A REAPPRAISAL OF THE
Prof. Dr. Rati Ram Sharma,
DSc, PhD(London), MD(MA), MSc, MAMS, FIAMP
Professor & Head (retired), Department of Biophysicsc,
Postgraduate Institute of Medical Education & Research, Chandigarh, India;
Present residence & mailing address:
615, Sector 10, Panchkula-134113, Haryana, India;
Phone: 0091-172-2563949; e-mail: email@example.com
(Last revised 3 December 2006)
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Here we will present the conceptual outlines of the String Theories based on ref. [1-3] leaving aside the mathematics and intricate parts of the theory. The UPT reappraisal will also be given where appropriate. This will suffice for the reader to make out that the String Theory is unrealistic and the UPT is realistic.
The string theory claims to be the unifying theory of everything. It predicts that everything in the universe, from galaxies and suns to atoms and subatomic particles, can be broken down to incomprehensibly small loops of vibrating strings 10-33 cm long. The building blocks of reality are merely a pattern of their vibrations, just as strings on a violin vibrate at different rates in different modes to produce different notes. The string theory has never been experimentally tested, however.
The string theory envisages a unity of quantum mechanics, particle physics and general relativity (gravitation). It postulates that there was a String Bang before the Big Bang. It also considers the black holes. In fact all the elements of major current theories are in it.
It is not true that string theory is the theory of every thing. Its scope of application is very narrow as compared to that of UPT presented in this book. The string theory, on the other hand, makes its own addition to the unreality of concepts, complexity of mathematics and to other shortcomings of the current theories, which it unifies, as brought out by the Unified Physical Theory in this book.
This can be appreciated from the following broad outlines of the string theory collected from the few references given at the end [1-5]. The interested reader can refer to these [1-5] and other published literature for details.
1.2 The basic outlines of the string theory
1.2.1 The strings as basic elements
The modern Standard Model of the relativistic electrodynamics considers fundamental particles like electron, proton & neutron as sizeless point-like 0-dimensional objects. In string theory the fundamental strings are 1-dimensional. They have no thickness but do have a length, typically 10-33 cm, which is the Planck length. In UPT zero thickness is inconceivable, unreal and non-existent.
The quantum numbers of the vibrational mode of the string deduce the physical parameters like mass, charge and spin of the test particle. But it is inconceivable for a string with zero thickness to exist as a physical entity and vibrate!
In fact the strings and their physical parameters were not deduced realistically from scientific logic but were induced by assumption following some mathematical equations in particle Physics.
The micromost elements in the UPT however are deduced and calculated in Chap.5 through scientific logic from the real physical sharmon medium in space. These are two cosminos: the ve positrino and –ve negatrino. A cosmino is found to have the diameter lp = 1.6x10-33 cm, mass = 2.596x 10-48 gm, electric charge =1.3729x10-30 esu, and spin = .
1.2.2 Two types of strings
Strings can be open or closed. As they move through the10-dimensional spacetime continuum they sweep out an imaginary surface called a worldsheet.
The above figure is adapted from ref..
1.2.3 Interaction of strings
Strings interact by splitting and joining. The two closed strings interact by joining into an intermediate closed string, which splits apart into two closed strings again:
The interaction of nonexistent strings is a mere speculation from the point of view of the UPT.
To compute quantum mechanical amplitudes using ‘perturbation theory’ contributions from higher order quantum processes are added. Perturbation theory provides good answers as long as the contributions get smaller and smaller as one goes to higher and higher orders. Then only the first few diagrams need computation to get accurate results. In string theory, higher order diagrams correspond to the number of holes (or handles) in the worldsheet.
In perturbation theory there is only one diagram for each order. [In point particle field theories the number of diagrams grows exponentially at higher orders.] But extracting the answers from diagrams with more than about two handles is very difficult due to the complexity of the mathematics involved. Perturbation theory is a very useful tool for studying the physics at weak coupling, and most of the current understanding of particle physics and string theory is based on it. However it is far from complete. The answers to many of the deepest questions will only be found from a complete non-perturbative description of the theory.
