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23/02/2017 Dr Ray Adams 1 a 6-dimensional Calabi–Yau manifold 23/02/2017 Dr Ray Adams 2 Maths can be easy Paul Dirac’s professor in his Masters program gave him a list of ten mathematical problems that had been considered "unsolvable" by mathematicians when Dirac had complained about not being challenged enough. 23/02/2017 Dr Ray Adams 3 Maths can be easy • A few months later, the professor checked in on him and jokingly asked how he was doing in solving the problems... to which, Dirac replied, "I got the first six, but the last four are proving to be a little difficult.” 23/02/2017 Dr Ray Adams 4 Astrophysics can be difficult • “There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. • There is another theory which states that this has already happened.” ― Douglas Adams, The Restaurant at the End of the Universe 23/02/2017 Dr Ray Adams 5 NASA News Announcement • NASA will present new findings today about exoplanets that orbit stars other than our sun. • NASA will hold a major press conference at 6pm GMT today (22 nd Feb.) • See the press conference at: https://www.nasa.gov/multimedia/nasatv/index.html#public 23/02/2017 Dr Ray Adams 6 String Theory SuperString Theory & M – theory One conjecture in superstring theory states that the extra dimensions of spacetime take the form of a 6-dimensional Calabi–Yau manifold. 23/02/2017 Dr Ray Adams 7 Superstrings in Fiction ‘Colliding Branes’ Rudy Rucker & Bruce Sterling “But why call this the end of the universe?” . . . “At dawn our universe’s two branes [will] collide in an annihilating sea of light.” . . . “The colliding branes will crush the stars and planets to a soup of hard radiation.” 23/02/2017 Dr Ray Adams 8 Superstring Theory 23/02/2017 Dr Ray Adams 9 Superstring Jargon • Strings: particles in string theory arise as excitations of the string, and included in the excitations of a string. A "1-brane," is a string. • Brane: Short for membrane. a higher-dimensional manifold moving in spacetime. • D-brane; D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named • Brane cosmology: One possibility would be that the visible Universe is in fact a very large D-brane extending over three spatial dimensions. Material objects, made of open strings, are bound to the D-brane, and cannot move "at right angles to reality" to explore the Universe outside the brane. • p-brane; A brane that has p dimensions. 23/02/2017 Dr Ray Adams 10 Boundaries The Dirichlet boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet. When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take on along the boundary of the domain. 23/02/2017 Dr Ray Adams 11 Superstrings • By the mid-1980s the new superstring theory had emerged as the hottest theoretical breakthrough since quantum mechanics, mainly because it seemed to show a way that quantum mechanics itself could be merged with Einstein’s general relativity. • For the first time, physicists had a unified theory that could accommodate both Einstein’s explanation for gravity and the quantum explanation for particles and other forces. • To make the math work you needed several additional dimensions of space. Instead of a four-dimensional universe — three space, one time — you needed 10 or 11. 23/02/2017 Dr Ray Adams 12 Superstrings • Another early problem was that there seemed to be more than one superstring theory — different mathematical versions of the basic idea. • If superstring theory offered the one true final theory describing all of fundamental physics, how could there be more than one? • But in 1995 Edward Witten showed that the various string theories were all just different views of a deeper theory — he called it M theory. • String theory offered various different descriptions of the same subatomic reality. 23/02/2017 Dr Ray Adams 13 Superstring Dimensions • In super string theory there are only 10 dimensions. • They consist of 9 space dimensions and 1 time dimension. • They all have the same three types of dimensions. 1. Three spatial dimensions. Known in common English as "depth", "width", "height". Or some similar grouping of three perpendicular angles that describe these spatial dimensions. 2. One temporal dimension. (We call the grouping of these first four dimensions "spacetime“) 3. The third type of dimension are those which are undergoing Compactification. The "size" of the dimensions is not apparent at the macro scale. These additional six or seven dimensions would be "curling" spatial dimensions that are bound to the three big spatial dimensions. 23/02/2017 Dr Ray Adams 14 Compactification • Compactification is the process or result of making a topological space into a compact space 23/02/2017 Dr Ray Adams 15 Topological Space • A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods i.e. no missing sets, unions or intersections. Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. 