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A General Look at the String Theory and its Variations

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String theory is a model of physics which intends to describe the matter in its tiniest building blocks. The fundamental building blocks of string theory is one-dimensional vibrating strings, which means that they have a spatial extent, unlike
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  A GENERAL LOOK AT THE STRING THEORY AND ITS VARIATIONS Page 1  of 8 A General Look at the String Theory and its Variations 26/11/2017 Serap Öğmen 1 1  Department of Physics, Koç University  Rumelifeneri, Sariyer, 34450  İstanbul, Turkey  e-mail address: sogmen16@ku.edu.tr  Abstract    String theory is a model of physics which intends to describe the matter in its tiniest building blocks. The fundamental building blocks of string theory is one-dimensional vibrating strings, which means that they have a spatial extent, unlike previous physical models based on the same point (zero-dimensional) particles. By using this model, physicists can avoid certain problems that arise. Deeper study of string theory has shown that string theory describes not only strings but also other items, both point the same items and items with higher dimensionality, known as branes. This paper aims to provide general knowledge about the String Theory and its variations F-theory and M-Theory. Throughout the text the history and development of the theory and the algebraic methods which are specific to ST will be introduced and discussed. Keywords:  String Theory, M-Theory, F-Theory, Algebraic methods in ST PACS:  11.25.-w, 11.25.Hf, 11.25.Mj   I.   INTRODUCTION In 1921, Theodor Kaluza presented the Academy in Germany a theory with several dimensions to get Maxwell's equations of Einstein's field equations. Oskar Klein showed in 1924 that the extra dimensions are rolled up with a length corresponding to a Planck length. This he realized as he tried to include both wave and particle aspects of quantum mechanically at high energies. Types of theories which have additional curled spatial dimensions called Kaluza-Klein theories. Oskar Klein was far ahead of his time and received no attention for his work. It was not until the late 1960s, before the discovery of the strings when they studied the strong nuclear force with the help of the dual resonance model. What you really wanted to look at was how hadrons behaved. Quantum field theory could not be used to describe the strongly interacting hadrons with large spin and why they developed the dual resonance model. In 1968 at the Weizmann Institute in Israel took Gabriele Venetian developed a function that declared large parts of hadrons behavior. The operation was the same as Leonhard Euler beta function. Around 1970 showed Leonard Susskind of Stanford University that the formula could also be used to describe hadrons that the quantum state of open strings. Although Yoichiro Nambu at the University of Chicago and T. Goto and Holger Nielsen at the Niels Bohr Institute in Copenhagen came to the same thing. The early versions of string theory called bosonisk string theory. Stanley Mandelstam, working at Princeton University, found that the strings can interact with each other, he meant that they could be merged and even split up. By joining their ends can be closed when the strings "rings". At first people did not know what to use strings to but had a variety of ideas, including the strings linking particles. In the early 1970s featured C Lovelace his idea that it required more than the usual three spatial dimensions we are accustomed to, and he meant it in string theory needed 25 pieces. Pierre M. Ramond constructed in 1971, two-dimensional supersymmetry. Not long after came Neveu, Scherk and Schwarz with its dual model based on Ramond’s ideas. This theory has nine spatial dimensions and three of them are our everyday dimensions. Partly explained the other six dimensions of Eugenio Calabi when he formulated what we now refer to Calabi-Yau manifold. This theory was proven completed by Shing-Tung Yau. This manifold or this space is the one that best describes the six coiled spatial dimensions in the ten-dimensional string theories. Supersymmetry with four dimensions discovers in 1971 in the then Soviet Union. Supersymmetry appeared also to explain the between heavy particle loads that mediate the weak nuclear force. W + W - and Z 0 have loads of about 10 11 electron volts. Planck mass is about 10 28 electron volts. Now, this could be explained by a single theory. Scherk and Schwarz did not give up what was string theory as so many others did after 1973, but they continued to investigate and discovered that the strings had to be reduced in size from having been 10 -15  m down to the Planck length are aged 10 -35  m. They presented the idea in 1974 but it took about 10 years for the rest of the scientific community to recognize and accept this.  A GENERAL LOOK AT THE STRING THEORY AND ITS VARIATIONS Page 2  of 8 1984 published Schwarz and Green an article showing that the chiral theory can be formulated in ten dimensions. The result was that string theory was a major area of research. Between the years 1984-1986 was published more than 1,000 articles in the area and hundreds of scientists engaged in theory. What happened during this time is usually called the first superstring revolution? In 1994-1995 Chris Hull, Paul Townsend and Edward Witten discovered that it was approximations methods in areas such as superstring theory of type IIB that hid an eleventh dimension, and this discovery led to what is called the second superstring revolution. A super-gravity theory with an eleventh dimension was something that Eugen Cremmer, Bernard Julia and Scherk discovered in 1978, but they then could not explain it. Edward Witten considered that one could describe all of the five superstring theories with fewer dimensions than has been possible before. He introduced the M-theory and the eleventh dimension. 