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MIT Physics 8.251: String Theory 8.251: String Theory for Undergraduates Lecturer: Professor Hong Liu Notes by: Andrew Lin Spring 2021 Introduction It’s good for all of us to have our videos on, so that we can all recognize each other in person and so that the pace can be adjusted based on face feedback. In terms of logistics, we should be able to find three documents on Canvas (organization, course outline, and pset 1). Sam Alipour-fard will be our graduate TA, doing recitations and problem set grading. • There are no official recitations for this class, but Sam has offered to hold a weekly Friday 11am recitation. • Office hours are Thursday 5-6 (Professor Liu) and Friday 12-1 (Sam). If any of those times don’t work for us, we should email both of the course staff for a separate meeting time. Professor Zwiebach’s “A First Course in String Theory” is the main textbook for this course, which we’ll loosely follow (and have assigned readings and problems taken from). We won’t have any tests or exams for this class – grading is based on problem sets (75%) and a final project (25%) where we read Chapter 23 and do calculations and problems on our own. The idea is that we’ll learn some material on our own so that we can understand the material deeply! (Problem sets will be due at midnight on Fridays, with 50% credit for late submissions within 2 weeks.) Fact 1 The “outline” document gives us a roadmap of what we’ll be doing in this class. In particular, we’ll be trying to answer the following four questions: • Why do quantum strings demand a particular spacetime dimension? • Why is string theory a theory of quantum gravity? • Why does string theory have the potential to unify all fundamental interactions? • How do we derive the AdS/CFT duality from string theory? The final topic here will require a bit of general relativity, but for those of us who have not studied it, we shouldn’t be too concerned. Finally, lectures will focus on main messages and concepts rather than mathematical calculations (unless those calculations are very important, we’ll be told to read them ourselves). Questions are particularly important in these times, because we need to make sure not to get lost. 2 February 17, 2021 We’ll start with the basic ideas before diving into the main questions. Before strings, we had field theory: it was believed that the building blocks of the universe are “fundamental particles” like electrons, photons, gravitons, and quarks (where “fundamental” means that they have no internal structure and can be described as zero-dimensional points). In principle, these different particles are completely independent – photons have nothing to do with gravitons. But in string theory, the fundamental objects are instead (one-dimensional) strings (which can be open or closed. We can still have different particles, like photons and gravitons, but they arise from different oscillation modes of strings. So different particles are different patterns for the same fundamental string, and this gives us a unified approach for all particles and interactions. (If photons are just a certain oscillatory pattern, and photons propagate through electromagnetic interactions, then E&M is then also part of string theory.) One of the biggest projects in physics is to unify quantum mechanics and gravity, and that’s one thing string theory aims to achieve. Fact 2 Our understanding of string theory is still incomplete, similar to the story of the blind man and the elephant. We see different aspects from different angles, but the global picture is still lacking today because the field is very young. There are a variety of other physical contexts in which strings arise, including the following: 0. Strings on a violin (not a very serious example), 1. Flux tubes that connect quarks in a proton, 2. Vortex strings in superfluids, 3. Cosmic strings in cosmology. But the strings that arise in these physical situations are all different from the fundamental string that will be used to obtain quantum gravity: the four examples above are all string-like objects with internal structure and internal physics, so they are composite strings. The point is that the fundamental string is a one-dimensional object with no internal structure – it can only vibrate. Fact 3 String theory has a reputation for being very mathematical and difficult to understand, but the motto we should keep in mind is that nothing is difficult if you understand it in the right way. If we find certain concepts unintuitive, we should find another