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Steven S. Gubser is professor of physics at Princeton University.

De la contraportada

"This is an engaging and concise introduction to the main ideas in string theory. Gubser gives us a quick tour of the basic laws of physics as we understand them today, and then demonstrates how string theory seeks to go beyond them. He serves as an artful and attentive guide, as the reader explores the mysteries of quantum mechanics, black holes, strings, branes, supersymmetry, and extra dimensions in the pages of this book."--Juan Maldacena, Institute for Advanced Study

"Steve Gubser has written an engaging and thought-provoking account of what was achieved in physics in the last century and how physicists are seeking to go farther in the ambitious framework known as string theory. This is one of the most thoughtful books on this much-discussed topic, and readers will find much to ponder."--Edward Witten, Institute for Advanced Study

"This book offers a very nice short introduction to some of the basic ideas and implications of string theory. Gubser knows his subject."--John H. Schwarz, coauthor ofSpecial Relativity: From Einstein to Strings

De la solapa interior

"This is an engaging and concise introduction to the main ideas in string theory. Gubser gives us a quick tour of the basic laws of physics as we understand them today, and then demonstrates how string theory seeks to go beyond them. He serves as an artful and attentive guide, as the reader explores the mysteries of quantum mechanics, black holes, strings, branes, supersymmetry, and extra dimensions in the pages of this book."--Juan Maldacena, Institute for Advanced Study

"Steve Gubser has written an engaging and thought-provoking account of what was achieved in physics in the last century and how physicists are seeking to go farther in the ambitious framework known as string theory. This is one of the most thoughtful books on this much-discussed topic, and readers will find much to ponder."--Edward Witten, Institute for Advanced Study

"This book offers a very nice short introduction to some of the basic ideas and implications of string theory. Gubser knows his subject."--John H. Schwarz, coauthor ofSpecial Relativity: From Einstein to Strings

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the LITTLE BOOK of STRING THEORY

PRINCETON UNIVERSITY PRESS

Contents

Introduction..........................................................1CHAPTER ONE energy...................................................11CHAPTER TWO Quantum Mechanics........................................19CHAPTER THREE gravity and Black holes................................34CHAPTER FOUR String Theory...........................................49CHAPTER FIVE Branes..................................................69CHAPTER SIX String Dualities.........................................99CHAPTER SEVEN Supersymmetry and the LHC..............................117CHAPTER EIGHT heavy ions and the fifth Dimension.....................140Epilogue..............................................................159Index.................................................................163

Chapter One

The aim of this chapter is to present the most famous equation of physics: E = mc². This equation underlies nuclear power and the atom bomb. It says that if you convert one pound of matter entirely into energy, you could keep the lights on in a million American households for a year. E = mc² also underlies much of string theory. In particular, as we'll discuss in chapter 4, the mass of a vibrating string receives contributions from its vibrational energy.

What's strange about the equation E = mc² is that it relates things you usually don't think of as related. E is for energy, like the kilowatt-hours you pay your electric company for each month; m is for mass, like a pound of flour; c is for the speed of light, which is 299,792,458 meters per second, or (approximately) 186,282 miles per second. So the first task is to understand what physicists call "dimensionful quantities," like length, mass, time, and speed. Then we'll get back to E = mc² itself. Along the way, I'll introduce metric units, like meters and kilograms; scientific notation for big numbers; and a bit of nuclear physics. Although it's not necessary to understand nuclear physics in order to grasp string theory, it provides a good context for discussing E = mc². And in chapter 8, I will come back and explain efforts to use string theory to better understand aspects of modern nuclear physics.

Length, mass, time, and speed

Length is the easiest of all dimensionful quantities. It's what you measure with a ruler. Physicists generally insist on using the metric system, so I'll start doing that now. A meter is about 39.37 inches. A kilometer is 1000 meters, which is about 0.6214 miles.

Time is regarded as an additional dimension by physicists. We perceive four dimensions total: three of space and one of time. Time is different from space. You can move any direction you want in space, but you can't move backward in time. In fact, you can't really "move" in time at all. Seconds tick by no matter what you do. At least, that's our everyday experience. But it's actually not that simple. If you run in a circle really fast while a friend stands still, time as you experience it will go by less quickly. If you and your friend both wear stopwatches, yours will show less time elapsed than your friend's. This effect, called time dilation, is imperceptibly small unless the speed with which you run is comparable to the speed of light.

