tag:blogger.com,1999:blog-23649030731692921342014-10-04T23:30:46.032-07:00Introduction to String TheoryThese lectures are part of a ten-hour course given during the Interterm at Smith College. They are designed to provide an introduction to some of the basic ideas involved in string theory. The course has no pre-requisites and assumes only curiosity and an open mind.
One of the primary sources of these lectures is Lisa Randall's book "Warped Passages," which I recommend to anyone wishing to go beyond what is presented here.sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-2364903073169292134.post-62850276385126419172010-03-30T20:44:00.000-07:002010-03-30T20:47:13.792-07:00The LHCToday, for the first time, two protons collided at 7 TeV in the Large Hadron Collider. Among other things, the Large Hadron Collider may detect supersymmetric particles. <br /><br /><a href="http://www.symmetrymagazine.org/breaking/2010/03/30/lhc-research-program-launched-with-7-tev-collisions/">http://www.symmetrymagazine.org/breaking/2010/03/30/lhc-research-program-launched-with-7-tev-collisions/ </a> !!sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0tag:blogger.com,1999:blog-2364903073169292134.post-87568330912325121102010-01-18T19:06:00.004-08:002010-01-19T13:52:49.135-08:00Lecture 1<div>The primary goal of physics is to determine the nature of the universe: its size, its structure, its beginning. The observable universe is vast and incredible, and certainly there is much more to it than we can observe.<br /><br />A main idea in physics is that elementary particles constitute the building blocks of matter.<br /><br />The goal of particle physics is to discover matter's most basic constituents and the most fundamental physical laws obeyed by those constituents.<br /><br />In a way, understanding nature at this level would mean understanding everything.<br /><br />I - What are the fundamental particles?<br />Every time more accurate technological tools are developed, more elementary constituents have emerged.<br /><div id="b2.q" style="text-align: left;"><img style="width: 583px; height: 231px;" src="https://docs.google.com/File?id=dfkz4s8v_24g8bbbnxh_b" /></div>This is as far as we have been able to break down matter so far.<br />Everything around you: the blackboard, the windows, your shoelaces, etc. is made up of electrons and up/down quarks.<br /><br />The Standard Model (well established theory of particle physics) includes many other fundamental particles.<br />There are additional heavier quarks and heavier election-like particles that are nowhere to be found in ordinary matter. These particles have been discovered in high energy particle colliders that simulate the early universe (immediately after the big bang). The most powerful one, which many of you have probably heard of, is the Large Hadron Collider in Geneva.<br /><br />THE STANDARD MODEL<br />Matter Particles<br /><div id="bmmm" style="text-align: left;"><img style="width: 593px; height: 114px;" src="https://docs.google.com/File?id=dfkz4s8v_26r854x2hb_b" /></div><br />Force-carrying particles<br /><div id="wxyq" style="text-align: left;"> <img style="width: 514px; height: 64px;" src="https://docs.google.com/File?id=dfkz4s8v_27gq9b8br3_b" /></div><br />The idea of force carrying particles comes out of quantum physics, which we will go over briefly later in this lecture. Notice that there is no particle associated with the force of gravity.<br /><br />Questions raised by the Standard Model:<br /><ul><li>Why heavy particles?</li><li>Why such different masses?</li></ul>It is very possible that matter may be broken down further; string theory postulates that there is a more fundamental constituent of matter: an elementary string whose vibrations determine the known particles.<br /><br /><a title="http://www.youtube.com/watch?v=7zR1bWQFyWM" href="http://www.youtube.com/watch?v=7zR1bWQFyWM" id="zy-l">http://www.youtube.com/watch?v=7zR1bWQFyWM</a><br /><br />II- The fundamental forces<br />It has been the belief of physicists for some time that all known interactions can be reduced to combinations of four fundamental forces:<br /><br />1. Gravity<br /><ul><li>A force between objects with mass that depends on the mass of the objects and the distance between those objects</li></ul>2. Electromagnetic<br /><ul><li>a unification of the electric and magnetic forces </li><li>Electric charges attract or repel one another with a force inversely proportional to the square of the distance between them: unlike charges attract, like ones repel. An electric current in a wire creates a circular magnetic field around the wire, its direction depending on that of the current.</li><li>holds electrons and protons together in atoms, and holds atoms together to make molecules.</li><li>The electromagnetic force is the one responsible for practically all the phenomena one encounters in daily life</li><li>all the forces involved in interactions between atoms can be traced to the electromagnetic force acting on the electrically charged protons and electrons inside the atoms</li></ul></div><div>3. The Weak Force<br /><ul><li>The weak force is indeed weak; well actually it's about 10^25 times stronger than gravity but it only holds on very very small scales, so it doesn't affect most matter.</li><li>Explains some forms of nuclear decay, and is essential to many other nuclear processes </li><li>plays a role in the creation of heavy elements</li><li>is essential for stars to shine: kicks off the chain of reactions that convert hydrogen to helium<br /></li></ul>4. The Strong Force<br /><ul><li>extremely powerful force</li><li>so strong that particles that experience the strong force never exist in isolation: quarks are always bound together in threes.<br /></li></ul><br /><div id="sepr" style="text-align: left;"><img style="width: 618px; height: 151px;" src="https://docs.google.com/File?id=dfkz4s8v_259nzf6xch_b" /><br /><br /></div>Notice how much weaker gravity is than the other three forces. This fact is something that intrigues physicists and may come out of string theory.