A Brief History of Superstrings
[Sejarah Singkat Teori Adi dawai,
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
By:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
Abstract
Introduction
Closing
Acknowledgement
References
Other Contents Resources
Superstring Theory (Teori Adidawai) merupakan teori tertinggi yang dicari oleh para ilmuwan saat ini untuk dapat memerikan segala kejadian di Alam Raya. Teori ini merupakan bagian dari Teori Segala hal, dan merupakan kandidat terkuat dalam menyangga kebangunan teori segala hal (The Theory of Everything).
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Superstring, Special relativity, The Quantum Theory, General Relativity, Quantum Field theory, Supersymetri, Superpartner.
Introduction
Hasrat manusia untuk menguak tabir rahasia alam raya sangatlah besar itu dibutikan dengan banyak peradaban umat manusia yang senantiasa mengamati dan melakukan perenungan yang mendalam terhadap keberadaan jagat raya yang kita huni ini. Inilah sebuah bukti bahwa manusia pada hakekatnya selalu ingin mencari tahu, dari mana ia berasal, untuk apa dia hidup dan akan kemana kelak pada suatu saat.
Contents
Resolution of Contradictions
Major advances in understanding of the physical world have been achieved during the past century by focusing on apparent contradictions between well-established theoretical structures. In each case the reconciliation required a better theory, often involving radical new concepts and striking experimental predictions.
Four major advances of this type are indicated in Figure 1.These advances were the discoveries of special relativity, quantum mechanics, general relativity, and quantum field theory. Let us briefly recall the contradictions that each of these resolved and some of the experimental confirmations that followed.
Special Relativity
Galilean invariance, which is embodied in Newton's mechanics, for example, was considered intuitively obvious and observationally successful for several centuries. Among other things it implies that velocities add; i.e., if A observes B moving with speed v1 and B observes C moving with speed v2 in the same direction, then A observes C moving with speed v12 = v1 + v2 in that direction.
A contradiction arose as a consequence of the development of a very successful theory of electricity and magnetism in the nineteenth century, which is embodied in a set of differential equations known as Maxwell's equation. One of the implications of Maxwell's equations is the existence of various waves, such as radio waves, all of which are just light waves at various different frequencies. The equations imply that all of these waves travel with the speed of light (about 300,000 km/sec) regardless of the motion of the observer. This is a violation of Galilean invariance. Einstein resolved this paradox by recognizing that Galilean invariance is just an approximation, valid for speeds much smaller than the speed of light. The principles that apply for all speeds are embodied in the special theory of relativity. In this theory the rule for the addition of velocities is
,
Where c denotes the speed of light.
The Quantum Theory
The theory of electromagnetism embodied in Maxwell's equations also conflicted with the understanding of thermodynamics -- the behavior of systems in thermal equilibrium. In particular, a hot object (a so-called `black-body') emits electromagnetic radiation with a certain well-defined spectrum (intensity as a function of the frequency of the radiation). The problem with this is that if one adds up the energy carried off by radiation at all the different frequencies, the formulas imply that the total is infinite, which is an absurd result.
Just before the turn of the century, Max Planck realized that if the energy was emitted in discrete packets (or `quanta'), rather than in a continuous distribution, the total energy would be finite. He postulated that radiation of frequency n comes in quanta of energy E = hn, where h is a fundamental constant of nature, known as Planck's constant. The individual quanta of light are called ``photons''. This was the beginning of the idea of wave-particle duality and the quantum theory.
Abstract
Superstring Theory (Teori Adidawai) merupakan teori tertinggi yang dicari oleh para ilmuwan saat ini untuk dapat memerikan segala kejadian di Alam Raya. Teori ini merupakan bagian dari Teori Segala hal, dan merupakan kandidat terkuat dalam menyangga kebangunan teori segala hal (The Theory of Everything).
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Perturbation Theory, Compactification of Extra Dimensions, A Theory of Everything?
Introduction
A string's space-time history is described by functions Xm(s,t) which describe how the string's two-dimensional "world sheet," represented by coordinates (s,t), is mapped into space-time Xm. There are also functions defined on the two-dimensional world-sheet that describe other degrees of freedom, such as those associated with supersymmetry and gauge symmetries. Surprisingly, classical string theory dynamics is described by a conformally invariant 2D quantum field theory. (Roughly, conformal invariance is symmetry under a change of length scale.) What distinguishes one-dimensional strings from higher dimensional analogs is the fact that this 2D theory is renormalizable (no bad short-distance infinities). By contrast, objects with p dimensions, called "p-branes," have a (p+1)-dimensional world volume theory. For p > 1, those theories are non-renormalizable. This is the feature that gives strings a special status, even though, as we will discuss later, higher-dimensional p-branes do occur in superstring theory.
Perturbation Theory
Compactification of Extra Dimensions
A Theory of Everything?
Contents
Perturbation Theory
As was briefly mentioned earlier, a useful way of studying theories that cannot be solved exactly is by computing power series expansions in a small parameter. For example, quantum electrodynamics has a small parameter, called the fine structure constant, which is given by
Thus, if T(a) denotes some physical quantity of interest, one computes
and the first few terms can give a very good approximation. This approach, which is called perturbation theory, is the way superstring theories were studied until recently. The problem is that in superstring theory there is no reason that the expansion parameter a should be small.
More significantly, there are important qualitative phenomena that are missed in perturbation theory. The reason is that there are non-perturbative contributions to many physically interesting quantities that have the structure
Such a contribution is completely invisible in perturbation theory.
Perturbative quantum string theory can be formulated by the Feynman sum-over-histories method. This amounts to associating a genus h Riemann surface, which can be visualized as a sphere with h handles attached to it, to the hth term in the string theory perturbation expansion. The genus h surface is identified as the corresponding string theory Feynman diagram. The attractive features of this approach are that there is just one diagram at each order of the perturbation expansion and that each diagram represents an elegant (though complicated) mathematical expression that is ultraviolet finite (no short-distance infinities). The main drawback of this approach is that it gives no insight into how to go beyond perturbation theory.
Compactification of Extra Dimensions
As has already been mentioned, to have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length Lst..
Since space-time geometry is determined dynamically (as in general relativity), only geometries that satisfy the dynamical equations are allowed.
The HE string theory, compactified on a particular kind of six-dimensional space, called a Calabi--Yau manifold, has many qualitative features at low energies that resemble the standard model. In particular, the low mass fermions (identified as quarks and leptons) occur in families, whose number is controlled by the topology of the CY manifold.
These successes have been achieved in a perturbative framework, and are necessarily qualitative at best, since non-perturbative phenomena are essential to an understanding of supersymmetry breaking and other important matters of detail.
A Theory of Everything?
In the euphoria following the first superstring revolution in 1985, some of the less experienced participants in the enterprise thought that we were on the verge of constructing a complete fundamental theory of the physical world. To put it mildly, I found this naive. In this setting, the phrase "Theory of Everything" was introduced and propagated by the public media. This was very unfortunate for several reasons.
The TOE phrase is very misleading on several counts. First of all, the theory is not yet fully formulated, and when it is (which might still take decades) it is not entirely clear that it will be the last word in fundamental physics.
