The integral in (6.1) is Fourier Transform; and therefore its lower and upper limits are minus infinity and plus infinity respectively. As we have already mentioned infinity is not meaningful in nature, therefore existence of infinity was Heisenberg’s second assumption. We show that, even if you assume (6.1), changing infinity to a finite number will eliminate uncertainty defined by (6.2), that is, the right hand side of (6.2) can be made arbitrarily small.
We also show that Uncertainty Principle violates a very fundamental concept or result in mathematics. This theory says that all continuous functions defined over finite or infinite interval has infinite dimensions. This is a very well accepted concept in QM, but it is not so in engineering. For more details please review the paper [Das, 2013-1] and search for the blog site ‘uncertaintyprincipleiswrong’ all one word. Do not hesitate to add your comments and you will get a feedback from this author.
To understand why (6.1) will lead to (6.2), we will explain Equivalence Principle, a very fundamental concept in mathematics, which we all know but probably we never took it seriously, noticed it, or thought about it.
3. Equivalence Principle
It is very fundamental to recognize that a function can be represented in many equivalent ways. For example the following are equivalent.
(a) It can be described by providing a table of values of x coordinates at fixed intervals and corresponding values of y coordinates, as numerical data;
(b) The above table of data can be interpolated and then represented by a smooth graph in the x-y plane;
(c) It is also possible to represent the same graph using an algebraic equation of y as a function of x; like y= 3x+x2;
(d) It can be represented by a combination of sinusoidal or exponential functions, like Fourier or Taylor series;
It is clear that all of the above representations of the numeric data in (a), are equivalent. This various representation methods may be called equivalence principle. In fact Ohanian [Ohanian] on page 33 admits that – “The free-particle Schrodinger equation does not produce any solutions except those that can be directly constructed by superposition – the equation gives us nothing new”. Any information derived from any one of the above items can be derived from all other representations. They only enhance the conveniences. Schrodinger equation is nothing but the derivative of a sine function.
A better and sophisticated look at the numerical data in (a) will definitely allow us to visualize the graph. Similarly we may be able to visualize the spectrum from the graphical representation (b) of the numeric data. The information content of the time function and its spectrum is same, because one can be derived from the other. Thus Fourier Transform cannot give any new information that is not present in the numeric data in (a) or the graphical content in (b). An experienced eye will be able to observe the harmonic contents of FT in the graph, although not numerical values. It will be possible to get the fine characteristics of the data from any one of the above (a-g) equivalent representations.
Thus there is no reason to believe, as is done in quantum mechanics, see Ohanian [Ohanian] pages 32-36, that if the time function represents the position of the particle then the corresponding spectrum function will represent the momentum. Position and momentum are two independent variables of a dynamical system. It was very unfortunate that Heisenberg linked them via the FT pair (6.1). FT pair cannot give any new information as explained by the equivalence principle. As a consequence of this FT property, Heisenberg has to introduce uncertainty principle causing more confusion. This position-momentum relationship (6.1) via FT does not satisfy engineering or physical intuitions. Both time and spectrum functions are different representations of same thing; on the other hand both position and momentum are independent properties of a particle. The uncertainty principle (6.2) is thus a mathematical consequence of equivalence principle and cannot be a law of nature.
about that engineering. Clearly in modern times this will not be the design of an experiment by any stretch of mind of any system engineer. Two unknowns cannot be found out by one measurement. One measurement will produce only one equation in two unknowns. At least two equations will be necessary to solve for both p and q variables. Heisenberg seems to believe that only one measurement will give values for both p and q. This is an assumption he used unconsciously. In reality, a significantly large volume of dynamic data should be collected, for a long period after hitting the electron, all simultaneously, and then eliminate all unknowns by least square curve fitting algorithm of dynamical systems, something like Kalman Filtering.
It is almost unbelievable that how much accuracy we can achieve using modern technology and with such simultaneous measurements. Simultaneity is a law of nature, more we encompass it better results we get. GPS satellites are about 20,000 km above earth. Yet we can measure distances on the surface of earth at the accuracy of sub-millimeter level, see [Hughes], in geodetic survey, by measuring the satellite distances. In one sense then we can measure a distance of 20,000 km at the accuracy of sub-millimeter. It is about one part in one trillion. This approach uses only non-military GPS signals. So we can think, how accurate the results can be, with the exact military signals from new generation of GPS satellites and receivers. Thus at this modern time, in retrospect, it is difficult to understand why Heisenberg thought about such an experiment involving one measurement to identify two variables.
engineering, see [Couch, p.93]. Clearly (6.19) is a characteristics of Fourier transform pair as we have seen in the proofs given above. Since the spectrum function is derived from the time function, the two functions are related, and (6.19) represents one such relation. More details of such relationship are described in a later section.