1.2.4 The Boundary conditions
Strings can have various kinds of boundary conditions. For example closed strings have periodic boundary conditions (the string comes back onto itself). Open strings can have two different kinds of boundary conditions called Neumann and Dirichlet boundary conditions. With Neumann boundary conditions the endpoint is free to move about but no momentum flows out. With Dirichlet boundary conditions the endpoint is fixed to move only on some manifold. This manifold is called a D-brane or Dp-brane ('p' is an integer which is the number of spatial dimensions of the manifold). For example we see open strings with one or both endpoints fixed on a 2-dimensional D-brane or D2-brane:
Incidently the suffix 'brane' is borrowed from the word 'membrane' which is reserved for 2-dimensional manifolds or 2-branes!
The D-branes can have dimensions ranging from -1 to the number of spatial dimensions in the spacetime continuum. For example superstrings live in a 10-dimensional spacetime, which has 9 spatial dimensions and one time dimension. Therefore the D9-brane is the upper limit in superstring theory. In this case the endpoints are fixed on a manifold that fills all of space. So it is really free to move anywhere and this is just a Neumann boundary condition!
The case p = -1 is when all the space and time coordinates are fixed. This is called an instanton or D-instanton. When p=0 all the spatial coordinates are fixed so the endpoint must live at a single point in space, therefore the D0-brane is also called a D-particle. Likewise the D1-brane is also called a D-string.
The D-branes are actually dynamical objects, which have fluctuations and can move around.
In UPT the space and time are not entities but mere concepts, which evolve from our direct percepts of ‘there, here, there’ and ‘then, now, then’ arising from the successive motions and changes in the surrounding objects. These cannot and actually do not fuse into any spacetime continuum of 4, 10 or 111 dimensions.
More significantly, the real Nature exhibits granularity right up to the micromost elements, the 10-33 cm cosminos. The 10- or 11-dimensional spacetime and the various branes and membranes, if existent as continua, would prevent or at least retard the motions of fundamental particles, material bodies, planets, galaxies and of even the photon, through them. But none of these expectations is supported by actual observation.
Therefore the 10- and/or 11-dimensional theories of nonexistent strings are mere speculations. So are the various branes and membranes.
1.2.5 The 11-dimensional supergravity & M-theory
The M-theory is described at low energies by an effective theory called 11-dimensional supergravity. This theory has a membrane and 5-branes as solitons, but no strings. The 11-dimensional M-theory can compactify on a small circle to get a 10-dimensional theory. If we take a membrane and wrap one of its dimensions on this compact circle, the membrane becomes a closed string!
We can also get a consistent 10-dimensional theory if we compactify M-theory on a small line segment. That is, take one dimension (the 11-th dimension) to have a finite length. The endpoints of the line segment define boundaries with 9 spatial dimensions. An open membrane can end on these boundaries. Since the intersection of the membrane and a boundary is a string, we see that the 9 1 dimensional world-volume of the each boundary can contain strings, which come from the ends of membranes.
To underline the importance of the theory the ‘M’ in the term ‘M-theory’ may also signify ‘mother’, ‘magic’, ‘master’ or ‘majestic’. The M-theory is the fundamental theory. It may deduce the five types of ‘string theories’, namely the Type I, Type IIA, Type IIB, E8XE8 Heterotic, SO(32) Heterotic.
But most significantly in UPT, the real Nature exhibits granularity right up to the micromost levels of the 10-33 cm cosminos. However, the multi-dimensional spacetime and membranes in the M-theory are continua. Their existence is therefore inconceivable. It is also difficult to visualize the inter-conversions between a continuous membrane and an isolated fundamental particle. These continuous spacetime and membranes should prevent, or at least retard, the motions of the material bodies through them, against the Newton’s three laws of motion. These should also rule out all kinds of ‘free motions’ including that of the ‘photon’ in the universe. But such possibilities are not actually observed.
Therefore the 10-dimensional string theories and the 11-dimensional M-theory are mere mathematical speculations, having no physical confirmation.
1.2.6 The multiple dimensions & compactification
Superstrings live in a 10-dimensional spacetime, but we observe a 4-dimensional spacetime. The extra 6 dimensions are curled up into a small compact space of the string scale (10-33 cm). So we cannot detect the presence of these extra dimensions directly - they're just too small. The end result is that we get back to our familiar (3 1)-dimensional world, but there is a tiny "ball" of 6-dimensional space associated with every point in our 4-dimensional universe.
Actually this idea dates back to the 1920's and the Kaluza and Klein’s [6,7] theory of compactification. Kaluza  showed that if we start with a theory of general relativity in 5-spacetime dimensions and then curl up one of the dimensions into a circle we end up with a 4-dimensional theory of general relativity plus electromagnetism!
