String Theory for Everyone | Prof. (Dr.) Shibaji Roy

String Theory for Everyone

~~ Shibaji Roy

To talk about string theory in a way that everyone can understand, I have to talk about some basics. String theory is a subject of science, or more specifically, theoretical physics, that physicists have been studying for more than half a century, and which is still a subject of much debate among physicists. There are two main reasons for this.

1. It is impossible to obtain direct evidence of this theory in the laboratory with current technology. The most powerful accelerator (a circular device) that we have been able to build so far, which is located on the border of France and Switzerland and is called the `Large Hadron Collider' (LHC), has a perimeter of 27 km. But to understand string theory, the perimeter of that device would have to be of the order of the size of a galaxy, i.e. 10^(18) km (1 followed by eighteen zeros, i.e. 1 billion billion), which is absolutely impossible (I will talk about this in more detail later). Although some of the experimental results obtained in the late 1960s at the `Stanford Linear Accelerator Center' (SLAC) laboratory in America, which is a much lower energy accelerator, proved the validity of some underlying string theory. But giving a mathematical form to that low energy string theory is not a very easy task. And that string theory still remains elusive to physicists.


2. So far, no conclusive indirect evidence has been found in the laboratory for this theory (high-energy string theory, the mathematical form of which is known) that it describes nature correctly, although there are indications that this theory does point in the right direction. Therefore, I will talk about the main topic without getting into various complications.

However, before going to the main topic, I will say a few words for the convenience of understanding of the students. The lexical meaning of the word `science' is: special knowledge. That special knowledge cannot be acquired by collecting only some information. `Science' is the acquisition of systematic knowledge by analyzing information backed by experimental evidence. The three main branches of `science' are physics, chemistry and biology. In addition, there are many other sub-branches of science. And the language of describing science is mathematics. Mathematics helps us understand science in a very simple and objective way. Therefore, mathematics is an essential tool of science. According to my favorite British physicist Paul Dirac "God is a mathematician of the highest order". This is how he regarded mathematics. And to do research (in various scientific subjects) means to `search' or to seek. What to search for? To search for or find the `reasons behind' the various phenomena that constantly occur in nature and to explain them. So the job of a scientist is to try to understand nature, that is, to search for the reasons for the occurrence of various phenomena and to give them a systematic explanation with the help of experimental evidence.

Among them, physics is the most fundamental branch of science, because with its help, scientists study the nature in terms of the very basic state of matter, their motion in space and time and their interactions. And with the help of the formulae derived from this study, they try to explain various natural phenomena. Chemistry mainly studies the structure, properties of basic or compound substances and the reactions between them. According to the classification of compounds, chemistry has two parts: inorganic and organic chemistry. In inorganic chemistry, the main subject of study is metals, non-metals, their compounds and their properties. And organic chemistry mainly discusses the structure, properties and reactions of carbon-containing compounds. Biology is the branch of science where the life processes of all kinds of living system, starting from the smallest microbes to large terrestrial, aquatic, air borne, plants, etc., are studied. However, just as physics and chemistry are applied to the details of nature beyond our habitable earth, biology is not. Because, life has not yet been found on any other nearby (within our reach) planet or satellite in the universe outside our earth (except for one or two weak signals). However, it cannot be said that it will not be found in the future, and on the contrary, the possibility of life existing elsewhere in the universe is much more reasonable. Another thing to note is that science, as the study of nature, often cannot be compartmentalized from one part to the other. That is why new branches of science are often created by combining one department with another, such as physical chemistry, biochemistry, biophysics, chemical biology, etc.

There are two aspects of scientific practice that complement each other. One is the theoretical aspect and the other is the experimental aspect. In some branches of science, the theoretical aspect is more dominant (such as physics), while in some branches, the experimental aspect is more dominant (such as chemistry and biology). The main reason for this difference is, as I said earlier, that in physics, various phenomena are explained with the help of the nature of the basic elements of matter and their interactions. On the other hand, in the fields of chemistry or biology, many complex and large molecules and atoms have to be studied, which makes it very difficult to theoretically explain various phenomena with the help of basic elements. In these cases, an attempt is made to understand the reasons or theories behind these phenomena based on the results obtained with the help of various experiments. However, it must be said that my last remarks are equally applicable to many branches of physics. In fact, in the practice of science, any theory is not accepted by the scientific community until it is experimentally proven that it describes nature correctly. Therefore, in scientific practice, the experimental results of any theory are given the greatest importance. The advantage of theoretical research is that a correct theory can explain many phenomena occurring in nature at once. Not only that, with the help of a correct theory, it is also possible to accurately predict the past and future of an event.