<br /><br />Questions raised by the Standard Model:<br /><ul><li>Why FOUR forces? Are there others?</li><li>Why is gravity so much weaker than the other three forces?<br /></li></ul>III- What do we know about how these forces work?<br />A- Quantum Physics<br />Quantum physics comes out of several theories:<br /><ul><li>quantization</li><li>wavefunction</li><li>wave-particle duality</li><li>uncertainty principle<br /></li></ul>look out for these in the following video:<br /><a title="http://www.youtube.com/watch?v=45KGS1Ro-sc" href="http://www.youtube.com/watch?v=45KGS1Ro-sc" id="j5qj">http://www.youtube.com/watch?v=45KGS1Ro-sc</a><br /><br />Quantum mechanics is very bizarre and unintuitive, mainly because quantum effects generally aren't significant at distances greater than the size of an atom. We can generally only see matter at larger scales than this, and further, particles are not isolated, forbidding many quantum effects to be noticed.<br />When large objects are involved, quantum mechanics agrees with classical physics, but classical physics will never generate quantum predictions.<br /><br />It turns out that all matter consists of fundamental quanta.<br />Quanta are the basis of particle physics.<br /><br />B- Relativity(Theory of Gravity)<br />Newton's laws allow accurate enough predictions to send men to the moon, satellites into orbit, and was responsible for the discovery of Neptune.<br />But, Newton's laws are not accurate enough for a functioning GPS system, which relies on General Relativity! Otherwise the system would accumulate errors of more than 10km/day.<br />(Einstein's theory is proof that abstract mathematical ideas can have incredibly practical applications)<br />Newton's theory fails for high speed/mass/energy<br /><br />Newton's law of gravity vs. Relativity<br /><a title="http://www.youtube.com/watch?v=O-p8yZYxNGc&feature=related" href="http://www.youtube.com/watch?v=O-p8yZYxNGc&feature=related" id="byif">http://www.youtube.com/watch?v=O-p8yZYxNGc&feature=related</a><br /><br />Newton's law is not wrong, it is an approximation; it's still used often.<br /><br />Special Relativity - comes from two assumptions<br /><ol><li>Physical laws are the same for all observers (as long as one is not accelerating)</li><li>The speed of light is a constant</li></ol>The theory is described in 4 dimensions - 1 time dimension and 3 spatial dimensions, although relativity works equally well in 3,4,or 10 dimensions.<br />These 4 dimensions make up a "spacetime fabric" which is curved or "warped" by large masses.<br />For distances to be equal in spacetime, the more one travels in space, the less one travels in time.<br /><a title="http://www.youtube.com/watch?v=NyR7s4BsGZw&feature=related" href="http://www.youtube.com/watch?v=NyR7s4BsGZw&feature=related" id="uu95">http://www.youtube.com/watch?v=NyR7s4BsGZw&feature=related</a><br /><br />Newton's theory suggests that to calculate the speed of light on a moving train, we would add the two speeds which violates (2.).<br /><br />Newton's theory is intuitive while many of relativity's implications are unintuitive;<br />for example, moving and stationary clocks tick at different rates.<br /><br />videos: simultaneity<br /><a title="http://www.youtube.com/watch?v=wteiuxyqtoM&feature=related" href="http://www.youtube.com/watch?v=wteiuxyqtoM&feature=related" id="jx8m">http://www.youtube.com/watch?v=wteiuxyqtoM&feature=related</a><br /><br />General Relativity -- involves Gravity; relativity with acceleration<br /><ul><li>the effects of acceleration cannot be distinguished from gravity</li><li>the theory predicted how much light should bend because of the sun's influence</li><li>matter tells spacetime how to curve and spacetime tells matter how to move<br /></li></ul>Evolution in Theory of Gravity:<br />Newton formulated an equation giving the strength of gravity<br />Einstein showed how gravity works<br />String theory may answer the question of "why gravity?"<br /><br />IV- Unification<br />Quantum mechanics allows the unification of all four fundamental forces EXCEPT gravity.<br />Using the standard model, there seems to be no way to unite all four forces.<br />String theory has the potential to do so.<br /><br />V- Introduction to string theory<br /><a title="http://www.youtube.com/watch?v=_B0Kaf7xYMk" href="http://www.youtube.com/watch?v=_B0Kaf7xYMk" id="c9jt">http://www.youtube.com/watch?v=_B0Kaf7xYMk</a><br /><br /><br />Primary source: Warped Passages by Lisa Randall<br /></div>sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0tag:blogger.com,1999:blog-2364903073169292134.post-39757977936233730222010-01-18T19:06:00.003-08:002010-01-20T09:56:43.114-08:00Lecture 2Questions from last time:<br />1. How do particle accelerators create massive particles from small, naturally occurring particles?<br />Every particle has an associated antiparticle with opposite charge. When particles and antiparticles meet, the charge adds up to zero and the mass can be converted to energy (E = mc^2). Likewise, energy can be converted to mass. In a high-energy particle collider, the accelerated beam of particles collides with an accelerated beam of antiparticles creating a huge amount of energy. In particle colliders, the energy created by these collisions allows new particle-antiparticle pairs to form. This energy is sometimes converted into mass in the form of heavy particles that have not occured naturally since the very beginning of the universe.<br /><br />2. Why strings?<br />The idea came out of work by Leonard Susskind:<br /><a title="http://www.youtube.com/watch?v=E7FV9aaiwKQ" href="http://www.youtube.com/watch?v=E7FV9aaiwKQ" id="q5rt">http://www.youtube.com/watch?v=E7FV9aaiwKQ</a><br /><br />-----------<br />A BIT OF REVIEW<br /><br />The two main goals of string theory are to unify the fundamental forces, and to find the most elementary form of matter.<br />The idea that all of matter could be made up of vibrating strings is elegant because a list of fundamental particles is replaced by a string, and a vibrating string is dynamic enough to allow many possible forms.