Strings when compactified can wind around a compact dimension, which leads to winding modes in the mass spectrum. A closed string can wind around a periodic dimension an integral number of times.
Similar to the Kaluza-Klein case they contribute a momentum p = w R (w=0,1,2,...). As the compact dimension becomes very small these winding modes are very light! To make contact with the 4-dimensional world the 10-dimensional superstring theory is compactified on a 6-dimensional compact manifold.
The Kaluza Klein picture described above becomes a bit more complicated. One way could simply be to put the extra 6 dimensions on 6 circles, which is just a 6-dimensional Torus. As it turns out this would preserve too much supersymmetry. It is believed that some supersymmetry exists in our 4-dimensional world at an energy scale above 1 TeV (this is the focus of much of the current and future research at the highest energy accelerators around the word!). To preserve the minimal amount of supersymmetry, N=1 in 4 dimensions, we need to compactify on a special kind of 6-manifold called a Calabi-Yau manifold.
The properties of the Calabi-Yau manifold can have important implications for low energy physics such as the types of particles observed, their masses and quantum numbers, and the number of generations. One of the outstanding problems in the field has been the fact that there are many Calabi-Yau manifolds (thousands upon thousands?) and we have no way of knowing which one to use.
In UPT the space and time are mere abstract concepts and spacetime continua of 4-, 5-, 10- and 11-dimensions are nonexistent. So the string theory, superstring theory and the M-theory based on them are unrealistic.
1.2.7 The black holes
The General Relativity as a theory of gravitation contains solutions, which are called "black holes". There are many different kinds of black hole solutions but they share some common characteristics. The event horizon is a surface in spacetime which, loosely speaking, divides the inside of the black hole from the outside. The gravitational attraction of a black hole is so strong that any object, including light, that crosses the event horizon can never escape out of the black hole. Classical black holes are therefore relatively featureless, but they can be described by a set of observable parameters such as mass, charge, and angular momentum.
Black holes aren't really "black" since they radiate! Stephen Hawking showed black holes emit a thermal spectrum of radiation at their event horizon. In fact there are black hole solutions, which satisfy the string equations of motion, similar to the equations of general relativity with some extra matter fields coming from string theory. Superstring theories also have some special black hole solutions, which are themselves supersymmetric, in that they preserve some supersymmetry.
The black holes obey an "area law", dM = K dA, where 'A' is the area of the event horizon and 'K' is a constant of proportionality. Since the total mass 'M' of a black hole is just its rest energy, Bekenstein realized that this is similar to the thermodynamic law for entropy, dE = T dS. Hawking later performed a semi-classical calculation to show that the temperature of a black hole is given by T = 4 k [where k is a constant called the "surface gravity"]. Therefore the entropy of a black hole should be written as S = A/4.
Andrew Strominger and Cumrin Vafa, showed that this exact entropy formula can be derived microscopically (including the factor of 1/4) by counting the degeneracy of quantum states of configurations of strings and D-branes which correspond to black holes in string theory. This is compelling evidence that D-branes can provide a short distance weak coupling description of certain black holes! Hawking radiation can also be understood in terms of the same configuration, but with open strings traveling in both directions. The open strings interact, and radiation is emitted in the form of closed strings.
UPT: A plausible physical mechanism under the UPT for the emission of low energy Hawking radiation is given in sec. 5.10.8 on the black holes in Chapter 5.
1.3 Some fundamental questions
On March 26, 20004 Brian Braiker of KeepMedia <NewsWeek>
interviewed Professor Brian Greene of Columbia University and published a report under the title “The World on a string” . The following are the edited excerpts along with the UPT comments side by side.
BRAIKER: Brian Greene likes to think he's got it all figured out. And it all boils down to string. Greene is one of the world's leading thinkers and writers on string theory, which purports to be the unifying theory of everything.
Greene did not invent string theory. But in 1999, he published "The Elegant Universe" (Norton) , a popular presentation of string theory that became a major best-seller and Pulitzer Prize finalist. Last fall he hosted a "Nova" television series based on that book. Now he's back with "The Fabric of the Cosmos: Space, Time, and the Texture of Reality" (Knopf) published last month February 2004.