Now let us return to the main topic, i.e. what is string theory and how did it originate? If any ordinary person observe around a little carefully, he/she can see many different natural phenomena. For example, lightning flashes in the sky, occurrence of day and night, planets revolve around the sun, water falling from a mountain, etc., etc. But the job of a physicist is not only to observe, but also to understand why and how it happens or to find the reasons behind them. A physicist always wants to find a formula with which these phenomena can be explained. The goal of a physicist is to explain the maximum number of phenomena with the minimum number of formulae. If it is possible to understand all the phenomena of the universe with the help of a single formula or a few formulae, that will be the ultimate success. String theory actually points in that direction.

If we think a little deeper, we will see that there are two things in all natural phenomena -- that is, some objects and some forces acting between them. For example, in the case of water falling from a mountain, there is water on the mountain and there is the surface of the earth. The gravitational force of the earth pulls water, so water falls from the mountain. Similarly, some electric charges are created due to the friction of clouds in the sky, and lightning flashes due to the electromagnetic force of the opposing electric charges created between two pieces of clouds. Here, the electric clouds and the electromagnetic force between them are working.

So naturally the question arises in mind what are the objects, how many types of objects are there in nature, what is their structure etc. and what are the forces or what are the types of forces that act between objects? By objects we mean bricks, wood, stones, water, sand, air etc. If we want to know their structure, we have to break them into pieces and turn them into the smallest pieces. The smallest building block of an object or substance in which the properties of that object is preserved is called a molecule. That is, an object is a collection of many molecules or a specially arranged form of them. So the molecules are the root of any object. However, molecules are not indivisible. If molecules are broken down again, it will be seen that they are made of one or more types of atoms. To get an estimate how small atoms are we can say that the size of an atom is one millionth of the thickness of a human hair. All the matter in our universe (discovered so far) is made up of 118 types of atoms (since atoms do not have a separate existence, they are found in nature as molecules or chemical elements, which are arranged in Mendeleev's periodic table). Of these, 94 are found naturally on Earth. The rest have been created in laboratories with difficult processes. Efforts are ongoing to create more and more new elements in laboratories. However, their number is not unlimited. The famous American physicist Richard Feynman showed by a very simple calculation that the maximal number of chemical elements could be 137. However, later, with the help of more precise calculations, it was shown that the number could increase a little more.

For a long time, it was believed that the atom was indivisible and that it was the smallest particle of matter. But in 1897, British scientist J. J. Thomson discovered an even smaller particle called an electron in the atom. Electrons have a negative charge and are about 100,000 times smaller than the atom. Since atoms have no charge, in 1911, New Zealand scientist Ernest Rutherford discovered a positively charged nucleus in the atom. Therefore, the atom is not indivisible, but rather contains a nucleus about 100,000 times smaller than the atom at its center, and the electrons revolve around the nucleus. Later, it was found that the nucleus is also not indivisible, but is made up of some positively charged protons and uncharged neutrons. Ernest Rutherford, whom I mentioned earlier, discovered the proton in 1909, and British scientist James Chadwick discovered the neutron in 1932. Afterwards, in 1964, two American scientists, Murray Gell-Mann and George Zweig, discovered that neither neutrons nor protons are indivisible, but they are made up of even smaller particles called quarks. However, as far as is known, quarks are indivisible.

So what has been found so far is that all the objects in our universe - asteroids, planets, stars, bricks, wood, stones, sand, etc. - are all made up of two types of particles - electrons and quarks. But it doesn't end here. Two more companion particles of electrons are found in cosmic rays. Their nature and charge are the same as electrons, only their masses are greater than that of electrons. Their names are muons and tauons. They are also indivisible like electrons. Muons were discovered in 1936 and tauons were discovered in 1974. Some new fundamental indivisible particles have been found from radioactive substances. They are uncharged and have tiny masses. Their names are neutrinos - electron neutrinos, mu neutrinos, tau neutrinos. Neutrinos were theoretically discovered by Austrian scientist Wolfgang Pauli in 1930 (Nobel Prize 1945). Three types of neutrinos were discovered in the laboratory in 1956, 1962 and 2000. Similarly, quarks are not of one type but six types. Up quark, down quark, charm quark, strange quark, top quark and bottom quark. These are just names, they have been given different names for their different properties. Further, each quark has three colors - red, blue, green. Quarks are invisible to the eye, so these names are purely symbolic. But when these three colors are mixed together, they become white or colorless. Protons and neutrons in the nucleus of an atom are made of two types of quarks - up and down. Protons are made of two up quarks and one down quark, and neutrons are made of two down quarks and one up quark. Protons and neutrons are colorless.