23/02/2017 Dr Ray Adams 16 Compact Space • Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space is called compact if it is closed and bounded. • A space is closed if and only if it contains all of its limit points. N.B. With respect to the usual Euclidean topology, the sequence of rational numbers has no limits. • A space is called bounded, if it is of finite size. 23/02/2017 The interval A = (-∞, -2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed. The interval B = [0, 1] is compact because it is both closed and bounded. Dr Ray Adams 17 M theory Dimensions • In M theory there are 11 dimensions - 9 space dimensions, 1 time dimension and 1 energy dimension. • M theory the same three types of dimensions plus one energy dimension . • A super string theory is a low energy special case of an M theory. • This is true for all 5 super string theories. • So as one goes from high energy to low energy, the 11 dimension disappears and one is left with a 10 dimensional theory. 23/02/2017 Dr Ray Adams 18 23/02/2017 Dr Ray Adams 19 Standard Texts for Superstring Theory (free downloads) http://ebooktop.biz/pdf/title/Polchinski-J-String-theory-Vol-1-An-introduction-t.html http://www.4shared.com/office/bD_hHk_0/String_Theory_V2_Joseph_Polchi.htm https://arxiv.org/pdf/hep-th/9611050.pdf Polchinski 23/02/2017 TASI Lectures on D-Branes, Joseph Dr Ray Adams 20 Quantum connection could revitalize superstrings • In a new paper, Bars and Rychkov (2014) point out that when string theory was developed, everybody assumed quantum mechanics was correct and designed string theory to observe the quantum rules. • Bars and Rychkov work out how to build string theory without any quantum restrictions in a simplified version of string theory, specifically a version when the strings are “open” (not closed to make a loop). • In string theory, the common interactions between fundamental particles that physicists study are described as strings joining or splitting. • In analysing the details of the splitting and joining process, Bars and Rychkov found that the basic rules of quantum mechanics naturally emerge. In other words, you don’t need to assume quantum mechanics to find string theory — it’s the physics of strings that makes the world quantum mechanical. • In this paper they present arguments that there may be a physical explanation for where the quantum mechanics rules come from. 23/02/2017 Dr Ray Adams 21 Is String Interaction the Origin of Quantum Mechanics? https://arxiv.org/pdf/1407.6833.pdf 23rd Sept. 2014 Bars and Rychkov (2014) 23/02/2017 Dr Ray Adams 22 Successes for Superstring Theory String theory dovetails beautifully with the previous ideas for explaining the patterns in the Standard Model, and does so with a structure more elegant and unified than in quantum field theory. In particular, if one tries to construct a consistent relativistic quantum theory of onedimensional objects one finds: 1. Gravity. Every consistent string theory must contain a massless spin-2 state, whose interactions reduce at low energy to general relativity. 2. A consistent theory of quantum gravity. As we have noted, this is in contrast to all known quantum field theories of gravity. 3. Grand unification. String theories include the Standard Model. Some of the simplest string theories lead to the same representations that arise in the unification of the Standard Model. 23/02/2017 Dr Ray Adams 23 Successes for Superstring Theory 4. Dimensions. String theory predicts ten dimensions, thus incorporating general relativity and quantum mechanics. Einstein’s field equations are claimed to have solutions with four large flat and six small curved dimensions. The four dimensions support the physics of the Standard Model of particle physics. 5. Supersymmetry: String theories predict spacetime supersymmetry. 6. The infinities of quantum field theory disappear when we go to string theory. This comes from a qualitative difference between a Feynman diagram and its stringy cousin. 7. Chiral bosonic interactions. String theory correctly predicts that bosonic interactions are chiral. 8. No free parameters. String theory has no arbitrarily, adjustable constants. Quantum field theory has twenty free parameters, for which values has to be added by hand. 23/02/2017 Dr Ray Adams 24 Why Superstring Theory? Superstring Theory proposes a new answer to the fundamental question: “what are the smallest, indivisible constituents of matter?” Particles are not point particles, with location but no dimensions, but instead are tiny, vibrating filaments or strings that are much smaller (billions of times smaller) than the nucleus of an atom i.e. 10-33 centimetres, or about a millionth of a billionth of a billionth of a billionth of a centimetre. The Official String Theory Web Site: http://www.superstringtheory.com/index.html “If superstring theory is proven correct, we will be forced to accept that the reality we have known is but a delicate chiffon draped over a thick and richly textured cosmic fabric.” (Brian Greene) “All of the different fundamental particles of the Standard Model are really just different manifestations of one basic object: a vibrating oscillating string." 