Furthermore, even if the theory is a complete description of quantum dynamics, it seems unlikely that it will also provide a theory of initial conditions, which is another key ingredient required to explain why we observe the particular universe that we do. But even if a theory of initial conditions is also obtained, there will still be much about this universe that cannot be explained. Many things, such as our very existence, are a consequence of the inherent quantum indeterminacy of nature. I believe that cannot be overcome. Maybe that is just as well, because if we had old-fashioned classical determinism, the future would be fully determined, which would undermine our humanity.
There is also a more mundane sort of unpredictability that is also to be expected. Many of the things that the theory predicts unambiguously in principle could require intractable calculations. Part of the art of physics is to identify those things that can be calculated. The other reason the TOE phrase upset me is that it alienated many of our physics colleagues, some of whom had serious doubts about the subject anyway. Quite understandably, it gave them the impression that people who work in this field are a very arrogant bunch. Actually, we are all very charming and delightful.
Closing
Acknowledgement
References
Abstract
Introduction
Contents
Just One Theory!
The second superstring revolution (1994-??) has brought non-perturbative string physics within reach. The key discoveries were the recognition of amazing and surprising "dualities." They have taught us that what we viewed previously as five distinct superstring theories is in fact five different perturbative expansions of a single underlying theory about five different points. It is now clear that there is a unique theory, though it may allow many different quantum mechanical solutions. For example, a sixth special quantum solution implies the existence of an 11-dimensional space-time. Another lesson we have learned is that, non-perturbatively, objects of more than one dimension (membranes and higher "p-branes") play a central role. In most respects they appear just as fundamental as the strings (which can now be called one-branes), except that a perturbation expansion cannot be based on p-branes with p >1.
Three kinds of dualities, called S,T, and U, have been identified. It can sometimes happen that theory A with a large strength of interaction (or `strong coupling') is equivalent to theory B at weak coupling, in which case they are said to be S dual. Similarly, if theory A compactified on a space of large volume is equivalent to theory B compactified on a space of small volume, then they are called T dual. Combining these ideas, if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling, they are called U dual. If theories A and B are the same, then the duality becomes a self-duality, and it can be viewed as a kind of symmetry.
T duality, unlike S or U duality, can be understood perturbatively, and therefore it was discovered between the two string revolutions.
T Duality
The basic idea of T duality(for a recent discussion see [5]) can be illustrated by considering a compact dimension consisting of a circle of radius R. In this case there are two kinds of excitations to consider. The first, which is not special to string theory, are Kaluza--Klein momentum excitations on the circle, which contribute (n/R)2 to the energy squared, where n is an integer. Winding-mode excitations, due to a closed string winding m times around the circular dimension, are special to string theory. If
Denotes the string tension (energy per unit length), the contribution to the energy squared is :
Em=2pmRT.
T duality exchanges these two kinds of excitations by exchanging m with n and
This is part of an exact map between a T-dual pair A and B.
One implication is that usual geometric concepts break down at short distances, and classical geometry is replaced by "quantum geometry," which is described mathematically by 2D conformal field theory. It also suggests a generalization of the Heisenberg uncertainty principle according to which the best possible spatial resolution Dx is bounded below not only by the reciprocal of the momentum spread, Dp, but also by the string scale Lst. (Including non-perturbative effects, it may be possible to do a little better and reach the Planck scale.) Two important examples of superstring theories that are T-dual when compactified on a circle are the IIA and IIB theories and the HE and HO theories. These two dualities reduce the number of distinct theories from five to three.
S Duality
Suppose now that a pair of theories A and B are S-dual. This means that if f denotes any physical observable and l denotes the coupling constant, then
(The expansion parameter a introduced earlier corresponds to l). This duality, whose recognition was the first step in the current revolution, [6] generalizes the electric-magnetic symmetry of Maxwell theory. Since the Dirac quantization condition implies that the basic unit of magnetic charge is inversely proportional to the unit of electric charge, their interchange amounts to an inversion of the charge (which is the coupling constant). S duality relates the type I theory to the HO theory and the IIB theory to itself. This explains the strong coupling behavior of those three theories.
M Theory
The understanding of how the IIA and HE theories behave at strong coupling, which is by now well-established, came as quite a surprise.
In each of these cases there is an 11th dimension that becomes large at strong coupling, the scaling law being
In the IIA case the 11th dimension is a circle, whereas in the HE case it is a line interval (so that the eleven-dimensional space-time has two ten-dimensional boundaries). The strong coupling limit of either of these theories gives an 11-dimensional space-time. The eleven-dimensional description of the underlying theory is called "M theory." As yet, it is less well understood than the five 10-dimensional string theories. The connections among theories that we've mentioned are sketched below.
(S1 denotes a circle and I denotes a line interval.) There are many additional dualities that arise when more dimensions are compactified, which will not be described here.
D-Branes
Another source of insight into non-perturbative properties of superstring theory has arisen from the study of a special class of p-branes called Dirichlet p-branes (or D-branes for short). The name derives from the boundary conditions assigned to the ends of open strings. The usual open strings of the type I theory satisfy a condition (Neumann boundary condition) that ensures that no momentum flows on or of the end of a string. However, T duality implies the existence of dual open strings with specified positions (Dirichlet boundary conditions) in the dimensions that are T-transformed. More generally, in type II theories, one can consider open strings with specified positions for the end-points in some of the dimensions, which implies that they are forced to end on a preferred surface. At first sight this appears to break the relativistic invariance of the theory, which is paradoxical. The resolution of the paradox is that strings end on a p-dimensional dynamical object -- a D-brane. D-branes had been studied for a number of years, but their significance was explained by Polchinski only recently!*)***[7]
The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many (but not all) of the previously identified p-branes are D-branes. Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many interesting applications, but the most remarkable of these concerns the study of black holes. Strominger and Vafa[8] (and subsequently many others) have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case, which was studied first, is static extremal charged black holes in five dimensions. Strominger and Vafa showed that for large values of the charges the entropy (defined by S = log N, where N is the number of quantum states that system can be in) agrees with the Bekenstein-Hawking prediction (1/4 the area of the event horizon). This result has been generalized to black holes in 4D as well as to ones that are near extremal (and radiate correctly) or rotating. In my opinion, this is a truly dramatic advance. It has not yet been proved that there is no breakdown of quantum mechanics due to black holes, but I expect that result to follow in due course.
I have touched on some of the highlights of the current revolution, but there is much more that does not fit here. For example, I have not discussed the dramatic discoveries of Seiberg and Witten [9] for supersymmetric gauge theories and their extensions to string theory. Another important development due to Vafa[10] (called F theory) has made it possible to construct large new classes of non-perturbative vacua of Type IIB superstrings. For a somewhat more technical discussion of these recent developments I recommend ref[11].
Despite all the progress that has taken place in our understanding of superstring theory, there are many important questions whose answers are still unknown. It is not clear how many important discoveries still remain to be made before it will be possible to answer the ultimate question that we are striving towards -- why does the universe behave the way it does? Short of that, we have some other pretty big questions. What is the best way to formulate the theory? How and why is supersymmetry broken? Why is the cosmological constant, which characterizes the energy density of the vacuum, so small (or zero)? How is a realistic solution of the theory chosen from the myriad of possibilities? What are the cosmological implications of the theory? What testable predictions can we make? Stay tuned.