British scientist J. J. Thomson's discovery of the electron has already been mentioned. But theoretically, quarks were discovered by American scientists Murray Gell-Manm and George Zweig in 1964, and six quarks were discovered in the laboratory between 1968 and 1995. So, all matter in the universe is made up of electrons and two similar particles, muons, tauons, neutrinos of each of them, and these different types of quarks. However, another type of fundamental particle called the Higgs particle is responsible for their mass. Theoretically, this particle was discovered in 1964 by six scientists, namely Peter Higgs, François Englert, Robert Brout, Carl Hagen, Tom Kibble, and Gerald Guralnik. And it was discovered experimentally in the "Large Hadron Collider" mentioned earlier in 2012. Also, every particle has an antiparticle. For example, the antiparticle of an electron is a positron, the antiparticle of a quark is an antiquark and so on. An antiparticle has the same mass as the particle but has opposite charges (and some other opposite properties). I have already written the estimate of the diameter of the nucleus and the proton. Roughly, the diameter of electrons is one-tenth of the diameter of the proton (but the mass of the electron is about two-thousandth of the mass of the proton) and the diameter of quarks is one-thousandth of the diameter of the proton. But since these particles are experimentally indivisible, theoretically they are considered to be a point. That is, elementary particles have no dimensions, that is, no length, breadth or thickness. They are zero-dimensional.

That's all about matter and its structure. But to explain worldly phenomena, we need to know about the forces acting between objects. We have already talked about lightning in clouds, water falling from a mountain, and the rotation of planets. All the natural phenomena that are visible around us mainly occur under the influence of two types of forces - electromagnetic force and gravitational force. The main reason for this is that the natural phenomena, that we usually encounter, occur between large objects. We cannot see very small particles such as molecules, atoms, or the events that are constantly happening inside them with the naked eye or in a normal laboratory. All the phenomena that we encounter belong to classical mechanics or classical theory. And these theories were mainly developed from the middle of the seventeenth century to the early nineteenth century. The main architects of this theory were scientists such as Isaac Newton, Charles Augustin de Coulomb, Michael Faraday, James Clerk Maxwell and others. Here's an interesting story: At the end of the 19th century (1878), when German scientist Max Planck decided to study and do research in physics, Professor Philipp von Jolly of the University of Munich discouraged him greatly and said that in physics everything is understood, and that the final theory would be written soon. So there would be nothing left to do there.

Planck did not give up. He started doing research in physics and later became known as the father of a new theory of physics (quantum mechanics or quantum theory). Physicists failed to explain some of the experimental results obtained in the late 19th century (such as the photoelectric effect, blackbody radiation, etc.) using the classical theory of physics. Later, it was found that a new theory was needed to explain these phenomena, one of the fathers of which was the German scientist Max Planck (who was discouraged from studying physics by his master). He explained blackbody radiation in 1900 (Nobel Prize 1918) with that theory, and in 1905, Albert Einstein explained the photoelectric effect with that theory (Nobel Prize 1921). Later, the theory was successfully implemented by Austrian scientist Erwin Schrödinger, German scientist Werner Heisenberg, and British scientist Paul Dirac (between 1920 and 1928). For this, Heisenberg received the Nobel Prize in 1932, and Schrödinger and Dirac received the Nobel Prize in 1933.

However, with the discovery of new theories, more experiments continued on small particles, atoms and molecules. As a result, two new forces were discovered that exist in the small range of the nucleus inside the atom - which we do not usually see - they are the weak force and the strong force. The weak force is used to explain the radioactivity of atoms, and the force under the influence of which the neutrons and protons in the nucleus remain stable (since protons have a positive charge, they repel each other) is the strong force. Since these two forces are applicable to very small distances, they belong to the new theory, that is, quantum theory. However, it must be said that since all objects, large or small, are actually made of indivisible particles such as electrons, quarks, etc., all types of forces must ultimately belong to quantum theory. Classical theory can actually be seen as an average effect of quantum theory. What has been discovered so far is that there are four types of forces behind all events in this universe - gravitational force, electromagnetic force, weak force and strong force. Among them, the strongest force is the strong force, then the electromagnetic force, then the weak force and the weakest is the gravitational force.

