complementary variables, such as position x and momentum p cannot be fixed simultaneously with higher precision. Instead of dealing with multiple variables, my observation is the limit to the precision of any single measurable analog value such as position x or angle a.


In fact, it is basically concerning about our human's incapability in realizing a irrational number such as .

For those who wander what realizing a number means, let me introduce one concept as follows.

I see two domains when I see a real number, one is ideal world and other is real world. And I see two domains need to be distinguished because a real number can be represented differently from each other domain. Mathematicians only care about the ideal domain for a real number but physicists care about only real part for it and it is obvious that there is a non-zero error in number representation when they convert it to the opposite domain.


Let's assume we want to prove something. In order to do that, we need to foretell something first and check it with the measurement later. One example is that I can predict that the sun will rise in the east tomorrow morning. In order to prove my prediction correct, then I need to wait till tomorrow morning and see if the sun rise in the east. the same process to prove something can be applied to the proof that for a number, representation in real domain is the sum of the representation in ideal domain Ni and non-zero error e. It should be noted that a famous irrational number is just a symbol indicating a location somewhere around 3.14... and must be converted to real domain if we want to calculate something output related to it such as circle length.


Back to the uncertainty principle, I know position x can be any real number and it means it could be an irrational number. So for physicist preferring living in the real domain, it is somewhat meaningless to talk about precision limit of knowing the complementary variables simultaneously because it is obvious that there is a fundamental limit in realizing position x itself. At least for me, it will be suffice to consider my uncertainty principle solely to confirm that we can not predict what will happen. Again I want to note that what Heisenberg's uncertainty principle suggests is that predicting an object's whole movement is impossible without knowing position and momentum simultaneously.






Posted by kevino
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