angle or perspective to use instead. In particular, we can ask ourselves two questions: • Do we understand the basic concept correctly? (Usually, confusions stem from misunderstandings in these fundamental ideas.) • If we’re confused about formalism, what are the questions that the formalism is trying to address? After all, theories in physics, including string theory, were initially invented to address concrete physical questions, even if there is lots of mathematical formalism. So understanding those questions also goes a long way towards resolving our difficulties (for example, why we’re using certain mathematical tricks). With that, we’re ready to jump into the first chapter of the class, and we’ll start by reviewing special relativity (partially to set up notation for relativistic systems). 3 1: Basics 1.1: Special relativity Proposition 4 (Important elements of SR) • A reference frame is a system of coordinates that we use to describe or label physical processes (events in spacetime). We often use notation like x µ = (x 0; x 1; x 2; x 3) = (ct; x; y; z). • An inertial frame is one where a freely moving body moves with constant velocity. Two inertial frames always move uniformly relative to each other. µ µ0 0 • The same event A may have different coordinates x ; x in two frames S; S (for both time and space). • (Principle of relativity) Physical laws, as well as the speed of light, are the same in all inertial frames (so we can’t tell inertial frames apart by doing experiments). Suppose we have two events A and B, labeled with coordinates x µ; y µ. Then the invariant interval 3 0 0 2 X i i 2 sAB = −(x − y ) + (x − y ) i=1 is the same in all inertial frames. (So if we describe this system in another frame S0 with coordinates x µ0 ; y µ0 , then 0 sAB = sAB.) Since sAB is agreed upon by all inertial frames, it can be a meaningful physical quantity, and in particular we say that the events A; B are spacelike separated if sAB > 0, timelike separated if sAB < 0, and null-like separated if sAB = 0.) If we consider two infinitesimally separated events A : x µ and B : x µ + dx µ, then we have 3 X ds2 = −(dx 0)2 + (dx i )2; i=1 and we write this in the shorthand notation µ ν = ηµν dx dx ; where repeated indices in a product are summed (Einstein convention) from 0 to 3, and 2 3 −1 6 7 6 1 7 ηµν = 6 7 : 6 1 7 4 5 1 When ds2 > 0, we write it as d`2 for the (infinitesimal) proper distance d`, and when ds2 < 0, we write it as −c2dτ 2 for the proper time dτ > 0. The point of all of these infinitesimal elements is that we can now assign a proper distance or proper time for any arbitrary curve in spacetime. If we have a particle which moves in spacetime from point A to point B along some curve C, it might take some arbitrary path (depending on the forces acting on that particle), but regardless of that path, we can find a proper time for that path. To do this, we divide this curve into many small 4 intervals and applying the ds2 = −(cdt)2 + d~x 2 definition to each one to find its proper length. We then have d~x 2 ~v 2 ds2 = −c2dt2 1 − = −c2dt2 1 − = −c2dτ 2 dt2 c2 (this quantity is indeed negative because v < c) and thus we’ve found the proper-time dτ, and we get our final answer by integrating τ = R dτ from A to B. Similarly, if we have two events that are spacelike separated along a curve AB C p 0 R 2 C , we can find the path’s proper length by integrating ` = C0 d`, where d` = ds . We’ll next turn to Lorentz transformations, which are ways of getting between coordinates x µ; x µ0 in different frames. We need to make sure that these transformations leave ds2 invariant: we can write down the new coordinates 0µ µ ν x = L ν x ; where x ν ; x 0µ can be thought of as column vectors and L as a matrix. If we substitute in our new coordinates into the 2 µ ν definition of ds = ηµν dx dx , it turns out that the relation that L must satisfy is LT ηL = η: This is a matrix identity, and we can fully classify Lorentz transformations by studying it. For example, if a frame S0 moves with velocity v in the x 1 direction relative to the frame S, then we can show the corresponding Lorentz transformation is 2 3 γ −γβ 0 0 6 7 6−γβ γ 0 07 L = 6 7 ; 6 0 0 1 07 4 5 0 0 0 1 where we use the standard notation v 1 β = ; γ = : c p1 − β2 The reason for switching between frames is that certain frames might be more convenient for doing mathematical calculations than others depending on the particular physics question that we’re trying to solve (even if the physical description is always the same).

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