<br /><br />By quantum mechanics, every force except gravity is given by a particle:<br /><a title="http://www.youtube.com/watch?v=WT_IVdCO4ZE&NR=1" href="http://www.youtube.com/watch?v=WT_IVdCO4ZE&NR=1" id="iqsz">http://www.youtube.com/watch?v=WT_IVdCO4ZE&NR=1</a><br /><br />The next video explains how string theory describes gravity at the quantum level and hence why string theory allows a "<a title="theory of everything" href="http://en.wikipedia.org/wiki/Theory_of_everything" id="x956">theory of everything</a>" and could in fact unify the forces by including a particle corresponding to gravity: the graviton.<br /><a title="http://www.youtube.com/watch?v=g7bakDI_TwA&feature=related" href="http://www.youtube.com/watch?v=g7bakDI_TwA&feature=related" id="p4tr">http://www.youtube.com/watch?v=g7bakDI_TwA&feature=related</a><br /><br />Recall: We will never be able to "see" a string. There is currently no way to PROVE string theory, although we may be able to find supporting evidence in the near future (possibly by using the LHC)<br />There are currently 5 versions of string theory and we don't know which one is right. It's possible that they all are, or that none of them are.<br /><br />So why the big deal about String theory?<br />Here are some of the problems string theory addresses/may potentially solve. Some of these we have already discussed.<br /><ul><li>quantum physics & general relativity, two theories accepted by modern physics, are incompatible at small scales and high energies. In particular, they allow no insight into what happened at the time of the big bang.<br /></li><li>why is gravity so weak compared to the other forces?<br /></li><li>why is gravity so different than the other forces? As it stands, we can unite all of the forces except for gravity by using quantum physics. It has long been a goal to unite all four of the fundamental forces: gravity, electromagnetic, weak and strong (say what each is, try to describe a little bit of how they're united).</li><li>what is the smallest unit? Throughout history, scientists have been chipping away at the smallest units of matter- from atoms to electrons to quarks (more detail) String theory provides a smallest unit- the String</li><li>how did the universe begin? </li><li>Is it possible that other universes may exist?<br /></li></ul> We will address all of these questions (or as many as we have time for) in this course.<br /><br />Goal of physics to move towards simplicity; many people perceive simplicity as beauty.<br /><br />----------------<br /><br />EXTRA DIMENSIONS<br /><br />I- String theory works best in 10 or 11 dimensions<br /><a title="http://www.youtube.com/watch?v=ehVpYqTuJLw&feature=related" href="http://www.youtube.com/watch?v=ehVpYqTuJLw&feature=related" id="gylp">http://www.youtube.com/watch?v=ehVpYqTuJLw&feature=related</a><br /><br />String theory makes the most sense in an extra dimensional setting; i.e. more than the 3+1 that we are aware of. For a long time, string theory was dismissed for exactly this reason. However, even though we cannot perceive more than 3+1 dimensions, there is no reason there couldn't be additional dimensions.<br /><br />No physical theory (we know of) dictates that there should only three dimensions of space.<br />Although we only perceive 3-dimensions, extra spatial dimensions are a logical possibility.<br />Einstein's theory of relativity still holds in 10 and 11 dimensions.<br /><br />There are currently 5 distinct theories of string theory. Each is similar in that it involves strings vibrating in 10 dimensions, but the mathematics of each theory is somewhat different. In 1995 Ed Witten developed a single theory, M-theory, involving 11 dimensions, to replace the existing 5-theories.<br />Not shown in class: <a title="http://www.youtube.com/watch?v=6PVjNlXj2WQ&NR=1" href="http://www.youtube.com/watch?v=6PVjNlXj2WQ&NR=1" id="m6dr">http://www.youtube.com/watch?v=6PVjNlXj2WQ&NR=1</a><br /><br />So how could it be that there are 10 or even 11 dimensions? And what are extra dimensions?<br /><br />II-There are multiple ways physicists explain extra dimensions that are "hidden" from view.<br />A- Small dimensions<br />B- Small curled up dimensions<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_-UlA_Aa6k5o/S1al7vXaVQI/AAAAAAAAAEw/LtSnYD9o5jA/s1600-h/curledup.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 258px;" src="http://1.bp.blogspot.com/_-UlA_Aa6k5o/S1al7vXaVQI/AAAAAAAAAEw/LtSnYD9o5jA/s400/curledup.gif" alt="" id="BLOGGER_PHOTO_ID_5428708846587237634" border="0" /></a><br /><img src="file:///C:/DOCUME%7E1/Sylvia/LOCALS%7E1/Temp/moz-screenshot-1.jpg" alt="" /><img src="file:///C:/DOCUME%7E1/Sylvia/LOCALS%7E1/Temp/moz-screenshot-2.jpg" alt="" /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_-UlA_Aa6k5o/S1ameDcz1wI/AAAAAAAAAE4/mrKQffXYuYk/s1600-h/curledup6.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 400px;" src="http://4.bp.blogspot.com/_-UlA_Aa6k5o/S1ameDcz1wI/AAAAAAAAAE4/mrKQffXYuYk/s400/curledup6.gif" alt="" id="BLOGGER_PHOTO_ID_5428709436094142210" border="0" /></a>C- The possibility that we are trapped on a "brane." An analogy is the way shower droplets are confined to the curtain. Branes make far larger extra dimensions possible.<br />(We will discuss branes more tomorrow)<br />D- We may be in a 3-dimensional "pocket" of space.<br /><br /><br />III- What can extra dimensions add to our current theories?<br />Extra dimensions could ultimately reveal connections that we miss in 3-dimensional space.<br />For example, consider viewing your hand in only two dimensions. You would see 2-d dimensional cross sections. Maybe 5 circles corresponding to your fingers, or to your knuckles; in 3-dimensions we see the entire picture and the way that these parts are connected.<br />Another example is quasicrystals- used on nonstick frying pans. Generally, a crystal is a highly symmetric structure of atoms and molecules with a basic form repeated. Quasicrystals lack the regularity you would see in a regular crystal, however, if we consider the pattern in five dimensions, there is such an ordered structure.<br /><br /><br />IV- How can we imagine these extra dimensions?