23/02/2017 Dr Ray Adams 25 What is wrong with physics? There are two powerful theories in physics; relativity and quantum mechanics. • Relativity focuses on large scale events whilst quantum mechanics focuses on very small events. • Both make very accurate predictions in their own domains that are supported by experimental evidence. • But when the equations of both are combined, the equations explode into infinite or undefinable terms! • Clearly, one solution is to subsume them in a bigger conceptual framework. Superstring theory shows potential to be such a bigger conceptual framework. Science Journal (1985) has commented that superstring theory is “no less profound than the transition from real numbers to complex numbers in mathematics.” 23/02/2017 Dr Ray Adams 26 What is wrong with physics? Both relativity and quantum theories are based on two, untested axioms: 1. The existence of point particles **that have location but no dimensions. 2. The continuous nature of space, such that you can take a line and repeatedly divide it into smaller and smaller lines for ever and ever. {It is possible that the unsolvable equations that result from combining the two theories may be the product of one or both of these axioms?} **The assumption of the point particle goes back as far as the Buddhist philosophers (6th century BC), Indian Vaisheshika school of philosophy (4 th century AD) and, of course, to Isaac Newton. 23/02/2017 Dr Ray Adams 27 Why Superstring Theory? The problem with point particles that have location but no dimensions. Newton’s law of gravity in scalar form 23/02/2017 Dr Ray Adams 28 Why Superstring Theory? The problem with point particles that have location but no dimensions. This is a point . 23/02/2017 This is a point also . Dr Ray Adams 29 Why Superstring Theory? The problem with point particles that have location but no dimensions i.e. r = 0. These are two points touching . But r2 must now also equal zero! This means that F1 and F2 are either equal to infinity or are not measurable! What would an infinite force do to two particles? 23/02/2017 Dr Ray Adams 30 Newton’s unsolved problems What happens when F = ∞? 23/02/2017 Dr Ray Adams 31 What about General Relativity? The same problem applies here too, though gravity is caused by the curvature of space and is not really a force. (As it acts like a force, it is called a “fictitious force”!) When a particle gets close to r = 0, of another particle, it will fall into it and never escape! When an external observer watches a particle falling into a black hole, it seems get slower and slower but never seems to cross the event horizon. If you were that particle, you would not experience time slowing at all, but once you have passed the event horizon, you are doomed to continue to r = 0. In Newtonian physics, gravity propagates instantaneously. In General Relativity, gravity propagates at the speed of light or less. But, of course, when the two point particles touch, this predicted difference disappears! So particles need dimensions after all? 23/02/2017 Dr Ray Adams 32 Feynman Diagrams and Strings • Here is a quote from Edward Witten from his article in Physics Today November 2015: “Thus string theory describes quantum gravity in spacetime”, from: What every physicist should know about string theory. • In this article Witten describes how transforming the **Feynman diagrams lines into tubes allows calculating the curvature of spacetime from that string while also describing other quantum field features of particles. • **Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. The scheme is named after its inventor, American physicist Richard Feynman, and was first introduced in 1948. The interaction of sub-atomic particles can be complex and difficult to understand intuitively, but Feynman diagrams provide simple visualisations. 23/02/2017 Dr Ray Adams 33 What about General Relativity? 23/02/2017 Dr Ray Adams 34 Feynman Diagrams and Strings To go from point particles to strings, each line in a Feynman diagram is replaced by a thin strip representing propagation of a string. The strings join smoothly to describe interaction events. Edward Witten (2001) 23/02/2017 Dr Ray Adams 35 Problems with Singularities and Infinities Singularities and infinities pose substantial problems for both maths and physics. “Our challenge as physicists is to discover . . . Infinity free equations” In general, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. Singularities are often also called singular points. Consider the following equation: If this equation cannot be differentiated at x0, then x0 is a singularity. In other words, a singularity is a point in which a given mathematical function is not defined or is not well behaved. For example, the function has a singularity at x = 0, where x explodes to ∞ Thus a singularity is a point at which the derivative does not exist for a given function. 