Closing
Acknowledgement
References
[Sejarah Singkat Teori Adi dawai,
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
By:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
Abstract
A theory of everything (TOE) is a hypothetical theory of theoretical physics that fully explains and links together all known physical phenomena. Initially, the term was used with an ironic connotation to refer to various overgeneralized theories. For example, a great-grandfather of Ijon Tichy — a character from a cycle of Stanisław Lem's science fiction stories of 1960s — was known to work on the "General Theory of Everything". Physicist John Ellis claims [1] to have introduced the term into the technical literature in an article in Nature in 1986 [2]. Over time, the term stuck in popularizations of quantum physics to describe a theory that would unify or explain through a single model the theories of all fundamental interactions of nature.
Introduction
The history of string theory is very fascinating, with many bizarre twists and turns. It has not yet received the attention it deserves from historians of science. Here we will settle for a very quick sketch.
String Theory of the Strong Nuclear Force
Superstring Unification
The First Superstring Revolution
String Theory of the Strong Nuclear Force
Superstring Unification
The First Superstring Revolution
There have been many theories of everything proposed by theoretical physicists over the last century, but none have been confirmed experimentally. The primary problem in producing a TOE is that the accepted theories of quantum mechanics and general relativity are hard to combine.
Based on theoretical holographic principle arguments from the 1990s, many physicists believe that 11-dimensional M-theory, which is described in many sectors by matrix string theory, in many other sectors by perturbative string theory is the complete theory of everything. Other physicists disagree.
Contents
String Theory of the Strong Nuclear Force
The subject of string theory arose in the late 1960's in an attempt to describe strong nuclear forces. This approach to the description of strong nuclear forces was a quite active subject for about five years, until it was abandoned because it ran into various theoretical difficulties and because a better theory came along.
There were two main problems that stymied us in our attempts to use string theory to describe the strong nuclear forces. One was that the theory required the existence of a kind of particle that we didn't want -- namely a particle with no mass and two units of spin. Its existence was very generic and very frustrating. The second problem was that the theory required that space-time have ten dimensions (nine space and one time), whereas the correct answer is clearly four (three space and one time).
As if all that weren't bad enough, around 1973 quantum chromodynamics (QCD) -- the SU(3) part of the standard model -- merged as a convincing theory of the strong nuclear force. One curious fact about string theory in the 1968--1973 period is that it took two years studying various complicated mathematical formulas before several people realized that these were formulas describing the interactions of extended one-dimensional objects, which were named ``strings.'' Once this was realized, it became clear that this type of theory was outside the framework of conventional quantum field theory, which as we have emphasized, is based on point-like elementary particles.
Another important development during this period (in 1971) was the discovery that to incorporate a class of elementary particles called fermions (electrons and quarks are examples) string theory requires a two-dimensional version of supersymmetry.[2] This led to the development of space-time supersymmetry, which was eventually recognized to be a generic feature of all consistent string theories (hence the name superstrings").
Superstring Unification
In 1974 Joel Scherk and I proposed that the problems that string theory had encountered could be turned into virtues if it were used as a framework for realizing Einstein's old dream of a ``unified theory'' of fundamental forces and elementary particles, rather than as a theory of hadrons (the strongly interacting nuclear particles).[3 ]
Specifically, we pointed out that it would provide a theory that incorporates general relativity without the characteristic short-distance infinities of quantum field theory. The massless spin two particle, which we had tried so hard to get rid of, would be identified as the graviton -- the quantum of gravitation!
One implication of this change in viewpoint was that, to account for the observed strength of the gravitational force, the characteristic size of a string had to be roughly the Planck length (the symbol h is Planck's constant.) This was a big change, since this distance is some 20 orders of magnitude smaller than the characteristic size of hadrons previously envisaged. More refined analyses lead to a string scale Lst that is about two orders of magnitude larger than the Planck length. In any case, experiments at existing accelerators cannot resolve distances shorter than about 10-16 cm, which explains why the point-particle approximation of ordinary quantum field theories is so successful.
The second problem -- the extra dimensions -- could also be addressed in this setting. Once one has a theory containing gravitation and generalizing general relativity, one knows that that the geometry of space-time is dynamically determined.
One could imagine that, as a consequence of the dynamics, the extra six dimensions form a small compact space attached to each point in ordinary four-dimensional space-time. If the size of the extra dimensions is sufficiently small, there would be no conflict with observations. I found these ideas very exciting and have been pursuing them ever since. However, for the ten year period 1974--1984 only a few colleagues and I pursued these ideas. One who did was Joel Scherk; tragically, he passed away in 1980.
The First Superstring Revolution
In 1984-85 there was a series of discoveries[4] that convinced many theorists that superstring theory is a very promising approach to unification. Almost overnight, the subject was transformed from an intellectual backwater to one of the most active areas of theoretical physics, which it has remained ever since.
By the time the dust settled in 1985, it seemed clear that there are five different superstring theories, each requiring ten dimensions (nine space and one time), and that each of them has a consistent description in term of a power series expansion in the coupling constant (perturbation expansion).
The five theories, about which I'll say more later, are denoted type I, type IIA, type IIB, E8 X E8 heterotic (HE, for short), and SO(32) heterotic (HO, for short). The type II theories have two supersymmetries in the ten-dimensional sense, while the other three have just one.
The type I theory is special in that it is based on unoriented open and closed strings, whereas the other four are based on oriented closed strings. The IIA theory is special in that it is non-chiral (i.e., it is parity conserving), whereas the other four are chiral (parity violating).
At this point I'll end the historical discussion and turn to superstring theory itself.
Closing
Acknowledgement
References
Other Contents Resources
- 1 Historical antecedents
- 2 Modern physics
- 3 With reference to Gödel's incompleteness theorem
- 4 Potential status of a theory of everything
- 5 Theory of everything and philosophy
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
An Introduction to the Super string Theory
[Sebuah Pendahuluan Terhadap Teori Adidawai,
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
By:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
AbstractArip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
Superstring Theory (Teori Adidawai) merupakan teori tertinggi yang dicari oleh para ilmuwan saat ini untuk dapat memerikan segala kejadian di Alam Raya. Teori ini merupakan bagian dari Teori Segala hal, dan merupakan kandidat terkuat dalam menyangga kebangunan teori segala hal (The Theory of Everything).
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Superstring, Special relativity, The Quantum Theory, General Relativity, Quantum Field theory, Supersymetri, Superpartner.
Introduction
Hasrat manusia untuk menguak tabir rahasia alam raya sangatlah besar itu dibutikan dengan banyak peradaban umat manusia yang senantiasa mengamati dan melakukan perenungan yang mendalam terhadap keberadaan jagat raya yang kita huni ini. Inilah sebuah bukti bahwa manusia pada hakekatnya selalu ingin mencari tahu, dari mana ia berasal, untuk apa dia hidup dan akan kemana kelak pada suatu saat.
