Quantum theory is modern physics which describes the behavior of sub-atomic particles. It was established by two great scientists , namely Albert Einstein and Max Planck. Max Planck showed that energy of a process is quantized. Any periodic process can carry energy that is integer multiple of hv [ its frequency times Planck’s constant h]. Einstein later showed that light is composed of energy packets called photon. Planck’s theory was applied by Bohr inside atom and a successful atomic model was created. Any electron in its quantum orbit can emit or absorb light of certain frequency and jump from it. Later Schrodinger was very frustrated by this weird quantum jumping of electron.

Luis De Broglie hypothesized that sub-atomic particles act like wave. He also showed exact relationship between matter wave and particle’s momentum which is mass times velocity. Every elementary particle like electron is a wave and every wave is a particle of some sort. In classical mechanics this effect is negligible due to short wavelength associated with the massive objects. In quantum world a tiny particle shows more waviness.

The last revolution was the Schrodinger’s wave equation which describes the wave function of particle in terms of probability. Schrodinger’s equation successfully explains Hydrogen and other simple atoms and shows how bizarre an atom looks. An electron does not exist in a definite place but it has only some probability to be in some place at a time. Electrons tend to exist at some place. This was the consequence of Heisenberg’s uncertainty principle. What we call electron is really a collection of radiations which try to be localized at some point in space.

Earlier Heisenberg stated his uncertainty principle which shocked everyone. In quantum world nothing has definite position and momenta. If we measure position more precisely the momentum becomes more uncertain and vice versa. We can not know both of them simultaneously with arbitrary precision. The nature has set a limit to our knowledge about it.

Secret of creation :

The beginning of our universe was a quantum event. General theory of relativity fails at this event when universe was very tiny and dense. Whenever small object is concerned quantum mechanics is necessary to describe its behavior. Spacetime is smooth and continuous on large scale. All the stars , galaxies and planets exist in a continuous spacetime manifold. What if spacetime may not be smooth in quantum level? It might have holes and gaps like the pretzels and doughnuts. The secret of the universe can only be traced in quantum level where everything is governed by chance and discontinuity. We do not know , in advance , when an electron will jump from one orbit to another. We only know if they do they will emit a photon of specific wavelength to go into another orbit. Nature, so it seems, is full of revolutionary occurrences , whose cause is never known. All we can predict is the probability of an outcome : like the chance of getting a six by throwing a six headed dice.

It is hard to reconcile quantum mechanics into the framework of general theory of relativity. There arise many mathematical problems while doing that. Quantum mechanics and general relativity become very antagonistic to each other. They do not shake hands at all. Quantum mechanics certainly holds the key to unlocking secrets of the universe. Three fundamental forces can be explained by the principles of quantum mechanics. Quantum mechanics has changed the view of our world completely when it comes to how these fundamental forces work. Electromagnetic force is the exchange of photons between two charge particles. Similarly strong nuclear and weak nuclear forces are carried by other sub-atomic particles like gluons and boson. Electromagnetic force is long range while strong and weak nuclear forces are short range where strong magnetic has the special property that when the distance in the nucleus increase the intensity of the force also increase. This has to do with another weird phenomena that quantum theory describes. This phenomena is quantum fluctuation.

Vacuum is not truly empty. It is full of activities dues to random generation of particle -antiparticle pairs all the time. This is also a consequence of Heisenberg’s uncertainty principle. The energy of vacuum can not be zero because in context of quantum mechanics zero is a precise number. Electron and positron pair can come into existence out of nothing. So vacuum is not truly passive . It is always seething with virtual particles and energy.

The greatest mystery is probably quantum entanglement which states that information can travel faster than light, which is also known as the spooky action at a distance. Einstein was deeply worried by this fact and unfortunately he could never come up with any valid counter argument to refute this. An electron when entangled can communicate instantly with its entangled pair over a larger distance; probably at the other side of the universe. Secret of creation or secrets of the universe deeply lies at the heart of quantum mechanics. I like to rephrase Nikola Tesla : If you want to find the secret of the universe think in terms of vibration , energy and frequency. In quantum mechanics all particles are vibrations or oscillations in their corresponding quantum field. Oscillation in the quantum field creates energy and that energy, in turns, create particles according to Einstein’s famous energy mass equivalence formula (E=mc^2). Einstein was a strong opponent of quantum theory throughout his whole life.