23/02/2017 Dr Ray Adams 36 The assumption that space is continuous? “The postulate of a space-time continuum is not a logical necessity, since it is possible to construct a theory, where the ultimate limit for the smallest measurable distance a is finite. This quantum of length is a universal constant, like the light velocity c and Planck’s constant h.” A. Meessen** http://www.meessen.net/AMeessen/STQ/STQ2.pdf W Heisenberg has also queried the continuity assumption: In “Die physikalischen Prinzipien der Quantentheorie“ (1930) 23/02/2017 Dr Ray Adams 37 The assumption that space is continuous? This would also solve the EPR paradox. “As superluminal velocities are not forbidden anymore when the smallest measurable distance a is not 0.” i.e. by the exchange of superluminal particles with velocity v such that v > c, the familiar statement that velocities v > c are forbidden is only true when a = 0. We can even show that it is necessary that a is not zero, to insure the internal coherence of physics. 23/02/2017 Dr Ray Adams 38 Space-Time Quantization The concept of Space-Time Quantization (STQ) is beyond the scope of this presentation. But, in future, its predictions can be tested by the use of particle accelerators. See: http://arxiv.org/ftp/arxiv/papers/1108/1108.4883.pdf 2011. 23/02/2017 Dr Ray Adams 39 String Theory versus Superstring Theory • String theory is the generic name for the whole field. It began with Bosonic string theory which was not supersymmetric, and had two shortcomings, it had tachyons in it, faster than light particles that would destabilize the vacuum, and it didn't have fermions, or matter particles, but only bosons, or force carriers. Its main virtue was that one of the force carriers it did have was identifiable as a graviton, the first time a particle theory had produced such a valid account. • Reference https://www.physicsforums.com/threads/string-theory-vssuperstring-theory.142799/ 23/02/2017 Dr Ray Adams 40 Supersymmetry • The word super in a super string theory stands for super symmetry that's predicted to exist but has not been found yet. • This symmetry requires a super partner for every fundamental particle. A particle and its super partner have identical properties such as mass and charge except spin before the super symmetry is broken. • So for example the super partner of an electron would have zero spin and is called a selectron. An electron is a fermion and selectron is a boson. • “Results from the Large Hadron Collider (LHC) have all but killed the simplest version of an enticing theory of sub-atomic physics.” • “Researchers failed to find evidence of so-called "supersymmetric" particles, which many physicists had hoped would plug holes in the current theory.” • http://www.bbc.co.uk/news/science-environment-14680570 • Pallab Ghosh, Science correspondent, BBC News 27 August 2011 23/02/2017 Dr Ray Adams 41 Supersymmetry 23/02/2017 Dr Ray Adams 42 “The totally Unthinkable” - In 2016, CERN’s LHC Could Unveil Unknown Dimensions of the Universe. May 2016 Two separate teams of scientists said they had discovered anomalies that could possibly hint at the existence of a mysterious new particle that could prove the existence of extra space-time dimensions, or explain the enigma of dark matter, scientists say. The lab might prove the exotic theory of supersymmetry, SUSY for short, which suggests the existence of a heavier "sibling" for every particle in the universe. http://www.dailygalaxy.com/my_weblog/2016/05/the-totally-unthinkable-in-2016cerns-lhc-may-unveil-unknown-dimensions-of-the-universe-1.html 23/02/2017 Dr Ray Adams 43 String Theory versus Superstring Theory • It was soon discovered that adding the property called supersymmetry solved both the tachyon and the fermion problems and modern string theory is almost entirely based on superstrings. It has greatly proliferated, so that the modern theory is much more detailed and mathematical than the original bosonic theory, which was already more detailed and mathematical than previous particle theories. Reference https://www.physicsforums.com/threads/string-theory-vs-superstringtheory.142799/ 23/02/2017 Dr Ray Adams 44 Supersymmetry • Supersymmetry links the two different classes of particles known as fermions and bosons. Particles are classified as fermions or bosons based on a spin. • Fermions all have half of a unit of spin, while the bosons have 0, 1 or 2 units of spin. Supersymmetry predicts that each of the particles in the Standard Model has a partner with a spin that differs by half of a unit. • So bosons are accompanied by fermions and vice versa. Linked to their differences in spin are differences in their collective properties. • Fermions must always be in a different state from each other. 