Mass measures an amount of matter. We're used to thinking of mass as the same as weight, but it's not. Weight has to do with gravitational pull. If you're in outer space, you're weightless, but your mass hasn't changed. Most of the mass in everyday objects is in protons and neutrons, and a little bit more is in electrons. Quoting the mass of an everyday object basically comes down to saying how many nucleons are in it. A nucleon is either a proton or a neutron. My mass is about 75 kilograms. Rounding up a bit, that's about 50,000,000,000, 000,000,000,000,000,000 nucleons. It's hard to keep track of such big numbers. There are so many digits that you can't easily count them up. So people resort to what's called scientific notation: instead of writing out all the digits like I did before, you would say that I have about 5x10²⁸ nucleons in me. The 28 means that there are 28 zeroes after the 5. Let's practice a bit more. A million could be written as 1x10⁶, or, more simply, as 106. The U.S. national debt, currently about $10,000,000,000,000, can be conveniently expressed as 10¹³ dollars. Now, if only I had a dime for every nucleon in me ...

Let's get back to dimensionful quantities in physics. Speed is a conversion factor between length and time. Suppose you can run 10 meters per second. That's fast for a person—really fast. In 10 seconds you can go 100 meters. You wouldn't win an Olympic gold with that time, but you'd be close. Suppose you could keep up your speed of 10 meters per second over any distance. How long would it take to go one kilometer? Let's work it out. One kilometer is ten times 100 meters. You can do the 100-meter dash in 10 seconds flat. So you can run a kilometer in 100 seconds. You could run a mile in 161 seconds, which is 2 minutes and 41 seconds. No one can do that, because no one can keep up a 10m/s pace for that long.

Suppose you could, though. Would you be able to notice the time dilation effect I described earlier? not even close. Time would run a little slower for you while you were pounding out your 2:41 mile, but slower only by one part in about 1015 (that's a part in 1,000,000,000,000,000, or a thousand million million). In order to get a big effect, you would have to be moving much, much faster. Particles whirling around modern accelerators experience tremendous time dilation. Time for them runs about 1000 times slower than for a proton at rest. The exact figure depends on the particle accelerator in question.

The speed of light is an awkward conversion factor for everyday use because it's so big. Light can go all the way around the equator of the earth in about 0.1 seconds. That's part of why an American can hold a conversation by telephone with someone in India and not notice much time lag. Light is more useful when you're thinking of really big distances. The distance to the moon is equivalent to about 1.3 seconds. you could say that the moon is 1.3 light-seconds away from us. The distance to the sun is about 500 light-seconds.

A light-year is an even bigger distance: it's the distance that light travels in a year. The Milky Way is about 100,000 light-years across. The known universe is about 14 billion light-years across. That's about 1.3x10²⁶ meters.

E = mc²

The formula E = mc² is a conversion between mass and energy. it works a lot like the conversion between time and distance that we just discussed. But just what is energy? The question is hard to answer because there are so many forms of energy. Motion is energy. Electricity is energy. Heat is energy. Light is energy. Any of these things can be converted into any other. For example, a lightbulb converts electricity into heat and light, and an electric generator converts motion into electricity. A fundamental principle of physics is that total energy is conserved, even as its form may change. in order to make this principle meaningful, one has to have ways of quantifying different forms of energy that can be converted into one another.

A good place to start is the energy of motion, also called kinetic energy. The conversion formula is K = 1/2mv², where K is the kinetic energy, m is the mass, and v is the speed. imagine yourself again as an Olympic sprinter. Through a tremendous physical effort, you can get yourself going at v = 10 meters per second. But this is much slower than the speed of light. Consequently, your kinetic energy is much less than the energy E in E = mc². What does this mean?

It helps to know that E = mc² describes "rest energy." rest energy is the energy in matter when it is not moving. When you run, you're converting a little bit of your rest energy into kinetic energy. A very little bit, actually: roughly one part in 10¹⁵. It's no accident that this same number, one part in 10¹⁵, characterizes the amount of time dilation you experience when you run. Special relativity includes a precise relation between time dilation and kinetic energy. It says, for example, that if something is moving fast enough to double its energy, then its time runs half as fast as if it weren't moving.

It's frustrating to think that you have all this rest energy in you, and all you can call up with your best efforts is a tiny fraction, one part in 10¹⁵. How might we call up a greater fraction of the rest energy in matter? The best answer we know of is nuclear energy.

Our understanding of nuclear energy rests squarely on E = mc². Here is a brief synopsis. Atomic nuclei are made up of protons and neutrons. A hydrogen nucleus is just a proton. A helium nucleus comprises two protons and two neutrons, bound tightly together. What I mean by tightly bound is that it takes a lot of energy to split a helium nucleus. Some nuclei are much easier to split. An example is uranium-235, which is made of 92 protons and 143 neutrons. It is quite easy to break a uranium-235 nucleus into several pieces. For instance, if you hit a uranium-235 nucleus with a neutron, it can split into a krypton nucleus, a barium nucleus, three neutrons, and energy. This is an example of fission. We could write the reaction briefly as

U + n → Kr + Ba + 3n + Energy,

where we understand that U stands for uranium-235, Kr stands for krypton, Ba stands for barium, and n stands for neutron. (By the way, I'm careful always to say uranium-235 because there's another type of uranium, made of 238 nucleons, that is far more common, and also harder to split.)