<br />Thinking about extra dimensions isn't too hard, it's trying to imagine them that's hard.<br />For example, just as we can describe a point in three coodinates (0,0,1), we can describe a point in 5 coordinates (1,0,0,1,1), but it would be substantially more difficult to plot.<br /><br /><br />A- Flatland<br />Flatland is a novel narrated by a character named "A. Square" who presents to the reader life in a two dimensional world. The story is designed to provoke thought concerning dimensions other than the ones with which we are familiar, and A. Square himself is forced to do the same. First when he visits a one dimensional world in a dream, and then when he is visited by a three dimensional sphere who appears to him as a circle increasing and then decreasing in size.<br /><a title="http://www.youtube.com/watch?v=E8ONUUlqqhs&feature=related" href="http://www.youtube.com/watch?v=E8ONUUlqqhs&feature=related" id="bfib">http://www.youtube.com/watch?v=E8ONUUlqqhs&feature=related</a><br /><br />Ask the class to break into groups for 10 minutes and discuss:<br />How would a 4-d sphere appear to us?<br />How about a 4-dimensional being?<br />What could a 4-dimensional being do that we couldn't? examples: perform surgery without making an incision<br />How would a 4-dimensional being see us?<br /><br />Here is what a 4-dimensional cube looks like:<br /><a title="http://www.youtube.com/watch?v=iXYXuHVTS_k&feature=related" href="http://www.youtube.com/watch?v=iXYXuHVTS_k&feature=related" id="d3ds">http://www.youtube.com/watch?v=iXYXuHVTS_k&feature=related</a><br /><br />B- Building up the dimensions<br />A line segment consists of two points connected by <b>one</b> one-dimensional line (1d)<br />By connecting two such line segments with <b>two</b> additional line segments, we have a square (2d)<br />By placing one square above the other and connecting them with <b>four</b> additional squares, we have a cube (3d)<br />By placing one cube above another and connecting them by adding <b>six</b> additional cubes, we have a hypercube(4d)<br />and so on...<br /><br />C- Imagining the Tenth Dimension<br /><a title="http://www.youtube.com/watch?v=NNvHdCosX08&feature=related" href="http://www.youtube.com/watch?v=NNvHdCosX08&feature=related" id="a08x">http://www.youtube.com/watch?v=NNvHdCosX08&feature=related</a><br /><br /><br />V- Can we verify extra dimensions?<br />The LHC could detect evidence of extra dimensions by finding particles called Kaluza-Klein modes which travel in extra dimensions but would leave traces in our 3-dimensional world.<br />Evidence for extra dimensions will likely be indirect; we will need to piece it together.<br /><br />VI- What do extra dimensions imply?<br />Extra dimensions permit theories of parallel universes, warped geometry, the multiverse...<br />We will be discussing some of these tomorrow.<br /><br /><br /><br />Additional Videos:<br />TED talk by Brian Greene introducing string theory: <a title="http://www.youtube.com/watch?v=YtdE662eY_M" href="http://www.youtube.com/watch?v=YtdE662eY_M" id="celu">http://www.youtube.com/watch?v=YtdE662eY_M</a><br />Physicist Brian Cox discusses the Large Hardron particle Collider: <a title="http://www.youtube.com/watch?v=_6uKZWnJLCM&feature=channel" href="http://www.youtube.com/watch?v=_6uKZWnJLCM&feature=channel" id="d:yp">http://www.youtube.com/watch?v=_6uKZWnJLCM&feature=channel</a><br />Flatland explanation: <a title="http://www.youtube.com/watch?v=BWyTxCsIXE4" href="http://www.youtube.com/watch?v=BWyTxCsIXE4" id="ubqj">http://www.youtube.com/watch?v=BWyTxCsIXE4</a>sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0tag:blogger.com,1999:blog-2364903073169292134.post-66957289810693505882010-01-18T19:06:00.001-08:002010-01-21T13:11:52.819-08:00Lecture 3Lecture 3: Extra Dimensions, Branes, and Parallel Universes<br /><br />I- Introduction to Branes<br />A- History of Branes<br />A brane is a domain that has fewer dimensions than the higher-dimensional space that surrounds or borders it.<br /><br />The idea of branes came before string theory. Physicists derived similar objects that extend infinitely far in only some directions using Einstein's theory of relativity.<br />Particle physicists suggested brane-like surfaces that would trap particles.<br />String theory branes were the first branes proposed that could also trap forces.<br /><br />Branes imply that even though our universe may have more than 3 spatial dimensions, particles and forces are trapped on (or confined to) such lower dimensional surfaces.<br /><br />B- What do Branes imply?<br />Extra small dimensions aren't necessarily curled up, they could be intervals: bounded between two "walls"<br /><br />Branes can be inside a space or at a boundary but either way they will trap the particles and forces along them.<br /><br />Branes could have any number of dimensions, as long as the number is less than the number of dimensions in the full higher-dimensional space of which they are a part.<br /><br />Consider light particles confined to a 3-dimensional brane: light rays would spread out only along the brane; that is, light would behave as it would in a three-dimensional universe, making it impossible to distinguish the number of true dimensions of space.<br /><br />Anything confined to the 3-d brane would appear to be 3-dimensional.<br /><br /><br />C- Branes and forces<br />Although matter and forces could be stuck on the brane, gravity wouldn't be.<br />By general relativity, gravity is woven into the framework of space and time so gravity holds in all dimensions of space and time.<br /><br />In fact, gravity would be the only force that could communicate between the world on the brane and the higher-dimensional world beyond the brane.<br /><br />Are there other particles/forces not confined to the brane? Possibly.<br /><br /><div id="g9wi" style="text-align: left;"><img src="https://docs.google.com/File?id=dfkz4s8v_31gs34fmdm_b" height="257" width="376" /></div>(From Warped Passages by Lisa Randall. Sorry for the image quality.)