The dimensionality theorem is stated in [Couch, p.93], as:
“When BT is large, a real waveform may be completely specified by N=2BT independent pieces of information that will describe the waveform over a T interval. N is said to be the number of dimensions required to specify the waveform, and B is the absolute bandwidth of the waveform”.
Comparing with equation (6.19) we can see that Δt = T and Δw = 2B. Thus the dimensionality theorem is same as uncertainty principle. We show that the dimensionality theorem is derived from FT and requires infinity assumption. Thus we can say, dimensionality theorem = time bandwidth product = uncertainty principle. They are all equivalent to FT and are derived from it, in different branches of science and engineering, by different people, at different periods.
However these theories violate a fundamental property of continuous functions – all such functions are infinite dimensional over finite or infinite domain. But the dimensionality theorem clearly says they have dimension N.
Before we show this relationship, we state the Nyquist Theorem, without proof.
“The minimum sample rate allowed to reconstruct a band limited waveform without error is given by fs = 2B”.
Here B is the bandwidth and fs is the sampling rate also known as the Nyquist rate.
The following proof of dimensionality theorem is based on [Shannon]. In case of a band limited waveform, that is, a waveform whose
|8. Characterizing Nature
We have described in another chapter about the complexity of embedded engineering systems. They represent nature, because they are created using objects of nature, they interact with nature, they implement many laws of nature like – simultaneity, finite time, boundedness. We can then imagine how complex the real nature is. If we cannot characterize embedded system using mathematics then we cannot definitely characterize nature.
By Using Money
Can we compare two human beings using money? Can we say this man should get $10 per hour and the other man should get $1000 per hour? The answer is no for all those questions, but yet we are doing that in our society. Our economic system is designed to do just that. Discrimination is the foundation of our economic system. Everybody in the world hates the cast system of India, yet we see discriminations are all pervading in the economics, and therefore in all societies.
Can we compare two Ph.D qualified persons by using money? Can we compare two physics persons, one specialized in Quantum Mechanics and the other one in Special Theory of Relativity? Can we even compare two specialists in Quantum Mechanics? Here again, the answers are No.
Just like we cannot compare two apples by using money, we cannot similarly compare two humans also. No two apples are same; they have different colors, shapes, sizes, and even different tastes. How can we then dare to compare two humans, who are billion times more complex than apples? We simply cannot.
By Using Numbers
Just like money cannot be used to measure or characterize an object of nature, in the same way real numbers cannot also be used to characterize any object of nature. After all money is a real number.
Every object of nature has almost infinite number of characteristics. Each characteristic is different in each dimension. Total number of dimensions of two objects can never be same also. Thus no two objects can be compared. Unless two vector spaces are identical, objects from two different spaces cannot be compared.
An Apple: Can you describe an apple using math? Can you locate a point correctly and uniquely on the surface of every apple or any one apple? No you cannot. Where will be the origin of your coordinate system on the apple? Every apple has different shape at all places. The shapes or contours of two apples are not same, even at their stems. So the origins cannot be precisely located at the stem. The same is true for all locations on the surface of the apple. If you cannot locate the origin then you cannot identify the locations of any point on the apple. If you cannot locate a point then obviously you cannot describe that point.
Did you notice that every apple has different colors on the surface? No two neighboring points have the same color. So apple cannot be described by its colors. Two apples are completely different in shapes and colors. They cannot be compared. None of them therefore can be described by mathematics. If you cannot describe an apple using your thoughts, language, etc., then how can you think of describing it using a symbolic language like mathematics? You simply cannot. More you think about it, more you realize how false the math can be.
I am sure you have noticed that different parts of the same apple have different tastes. That means different molecules of the same apple have different tastes. Since molecules have different tastes, then different atoms must also have different tastes. Thus all finer particles of atoms, like electrons, neutrons must also have different tastes.
Thus electrons protons, atoms etc. are not just mechanical objects with specific structures, they have different tastes. Thus the descriptions of atoms, molecules, given by physics are incomplete, because they do not include the taste property. Similarly there are many other properties of particles that are beyond the scope of modern science.