As I said earlier, most of the natural phenomena happening around us can be explained by Newton's laws of motion, Newton's law of gravity and Maxwell's law of electromagnetism. These do not apply in two cases. One, if the object is a very small particle (molecule, atom, electron, proton, etc.), and two, if the speed of the object or particle is comparable to the speed of light. As I said earlier, in the case of small particles, quantum theory is used by refining the classical theory. In the case of particles with a speed comparable to light, Einstein's special theory of relativity, created in 1905, is used by refining Newton's laws of motion. In the early twentieth century, these two theories (the theory of relativity, which is a classical theory, and quantum theory) developed in completely different ways. Einstein took his special theory of relativity further and proposed the general theory of relativity in 1915. In fact, Newton's law of gravity works for low-mass planets and satellites, but it does not give correct results for high-mass planets and stars. Then general relativity comes into play (it is necessary to say here that general relativity is also a classical theory of gravity). In fact, it is important to note here that since other forces act on small particles and their masses are also negligible, the gravitational force acting between them can be ignored at the laboratory energy. And that is the main reason why the theory of gravity and quantum theory flourished in completely different directions. The quantum theory of the electromagnetic force was proposed in 1965, and for that year the Japanese scientist Shin'ichiro Tomonaga and two American scientists Julian Schwinger and Richard Feynman received the Nobel Prize. Quantum theories of the soft and strong forces also gradually developed thereafter. But general relativity continued to develop in its own way. It is not that there have been no attempts to create a quantum theory of general relativity, but they have not been successful, and the conflict between the two still exists (more on this later).

According to quantum theory, when a force acts between two particles, a third fundamental particle is exchanged between them and the particle is called the carrier particle of that force or a mediator particle. Therefore, in addition to the matter particles (which I have already mentioned), there are some other fundamental particles that transport various forces acting between objects. For example, in the case of electromagnetic forces, the mediator particle is the photon (chargeless) or light particle. There are three weak force mediator particles and they are -- W+, W-, Z particles (the first two are, respectively, positively and negatively charged whereas the last one is charge neutral). The strong force mediator particle is called the gluon (chargeless). There are eight types of gluons. Assuming the existence of a quantized version of general relativity or the quantum theory of gravity, the gravitational force mediator particle is called the graviton (chargeless). All of these mediator particles are different in characteristics from the matter particles. The force carrier particles belong to the Boson class -- Albert Einstein and our Bengali and Indian scientist Satyendranath Bose are proponents of this type of particles. And the particles of matter are Fermions -- Italian scientist Enrico Fermi and British scientist Paul Dirac are proponents of this type of particles. However, the previously mentioned Higgs particle is a Boson particle.

So, now we have a rough idea - which fundamental particles make up all the objects in our universe and which fundamental particles are responsible for the forces acting between them. But physicists are not satisfied with this. Because they want to explain everything with the help of a minimum or only one theory. Such attempts have been made before. For example, Maxwell combined theory of electricity and magnetism to create the electromagnetic theory. Electromagnetic theory and the theory of weak forces are combined to write the electro-weak theory. The main proponents of this theory are two American scientists Sheldon Glashow, Steven Weinberg and a Pakistani scientist Abdus Salam. For this, they received the Nobel Prize in 1979. The Standard Model is written by combining the electro-weak theory with the strong force. As I said earlier, this theory is a quantum theory. Various aspects of this theory have been thoroughly proven in the laboratory and it is the most successful theory of the forces under one umbrella. However, the Standard Model has several weaknesses. The biggest weakness is that the Standard Model does not include the theory of gravity. Since it has not been possible to combine quantum theory with the theory of gravity, this theory has remained separate from the Standard Model. Another weakness of the Standard Model is that this model has 19 (later 7 more were added due to the small masses of neutrinos. In the original Standard Model, neutrinos were considered massless) independent and free parameters whose values cannot be derived from the theory but rather determined in the laboratory. This is not expected in any fundamental theory. Research is still ongoing on some other weaknesses of the Standard Model, although the basic structure of this model is built on a very solid foundation. However, Einstein was the first who dreamt of writing a single unified theory by combining all the different theories. He attempted to unify the prevailing theories of his time - the gravitational theory and the electromagnetism (which later became useful in string theory). But his attempt was unsuccessful because other forces and quantum theories were not well understood at the time.