1985, David Gross, Jeff Harvey, Emil Martinec, and Ryan Rohm, then working at Princeton University until the next version of string theory and this was called the Straight Monastic string theory. While this theory is free from abnormalities and also promising. This is the wave that travels clockwise on the string ten-dimensional under the super crowding theory while the waves traveling counterclockwise is 26-dimensional according to the srcinal 26-dimensional string theory. The particles string represents are scattered on the string, rather than just be in the string ends. [1] String theory has as a goal to solve one of the major problems in theoretical physics: to unite quantum mechanics with general relativity. Quantum mechanics is needed to describe the physics of the very small scale and has so far been used to describe three of the four fundamental forces of nature: the electromagnetic force and the strong and weak nuclear forces, in the standard model of particle physics. The general theory of relativity, however, is a classical (non-quantum) theory which describes the remaining force, gravity, but when trying to formulate a quantum mechanical version of the theory in the same way as they do for the other forces must not be a reasonable theory. String theory has been shown to reduce the particle physics and general relativity when the energy of a thought experiment is low enough, i.e. when the "resolution" is too low to "see" the strings. As shown below, the string theory however not yet received experimental confirmation at "higher resolution". The term string theory srcinally referred to the 26-dimensional bosonic string theories but later also included the 10-dimensional superstring theories, obtained by supersymmetry added to the image. Nowadays, the term "string theory" usually refers to superstring theory, while the older model is called "bosonic string theory". In the 1990s, Edward Witten discovered circumstances that strongly suggest that the various superstring theories are various special cases of a hitherto unknown 11-dimensional theory called M-theory. These discoveries gave rise to what is called the second superstring revolution. [2]  The great interest in superstring theories will thus largely form by the hope that these bring about a theory that explains everything. Super Strings can explain quantum gravity, and moreover, one can describe the other forces of nature and fermions, the building blocks. It is not yet known whether string theory could describe a universe with exactly the forces and particles we can observe, that is, the various fields that make particle physics Standard Model. It is not clear how much freedom to choose those details that the theory allows - if string theory will serve as a Theory of Everything must be able to come up with the Standard Model as a unique theory. Figure 1: String Theory the Basic Idea [3]   On a more concrete level string theories have led to advances in the mathematics of topology, algebraic geometry, differential geometry, representation theory of groups, knots, Calabi-Yau manifold and several other fields. A significant part of mathematics’ recent advances is based on string theory. String theory has also led to improved insight into supersymmetric gauge theory, and by the so-called ADS / CFT duality led to opportunities to study gauge theories at strong coupling. The duality in turn provides an opportunity to understand quantum gravity from weakly coupled gauge theory. [4]  A GENERAL LOOK AT THE STRING THEORY AND ITS VARIATIONS Page 3  of 8 II.   PROPERTIES OF THE STRING THEORY String theory says that our elementary particles consist of strings. These strings vibrate in different ways and it is these vibrations that determines an observed particle flavor, charge, mass and spin, and thus the type of particle that string causes. The string is one-dimensional and has a length which is likely a Planck length (about 10 -35  m). A string is not just a property, it is fundamental. There is thus no less than one string. We have two main types of strings, the open and the closed. The ends of the open string can either be free or be attached to a surface (brane). The open string can then move over the surface but not let go of it. A point particle moving in space and time will create a world line. This world line is the particle's "history". A string correspondingly moves will instead create a worldsheet. The string creates a surface - a diversity. If the string is closed, the surface can be likened to a pipe (a two-dimensional diversity) without endpoints. An open string, however, will create a surface that can be likened to a strip. Figure 2: World line, worldsheet, and world volume, as they are derived from particles, strings, and branes. [5] A.   Duality Duality is about two physical systems that are sides of the same coin. They differ mathematically, but not physically. One example is in quantum mechanics where the Schrödinger wave functions and the Heisenberg matrices were two completely different methods described the same phenomenon. As mentioned earlier, there is not just one string theory, but a large number of different string theories. These are compressed to five superstring theories that could be possible. The srcinal string is known as the bosonic string that requires 26 dimensions of space-time, but first strand revolution (around 1985) found five different functional 10-dimensional string theories. Five theories were still too many when looking for a theory of everything. The second superstring revolution sparked by Edward Witten, who found a way to connect the five different theories with each other by duality. Previously they had tried using theories and string coupling constant using the interference method to find correlations between the different theories. Interference method can be used only if string theory the string coupling constants are small, which means they are the one that is the border between small and large according to the mathematics behind this phenomenon. Which coupling constants theories have however, is not yet, which means that you cannot use the interference method directly since this is only relevant when the constants are small. At first, the five superstring theories out to be completely different theories, but with the help of Witten’s idea of dualities success we are now joining them into a single theory called M-theory which will be discussed in the latter sections. With the help of T-duality, in which, for example, looks at a dimensionally small or large size of the circular, we can connect theory Type IIA with type IIB (which then are dual to each other) and straight athletic E8 × E8 heterojunction Semitic SO (32). S-duality based on string theory, the coupling constant connecting theory IIB with himself and then called self-dual. It also joins the string theory type I heterojunction Semitic SO (32). In the S-duality may coupling constant be great, and the links a theory with low coupling constant with having high coupling constant. When the string coupling constant is low on one side and high on the other are the two theories identical.  