23/02/2017 Dr Ray Adams 45 Supersymmetry • The Pauli exclusion principle** states that two identical fermions cannot occupy the same quantum state simultaneously. • There is no exclusion property for bosons, which are free to crowd into the same quantum state. • Bosons are inclined to be in the same state. • Fermions and bosons are very different but Supersymmetry brings these two types of particle together. ** The Pauli exclusion principle is part of one of our most basic observations of nature. is generally taken as a `brute fact', i.e. as a defining characteristic of fermions or as a feature of nature that cannot be otherwise explained. The exclusion principle acts primarily as a selection rule for non-allowed quantum states and cannot be deduced as a theorem from the axioms of Orthodox Quantum Theory. http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/towler_pauli.pdf 23/02/2017 Dr Ray Adams 46 Supersymmetry • The history of a string is a surface in space-time, called a "world-sheet". There are also space and time directions inside the world-sheet. So the world-sheet is like a little two-dimensional space-time, embedded in the n-dimensional space-time that the string is moving through. In the larger space-time, every point on the world-sheet has an n-dimensional position vector. So it's as if there are n fields on the string - on the world-sheet - corresponding to the space-time coordinates in which the string is moving . • Each particle from one group is associated with a particle from the other, known as its superpartner, the spin of which differs by a half-integer. In a theory with perfectly "unbroken" supersymmetry, each pair of superpartners would share the same mass and internal quantum numbers besides spin. The development of supersymmetry is beyond the scope of this presentation, but see: https://www.physicsforums.com/threads/string-theory-vs-superstringtheory.142799/ 23/02/2017 Dr Ray Adams 47 Superstring Theory This theory aims to be the 23/02/2017 “TOE”, the “Theory of Everything” Dr Ray Adams 48 Superstring Theory An Introduction to the String Wars “Some string theorists prefer to believe that string theory is too arcane to be understood by human beings, rather than consider the possibility that it might just be wrong.” ― Lee Smolin “String theory is an attempt at a deeper description of nature by thinking of an elementary particle not as a little point but as a little loop of vibrating string.” ― Edward Witten versus 23/02/2017 Dr Ray Adams 49 Superstring Theory An Introduction to the String Wars “The genesis of a new physics”. “The theory could be the ultimate ‘theory of the universe’”. ― Michio Kaku (1999) “We don't know what we are talking about. Many of us believed that string theory was a very dramatic break with our previous notions of quantum theory. But now we learn that string theory, well, is not that much of a break.” — David Gross versus 23/02/2017 Dr Ray Adams 50 Superstring Theory An Introduction to the String Wars Columbia University mathematical physicist Peter Woit maintains the popular multiverse-critical blog Not Even Wrong. “After a decade, Peter Woit still thinks string theory is a gory mess.” Max Tegmark “String theory is currently the most popular candidate for a consistent theory of quantum gravity . . .” http://arxiv.org/pdf/0709.0002v3.pdf versus 23/02/2017 Dr Ray Adams 51 Superstring Theory An Introduction to the String Wars “A central point to understanding string theory is that it cannot be formulated the way all other fundamental theories are, by giving the dynamical variables and the equations they obey. We do not know what the fundamental dynamical variables of string theory are, nor the equations they obey.” Richard Woodard, Physics Professor, University of Florida. versus 23/02/2017 Dr Ray Adams 52 Three revolutions in Superstring Theory 1. Yoichiri Nambu, developed the first string theory 2. Michael Green & John Schwarz, developers of superstring theory 23/02/2017 3. Edward Witten developer of M theory Dr Ray Adams 53 What do Superstrings look like? 23/02/2017 Dr Ray Adams 54 What do Superstrings look like? 23/02/2017 Dr Ray Adams 55 How big are Superstrings? If a superstring were the size of an ant, then a hydrogen atom would be as large as the observable universe. 23/02/2017 Dr Ray Adams 56 How big are Superstrings? One centimetre contains 1033 superstrings. i.e. 1,000,000,000,000,000,000,000,000,000,000,000. According to Brian Greene, a Columbia University physicist educated at Harvard and Oxford, “If an atom were enlarged to the size of the solar system, a string would only be as large as a tree.” 23/02/2017 Dr Ray Adams 57 Introducing M - theory “Superstring theory is now being reincarnated as a new theory, something called M – theory . . .” M apparently stands for magic, majesty, mother, mystery or membrane. Witten and Townsend argued that the five mathematically consistent superstring theories in ten dimensions, were all the same when viewed in eleven dimensions. M theory thus exists in eleven dimensions and is based on both strings and membranes. Artist’s impression of a rippling membrane 23/02/2017 Dr Ray Adams 58 The mathematics of time Dr Ray Adams 23/02/2017 59 String Theory & Infinite Series “Physicists believe that the best hope for a fundamental theory of nature – including unification of quantum mechanics with general relativity and elementary particle theory - lies in string theory.” American Scientist “From the beginning it was clear that . . . the Standard Model of elementary particles would have to be embedded in a broader theory . . . the theory of strings, which started in the 1960s as a not – very – successful model of hadrons, and only later emerged as a possible