Once again, assuming an audience of lay readers, Greene explains some of the more mind-melting features of today's cutting-edge physics in a language that is easy to understand. For example, one feature of string theory--also known as superstring theory--is that it suggests the universe has more than three, and possibly up to 11, spatial dimensions. Reality as we perceive it may in fact just be an approximation of the universe we inhabit. Time, which is relative to space, may not allow us to ever visit the past, but jumping into the future is possible within the laws of physics.
Greene recently spoke with NEWSWEEK's Brian Braiker, who is--under this theory--just a vibrating mass of tiny string, about the ideas he explores in the new book. Excerpts:
GREENE: At first sight it wouldn't be a revolutionary idea to go from the concepts of atoms and subatomic particle to vibrating little lines. But when you study it in detail, you find that it is for a number of reasons. The first is that we believe that it gives a uniform description of all matter and radiation and all the forces of nature in one unified language. Prior to string theory, when you spoke about the elementary constituents of matter, you had to talk about electrons, you had to talk about protons and neutrons, you had to talk about the quarks that make them up. When string theory comes on the scene, everything simplifies because you have one entity: the string. And like any other string that you're more familiar with, like on a violin or a cello, the string in string theory can vibrate in different patterns. When a string on a violin vibrates differently, it produces different musical notes. Here when the little strings vibrate, they produce different particles.
UPT: In fact the strings and their physical parameters were not deduced realistically from scientific logic but were induced by assumption following some mathematical equations in particle Physics. It is inconceivable for a string with zero thickness to exist as a physical entity and vibrate!
The unified Physical Theory presented in this book however is realistic and explanatory. The micromost elements and their parameters (mass, charge, radius) in the UPT were deduced from the real physical sharmon medium in space. These are two cosminos: the ve positrino and –ve negatrino. A cosmino has the diameter lp = 1.6x10-33 cm, mass = 2.596x 10-48 gm, electric charge =1.3729x10-30 esu, and spin = .
GREENE’s reply to a question: We don't know for certain what the strings are made of, or what is it that is vibrating. This may be the one time when that question fails to make sense. When you look at anything around you--the mug on your desk or the tabletop that you're working on--you can say, "What's it made of?" You can ask what the atoms that you hypothesize and prove by experiment make up the entity are them selves made of. You can ask what the nucleus of an atom is made of and get to the neutrons and protons. And you can ask what they're made of: quarks. But when you get down to strings, it may be that that's where the story ends. It may be that they are the fundamental entity.
UPT: According to UPT and common sense no object with zero size can or does exist. So the strings having zero thickness are non-existent. And there is nothing to vibrate in the inconceivable strings of zero thickness. The vibrations of non-existent strings are mere speculations.
BRAIKER: You keep saying, "may," which means none of this is certain at all.
GREENE: Oh, string theory is definitely a work in progress. It's definitely a theory; it has yet to be experimentally confirmed. [But] if string theory is right--and again, I always emphasize the "if"--it is a unified theory. I think the important thing to bear in mind is that general relativity has been tested up the wazoo to incredible accuracy. Quantum mechanics has been tested; it works with fantastic precision. String theory simply puts them together in the first consistent framework.
UPT: Braiker is forthright in asking and Greene is right to imply that the existence of strings and the theory based on them cannot be supported with certainty. In UPT too no real object has zero thickness. And therefore the strings with zero thickness and their vibrations cannot be basic to all in the Cosmos.
Moreover, as repeatedly shown in this book the special & general theories of relativity as well as the quantum theory are based on unsustainable unrealistic assumptions, and so is the string theory, which relies on the relativity and quantum theories.
GREENE: The general relativity is a theory of gravity. Gravity becomes most relevant when things are big--the earth's gravity, the sun's gravity--but rarely do we talk about the gravity of a coffee cup because it's too small. Quantum theory is most relevant on the opposite end of the spectrum; it comes into play when you are talking about atoms and subatomic particles. But we have come upon realms where you need both theories at the same time, such as black holes and the big bang--two examples where you have a lot of materials crushed to a very small size. Therefore you can't live your life keeping general relativity and quantum mechanics permanently separate. That's not how the world works, so you need a theory that can put them together in a consistent manner. String theory is the first theory to do that.
UPT: The fatal flaw in the General Relativity, as a theory of gravitation, is its application of the Reimannian differential geometry to the non-existent 4-D spacetime continuum. Its strong point lies in its prediction of the bending of light rays in a gravitational filed, which was later observed Eddington in 1919. But in UPT, its formula for the bending of light ray in a gravitational field is exactly derived from the realistic sharmon medium in space. And the UPT does not support the violation of the conservation of energy and momentum postulated under the Uncertainty Principle of the Quantum Theory. In fact the UPT as presented in this book brings out the unnecessity and irrelevance of both the relativity and quantum theories. Because the UPT, with its realistic sharmon medium, does all that they both did and/or even that they were to but could not do. Putting the relativity and quantum theories together in string theory also means adding their conceptual flaws.