A GENERAL LOOK AT THE STRING THEORY AND ITS VARIATIONS Page 4  of 8 Figure 3: A diagram of string theory dualities. Yellow arrows indicate S-duality. Blue arrows indicate T-duality. [6] B.   Extra Dimensions An interesting feature of string theory is that they predict how many dimensions of the universe should have. Nothing in Maxwell's electromagnetic theory or Einstein's relativity theories make such predictions, but presupposes that the observer "fills in" the number of dimensions "by hand". String Theories on the other hand allows a calculation of the number of dimensions based on the basic premise. Technically, this will result in Lorentz invariance only to be met for some number of dimensions. This is like saying "if an observer measures the distance between two points and then rotated to any angle and measure the distance again, so the distance will be the same at both measurement occasions only if the universe has any certain number of dimensions". The main problem proves that the universe is not all they have waited four dimensions - three of space and one-time axis - but 10 or 11. More precisely, bosonic string theories 26 dimensions, while the super string and M-theory is found to contain 10 or 11 dimensions. This seems at odds with observable facts. Physicists use mainly two ways to handle this problem. The first is to compactify the extra dimensions - in other words, the 6 or 7 extra dimensions are so small that they cannot be detected. For the six dimensions used Calabi-Yau spaces. For seven dimensions is called the G 2 -manifold. In short, "compactifies" the extra dimensions by the short-circuited with themselves. Figure 4: A cross section of a quintic Calabi–Yau manifold   A common analogy for compactifiying is to emulate a multi-dimensional space with a garden hose. Viewed from a sufficient distance it appears to have only one dimension, its length. This dimension corresponds to our usual four observable dimensions. If, however, approach the garden hose, there would be a second dimension, the hose periphery. This "extra dimension" can only be observed from close enough quarters, like extra dimensions of a Calabi-Yau- manifold only visible at extremely small distances and therefore not easily detected. The Swedish physicist Oskar Klein had already in the 1920s worked with an extra fifth dimension to describe the Maxwell equations. Figure 5: An example of compactification: At large distances, a two dimensional surface with one circular dimension looks one-dimensional.    A GENERAL LOOK AT THE STRING THEORY AND ITS VARIATIONS Page 5  of 8 “A real garden hose naturally has more dimensions (three spatial) but for the sake of the analogy, we ignore these and instead consider only the dimensions of the tube surface. A point on the surface can be specified by two parameters, the distance of the tube length and position on the perimeter,  just as a point on the earth's surface can be specified by latitude and longitude. In both cases, we say, because the object has two spatial dimensions.” Another possibility is that we are in a "3 + 1-dimensional in space" to the universe, where you write "3 + 1" to show that the time is a different kind of dimension than spatial dimensions. Because this theory uses objects called " D-branes " (a pun, of the membrane, diaphragm) is called the " braneworld " theory. An interesting side effect of the theory is that if it is true it should be able to allow (but not fully predict or necessitate) observation of quantum gravity effects, and at a level that should be observed at CERN's Large Hadron Collider. Despite the possibility there are not many who believe that the wish will be fulfilled. Whether this gravity acting in these hidden dimensions produce other non-gravitational forces, such as electromagnetism. In principle, it is possible to derive the extra dimension to the identity by requiring consistency with the standard model but even this is not practical. C.   D-Brane After multiplication with appropriate units to make the functional consistent with a physical effect, now give the Nambu-Goto action for closed and open relativistic strings in a d-dimensional space-time, with the speed of light describes. In which With . The pulse densities result in; can also be written with in detail: this leads to: The shape of this effect is also suitable for generalization of objects which have a higher dimensionality than strings have such as D-brane. A D-brane is a membrane that has open strings stationed on it. A premise of open strings should exist in string theories of type II. The voltage of the D membrane obtained T Dp   = 2πm   's 2  / g s  (m s   is the string mass and g 's are coupling constant of the theory). The voltage depends on the coupling constant is a typical feature of a D-brane. When the open strings are attached to the D-membrane, they cannot leave it, the strings can move along branes dimensions but not detach from it. In that case, there would be multiple parallel D-branes can have its strings, each with two ends secured in two different D-brane. The distance between the D-brane is then coupled to the strand length. D-brane does not have unlimited number of dimensions; they depend on which of the five superstring theories D membrane operates in. In theory type I, D-membrane have 1, 3 or 9 dimensions, the theory of type IIA 0, 2, 4, 6 or 8 dimensions while in theory type IIB may have -1, 3, 5 or 7 dimensions (time dimension is not included). D.   Mathematical Approach Let’s look at the description of a string by a spring chain consisting of N mass points of Mass m, N springs corresponding N coupled harmonic oscillators. Hamiltonian and equations of motion can be easily decoupled (Fourier transform) quantization so the ordinary harmonic oscillator is
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