theory of all forces.” From: Joseph Polchinski (1998) Dr Ray Adams 23/02/2017 60 Starting with String Theory * “This leads to the odd result:” Polchinski v1 p 22 *From: Joseph Polchinski (1998) String Theory Volume 1. An Introduction to the Bosonic String. Cambridge : CUP Part of Cambridge Monographs on Mathematical Physics Available from: http://stringworld.ru/files/Polchinski_J._String_theory._Vol._1._An_introduction_to_the_bosonic_string.pdf http://stringworld.ru/files/Polchinski_J._String_theory._Vol._2._Superstring_theory_and_beyond.pdf Dr Ray Adams 23/02/2017 61 Starting with String Theory Dr Ray Adams 23/02/2017 62 Starting with String Theory * “This leads to the odd result:” Polchinski v1 p 22 Dr Ray Adams 23/02/2017 63 Looking at Super String Theory Relativistic quantum field theory has worked very well to describe the observed behaviours and properties of elementary particles. But the theory itself only works well when gravity is so weak that it can be neglected. Particle theory only works when we pretend gravity doesn't exist. General relativity has yielded a wealth of insight into the Universe, the orbits of planets, the evolution of stars and galaxies, the Big Bang and recently observed black holes and gravitational lenses. However, the theory itself only works when we pretend that the Universe is purely classical and that quantum mechanics is not needed in our description of Nature. That is where String Theory comes in. Dr Ray Adams 23/02/2017 64 Looking at Super String Theory Originally, string theory was proposed as an explanation for the observed relationship between mass and spin for certain particles called hadrons, which include the proton and neutron. But that proved to be less successful than other theories, such as Quantum Chromodynamics. String Theory includes a particle with zero mass and two units of spin. This theorized particle is called the graviton and opened the possibility that String Theory could also deal with quantum gravity. For gravitons as point particles, the mathematics behaves so badly at zero distance that the answers just don't make sense. In string theory, the strings collide over a small but finite distance and the answers do make sense. Dr Ray Adams 23/02/2017 65 Looking at Super String Theory • Therefore quantum mechanics was not a reliable description of nature when the system contained particles that would move at or near the speed of light. • Everything that is a particle is also a wave. Thus matter particles are also waves, the so-called ‘matter waves’. • So we should be able to describe matter waves changing with time. Erwin Schrödinger came up with this equation in 1926. • Once special relativity was on firm observational and theoretical footing, it was seen that the Schrödinger equation of quantum mechanics was not Lorentz invariant. If so, the equation was not consistent with Special Relativity. • For a single particle moving around in three dimensions the equation can be written as: Dr Ray Adams 23/02/2017 66 String Theory as Quantum Gravity Dr Ray Adams 23/02/2017 67 String Theory as Quantum Gravity Dr Ray Adams 23/02/2017 68 Introduction to M-theory M-theory presents an idea about the basic substance of the universe. So far no experimental evidence exists showing that M-theory is a description of the real world. Interest in this theory is mainly driven by mathematical elegance. Bertrand Russell expressed his sense of mathematical beauty in these words: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” Dr Ray Adams 23/02/2017 69 Introduction to M-theory • However, to make string theory mathematically consistent, the universe the strings exist in must have ten dimensions. • This contradicts the experience that our real universe has four dimensions: three space dimensions (height, width, and length) and one time dimension. • To "save" their theory, string theorists therefore added the explanation that the additional six dimensions exist but cannot be detected directly; this was explained by sophisticated mathematical objects called Calabi–Yau manifolds. • The number of dimensions was later increased to 11 based on various interpretations of the 10-dimensional theory that led to five partial theories, as described below. Supergravity theory also played a significant part in establishing the necessity of the 11th dimension. • Supergravity theory is a type of quantum field theory of elementary subatomic particles and their interactions that is based on supersymmetry and includes the gravitational force along with the other fundamental interactions of matter— the electromagnetic force, the weak force, and the strong force. Dr Ray Adams 23/02/2017 70 Introduction to M-theory • These "strings" vibrate in multiple dimensions and depending on how they vibrate, they might be seen in three-dimensional space as matter, light or gravity. It is the vibration of the string which determines whether it appears to be matter or energy, and every form of matter or energy is the result of the vibration of strings. • String theory ran into a problem: another version of the