Contents
Resolution of Contradictions
Major advances in understanding of the physical world have been achieved during the past century by focusing on apparent contradictions between well-established theoretical structures. In each case the reconciliation required a better theory, often involving radical new concepts and striking experimental predictions.
Four major advances of this type are indicated in Figure 1.These advances were the discoveries of special relativity, quantum mechanics, general relativity, and quantum field theory. Let us briefly recall the contradictions that each of these resolved and some of the experimental confirmations that followed.
Special Relativity
Galilean invariance, which is embodied in Newton's mechanics, for example, was considered intuitively obvious and observationally successful for several centuries. Among other things it implies that velocities add; i.e., if A observes B moving with speed v1 and B observes C moving with speed v2 in the same direction, then A observes C moving with speed v12 = v1 + v2 in that direction.
A contradiction arose as a consequence of the development of a very successful theory of electricity and magnetism in the nineteenth century, which is embodied in a set of differential equations known as Maxwell's equation. One of the implications of Maxwell's equations is the existence of various waves, such as radio waves, all of which are just light waves at various different frequencies. The equations imply that all of these waves travel with the speed of light (about 300,000 km/sec) regardless of the motion of the observer. This is a violation of Galilean invariance. Einstein resolved this paradox by recognizing that Galilean invariance is just an approximation, valid for speeds much smaller than the speed of light. The principles that apply for all speeds are embodied in the special theory of relativity. In this theory the rule for the addition of velocities is
,
Where c denotes the speed of light.
The Quantum Theory
The theory of electromagnetism embodied in Maxwell's equations also conflicted with the understanding of thermodynamics -- the behavior of systems in thermal equilibrium. In particular, a hot object (a so-called `black-body') emits electromagnetic radiation with a certain well-defined spectrum (intensity as a function of the frequency of the radiation). The problem with this is that if one adds up the energy carried off by radiation at all the different frequencies, the formulas imply that the total is infinite, which is an absurd result.
Just before the turn of the century, Max Planck realized that if the energy was emitted in discrete packets (or `quanta'), rather than in a continuous distribution, the total energy would be finite. He postulated that radiation of frequency n comes in quanta of energy E = hn, where h is a fundamental constant of nature, known as Planck's constant. The individual quanta of light are called ``photons''. This was the beginning of the idea of wave-particle duality and the quantum theory.
General Relativity
Einstein was not content with the special theory of relativity, because it was in contradiction with Newton's theory of gravitation. Newtonian gravitation was a very successful theory. In particular, it accounted for the motion of the planets to high precision. However, it has a peculiar property that had even bothered Newton. It implies instantaneous transmission of the gravitation force between two objects across great distances. Einstein knew this had to be wrong, because special relativity implies that no signal can be transmitted faster than the speed of light. Einstein provided the resolution around 1915 with a new theory of gravitation, which he called the general theory of relativity. It agrees with the Newtonian theory for low speeds and weak gravitational fields, but differs from it at high speeds and strong fields.
The theory had several immediate observational successes. First it implied a small correction to the orbit of the planet Mercury that accounted for a small discrepancy between the orbit implied by the Newtonian theory and the observed orbit. (The effect is too small to be observed for the other planets.) Second, it predicted that light from a distant star passing near to the limb of sun would be bent by a small but measurable angle. The measurement was made by Eddington during a solar eclipse.
The observations confirmed the theoretical prediction, and Einstein became an international celebrity. Other predictions of the theory -- such as the existence of black holes and of gravitational radiation -- have been confirmed much more recently. The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles -- the equivalence principle and general coordinate invariance -- and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is
In this equation Gmn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum.
Quantum Field Theory
The next contradiction that physicists faced was between quantum mechanics (which had been developed over the thirty years following Planck's seminal insight) and the special theory of relativity. Most of the work in quantum mechanics was in the Galilean (or non-relativistic) approximation.
To be sure, Dirac had developed a relativistic wave equation for the electron, which was an important advance, but there was still a basic contradiction that needed to be resolved. The new feature that is required in a successful union of quantum mechanics and special relativity is the possibility of the creation and annihilation of quanta (or `particles'). The non-relativistic theory does not have this feature.
The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. It is based on three basic principles: two of them, of course, are quantum mechanics and special relativity. The third one, which I wish to emphasize, is the postulate that elementary particles are point-like objects of zero intrinsic size. In practice, they are smeared over a region of space due to quantum effects, but their descripton in the basic equations is as mathematical points.
Now the general principles on which quantum field theory are based actually allow for many different consistent theories to be constructed. (The consistency has not been established with mathematical rigour, but this is not a concern for most physicists.) Among these various possible theories there is a class of theories, called `gauge theories' or `Yang-Mills theories' that turn out to be especially interesting and important. These are characterized by a symmetry structure (called a Lie group) and the assignment of various matter particles to particular symmetry patterns (called group representations). There is an infinite set of possibilities for the choice of the symmetry group, and for each group there are many possible choices of group representations for the matter particles.
One of this infinite array of theories has been experimentally singled out. It is called the ``standard model''. It is based on a Lie group called SU(3) X SU(2) X U(1)!*)***. The matter particles consist of three families of quarks and leptons. (I will not describe the representations that they are assigned to here.) There are also addition matter particles called ``Higgs particles'', which are required to account for the fact that part of the symmetry is spontaneously broken. The standard model contains some 20 adjustable parameters, whose values are determined experimentally. Still, there are many more things that can be measured than that, and the standard model is amazingly successful in accounting for a wide range of experiments to very high precision. Indeed, at the time this is written, there is only one clear-cut piece of experimental evidence that the standard model is not an exactly correct theory. This evidence is the fact that the standard model does not contain gravity!
The Final Contradiction
The results described above constitute quite an achievement for one century, but it leaves us with one fundamental contradiction that still needs to be resolved. General relativity and quantum field theory are incompatible. Many theoretical physicists are convinced that superstring theory will provide the answer. There have been major advances in our understanding of this subject, which I consider to constitute the ``second superstring revolution,'' during the past few years.
After presenting some more background, I will describe the recent developments and their implications.
There are various problems that arise when one attempts to combine general relativity and quantum field theory. The field theorist would point to the breakdown of the usual procedure for eliminating infinities from calculations of physical quantities. This procedure is called "renormalization", and when it fails the theory is said to be "non-renormalizable." In such theories the short-distance behavior of interactions is so singular that it is not possible to carry out meaningful calculations. By replacing point-like particles with one-dimensional extended strings, as the fundamental objects, superstring theory overcomes the problem of non-renormalizability.
An expert in general relativity might point to a different set of problems such as the issue of how to understand the causal structure of space-time when the geometry has quantum-mechanical excitations. There are also a host of problems associated to black holes such as the fundamental origin of their thermodynamic properties and their apparent incompatibility with quantum mechanics. The latter, if true, would mean that a modification in the basic structure of quantum mechanics is required.
In fact, superstring theory does not modify quantum mechanics; rather, it modifies general relativity. The relativist's set of issues cannot be addressed properly in the usual approach to quantum field theory (perturbation theory), but the recent discoveries are leading to non-perturbative understandings that should help in addressing them. Most string theorists expect that the theory will provide satisfying resolutions of these problems without any revision in the basic structure of quantum mechanics. Indeed, there are indications that someday quantum mechanics will be viewed as an implication (or at least a necessary ingredient) of superstring theory.