At the very beginning of this article, I mentioned that in the late 1960s, some experimental results obtained at the Stanford Linear Accelerator Center (SLAC) laboratory proved the validity of low-energy string theory. In fact, nothing much was known about the theory of the strong force at that time. At that time, various experiments were being conducted at SLAC to learn about that theory and many new types of composite particles were being discovered. While researching their different properties, the first string theory was introduced. An Italian scientist Gabriele Veneziano first spoke about string theory in 1968. From his theory, it was possible to explain the properties of various composite particles discovered at SLAC. That was roughly the beginning of string theory. However, later, as the quantum theory of the strong force, known as quantum chromodynamics (QCD), was able to better explain the experimental results obtained at SLAC, string theory was abandoned as it was, and QCD became known as the correct theory of the strong force.

Since string theory had already been proposed, a handful of scientists (such as Leonard Susskind (US), Holger Nielsen (Danish), Yoichiro Nambu (Japanese)) took Gabriele Veneziano's proposal seriously, and later a few other scientists, such as John Schwarz (US), Joel Scherk (French) and Tamiaki Yoneya (Japanese), joined and established string theory more strongly. According to string theory, the previously proposed quantum theory, which says that elementary particles are zero-dimensional, is fundamentally changed and it is said that if we observe elementary particles more closely, we will see that they have a one-dimensional or string-like structure. But these strings are so small that even if they are placed under the most powerful `microscope' (i.e., the highest-energy accelerator), they will look zero-dimensional or point-like. The typical length of strings is one hundred million-billionth of the diameter of a quark. So it turns out that they are very difficult to detect experimentally. In fact, string theory is a theoretical result of physics - not an experimental result (although it was first proposed to explain experimental results). Strings are in the form of closed loops and can be torn into open strings. Just as the strings of a sitar vibrate at different frequencies and produce different tones, these tiny strings also vibrate at different frequencies and produce different particles. That is, one string may vibrate in a particular way, which we will see as an electron if we do not look very carefully. Another string may vibrate in another particular way, which we will see as an up quark. Similarly, a photon is another particular vibrating string - this shows that different fundamental particles are actually the result of different vibrations of the string. That is, according to string theory, different particles originate from single object called string. In other words, a fundamental object (string) created the particle world we know through its different vibrations, and they originate from those vibrations.

As I have said before, string theory was first proposed to explain the results of some experiments related to the strong force. Later, it was seen that in many cases there were some deviations between the calculations from string theory and the results of the experiments, but the theory of the strong force QCD was able to explain the results of the experiments better. In such a situation, except for a few scientists, most scientists lost confidence in string theory. Few scientists continued to practice string theory. After understanding the classical theory of strings, attempts were made to understand quantum theory. At this time, scientists accidentally discovered that this theory, in addition to the standard model, also contained a quantum theory of gravitational force. In other words, string theory was the first to combine the theory of general relativity and quantum theory, which had been elusive until that time. The difficulties that scientists were facing in uniting these two theories separately seemed to disappear in this string theory. The first mathematical demonstration that string theory could reconcile these two theories in a coherent way was made by a Spanish scientist, Luis Álvarez-Gaume, and an American scientist, Edward Witten (1983). Later, it was further established by a British scientist, Michael Green, and an American scientist, John Schwarz (1984). Scientists believed that string theory was Einstein's dream theory that could unify all the different forces - the electromagnetic, weak, strong, and the gravitational forces. String theory could unify the fundamental particles of all matter and the fundamental particles that mediate the forces. All are created by the different vibrations of strings. However, as I said before, detecting strings is not easy, because they are so small that it would require a very powerful accelerator to detect them, which is currently beyond our reach.