<br /><br /><br />II- The Multiverse<br />A multiverse describes a space with more than one brane, that is, multiple universes.<br />It's possible that the universe contains multiple branes that only interact via gravity, or don't interact at all (if the branes were too far apart to ever communicate with each other).<br /><div id="p4m0" style="text-align: left;"><img style="width: 415px; height: 259px;" src="https://docs.google.com/File?id=dfkz4s8v_33fb5dsjds_b" /><br />(Warped Passages)<br /></div><br />Particles on distinct branes could be ENTIRELY different than the ones we know; these universes could be made up of a completely different set of forces and set of matter particles.<br /><br />Branes could be parallel or intersect. If they intersected, particles could be shared between intersecting branes.<br />The only force that would necessarily be shared is gravity, although the strength of gravity could vary, that is, the gravitational constant could be different.<br /><br /><br />III- Brane Theory<br />An interesting feature of branes is that even though physics is set up so that the laws should be the same at any point in space, branes don't respect these symmetries.<br /><br />We will not go into this but branes actually play a huge role in string theory and are just as important as the strings!<br />In fact, in addition to fundamental strings, some force particles may be a result of strings attached to branes; that is, it is not strings alone that make up everything.<br /><br />It was branes that allowed the different versions of string theory to become a single inclusive theory (M-theory).<br /><br />IV- Gravity and Branes<br />A- Open and closed strings<br />If strings are like Leonard Susskind described them (open), there should exist closed strings; the strings should be able to close into loops.<br />Closed loops correspond to the graviton (the particle associated with gravity).<br />This is what singles out gravity as a force: since the strings are closed, they are not confined to branes.<br />Therefore gravitons can travel throughout the entire space while all other force particles are confined to the brane.<br /><br />B- Why is Gravity so weak?<br />The fact that gravity is distributed throughout 9 or 10 spatial dimensions whereas all of the other forces may only interact on a 3 dimensional brane could account for gravity's being so much weaker than the other forces.<br /><br />Another possibility is warped Geometry:<br />In relativity, space and time are warped by matter and energy, Randall and Raman applied this idea in an extra-dimensional context and found a configuration in which spacetime warps so severely that gravity could be strong in one region of space and weak elsewhere.<br />Warped geometry also allows an invisible extra dimension infinite in size that could be hidden by the distortions in space.<br /><br /><br />V- Parallel Universes<br /><a title="The Universe: Parallel Universes" href="http://www.megavideo.com/?v=RSDUCP48" id="aqo3">The Universe: Parallel Universes</a><br />(44 minutes)<br /><br />VI- Different sizes of infinity<br />In the parallel universes video, the claim is that since there may be an infinite number of universes, any possible universe one could imagine would necessarily exist. We could imagine a universe where Bush had never been president; where some world wars had never taken place, but other ones had; one in which your hero or icon was your best friend; one in which your dreams were a reality; and so on.<br />While there may be an infinite number of parallel universes, and in fact, Brane worlds allow such a possibility, it is mathematically sloppy to say that this implies that any conceived universe can and must exist. The reason for this is that there are different sizes of infinity. That is, some infinities are larger than other infinities.<br /><br />A- How do we measure infinity?<br />Mathematicians measure the size of any set of elements with something called <span style="font-style: italic;">cardinality</span>.<br /><br />A set is a collection of elements. Sets can be finite or infinite in size and can include numbers, colors, shapes, objections, operations, you name it.<br />For example the set {1, 2, 3} has three elements 1, 2, and 3. This notation is pretty standard. Likewise, the set {x, y, z} has three elements. Meanwhile, the set {x, y, z, w} has four elements. If we want to compare sets, we can consider a map from one set to another; that is, we can associate each element in one set with a single element of another set.<br /><br />Let's take the sets A= {1, 2, 3} and B= {x, y, z} and say 1 --> x, 2 --> z, 3--> y. Every element in set A is mapped to a unique element in B, so our map is valid. But there is more going on. Notice that every element in the set B is mapped to. That is, for each element in B: x, y, and z, there is some element that maps to it. In math, such a map is called "onto" or "sujective." Note also that no two elements in A map to the same element in B. If 1 and 2 both mapped to x our map would still be valid, but, in this case, they don't. We call such a map "one-to-one" or "injective", meaning that each element maps to a unique element. If a map satisfies both of these properties (it is injective and surjective), we say it is bijective, and consequently the sets are the same size! This is certainly true of our example: both sets have three elements.<br /><br />Let's consider two infinite sets and compare their sizes.<br />Let N = {1, 2, 3, 4, ...}, that is every positive whole number, and let<br />E = {2, 4, 6, 8, ...}, or every positive even number.<br /><br />Are these sets the same size?<br /><br />We let x denote an arbitrary element in the set N. Our map will be given by the function f(x) = 2x. To find out whether the sets are the same size, let's check if the map is injective and surjective. If it's not, maybe there's another map that is. If it is, then we can stop there: our sets are the same size.<br /><br />First we check if the function is onto (surjective). Can we find an element in E that is not mapped to; that is, is there a positive even number for which there is no whole number that can be multiplied to two to obtain that number. Try to think of one!