Theory of relativity throws a little light on the secrets of creation. It has some philosophical consequences and some applications can be found in causal theory of perception and analysis of mind. Mind and matter are not very different from relativistic point of view. What we call mind is certain group of events connected with specific relation. These are all the events that have space-like interval between them so any other observer can judge any of two to be simultaneous. What we call matter is also a group of time-like events. What happens when we look into another people’s eye are the events connected with one light rays. These events has light-like or zero interval. Quantum theory might be more fundamental than theory of relativity but relativity is able to describe four fundamental forces. Special theory of relativity describes three forces whereas general relativity describes one force.

Quantum theory basics :

The most important basics of quantum theory may be coherence and de-coherence. In an isolated quantum system there is quantum coherence. That is , the states remain uniform and in phase. Schrodinger’s equation describes a quantum system as an smeared out probability of finding a particle like electron in space. These probability wave evolves through time. Schrodinger’s equation also states that particle can exist at different places at the same time , which is known as quantum superposition. That is what we better know as the Schrodinger’s cat. Schrodinger cat is alive and dead at the same time unless we look at it. That means, there is coherence in the system unless we observe any such quantum system. After the system is observed, the system collapsed to one of the states of the system. This can be explained as the particular state of the Schrodinger cat , which can either be dead or alive after box is opened. Decoherence is the act of observation which collapses the system. Due to decoherence the quantum system no longer remains coherence and this happens through the interaction with the environment.

Common sense and quantum theory:

]]>The diagonal of this square can not be multiple of any integer. On the other hand you can see that on a number line square root 2 can be represented as the distance which is the same as the diagonal of the square. All this was very troublesome for mathematics and mathematician in the ancient time. However we will proceed to the proof of the irrationality of the square root of two (2) :

Suppose root 2 is equal to the ratio of m to n. So

root 2 = m/n , if we square it we get 2n^2 = m^2 . That means that m^2 is an even number. Now let m = 2p so m^2 = 4p^2 . Thus we have 4p^2 = 2n^2 where p is the half of m. Hence 2p^2 = n^2 , and therefore n/p will also be equal to square root of 2. But we can repeat the argument : if n = 2q, p/q will also be the square root of 2 , and so on , through an unending series of numbers each of which is half of its predecessors. But this is clearly impossible, if we divide a number continuously by two 2 we must reach an odd number after a finite number of steps. Or we may put the argument in this way by assuming that the m/n we start with is in its lowest terms; in that case m and n can not both be even as we saw they were supposed to be.

So we must conclude that square root of 2 can not be represented as a ratio of two numbers. If we do we come to a contradiction as we have shown.

square root of two(2) can be expressed using circular and hyperbolic angle too. Here is an example .

Where the symbol ^ is the exponential.

Real numbers and irrational number:

As stated, no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This problem was purely geometrical but the same problem arose as regards the solution of equations, though here it took a wider form, since it involved complex numbers.

It is clear that fractions can be found which approach nearer and nearer to having their square equal to 2. We can form an ascending series of fractions all of which have their squares less than 2, but differing from 2 in their later members by less than any assigned amount. This is to say, suppose I assign some small amount in advance, say one-billionth, it will be found that all the terms of our series after certain one, say, the tenth, have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have reached sooner or later a term in the series, say, twentieth, after which all terms would have had squares differing from 2 by less than this still smaller amount. If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-an-so many places, exactly fulfils the above conditions. We can equally form an descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to the later terms of the series, and differing , sooner or later, by less than any assigned amount. In this way we seem to be drawing a cordon around the square root of 2, and , it seems difficult to prove that, it can permanently escape us. But , it is not by this method that we will actually reach the square root of 2.

DEDEKIND’S CUT AND IRRATIONAL NUMBER:

If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that , among those whose squares are not less than 2 , all have their squares greater than 2. There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. There is no lower limit short of zero to the difference between the numbers whose square is little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one class, and there is no minimum to the other. Between these two classes , where root 2 ought to be there, there is actually nothing. We have not actually caught root 2. The cordon, though we have drawn as tight as possible , has been drawn in the wrong place and has not caught 2 .

The above method of dividing all the terms of a series into two classes, of which the one wholly precedes the other, was brought into prominence by Dedekind, and is therefore called a “Dedekind cut” . With respect to what happens at the point of section, there are four possibilities.

If we divide all ratios into two classes such that all of the terms of one class is greater than all the terms of the other then there may arise four case: 1) there may be a minimum to upper the upper section and a maximum to the lower section 2) there may be a maximum to the one and no minimum to the other 3) there may be no maximum to the one , but a minimum to the other 4) there may be neither a maximum to the one nor a minimum to the other. Of these four cases, the first can be illustrated by any series in which there are consecutive terms: in the series of integers, the lower class may contain a maximum number n and the upper class can contain a minimum n+1 . The second of these cases will be illustrated in the series of ratios if we take as our lower section all the ratios up to and including 1, and in our upper section all the ratios greater than 1. The third case is illustrated if we take for our lower section all ratios from 1 upward (including 1 itself) . The fourth case, as we have seen , is illustrated if we put in our lower section all the ratios whose square is less than 2 , and in our upper section all the ratios whose square is greater than 2.