You must have also noticed that water tastes different at different sources of its origin. Water from lake, rivers, oceans, seas all taste differently. Therefore Hydrogen and Oxygen molecules have also different tastes. Such tastes are obvious properties of humans and animals also. How can you describe such humans using money? How can you compare two humans? In short form we say – all objects of nature are like apples and oranges. Any attempt to describe them using math is just outright impossible. They only represent waste of time and misguided effort or energy.
Vedas have described every object, from tiny un-manifested particles to humans and to galaxies, as having properties which is a result of combination of three fundamental characteristics (gunas), called sattwa (enlightment, light, knowledge), tamas (ignorance, darkness, ego), and rajas (activity, controller, combiner). Such combinations are manifested in all objects as tastes, colors, anger, love, hate, etc. There is no object that does not have these three gunas or properties. These three gunas are analog properties; their proportions continuously change with time, inside every object. You may think of these gunas as RGB colors of your computer screen, each one of which is controlled by 16 or 32 bit registers on your computer.
Thus you can see that engineering technology knows the nature much better than math and science. Math and science are lagging at least 200 years behind embedded engineering technology, and therefore nature.
Electrons: Now we know why and how two electrons are never same, just like two human beings are never identical. Each electron comes from different orbits or from different locations, identified by their quantum numbers or by some other characteristics of their orbits or neighborhoods. Thus two electrons are different; they not only have different features, their total number of features is different also.
Therefore position of one electron cannot be compared with the position of another electron. Same component of two different sized vector spaces cannot be compared. The vector spaces are different and therefore the objects are different.
You must know pure mathematics to know how to use applied mathematics.
Vector Spaces: Therefore algebra cannot be used on electrons or humans. Like two human beings, two electrons are not also same. Use of algebra requires the assumptions that both are real numbers. You cannot convert any characteristics of any human being to numbers, in the same way any characteristic, like position, velocity of an electron cannot be characterized by numbers and therefore algebra cannot be used. Such methods impose isolated environment, an impossible assumption for nature.
Operators: The operator theory requires two same linear vector spaces. But in nature there is no linearity . Every object has bounded characteristics, as we have discussed in chapter one on Truth. There is no infinity in nature and for any object. Thus linear vector space is not an applicable and meaningful concept for nature.
The inner product is also defined using infinity. Thus entire operator theory is not meaningful for real objects of nature. Conceptually, physically, and philosophically operator theory is incompatible with nature.
The Swartz Inequality which is used for the proof of uncertainty principle is thus meaningless for nature. It requires identical vectors for two different characteristics of objects, which is not possible. It requires inner product, which requires infinity, a meaningless concept for nature.
Cascading two operators are completely confusing. The output of one operator is a completely different type of an object from its input. The operator completely changes the characteristics of the input variables. This output variable cannot be used as input for another operator.
Imagine that you are using ECG data from one person. This data capture-process can be considered as an operator. Observe the falsity: the heart or health of a person cannot be completely specified by a finite number of ECG channel data. Besides this falsity, once you get this data, it cannot be fed into another operator or another ECG machine. The ECG output is a measured data and cannot be input for any operator, even of same type. Thus cascading two operators, as is done in operator theoretic methods of UP is not feasible by any engineering concepts.
It is a fact that engineering can be implemented completely without using any kind mathematics. Just like we humans do not use mathematics when we do our daily activities, just like god does not use mathematics, in the same way we can implement engineering without even using complex algebra [Das, 2012-2].
In this sense, mathematics is completely incompatible for the investigation of nature or characterization of any object of nature. Nature must be studied in a completely different way. It seems yogic methods are the best possible method. The chapter on yogic power shows there is nothing that we cannot do using yogic power.
Uncertainty Principle (UP) is an anti-destiny theory. So we investigated the internal details in the proof of UP. Heisenberg invented this theory, and has given two proofs of it. We have shown that both are wrong and have very basic and fundamental errors. We have also shown that UP violates another theory of mathematics, known as infinite dimensionality of functions over finite duration. Surprisingly this infinite dimensionality is widely used in QM.
Engineering does not need mathematics. We have briefly discussed, that engineering can be made robust and reliable if we avoid mathematics in the design of embedded engineering systems. We should recognize that god does not use mathematics, then why should we.
We have shown that mathematics is in general not at all suitable for analyzing and characterizing anything about nature. Yogic method as described in Vedas is the best approach.
For more details please visit the blog site on Uncertainty Principle by this author.
You may also want to download the peer reviewed published paper on Quantum Mechanics by this same author.