<br />14? Nope, 7 maps to 14.<br />1028? Nope, 514 maps to 1028.<br />There is in fact no number in E that is not twice some number in N. Try to think of how you might verify this. A trick is to think of a function that maps from E to N.<br /><br />Next, let's check to see that our function is one-to-one (injective). Are there any two numbers in the set N that map to the same number in E? Let's suppose there are. We'll call them x, and y, and we'll suppose that they both map to an element k in E. Then we have 2x = k = 2y. Division by 2 tells us that x = y, so in fact these must be the same number. The function is injective!<br /><br />This means that even though N seems like it might be a bigger set, these sets are actually the same size.<br /><br />But what about another infinite set, such as the real numbers? The real numbers, R, is the set of all positive numbers, negative numbers, whole numbers, fractions, and irrational numbers (numbers that can't be expressed as fractions such as pi (3.1415...)). There is no bijective map from N to R because there cannot be a map that is onto. For any map, we can always find a real number that is not mapped to. These infinite sets are not the same size!<br /><br />B- What does this imply about parallel universes?<br />We can see that since there are different sizes of infinity, there is not always a one to one correspondence between elements in infinite sets.<br />If we let<br />P = {all possible universes}, and<br />A = {all universes that have ever existed or will ever exist},<br /><br />the fact that both of these sets are infinite does not imply that any universe we can conceive of must exist. Although, it is possible that we can think of an infinite number of universes that do exist. Some possible universes must exist, in fact an infinite number of possible universes must exist. But we cannot be sure that all possible universes must exist. If you can think of a possible universe. Maybe one in which you are reading this with four eyes, there is no way to be sure whether this universe could or could not exist.<br /><br /><br /><br />A video on branes (not shown in class): <a title="http://www.youtube.com/watch?v=ZuK9Rb-tBBg" href="http://www.youtube.com/watch?v=ZuK9Rb-tBBg" id="mn14">http://www.youtube.com/watch?v=ZuK9Rb-tBBg</a><br />Additional sources: <a title="http://www.edge.org/3rd_culture/randall03/randall03_print.html" href="http://www.edge.org/3rd_culture/randall03/randall03_print.html" id="d9jk">http://www.edge.org/3rd_culture/randall03/randall03_print.html</a><br />For more detail, I highly recommend Lisa Randall's book Warped Passages.sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0tag:blogger.com,1999:blog-2364903073169292134.post-9599356226121776012010-01-18T19:05:00.001-08:002010-01-22T09:51:19.399-08:00Lecture 4Today's lecture is in two parts: First we will take a closer look at relativity, and then we will discuss a theory called supersymmetry which is an integral part of string theory.<br /><br />Part One: Relativity<br />I - Special Relativity<br />A- What is relativity?<br />Galileo's principle of Relativity was that two observers moving uniformly relative to one another must formulate the laws of nature in exactly the same way.<br />No observer can distinguish between absolute rest and absolute motion by appealing to any law of nature;<br />there is no such thing as absolute motion, but only relative motion (of one observer with respect to another).<br /><br />Examples:<br />Imagine two people in space floating towards each other. Who is moving? It could be that both are moving, or that only one is moving? Each person may think she is stationary while the other is moving. In fact, there is no objective truth, other than that they are both moving relative to one another.<br /><br />Consider yourself in a speeding car. You are going somewhere around 50 mph. And so is the t-shirt you are wearing. But relative to you, the t-shirt is not in fact moving, and this is what you experience.<br /><br />B- Einstein's Relativity<br />Einstein made relativity a universal principle that applies to all physical phenomena: electricity, magnetism, light, etc.<br /><br />From this assumption, and the assumption that the speed of light is a constant, Einstein derived his special theory of relativity: a description of space and time as a fabric of interwoven dimensions, a description of events with respect to observers.<br /><br />Special relativity describes linear motion. That means anything that is a constant speed. A change in speed or direction corresponds to an acceleration and is within the realm of general relativity.<br />It is general relativity therefore that describes gravity, which cannot be distinguished from acceleration. <br /><br />C- Spacetime<br />Usually we consider an event with respect to time. One could plot her height on the x-axis, and time on the y-axis. Or one could plot spatial position with respect to time, possibly choosing to include all three spatial coordinates, thus giving four coordinates (t, x, y, z).<br />Since we can't visualize a 4-dimensional space, it's typical to consider a 2-dimensional spacetime (t,z).<br /><br />The speed of light is approximately c = 3 x10^8 meters/second.<br /><br />We normalize c to 1, so that any other speed is some percentage of c.<br /><br /><div id="ujoz" style="text-align: left;"><img style="width: 314px; height: 290px;" src="https://docs.google.com/File?id=dfkz4s8v_34vr398c39_b" /></div><br /><br />Let's look at spacetime with two spatial directions. The boundary given by the speed of light is called a light cone. A photon (or light particle) must travel somewhere on this cone, and any thing else will travel inside the cone.<br /><br />Newton's second law is F= ma (i.e., force is mass times acceleration).<br />If we had a constant force k = ma or a = k/m.<br />If we take the derivative of both sides, we get the rate of change of acceleration: k/m.