BRAIKER: In your new book you talk about the possibility of there being up to 11 possible dimensions, where are they? What do they look like?
GREENE: I think this is the most stunning and surprising element of this theory. One approach, which is one that I've actually worked on for maybe 15 years now, is that they're all around us, they're just tightly curled up. If you imagine a garden hose that's unfurled long and horizontal stretched out, but it's far in the distance, it'll look like a straight line because you won't be able to see the thickness. You'll think that it only has a left-right dimension along its horizontal extent and nothing else. But then you take a pair of binoculars and zoom in, you know that there's more to the surface than just left-right. It also has clockwise and counterclockwise, the circular girth of the hose, which is a curled-up circular dimension you don't see without binoculars. We think the same may be true of the universe, namely there may be big easy-to-see dimensions--the horizontal extent of the hose--but there might also be tiny, curled-up dimensions all around us--like the circular part of the hose--but so tiny that we don't have the magnifying equipment like binoculars adequate to reveal the existence of these extra dimensions.
UPT: The space and time are not entities having substance. These are abstract concepts, which evolve from our direct percepts of ‘there, here, there’ and ‘then, now, then’ arising from the successive motions and changes in the surrounding objects. These abstract concepts cannot and do not actually fuse to constitute any 4-, 5-, 10-, or 11-dimensional spacetime continuum. These continua are mere mathematical construct bereft of real physical existence. And the string theories based on the 10- and 11-dimensional spacetime are unrealistic speculations.
The 10- or 11-dimensional spacetime, if existent would at least retard the motions of material bodies, planets, and galaxies and of even of the photon, through them. But such speed-retardations have never been actually observed.
BRAIKER: What's another way of envisioning the possibility of other dimensions?
GREENE: The idea is that the extra dimensions that we don't see might be big like the ones we know about, but we don't see them because we see with light. It might be that light is trapped in our three dimensions and it can't escape into the other dimensions and that's why they remain invisible to us. Imagine a universe as a big loaf of bread and what we've always thought to be the universe is merely one slice of bread in this huge cosmic loaf. Light in this picture will be trapped in our slice of bread; it can't travel to the other slices. The only force that wouldn't be trapped, it turns out, is gravity. So it might be possible one day to detect these extra dimensions through gravity. Experiments are actually underway today in an attempt to do that.
UPT: This is the most unrealistic proposal and the experiments, if any, planned are mere waste of funds, laboratory facilities and intellectual wealth of the brilliant scientists.
BRAIKER: But we still wouldn't be able to see them.
GRRENE: We wouldn't see them with light. But we would see them indirectly with gravity. The experiment is so simple: at the Large Hadron Collider, which is an atom-smasher being built [at the particle-physics laboratory] CERN in Switzerland, they're going to take protons and send them circling around a huge tunnel in opposite directions right near the speed of light. Every so often they'll use magnets to direct the beams of protons to smash into each other in head-on collisions. The idea is that in the collision, a certain amount of gravity will be produced. It might be that some of that gravity can leak off our dimensions, off of our slice of bread, and disappear into the other dimensions. If that happens, the amount of energy before the collisions will be a little bit bigger than the amount of energy after the collision because some of it will have seeped away. So the scientists are going to look for missing energy. If it's missing in just the right pattern, it could be very strong evidence that the extra dimensions are real, that it has gone into other dimensions.
UPT: Greene’s new proposition or hope is based on the paper, which the 32-year-old Nima Arkani-Hamed and his colleagues at the Harvard University published in Phys. Lett. B in 1998 . It presupposes and validates violation of the inviolable conservation of energy-mass. Therefore Greene’s answer is more unrealistic than the question he is trying to answer. Moreover, the UPT forestalls the predicted hopes of Greene and Arkani.Hamed. The collision of energetic protons may also produce some sharmon-cosmino dust, vide sec. 13.5 of Chap-13. It will account for the undetected missing energy, if any, rather than confirming extra dimensions.