When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made.
In my opinion, success in such enterprises requires a better understanding of the theory than has been achieved as yet. It is very difficult to assess whether this level of understanding is just around the corner or whether it will take many decades and several more revolutions. In the absence of this kind of confirmation, we can point to three qualitative "predictions" of superstring theory. The first is the existence of gravitation, approximated at low energies by general relativity. No other quantum theory can claim to have this property (and I suspect that no other ever will).
The second is the fact that superstring solutions generally include Yang--Mills gauge theories like those that make up the "standard model" of elementary particles. The third general prediction is the existence of supersymmetry at low energies (the electroweak scale). Since supersymmetry is the major qualitiative prediction of superstring theory not already known to be true before the prediction, let us look at it a little more closely. (One could imagine that in some other civilization, the sequence of discoveries is different.)
Supersymmetry
Supersymmetry is a theoretically attractive possibility for several reasons. Most important from my viewpoint, is the fact that it is required by superstring theory. Beyond that is the remarkable fact that it is the unique possibility for a non-trivial extension of the known symmetries of space and time (which are described in special relativity by the Poincare group). Mathematically, it can be described in terms of extra dimensions that are rather peculiar. Whereas ordinary space and time dimensions are described by ordinary numbers, which have the property that they commute: X·Y = Y·X, the supersymmetry directions are described by numbers that anti-commute: X·Y = -Y·X.
Superpartners
As we have said, supersymmetry (also known as SUSY) is the major prediction of superstring theory at experimentally accessible energies that has not yet been confirmed. If correct, it implies that every known elementary particle must have a "superpartner." We are quite sure, for reasons I won't go into, that no pair of the known particles are supersymmetry partners of one another. So supersymmetry requires the existence of a new elementary particle for every known one.
As ofter happens, the names are somewhat whimsical: the partners of quarks are called "squarks", the partners of electrons are called "selectrons", the partners of gluons (the Yang-Mills particles that carry the strong nuclear force) are called "gluinos" and so forth. It is believed that the reason that these particles have not yet been observed is because supersymmetry is a broken symmetry, and as a result the superpartners are heavier than the known elementary particles. Experiments carried out so far have not had particle beams of sufficient energy and intensity to produce them in observable numbers.
Unfortunately, current theoretical ideas are insufficient to accurately predict the superpartner masses, though the way in which these particles interact with one another and with the known particles is predicted precisely.
Even though accurate predictions of the the superpartner masses do not exist, there are three distinct arguments that make qualititative predictions of the masses. All three of them lead to the conclusion that a typical superpartner mass should be in the range of 100 GeV to 1000 GeV. In other words, they should be about 100--1000 times heavier than a proton. The three arguments are the following:
First, supersymmetry leads to a softening of the short distance singularities of quantum field theory. If we require a sufficient softening so that the Higgs mechanism can break the electroweak symmetry (SU(2) X U(1)) at the observed 300 GeV scale, in which case the Higgs particles have masses of the same order of magnitude, then the scale of supersymmetry breaking must also be approximately the same.
The second argument concerns the unification of the electroweak and strong nuclear forces at very high energy (around $10^{16}$ GeV). One can argue that such a unification is inconsistent with the current experimental data, if one includes the effects of only known particles in the extrapolation, but that it works if supersymmetry partner particles with masses in the 100 GeV to 1000 GeV range are included.
The third argument concerns the possibility that the lightest SUSY particle could be a form of dark matter accounting for a substantial fraction of the mass of the universe. This also requires the same range of masses!
Experimental Prospects
All of the above makes a very suggestive case for SUSY. It is also very exciting because the mass range that is predicted is just what the new generation of particle accelerators is beginning to explore.
Regrettably, the American superconducting supercollider project (SSC) was cancelled, but a European accelerator called the large hadron collider (LHC) will begin operating at a lab in Geneva, Switzerland (called CERN) around 2005. Its energy will be about 8000 GeV per beam, whereas the SSC energy would have been 20,000 GeV per beam.
In my opinion, The LHC energy is high enough so that if it does not find supersymmetry after a few years of operation, we can safely conclude that it does not exist in the vicinity of the electroweak scale.
If the LHC (or another machine) does find SUSY, on the other hand, this would be one of the most profound achievements in the history of humankind. It would be more profound, in my opinion, than the discovery of life on Mars, for example.
Closing
Untuk memahami teori ini teryata ada beberapa langkah yang harus ditempuh dahulu seperti pengertian yang memadai mengenai relativitas khusus, relativitas umum, Teori medan quantum, Teori meknika kuantum, supersimetri dan superpartner, tapi harus diingat bahwa hipotesis superstring tidak hanya berupa teori tanpa sebuah eksperimen yang memadai. Saat ini CERN (Lembaga Penelitian Partikel Nuklir Eropa) dan NBL (National Brokeheaven Laboratory ) telah meakukan beberapa percobaan pendahuluan untuk membuktikan hipotesisi ini.
Acknowledgment
Thanks to my friends (Quantum Study Club especially “Cecep” who was read and give some correction to my paper
Further Sources
Einstein was not content with the special theory of relativity, because it was in contradiction with Newton's theory of gravitation. Newtonian gravitation was a very successful theory. In particular, it accounted for the motion of the planets to high precision. However, it has a peculiar property that had even bothered Newton. It implies instantaneous transmission of the gravitation force between two objects across great distances. Einstein knew this had to be wrong, because special relativity implies that no signal can be transmitted faster than the speed of light. Einstein provided the resolution around 1915 with a new theory of gravitation, which he called the general theory of relativity. It agrees with the Newtonian theory for low speeds and weak gravitational fields, but differs from it at high speeds and strong fields.
The theory had several immediate observational successes. First it implied a small correction to the orbit of the planet Mercury that accounted for a small discrepancy between the orbit implied by the Newtonian theory and the observed orbit. (The effect is too small to be observed for the other planets.) Second, it predicted that light from a distant star passing near to the limb of sun would be bent by a small but measurable angle. The measurement was made by Eddington during a solar eclipse.
The observations confirmed the theoretical prediction, and Einstein became an international celebrity. Other predictions of the theory -- such as the existence of black holes and of gravitational radiation -- have been confirmed much more recently. The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles -- the equivalence principle and general coordinate invariance -- and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is
In this equation Gmn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum.
Quantum Field Theory
The next contradiction that physicists faced was between quantum mechanics (which had been developed over the thirty years following Planck's seminal insight) and the special theory of relativity. Most of the work in quantum mechanics was in the Galilean (or non-relativistic) approximation.
To be sure, Dirac had developed a relativistic wave equation for the electron, which was an important advance, but there was still a basic contradiction that needed to be resolved. The new feature that is required in a successful union of quantum mechanics and special relativity is the possibility of the creation and annihilation of quanta (or `particles'). The non-relativistic theory does not have this feature.