<br />Hence v(t) = k/m *t + b (some constant)<br />This linear function increases to infinity, violating the condition that nothing can travel faster than the speed of light.<br /><br />The reason this condition still holds is that mass is a function of velocity, and is not constant (although for all of the speeds we experience, it may as well be).<br /><br />As an object's velocity grows larger and larger, it must be harder and harder to increase the velocity still further.<br />In fact, mass must become infinite as the velocity approaches the speed of light. (Light particles are massless).<br />The function of mass with respect to velocity, must look something like this:<br /><div id="e6rn" style="text-align: left;"><img style="width: 308px; height: 234px;" src="https://docs.google.com/File?id=dfkz4s8v_35hjzwkrhd_b" /><br /></div><br />Since mass depends on velocity, mass too is relative; mass is not an absolute quantity.<br /><br /><br />II- General Relativity<br />Since special relativity describes linear motion, we can use it to describe accelerated motion.<br /><br />Something that accelerates changes speed.<br /><br /><div id="z7x:" style="text-align: left;"><img style="width: 257px; height: 228px;" src="https://docs.google.com/File?id=dfkz4s8v_36ct9znjf9_b" /></div>As opposed to a linear function (like the blue one), many functions are not linear. However, if we zoom in on them, they would appear linear.<br /><br />If we look at the slope of a function at any instant (the derivative), we always get a linear line. This line corresponds to the rate of change in speed; at any instant, motion is linear.<br /><br />This means that special relativity can be used to describe all motion; it can be extended to general relativity.<br /><br />By general relativity, gravity is a result of the warping of space and time by mass.<br /><br />It is general relativity that allows us to understand and consider black holes, regions of space where gravitational attraction is so strong that not even light can escape.<br /><br />Primary source: The Geometry of Spacetime by James Callahan (a Smith professor!)<br /><br /><br />Part Two: Supersymmetry<br />I- History<br />Supersymmetry is an important part of string theory because there is no way string theory could possibly describe the known universe without it.<br />Supersymmetry is a physical theory in its own right and has been around longer than string theory. It is possible that supersymmetry holds and string theory doesn't, but less likely that string theory is true and supersymmetry isn't.<br /><br />II- What is symmetry exactly?<br />Just as space and time are symmetrical under rotation and translation (by Relativity), supersymmetry suggests that particles may possess a powerful symmetry.<br /><br />If something is symmetrical, that means you can interchange the parts and be left with an identical object.<br />Consider the letter "Y". It has a reflective symmetry. The mirror image of Y cannot be distinguished from Y itself. The letter "O" has even more symmetries. "O" can be rotated or flipped to one's liking and still appear the same.<br /><br /><br />III- Bosons and Fermions<br />Supersymmetry describes a type of symmetry: a symmetry between two classes of particles: bosons and fermions.<br /><br />Bosons and fermions are in fact the only two types of particles. All particles are either bosons or fermions.<br /><br />Both fundamental particles such as the electron and quarks, and composite objects such as a proton or the nucleus of an atom can be classified distinctly as either a boson or a fermion.<br /><br />The classification is given by the particle's "spin." I use quotes because the particles are not actually spinning, but interestingly enough, they move as if they were. This property therefore called intrinsic spin. It is a fixed property for any particle. Further it's value is either a whole number 0, 1, or 2 (known particles), or a multiple of a half such as 1/2 or 3/2.<br /><br />What does spin mean? If a particle has a spin 1, that means the particle would rotate exactly once before returning to its original state.<br /><br />Remark: A graviton has spin 2.<br /><br />The particles are then classified as having either integer spin (0,1,2) as BOSONS or half-integer spin (1/2, 3/2) as FERMIONS. The fact that particles can only have integer or half integer spins is yet another example of quantization, roughly, an idea in Quantum physics that some things come in units.<br /><br />Protons, neutrons, and electrons are all fermions (and in fact, all have spin 1/2). In fact, all familiar matter seems to have spin 1/2.<br /><br />Bosons and fermions behave differently.<br /><br />By something called the Pauli exclusion principle, two fermions of the same type can never occupy the same space. This is why there is only one electron for each orbit around the nucleus of an atom.<br /><br />This principle is intuitive because since all familiar matter is made up of fermions, we experience this principle first-hand daily. It is the reason the matter around you can hold a solid structure; that you will not fall through the seat of your chair. It is also the reason you can't walk through walls: the electrons in your body cannot occupy the same space as those in the wall.<br /><br />Bosons on the other hand can and will be found in the same place, which seems to totally go against our intuition. However, consider light. Light is a bosonic particle and light can indeed pass through light.<br /><br />Also, of the known particles, bosons are always force carrying particles, whereas fermions are always matter particles.<br /><br />Supersymmetry exchanges bosons and fermions!<br /><br />IV- Superstring Theory<br />It's only theoretical and we don't have proof, but it has great potential as a theory. For one,<br />in order to use string theory to reproduce the fundamental particles in the Standard Model, supersymmetry must be incorporated into the theory.<br />String theory that incorporates supersymmetry is called "Superstring theory"<br /><br />Superstring theory eliminates the tachyon (a problem particle that could travel faster than the speed of light). Phew.