Let us all frankly and unanimously admit the unreality and non-existence of the extra dimensions. In the same stroke let us with UPT dismiss the existence of time and 3-D space as real entities.
BRAIKER: Why is time forward moving when everything else seems to tend toward randomness?
GREENE: It's a real big puzzle as to why time seems to be different. You stand in space and can move at will--left or right, back or forth, up or down, no constraint--but with time we seem to be relentlessly dragged forward. Why is that? We think that the answer, surprisingly, is to be found in the big bang itself. The big bang started the universe off in an incredibly ordered state and things tend to become more disordered over time. For that disorder to happen, you've got to begin highly ordered. There are sequences of events that we only see happen in one order--eggs splatter, they never unsplatter; glasses shatter, they never unshatter. I think it's a wonderful idea--every time you drop an egg and it splatters, it's actually telling you something very deep about the big bang itself.
UPT: No. It's NOT a real big puzzle as to why time seems to be different. The UPT shows that the concept of time evolves from our direct percepts of ‘then, now, then’ arising from the successive motions and changes in the surrounding objects, which we all receive from our childhood. The ‘only-forward’ and ‘never-backward’ movement of the “time arrow” is due to the irreversibility of successive natural processes of motion and change in the surrounding objects, which give rise to the abstract concept of time.
BRAIKER: So time travel may never be possible?
GREENE: I firmly believe that one day we will rule out the possibility of time travel to the past. On the same point, it's worth emphasizing that time travel to the future is a completely different ballgame. That is within the laws of physics, as we understand them. Einstein himself showed us how to accomplish time travel to the future. If you want to see what the earth is like ten or 100 or a million years into the future we know in principle how to do it: You build a spaceship; you travel out into space at near the speed of light; you turn around and come back. Perhaps a year may elapse for you, but because time slows down when moving at high speeds, 1,000 or a million years may have elapsed when you return to earth. That means you have jumped into earth's future. We can't build such ships yet, but these are technological issues. But physics definitely shows that this kind of leapfrog into the future is within the laws of physics.
UPT: Greene’s reply is based on the relativistic ‘dilatation of time’ as concluded from the theory of Special Relativity. But the Chapters 2 & 10 of this book show that the Lorentz transformation formulae of the Special Relativity are unrealistic and do not describe any real natural motion because the velocity of light in free space c is but the velocity v of a frame or an object is not invariant to source-observer motion. It is inconceivable that any natural velocity may vary (like v) and be invariant (like c) to the source-observer motion at the same time. Therefore the Lorentz transformation formulae do not describe any real motion in objective reality. And relativistic ‘contraction of space’ and ‘dilatation of time’ deduced from them are non-existent in real Nature. This rules out the possibility of any time travel into the future.
BRAIKER: When you really think about this--that we're just made up of vibrating strings, that we fail to perceive everything that constitutes reality--is it hard not to feel a little despair?
GREENE: When we recognize that the same laws that govern us are the laws that govern the molecules and atoms in interstellar space, the processes in the sun and the countless other stars in the heavens; when we learn, for instance, that the very atoms that make up our body were produced in stars and spewed out into space through supernova explosions, I think it makes us feel more connected to the cosmos. We may not be anything particularly special, but we're definitely part of the grand scheme. I think that can be very uplifting
UPT: Yes, all in the universe falls into a grand unified scheme as presented in this book under the UPT but not in accord with the string theory. And the same laws hold inviolably in the subatomic microcosm as they do in the macro-universe. This rules out the violation of conservation of energy-mass and momentum underlying the Heisenberg relations of Uncertainty Principle in the quantum theory, which string theory relies on. This also does away with the ‘creation of matter from nothing’ in the expanding universe, which string theory supports. The Chapter-18 on the Science-Spirituality Symbiosis in this book extends and enlarges the grand scheme of unity in the compositions from the micromost 10-33 cm cosminos through fundamental particles to the largest galaxies.
.Brian Greene,The Elegant Universe: Superstrings, Hidden Dimensions, And The Quest For The Ultimate Theory, NORTON, 1999.
.Brian Braiker of KeepMedia<NewsWeek> with Professor Brian Greene: “The World on a String”, 26 March, 2004.
.Kaluza, Th., Sitz.d .Preuss. Akad .d. Wiss. (1921) 966.
.Klein, O., Z. Phys. 37 (1926) 895.
.N. Arkani-Hamed, S. Dimopoulos and G. Dvali Phys. Lett. B 429 (1998) 263–272.