The framework in which quantum mechanics and special relativity are successfully reconciled is called quantum field theory. It is based on three basic principles: two of them, of course, are quantum mechanics and special relativity. The third one, which I wish to emphasize, is the postulate that elementary particles are point-like objects of zero intrinsic size. In practice, they are smeared over a region of space due to quantum effects, but their descripton in the basic equations is as mathematical points.
Now the general principles on which quantum field theory are based actually allow for many different consistent theories to be constructed. (The consistency has not been established with mathematical rigour, but this is not a concern for most physicists.) Among these various possible theories there is a class of theories, called `gauge theories' or `Yang-Mills theories' that turn out to be especially interesting and important. These are characterized by a symmetry structure (called a Lie group) and the assignment of various matter particles to particular symmetry patterns (called group representations). There is an infinite set of possibilities for the choice of the symmetry group, and for each group there are many possible choices of group representations for the matter particles.
One of this infinite array of theories has been experimentally singled out. It is called the ``standard model''. It is based on a Lie group called SU(3) X SU(2) X U(1)!*)***. The matter particles consist of three families of quarks and leptons. (I will not describe the representations that they are assigned to here.) There are also addition matter particles called ``Higgs particles'', which are required to account for the fact that part of the symmetry is spontaneously broken. The standard model contains some 20 adjustable parameters, whose values are determined experimentally. Still, there are many more things that can be measured than that, and the standard model is amazingly successful in accounting for a wide range of experiments to very high precision. Indeed, at the time this is written, there is only one clear-cut piece of experimental evidence that the standard model is not an exactly correct theory. This evidence is the fact that the standard model does not contain gravity!
The Final Contradiction
The results described above constitute quite an achievement for one century, but it leaves us with one fundamental contradiction that still needs to be resolved. General relativity and quantum field theory are incompatible. Many theoretical physicists are convinced that superstring theory will provide the answer. There have been major advances in our understanding of this subject, which I consider to constitute the ``second superstring revolution,'' during the past few years.
After presenting some more background, I will describe the recent developments and their implications.
There are various problems that arise when one attempts to combine general relativity and quantum field theory. The field theorist would point to the breakdown of the usual procedure for eliminating infinities from calculations of physical quantities. This procedure is called "renormalization", and when it fails the theory is said to be "non-renormalizable." In such theories the short-distance behavior of interactions is so singular that it is not possible to carry out meaningful calculations. By replacing point-like particles with one-dimensional extended strings, as the fundamental objects, superstring theory overcomes the problem of non-renormalizability.
An expert in general relativity might point to a different set of problems such as the issue of how to understand the causal structure of space-time when the geometry has quantum-mechanical excitations. There are also a host of problems associated to black holes such as the fundamental origin of their thermodynamic properties and their apparent incompatibility with quantum mechanics. The latter, if true, would mean that a modification in the basic structure of quantum mechanics is required.
In fact, superstring theory does not modify quantum mechanics; rather, it modifies general relativity. The relativist's set of issues cannot be addressed properly in the usual approach to quantum field theory (perturbation theory), but the recent discoveries are leading to non-perturbative understandings that should help in addressing them. Most string theorists expect that the theory will provide satisfying resolutions of these problems without any revision in the basic structure of quantum mechanics. Indeed, there are indications that someday quantum mechanics will be viewed as an implication (or at least a necessary ingredient) of superstring theory.
When a new theoretical edifice is proposed, it is very desirable to identify distinctive testable experimental predictions. In the case of superstring theory there have been no detailed computations of the properties of elementary particles or the structure of the universe that are convincing, though many valiant attempts have been made.
In my opinion, success in such enterprises requires a better understanding of the theory than has been achieved as yet. It is very difficult to assess whether this level of understanding is just around the corner or whether it will take many decades and several more revolutions. In the absence of this kind of confirmation, we can point to three qualitative "predictions" of superstring theory. The first is the existence of gravitation, approximated at low energies by general relativity. No other quantum theory can claim to have this property (and I suspect that no other ever will).
The second is the fact that superstring solutions generally include Yang--Mills gauge theories like those that make up the "standard model" of elementary particles. The third general prediction is the existence of supersymmetry at low energies (the electroweak scale). Since supersymmetry is the major qualitiative prediction of superstring theory not already known to be true before the prediction, let us look at it a little more closely. (One could imagine that in some other civilization, the sequence of discoveries is different.)
Supersymmetry
Supersymmetry is a theoretically attractive possibility for several reasons. Most important from my viewpoint, is the fact that it is required by superstring theory. Beyond that is the remarkable fact that it is the unique possibility for a non-trivial extension of the known symmetries of space and time (which are described in special relativity by the Poincare group). Mathematically, it can be described in terms of extra dimensions that are rather peculiar. Whereas ordinary space and time dimensions are described by ordinary numbers, which have the property that they commute: X·Y = Y·X, the supersymmetry directions are described by numbers that anti-commute: X·Y = -Y·X.
Superpartners
As we have said, supersymmetry (also known as SUSY) is the major prediction of superstring theory at experimentally accessible energies that has not yet been confirmed. If correct, it implies that every known elementary particle must have a "superpartner." We are quite sure, for reasons I won't go into, that no pair of the known particles are supersymmetry partners of one another. So supersymmetry requires the existence of a new elementary particle for every known one.
As ofter happens, the names are somewhat whimsical: the partners of quarks are called "squarks", the partners of electrons are called "selectrons", the partners of gluons (the Yang-Mills particles that carry the strong nuclear force) are called "gluinos" and so forth. It is believed that the reason that these particles have not yet been observed is because supersymmetry is a broken symmetry, and as a result the superpartners are heavier than the known elementary particles. Experiments carried out so far have not had particle beams of sufficient energy and intensity to produce them in observable numbers.
Unfortunately, current theoretical ideas are insufficient to accurately predict the superpartner masses, though the way in which these particles interact with one another and with the known particles is predicted precisely.
Even though accurate predictions of the the superpartner masses do not exist, there are three distinct arguments that make qualititative predictions of the masses. All three of them lead to the conclusion that a typical superpartner mass should be in the range of 100 GeV to 1000 GeV. In other words, they should be about 100--1000 times heavier than a proton. The three arguments are the following:
First, supersymmetry leads to a softening of the short distance singularities of quantum field theory. If we require a sufficient softening so that the Higgs mechanism can break the electroweak symmetry (SU(2) X U(1)) at the observed 300 GeV scale, in which case the Higgs particles have masses of the same order of magnitude, then the scale of supersymmetry breaking must also be approximately the same.
The second argument concerns the unification of the electroweak and strong nuclear forces at very high energy (around $10^{16}$ GeV). One can argue that such a unification is inconsistent with the current experimental data, if one includes the effects of only known particles in the extrapolation, but that it works if supersymmetry partner particles with masses in the 100 GeV to 1000 GeV range are included.
The third argument concerns the possibility that the lightest SUSY particle could be a form of dark matter accounting for a substantial fraction of the mass of the universe. This also requires the same range of masses!
Experimental Prospects
All of the above makes a very suggestive case for SUSY. It is also very exciting because the mass range that is predicted is just what the new generation of particle accelerators is beginning to explore.