We may neglect the first of our four cases, since it only arises in series where there are consecutive terms. In the second of our four cases, we say that the maximum of the lower section is the lower limit of the upper section, or of any set of terms chosen out of the upper section in such a way that no term of the upper section is the upper limit of the lower section, or of any set of terms chosen out of the lower section is after all of them. In the fourth case, we say that there is a “gap”: neither the upper section or lower section has a limit or last term. In this case, we may say that we have an irrational section, for the class of ratios which do not have limit corresponds to irrationals.

What delayed the true theory of irrational was a erroneous belief that there must be “limits” of series of ratios. The notion of “limit” is of the utmost importance , and before proceeding further it will be well to define it.

A term x is said to be an “upper limit” of a class s with respect to a relation P if (1) s has no maximum in P , (2) every member of s which belongs to the field of P precedes x , (3) every member of the field of P which precedes x precedes some member of s . ( by “precedes ” we mean “has the relation P to”).

This presupposes the following definition of “maximum” : –

A term x is said to be maximum of a class s with respect to a relation P if x is a member of s and of the field P and does not have the relation P to any other member of s.

These definitions do not demand that the terms to which they are applied should be quantitative. For example, given a series of moments of time arranged by earlier and later, their maximum (if any) will be the last of the moments.; but if they are arranged by later and earlier, their maximum (if any) will be the first of the moments.

The “minimum” of a class with respect to P is its maximum with respect to the converse of P; and the “lower limit” with respect to P is the “upper limit” with respect to the converse of P.

The notions of limit and maximum do not essentially demand that the relation in respect to which they are defined should be serial, but they have few important applications except to cases when the relation is serial and quasi-serial. A notion which is often important is the notion “upper limit or maximum” to which we may give the name “upper boundary”. Thus the “upper boundary ” of a set of terms chosen out of a series is their last member if they have one, but , if not it is the first term after all of them, if there is such a term. If there is neither a maximum nor a limit, there is no upper boundary. The “lower boundary” is the lower limit or minimum. Readers at this moment, should be aware that limit and maximum are not the same concept or the same thing.

Reverting to the four kinds of Dedekind section, we see that in the case of the first three kinds each section has a boundary( upper or lower as the case may be), while in the fourth kind neither has a boundary. It is also clear that, whenever the lower section has an upper boundary , the upper section has a lower boundary. In the second and third cases, the two boundaries are identical; in the first , they are consecutive terms of the series.

A series is called “Dedekindian” when every section has a boundary, upper or lower as the case may be.

We have seen that the series of ratios in order of magnitude is not Dedekindian.

From the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving that there was no rational limit to the ratios whose square is less than 2 , they allowed themselves to “postulate” an irrational limit, which was to fill the Dedekindian gap. Dedekind, in the above mentioned work, set up the axiom that the gap must always be filled, i.e that every section must have a boundary . It is for this reason that series where his axiom is verified are called “Dedekindian.” But there are an infinite number of series for which it is not verified.

It is clear that an irrational Dedekind cut in some way “represents” an irrational. But this is not true to say that an irrational must be the limit of a set of ratios. Just as ratios whose denominator is 1 is not identical with integers, so those rational numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios. We have to define a new kind of numbers called “real numbers” of which some will be rational and some irrationals. Those that are rationals correspond to ratios, in the same kind of way in which the ration n/1 corresponds to the integer n; but they are not the same as ratios. In order to decide what they are to be , let us observer that an irrational is represented by an irrational cut, and a cut is represented by its lower section. Let us confine ourselves to cuts in which the lower section has no maximum; in this case we call the lower section a “segment”. Then those segments that correspond to ratios are those that consist of all ratios less than a certain ratio that they correspond to, which is their boundary; while those that represent irrationals are those that have no boundary. Segments , both those that have boundaries and those that do not, are such that , of any two pertaining to one series one is the part of the other. In that case all those segments can be arranged by the relation of whole and part. A series in which there are Dedekind gaps, i.e in which there are segments that have no boundary, will give rise to more segments than its terms, since each term will define a segment having that term for boundary , and the segments without boundaries will be extra.

We are now in a position to define a real number and an irrational number.

A real number is a segment of the series of ratios in order of magnitude.