Finally, let me mention some of the features of string theory that are absent in our known particle theory. First, string theory provides a quantum theory of gravity, but it does not have any free parameter (similar to the weakness of the Standard Model). That is, it is a truly fundamental theory and everything can be determined from this theory in principle. Second, we know that our visible world is three-dimensional -- front-back, top-bottom, and left-right. And if we consider time as a dimension like space (which is consistent with Einstein's special theory of relativity), our world is four-dimensional. Particle theory is built in these four dimensions, although particle theory is compatible in any other dimension. String theory is compatible only in a world of ten dimensions (nine space dimensions and one time). It cannot be built in any other dimension. So is string theory wrong? In fact, it is said that when our universe evolved, the remaining six dimensions, except for the three space dimensions (and time), remained very small (compact) and did not expand. Therefore, they are not visible. An example will make the matter a little clearer. If we look at an orange peel, it will appear two-dimensional from a distance. But if we observe it more closely, we will see small bumps on the peel. Those bumps are actually an extra dimension -- that is, the orange peel is actually three-dimensional, but the third dimension is not easily visible because it is very small. The same is true in the case of string theory, that is, the six space dimensions are not visible because they are very small, only the three space dimensions and time are visible making our world four dimensional. Thirdly, another aspect is that for all the particles of the Standard Model that we have discussed, string theory says that each of these particles has a companion particle -- these are called supersymmetric particles. The supersymmetric particles have the properties that if the particle is Fermionic then its supersymmetric partner is Bosonic and vice-versa. The existence of these companion particles has not yet been proven in the laboratory. However, efforts are underway, perhaps in the future the truth of the existence of those particles will be known.

The success of string theory in constructing a quantum theory of gravity was understood after the Green-Schwarz discovery in 1984, and it caused a special attention in the physics community. At that time, physicists thought that string theory could show us the right direction to create a single unified theory. As a result, many physicists started understanding and researching on string theory at the same time. This event was called the "First String Revolution". After that, research on different aspects of string theory continued. Many new things were understood and many new puzzles were created. During this time, a total of five different string theories were created, all of which were mathematically consistent. As a result, scientists were disappointed again. In fact, a unified theory cannot have five different theories. At this time (around 1994/95), a famous Bengali and Indian scientist Ashoke Sen pioneered in shedding light in this puzzle. He noticed some features in string theory with which he proved that the five string theories are not actually independent but are closely related to each other, only their forms are different. As a result of this discovery, many physicists were again attracted to string theory, which is called the "Second String Revolution". Therefore, it can be said that the "Second String Revolution" began at the hands of our Bengali and Indian scientist Ashoke Sen, which is a matter of great pride for us, the Indians. Inspired by Ashoke Sen's work, the famous American scientist Edward Witten proposed a single new theory, which he called M-theory (where he said that M means "Magic", "Mystery" or "Membrane") which can be formed only in 11 dimensions (1 time and the remaining 10 space dimensions). Not only that, he also showed mathematically how five string theories can be obtained from that M-theory. As a result, scientists believe that M-theory is the Einstein's dream theory from which it is possible to explain various phenomena in the universe. It is worth noting that M-theory has unified all the different string theories by adding a new space dimension. M-theory has been studied a lot in the past, but the mathematical techniques to understand this theory are still unknown to us. As a result, studies are ongoing, and perhaps this theory will be understood more clearly in the future.

Finally, this article would be incomplete without a few words. Over the past two decades, string theory research has taken many twists and turns. The main reason for this is that understanding string theory in the conventional way is very challenging. Sometimes string theory requires mathematical techniques that are still unknown even to mathematicians. As a result, new generations of physicists from various angles (using the experience of older generations of physicists) are trying to understand this theory through new ideas and many new researches and information are being born that have truly made the practice of high energy physics particularly interesting. However, many staunch critics of string theory believe that this theory is nothing more than an academic exercise and has nothing to do with describing nature. The main reason for this criticism is that no experimental basis for string theory has yet been found. While this is not entirely incorrect, it is important to understand that the only way to know whether a theory, that cannot be proven in a laboratory due to technological limitations, is correct or not is through its internal mathematical consistencies, which string theory has proven time and again. Therefore, there is some truth in this theory, and it will continue to fascinate future generations until it is fully understood.

Reference:"String Theory and Einstein's Dream," Ashoke Sen, Current Science 88 (2005), 2045-2053, e-print arXiv: physics/0609062.

---

About the Author:

Dr. Shibaji Roy was a student of all the Kolkata prestigious academic institutions. He did his schooling from Hindu School, masters in Physics from Presidency college/Kolkata University. He completed doctorate from University of Rochester in 1991. Then after post doctorateship he joined Saha Institute of Nuclear Physics and retired as Senior Professor from there.