Regrettably, the American superconducting supercollider project (SSC) was cancelled, but a European accelerator called the large hadron collider (LHC) will begin operating at a lab in Geneva, Switzerland (called CERN) around 2005. Its energy will be about 8000 GeV per beam, whereas the SSC energy would have been 20,000 GeV per beam.
In my opinion, The LHC energy is high enough so that if it does not find supersymmetry after a few years of operation, we can safely conclude that it does not exist in the vicinity of the electroweak scale.
If the LHC (or another machine) does find SUSY, on the other hand, this would be one of the most profound achievements in the history of humankind. It would be more profound, in my opinion, than the discovery of life on Mars, for example.
Closing
Untuk memahami teori ini teryata ada beberapa langkah yang harus ditempuh dahulu seperti pengertian yang memadai mengenai relativitas khusus, relativitas umum, Teori medan quantum, Teori meknika kuantum, supersimetri dan superpartner, tapi harus diingat bahwa hipotesis superstring tidak hanya berupa teori tanpa sebuah eksperimen yang memadai. Saat ini CERN (Lembaga Penelitian Partikel Nuklir Eropa) dan NBL (National Brokeheaven Laboratory ) telah meakukan beberapa percobaan pendahuluan untuk membuktikan hipotesisi ini.
Acknowledgment
Thanks to my friends (Quantum Study Club especially “Cecep” who was read and give some correction to my paper
Further Sources
Basic Ideas of Superstring Theory
[Pemikiran Dasar dari Teori Adidawai ,
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
By:
Arip Nurahman
Department of Physic, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
Lanjutan dari Makalah Teori Segala Hal (The Theory of Everything)]
By:
Arip Nurahman
Department of Physic, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com)
Abstract
Superstring Theory (Teori Adidawai) merupakan teori tertinggi yang dicari oleh para ilmuwan saat ini untuk dapat memerikan segala kejadian di Alam Raya. Teori ini merupakan bagian dari Teori Segala hal, dan merupakan kandidat terkuat dalam menyangga kebangunan teori segala hal (The Theory of Everything).
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Perturbation Theory, Compactification of Extra Dimensions, A Theory of Everything?
Introduction
A string's space-time history is described by functions Xm(s,t) which describe how the string's two-dimensional "world sheet," represented by coordinates (s,t), is mapped into space-time Xm. There are also functions defined on the two-dimensional world-sheet that describe other degrees of freedom, such as those associated with supersymmetry and gauge symmetries. Surprisingly, classical string theory dynamics is described by a conformally invariant 2D quantum field theory. (Roughly, conformal invariance is symmetry under a change of length scale.) What distinguishes one-dimensional strings from higher dimensional analogs is the fact that this 2D theory is renormalizable (no bad short-distance infinities). By contrast, objects with p dimensions, called "p-branes," have a (p+1)-dimensional world volume theory. For p > 1, those theories are non-renormalizable. This is the feature that gives strings a special status, even though, as we will discuss later, higher-dimensional p-branes do occur in superstring theory.
Perturbation Theory
Compactification of Extra Dimensions
A Theory of Everything?
Contents
Perturbation Theory
As was briefly mentioned earlier, a useful way of studying theories that cannot be solved exactly is by computing power series expansions in a small parameter. For example, quantum electrodynamics has a small parameter, called the fine structure constant, which is given by
Thus, if T(a) denotes some physical quantity of interest, one computes
and the first few terms can give a very good approximation. This approach, which is called perturbation theory, is the way superstring theories were studied until recently. The problem is that in superstring theory there is no reason that the expansion parameter a should be small.
More significantly, there are important qualitative phenomena that are missed in perturbation theory. The reason is that there are non-perturbative contributions to many physically interesting quantities that have the structure
Such a contribution is completely invisible in perturbation theory.
Perturbative quantum string theory can be formulated by the Feynman sum-over-histories method. This amounts to associating a genus h Riemann surface, which can be visualized as a sphere with h handles attached to it, to the hth term in the string theory perturbation expansion. The genus h surface is identified as the corresponding string theory Feynman diagram. The attractive features of this approach are that there is just one diagram at each order of the perturbation expansion and that each diagram represents an elegant (though complicated) mathematical expression that is ultraviolet finite (no short-distance infinities). The main drawback of this approach is that it gives no insight into how to go beyond perturbation theory.
Compactification of Extra Dimensions
As has already been mentioned, to have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length Lst..
Since space-time geometry is determined dynamically (as in general relativity), only geometries that satisfy the dynamical equations are allowed.
The HE string theory, compactified on a particular kind of six-dimensional space, called a Calabi--Yau manifold, has many qualitative features at low energies that resemble the standard model. In particular, the low mass fermions (identified as quarks and leptons) occur in families, whose number is controlled by the topology of the CY manifold.
These successes have been achieved in a perturbative framework, and are necessarily qualitative at best, since non-perturbative phenomena are essential to an understanding of supersymmetry breaking and other important matters of detail.
A Theory of Everything?
In the euphoria following the first superstring revolution in 1985, some of the less experienced participants in the enterprise thought that we were on the verge of constructing a complete fundamental theory of the physical world. To put it mildly, I found this naive. In this setting, the phrase "Theory of Everything" was introduced and propagated by the public media. This was very unfortunate for several reasons.
The TOE phrase is very misleading on several counts. First of all, the theory is not yet fully formulated, and when it is (which might still take decades) it is not entirely clear that it will be the last word in fundamental physics.
Furthermore, even if the theory is a complete description of quantum dynamics, it seems unlikely that it will also provide a theory of initial conditions, which is another key ingredient required to explain why we observe the particular universe that we do. But even if a theory of initial conditions is also obtained, there will still be much about this universe that cannot be explained. Many things, such as our very existence, are a consequence of the inherent quantum indeterminacy of nature. I believe that cannot be overcome. Maybe that is just as well, because if we had old-fashioned classical determinism, the future would be fully determined, which would undermine our humanity.
There is also a more mundane sort of unpredictability that is also to be expected. Many of the things that the theory predicts unambiguously in principle could require intractable calculations. Part of the art of physics is to identify those things that can be calculated. The other reason the TOE phrase upset me is that it alienated many of our physics colleagues, some of whom had serious doubts about the subject anyway. Quite understandably, it gave them the impression that people who work in this field are a very arrogant bunch. Actually, we are all very charming and delightful.
Closing
Acknowledgement
References
The Second Superstring Revolution
[Revolusi ke-2 Adidawai, Lanjutan dari Makalah Teori Segala Hal
(The Theory of Everything)]
By:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com/)
[Revolusi ke-2 Adidawai, Lanjutan dari Makalah Teori Segala Hal
(The Theory of Everything)]
By:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education
(http://astrophysicsblogs.blogspot.com/)
Abstract
Superstring Theory (Teori Adidawai) merupakan teori tertinggi yang dicari oleh para ilmuwan saat ini untuk dapat memerikan segala kejadian di Alam Raya. Teori ini merupakan bagian dari Teori Segala hal, dan merupakan kandidat terkuat dalam menyangga kebangunan teori segala hal (The Theory of Everything).