An ‘irrational number” is a segment of the series of ratios which has no boundary.

A “rational real number ” is a segment of the series of ratios which has a boundary.

Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance , is the class of proper fractions.

In this way we come up with a new theory of real numbers.

]]> Have you ever seen the movie matrix? if you like science you must watch this. This movie tells us that our world is in a computer simulation or inside a matrix. what if our world is a complex computer program. We are nothing but the strings of binary digits 1 and 0. It is hard to believe but many scientists believe it to be so. According to the simulation theory even our consciousness is simulated by computer program. I do not know whether this theory is right or wrong but such theory has many advantages. ** yes we are inside the matrix. We are living in a dream world. About 14.4 billion years ago an accidental explosion happened. Everything including spacetime was created through that explosion we call big bang. What exists now is a consequence of that. We are here because our universe had no boundary at that special event. The universe started expanding rapidly ever since big bang happened. Everything follows simple rules which are universal and eternal. The grand design of the universe reflects this fact. Both small things and the large structure have to obey these laws which are scientific. Small world of atom is governed by chance and God plays dice all the time. Our universe can appear from nothing due to chance taking quantum leap from eternity. Our reality is a model which can be described by fundamental particles and forces. Without these forces you can not see this sentence and our earth would not stay on its orbit around the sun. The matrix is everywhere even when you look into the very room. It is the chair you are sitting on, it is the air you breath and electricity that is powering you and your computer. It is possible that our universe is not the only universe. There is a multiverse of universes each of which has fundamental laws and constants. This universe is a gigantic simulation. **

New theory of the universe:

Where does the law come from? Physics has a lot universal laws that describe our reality. But the provenance of these laws is unknown. The laws are supposed to exist all the time. So my contention is that laws exist as a logical necessity. The laws exist all by themselves. Mathematics is the only language that describes these laws precisely. Every physical law is a mathematical law. Mathematics can be built from purely logical grounds. Logic can be very abstract and removed from empirical world. So it can be assumed that physical laws exist all by themselves like logic and mathematics. The question of how the laws came from is solved apparently. Geometry is a priori ( that does not depend on experiment) from of human intuition. All possible worlds must confront to mathematics , not only this accidental world that we live in. Otherwise the idealism states that reality is our thoughts and imagination. Better still, it is the figment of our imagination. This is also consistent with this interpretation of the existence of physical laws.

The edge and boundary of the universe :

The universe that we live in may be finite without any boundary. It can be like the surface of the earth. The surface of the earth has no boundary but it is finite. There is no report of someone falling off . So we might be trapped inside a three-dimensional surface. A surface of an n-dimensional space is a n-1 dimensional space. So our universe can itself a surface of four dimensional space, which we can not perceive. You can not also ask what happened before the big bang. There was no notion of time prior to big bang. Space and time were created at the instant of big bang. It is like asking what is there at the south of south pole. There is nothing south of south pole of the earth. This issue can be tackled in another way:

Theory of relativity has improved our understanding of space and time. In theory of relativity space-time has dynamical characteristics. Time as well as space are relative. What we can observe is space and time. Footrule and a chronometer give the description of all the events in our universe. Now question may arise what is outside our visible universe. There is nothing actually. It is the same as the question of what happened before the big bang. We only know the universe through space and time. When there is no existence of space and time , there is no issue regarding what is outside our universe. We think that we can reach the boundary or horizon of our universe. But this is not only theoretically impossible but also implausible. Space and time must cease to exist as soon as you reach the edge or boundary. We should not think ourselves outside the framework of relativity or Newtonian absolute space and time.

Weirdness of the world:

Life may not have a purpose but when you try to observe deeply everything is mysterious and interesting. Take for example our very mind and its nature. What is the ground that other people are thinking like you are? We can not perceive other people’s mind like we perceive our own minds. It is only the analogy by using which we can infer that other people are thinking like me. We can not directly experience their thoughts and emotions. We can not be acknowledge of anything happening inside their head. This is very consistent with the concept of solipsism. Solipsism states that only you are sure to exist and everything else is just your imagination. What is happening to us is very occult and arcane. We are being played by the laws of physics. Physics only reveals the causal structure of the world. That is to say, it can only say that something is the cause of any other thing. It only can say something is going to happen after some other thing has happened. This is the nature of the world; cause-effect, action and reaction . We are forever slave to it as the Merovingian’s quotes go. What goes on inside the mind might also have causal relationships with one another. This will need a detail analysis of the causal theory of perception.

Anomaly or glitch in the system:

This world or the universe is a gigantic machine. It is better to call it a physical system.

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