equations was discovered, then another, and then another. Eventually, five major string theories were developed. The main differences between the theories were principally the number of dimensions in which the strings developed, and their characteristics (some were open loops, some were closed loops, etc.). • Furthermore, all these theories appeared to be workable. Scientists were not comfortable with five seemingly contradictory sets of equations to describe the same thing. Dr Ray Adams 23/02/2017 71 M-theory • In 1994, Edward Witten of the Institute for Advanced Study and other researchers suggested that the five different versions of string theory might be describing the same thing seen from different perspectives. • They proposed a unifying theory called "M-theory", in which the "M" is not specifically defined but is generally understood to stand for "membrane". • The words "matrix", "master", "mother", "monster", "mystery" and "magic" have also been claimed. M-theory brought all of the string theories together. • It did this by asserting that strings are really one-dimensional slices of a twodimensional membrane vibrating in 11-dimensional space. • These membranes are called Dirichlet branes or D – branes. Dr Ray Adams 23/02/2017 72 M-theory • In 1994, Edward Witten of the Institute for Advanced Study and other researchers suggested that the five different versions of string theory might be describing the same thing seen from different perspectives. • They proposed a unifying theory called "M-theory” that brought all of the string theories together. • Andre Miemiec and Igor Schnakenburg (2005) argue that :The etymology of the name “M-Theory” is explained in and traced back to Membranes. Nowadays the “M” is thought to refer to the word Mother due to the pivotal role it seems to claim in the unification of string theories. • M theory combined the five string theories by asserting that strings are really one-dimensional slices of a two-dimensional membrane vibrating in 11dimensional space. Dr Ray Adams 23/02/2017 73 M Theory and Superstring Theories A schematic illustration of the relationship between M-theory, the five superstring theories, and elevendimensional supergravity. The shaded region represents a family of different physical scenarios that are possible in M-theory. In certain limiting cases corresponding to the cusps, it is natural to describe the physics using one of the six theories labeled there. Dr Ray Adams 23/02/2017 74 The benefits of M Theory Recently there has been much progress in understanding a more fundamental description of the theory that has become known as M-theory. M-theory seem to be the most complex and richest mathematical object so far in physics. It seems to unify three great ideas of twentieth century theoretical physics: 1) General relativity – the idea that gravity can be described by the Riemannian geometry of space-time. 2) Gauge theory – the description of forces between elementary particles using connections on vector bundles. In mathematics this involves Ktheory and index theorems. 3) Strings, or more generally extended objects (branes), as a natural generalization of point particles. Mathematically this means that we study spaces primarily through their (quantized) loop spaces. Dr Ray Adams 23/02/2017 75 A Unified Theory? More speculatively, M-theory may provide a framework for developing a unified theory of all of the fundamental forces of nature. Attempts to connect M-theory to experiment typically focus on compactifying its extra dimensions to construct candidate models of our four-dimensional world, although so far none have been verified to give rise to physics as observed at, for instance, the Large Hadron Collider. 23/02/2017 Dr Ray Adams 76 A Unified Theory? M-theory will be considered as a two parameter family of deformations of “classical” Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric. 23/02/2017 Dr Ray Adams 77 M theory First, in perturbative string theory we study the loops in a space-time manifold. These loops can be thought to have an intrinsic length , the string length. At least at an heuristic level it is clear that in the limit the string degenerates to a point, a constant loop. The parameter ls controls the “stringiness” of the model. We will see how the quantity plays the role of Planck’s constant on the worldsheet of the string. That is, it controls the quantum correction of the twodimensional field theory on the world-sheet of the string. An important example of a stringy deformation is quantum cohomology, which is a type of algebra that quantizes topological spaces (manifolds).(This is a simplification by me.) 23/02/2017 Dr Ray Adams 78 M theory Second, strings can split and join, sweeping out a surface Σ of general topology in space-time. According to the general rules of quantum mechanics we have to include a sum over all topologies. Such a sum over topologies can be regulated if We can introduce a formal parameter λ ∈ R+, the string coupling, such that a surface of genus g gets weighted by a factor λ2g−2. Higher genus topologies can be interpreted as virtual processes wherein strings split and join —a typical quantum phenomenon. Therefore the parameter λ controls the quantum corrections. In fact we can equate λ2 with Planck’s constant in spacetime. Only for small values of λ can string theory be described in terms of loop spaces and sums over surfaces. 