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Just One Theory!, T Duality, S Duality, M Theory, D-Branes,
Teori ini mengasumsikan bahwa bila kita meninjau benda hingga ke bentuk yang sangat kecil, maka kita harus melihat benda itu sebagai dawai-dawai yang bergetar. Dalam makalah ini dipaparkan mengenai pendahuluan untuk memasuki relung-relung teori Adidawai dimana ada beberapa aspek yang harus dikuasai terlebih dahulu agar lebih memudahkan kita dalam memahami teori Adidawai yang akan di paparkan.
Keyword: Just One Theory!, T Duality, S Duality, M Theory, D-Branes,
Introduction
Contents
Just One Theory!
The second superstring revolution (1994-??) has brought non-perturbative string physics within reach. The key discoveries were the recognition of amazing and surprising "dualities." They have taught us that what we viewed previously as five distinct superstring theories is in fact five different perturbative expansions of a single underlying theory about five different points. It is now clear that there is a unique theory, though it may allow many different quantum mechanical solutions. For example, a sixth special quantum solution implies the existence of an 11-dimensional space-time. Another lesson we have learned is that, non-perturbatively, objects of more than one dimension (membranes and higher "p-branes") play a central role. In most respects they appear just as fundamental as the strings (which can now be called one-branes), except that a perturbation expansion cannot be based on p-branes with p >1.
Three kinds of dualities, called S,T, and U, have been identified. It can sometimes happen that theory A with a large strength of interaction (or `strong coupling') is equivalent to theory B at weak coupling, in which case they are said to be S dual. Similarly, if theory A compactified on a space of large volume is equivalent to theory B compactified on a space of small volume, then they are called T dual. Combining these ideas, if theory A compactified on a space of large (or small) volume is equivalent to theory B at strong (or weak) coupling, they are called U dual. If theories A and B are the same, then the duality becomes a self-duality, and it can be viewed as a kind of symmetry.
T duality, unlike S or U duality, can be understood perturbatively, and therefore it was discovered between the two string revolutions.
T Duality
The basic idea of T duality(for a recent discussion see [5]) can be illustrated by considering a compact dimension consisting of a circle of radius R. In this case there are two kinds of excitations to consider. The first, which is not special to string theory, are Kaluza--Klein momentum excitations on the circle, which contribute (n/R)2 to the energy squared, where n is an integer. Winding-mode excitations, due to a closed string winding m times around the circular dimension, are special to string theory. If
Denotes the string tension (energy per unit length), the contribution to the energy squared is :
Em=2pmRT.
T duality exchanges these two kinds of excitations by exchanging m with n and
This is part of an exact map between a T-dual pair A and B.
One implication is that usual geometric concepts break down at short distances, and classical geometry is replaced by "quantum geometry," which is described mathematically by 2D conformal field theory. It also suggests a generalization of the Heisenberg uncertainty principle according to which the best possible spatial resolution Dx is bounded below not only by the reciprocal of the momentum spread, Dp, but also by the string scale Lst. (Including non-perturbative effects, it may be possible to do a little better and reach the Planck scale.) Two important examples of superstring theories that are T-dual when compactified on a circle are the IIA and IIB theories and the HE and HO theories. These two dualities reduce the number of distinct theories from five to three.
S Duality
Suppose now that a pair of theories A and B are S-dual. This means that if f denotes any physical observable and l denotes the coupling constant, then
(The expansion parameter a introduced earlier corresponds to l). This duality, whose recognition was the first step in the current revolution, [6] generalizes the electric-magnetic symmetry of Maxwell theory. Since the Dirac quantization condition implies that the basic unit of magnetic charge is inversely proportional to the unit of electric charge, their interchange amounts to an inversion of the charge (which is the coupling constant). S duality relates the type I theory to the HO theory and the IIB theory to itself. This explains the strong coupling behavior of those three theories.
M Theory
The understanding of how the IIA and HE theories behave at strong coupling, which is by now well-established, came as quite a surprise.
In each of these cases there is an 11th dimension that becomes large at strong coupling, the scaling law being
In the IIA case the 11th dimension is a circle, whereas in the HE case it is a line interval (so that the eleven-dimensional space-time has two ten-dimensional boundaries). The strong coupling limit of either of these theories gives an 11-dimensional space-time. The eleven-dimensional description of the underlying theory is called "M theory." As yet, it is less well understood than the five 10-dimensional string theories. The connections among theories that we've mentioned are sketched below.
(S1 denotes a circle and I denotes a line interval.) There are many additional dualities that arise when more dimensions are compactified, which will not be described here.
D-Branes
Another source of insight into non-perturbative properties of superstring theory has arisen from the study of a special class of p-branes called Dirichlet p-branes (or D-branes for short). The name derives from the boundary conditions assigned to the ends of open strings. The usual open strings of the type I theory satisfy a condition (Neumann boundary condition) that ensures that no momentum flows on or of the end of a string. However, T duality implies the existence of dual open strings with specified positions (Dirichlet boundary conditions) in the dimensions that are T-transformed. More generally, in type II theories, one can consider open strings with specified positions for the end-points in some of the dimensions, which implies that they are forced to end on a preferred surface. At first sight this appears to break the relativistic invariance of the theory, which is paradoxical. The resolution of the paradox is that strings end on a p-dimensional dynamical object -- a D-brane. D-branes had been studied for a number of years, but their significance was explained by Polchinski only recently!*)***[7]
The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many (but not all) of the previously identified p-branes are D-branes. Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control. D-branes have found many interesting applications, but the most remarkable of these concerns the study of black holes. Strominger and Vafa[8] (and subsequently many others) have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case, which was studied first, is static extremal charged black holes in five dimensions. Strominger and Vafa showed that for large values of the charges the entropy (defined by S = log N, where N is the number of quantum states that system can be in) agrees with the Bekenstein-Hawking prediction (1/4 the area of the event horizon). This result has been generalized to black holes in 4D as well as to ones that are near extremal (and radiate correctly) or rotating. In my opinion, this is a truly dramatic advance. It has not yet been proved that there is no breakdown of quantum mechanics due to black holes, but I expect that result to follow in due course.
Conclusion
I have touched on some of the highlights of the current revolution, but there is much more that does not fit here. For example, I have not discussed the dramatic discoveries of Seiberg and Witten [9] for supersymmetric gauge theories and their extensions to string theory. Another important development due to Vafa[10] (called F theory) has made it possible to construct large new classes of non-perturbative vacua of Type IIB superstrings. For a somewhat more technical discussion of these recent developments I recommend ref[11].
Despite all the progress that has taken place in our understanding of superstring theory, there are many important questions whose answers are still unknown. It is not clear how many important discoveries still remain to be made before it will be possible to answer the ultimate question that we are striving towards -- why does the universe behave the way it does? Short of that, we have some other pretty big questions. What is the best way to formulate the theory? How and why is supersymmetry broken? Why is the cosmological constant, which characterizes the energy density of the vacuum, so small (or zero)? How is a realistic solution of the theory chosen from the myriad of possibilities? What are the cosmological implications of the theory? What testable predictions can we make? Stay tuned.
Closing
Acknowledgement
References