<br /><br />Supersymmetry also has the potential to solve another problem called the Hierarchy problem, although we will not discuss this problem here.<br /><br />V- Supersymmetry<br />In supersymmetric theory, every existing particle can be paired with a partner particle with the same mass and charge that interacts via the same forces, but is of the opposite class. That is, a boson can be replaced with a fermion.<br /><br />Two particles that are fundamentally different can be exchanged and physics cannot tell the difference!<br /><br />This is an amazing idea because it could mean that matter particles and force particles are symmetrical (or interchangeable) by supersymmetry transformations.<br /><br />Supersymmetry would be nicer if it paired known particles but the standard model does not contain equal numbers of fermions and bosons, and certainly doesn't contained matching pairs as would be desired. If supersymmetry is true, the universe must contain at least twice the number of known particles, which means we've got plenty more particles to find!<br /><br />We would expect that if there were particles out there with the same mass and charge as the ones we know of, that we would be able to observe them, however, because of something called broken symmetry, it is expected that these superpartners would actually have to be much more massive and therefore could only be produced at super high energies (in particle accelerators).<br /><br /><br />Additional videos/sources:<br />Review on a lot of what we've covered in an interview with Ed Witten (creator of M-theory): <a title="http://www.youtube.com/watch?v=iLZKqGbNfck&feature=related" href="http://www.youtube.com/watch?v=iLZKqGbNfck&feature=related" id="o82f">http://www.youtube.com/watch?v=iLZKqGbNfck&feature=related</a><br /><br />Wikipedia article on special relativity: <a title="http://en.wikipedia.org/wiki/Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited" href="http://en.wikipedia.org/wiki/Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited" id="l6tf">http://en.wikipedia.org/wiki/Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited</a><br /><br />An important part of general relativity is the equivalence principle: <a title="http://en.wikipedia.org/wiki/Equivalence_principle" href="http://en.wikipedia.org/wiki/Equivalence_principle" id="xy1m">http://en.wikipedia.org/wiki/Equivalence_principle</a><br /><br /><br />I HOPE YOU ALL ENJOYED THE COURSE!<br /><br />String theory topics not explored in this class that you may wish to pursue on your own:<br />(nearly all of these are a bit more advanced than what we covered)<br />The Higgs mechanism<br />broken symmetries<br />uncertainty principle<br />other Grand Unification Theories (GUTs)<br />virtual particles<br />Kaluza-klein modes<br />tachyons<br /><br /><br /><br />QUIZ<br /><br />1. What are the fundamental particles of all known matter as far as we know?<br /><br />2. What are the four fundamental forces? Which ones are currently part of a single theory, and how are those forces unified?<br /><br />3. What are some of the ideas involved in Quantum mechanics. Explain any or all of these to the best of your ability.<br /><br />4. What is the difference between special relativity and general relativity?<br /><br />5. What does relativity imply about how gravity works?<br /><br />6. What is a string?<br /><br />7. What are three of the problems in modern physics that string theory may solve?<br /><br />8. What are some of the reasons we may not be aware of extra dimensions?<br /><br />9. Why would gravity not be confined to a brane?<br /><br />10. What is supersymmetry?sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0tag:blogger.com,1999:blog-2364903073169292134.post-75628516288350427912010-01-17T18:53:00.000-08:002011-03-22T14:10:36.106-07:00Quiz1. What are the fundamental particles of all known matter as far as we know?<br />electrons, quarks<br /><br />2. What are the four fundamental forces? Which ones are currently part of a single theory, and how are those forces unified?<br />Strong, Weak, Electromagnetic, Gravity. All but Gravity are part of Quantum mechanics by which each force is transmitted by a particle (a boson)<br /><br />3. What are some of the ideas involved in Quantum mechanics. Explain any or all of these to the best of your ability.<br />Quantization, uncertainty principle, wave-particle duality, force carrying particles.<br /><br />4. What is the difference between special relativity and general relativity?<br />Special relativity refers to observers in uniform motion whereas general relativity compensates for accelerated motion as well; it is general relativity that describes gravity.<br /><br />5. What does relativity imply about how gravity works?<br />By general relativity, space and time are warped by mass and energy giving the effect of gravity.<br /><br />6. What is a string?<br />A string is a theoretical fundamental object. It is proposed that every fundamental particle in the standard model is a vibrational mode of a string vibrating in ten dimensions.<br /><br />7. What are three of the problems in modern physics that string theory may solve.<br />See lecture 2 for a list.<br /><br />8. What are some of the reasons we may not be aware of extra dimensions?<br />Small/ curled up/ branes/ warped geometry<br /><br />9. Why would gravity not be confined to a brane?<br />Relativity says that gravity should take place in all dimensions of space and time; also gravitons, the theoretical force particle corresponding to gravity, are given by closed strings which would not be trapped on a brane as open strings are.<br /><br />10. What is supersymmetry?<br />Supersymmetry is a symmetrical theory by which bosons (integer spin /force carrying particles) have supersymmetric partner particles that are fermions (half-integer spin/ matter particles) that can be exchanged. Supersymmetry is an important part of string theory. It's necessary if string theory is to replicate the known universe.sylvierhttp://www.blogger.com/profile/15171723126094294915noreply@blogger.com0