23/02/2017 Dr Ray Adams 79 M theory In fact, in the case of particles we know that for large values of λ it is better to think in terms of waves, or more precisely quantum fields. So we expect that for large λ and α the right framework is string field theory. This is partly true, but it is in general difficult to analyse this string field theory directly. In particular the occurrence of branes, higher-dimensional extended objects that will play an important role in the subsequent, is obscure. 23/02/2017 Dr Ray Adams 80 M theory General Relativity can be derived from M theory, though that derivation is too big to be developed in this presentation. M theory is not just about strings but the higher dimensional objects, called branes. Here is one way in which a closed string (graviton) can interact with a D brane. 23/02/2017 Dr Ray Adams 81 D-brane The interesting property of D-brane is that the segments of string on the brane behave just like elementary particles. The only thing missing on the D-brane is gravity. That's because the graviton is a closed string - a string with no ends would not be stuck to the brane at all. Instead, they can travel freely through all space. They can interact with other strings by moving in and out of the brane. According to string theorists we are most likely living in a D brane with six dimensions tightly rolled up. Such configuration would prevent gravity from spreading out too much. 23/02/2017 Dr Ray Adams 82 D branes • They are also objects that possess finite tension and carry charges. • Thus, they can be distorted and can interact with other charged objects and gravitational field. • They can move, collide, annihilate, and even form systems of branes orbiting around one another. • On the other hand, brane can provide an environment for the strings to play their roles. 23/02/2017 Dr Ray Adams 83 D branes • Since the bosons are also open strings, they would communicate a force that would act on the other brane-bound open strings with charge. • The photon being one of the bosons is also trapped within the D3 brane; thus guarantees that the principle of special relativity (about the constant speed of light in our three dimensional space) is not violated. • In short, the D brane would contain all the particles and forces in the Standard Model. From the perspective of brane-bound particles, if it weren't for gravity or other bulk particles with which they might interact, the world might as well have only the dimensions of the branes. 23/02/2017 Dr Ray Adams 84 D branes • Since the bosons are also open strings, they would communicate a force that would act on the other brane-bound open strings with charge. • The photon being one of the bosons is also trapped within the D3 brane; thus guarantees that the principle of special relativity (about the constant speed of light in our three dimensional space) is not violated. • In short, the D brane would contain all the particles and forces in the Standard Model. From the perspective of brane-bound particles, if it weren't for gravity or other bulk particles with which they might interact, the world might as well have only the dimensions of the branes. 23/02/2017 Dr Ray Adams 85 Quantum Mechanics and Particles 23/02/2017 Dr Ray Adams 86 Scientists find a practical test for string theory January 6, 2014, https://phys.org/news/2014-01-scientists-theory.html Scientists at Towson University in Towson, Maryland, have identified a practical, yet overlooked, test of string theory based on the motions of planets, moons and asteroids https://phys.org/news/2014-01-scientists-theory.html#jCp String theory is infamous as an eloquent theoretical framework to understand all forces in the universe —- a so-called "theory of everything" —- that can't be tested with current instrumentation because the energy level and size scale to see the effects of string theory are too extreme. Towson University scientists say that precise measurements of the positions of solarsystem bodies could reveal very slight discrepancies in what is predicted by the theory of general relativity and the equivalence principle, or establish new upper limits for measuring the effects of string theory. At the moment, this is work in progress. 23/02/2017 Dr Ray Adams 87 Some Definitions • Supersymmetry: Two basic classes of elementary particles: bosons and fermions are related by supersymmetry. Each particle from one group is associated with a particle from the other, known as its superpartner. • Chirality: a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral. • Bosons: Bosons are force carrier particles. They have integral spins and do not obey the Pauli exclusion principle, so that any number of identical particles may occupy the same quantum state. 23/02/2017 Dr Ray Adams 88