E = mc² allows you to calculate the amount of energy that is released in terms of the masses of all the participants in the fission reaction. It turns out that the ingredients (one uranium-235 nucleus plus one neutron) outweigh the products (a krypton atom, a barium atom, and three neutrons) by about a fifth of the mass of a proton. It is this tiny increment of mass that we feed into E = mc² to determine the amount of energy released. Tiny as it seems, a fifth of the mass of a proton is almost a tenth of a percent of the mass of a uranium-235 atom: one part in a thousand. So the energy released is about a thousandth of the rest energy in a uranium-235 nucleus. This still may not seem like much, but it's roughly a trillion times bigger as a fraction of rest energy than the fraction that an Olympic sprinter can call up in the form of kinetic energy.

I still haven't explained where the energy released in nuclear fission comes from. The number of nucleons doesn't change: there are 236 of them before and after fission. And yet the ingredients have more mass than the products. So this is an important exception to the rule that mass is essentially a count of nucleons. The point is that the nucleons in the krypton and barium nuclei are bound more tightly than they were in the uranium-235 nucleus. Tighter binding means less mass. The loosely bound uranium-235 nucleus has a little extra mass, just waiting to be released as energy. To put it in a nutshell: nuclear fission releases energy as protons and neutrons settle into slightly more compact arrangements.

One of the projects of modern nuclear physics is to figure out what happens when heavy nuclei like uranium-235 undergo far more violent reactions than the fission reaction I described. For reasons I won't go into, experimentalists prefer to work with gold instead of uranium. When two gold nuclei are slammed into one another at nearly the speed of light, they are utterly destroyed. Almost all the nucleons break up. In chapter 8, I will tell you more about the dense, hot state of matter that forms in such a reaction.

In summary, E = mc² says that the amount of rest energy in something depends only on its mass, because the speed of light is a known constant. It's easier to get some of that energy out of uranium-235 than most other forms of matter. But fundamentally, rest energy is in all forms of matter equally: rocks, air, water, trees, and people.

Before going on to quantum mechanics, let's pause to put E = mc² in a broader intellectual context. It is part of special relativity, which is the study of how motion affects measurements of time and space. Special relativity is subsumed in general relativity, which also encompasses gravity and curved spacetime. String theory subsumes both general relativity and quantum mechanics. In particular, string theory includes the relation E = mc². Strings, branes, and black holes all obey this relation. For example, in chapter 5 I'll discuss how the mass of a brane can receive contributions from thermal energy on the brane. It wouldn't be right to say that E = mc² follows from string theory. But it fits, seemingly inextricably, with other aspects of string theory's mathematical framework.

Chapter Two

After I got my bachelor's degree in physics, I spent a year at Cambridge University studying math and physics. Cambridge is a place of green lawns and grey skies, with an immense, historical weight of genteel scholarship. I was a member of St. John's College, which is about five hundred years old. I particularly remember playing a fine piano located in one of the upper floors of the first court—one of the oldest bits of the college. Among the pieces I played was Chopin's Fantasie-Impromptu. The main section has a persistent four-against-three cross rhythm. Both hands play in even tempo, but you play four notes with the right hand for every three notes in the left hand. The combination gives the composition an ethereal, liquid sound.

It's a beautiful piece of music. And it makes me think about quantum mechanics. To explain why, I will introduce some concepts of quantum mechanics, but I won't try to explain them completely. Instead, I will try to explain how they combine into a structure that is, to me, reminiscent of music like the Fantasie-Impromptu. In quantum mechanics, every motion is possible, but there are some that are preferred. These preferred motions are called quantum states. They have definite frequencies. A frequency is the number of times per second that something cycles or repeats. In the Fantasie-Impromptu, the patterns of the right hand have a faster frequency, and the patterns of the left hand have a slower frequency, in the ratio four to three. In quantum systems, the thing that is cycling is more abstract: technically, it's the phase of the wave function. You can think of the phase of the wave function as similar to the second hand of a clock. The second hand goes around and around, once per minute. The phase is doing the same thing, cycling around at some much faster frequency. This rapid cycling characterizes the energy of the system in a way that I'll discuss in more detail later.

Simple quantum systems, like the hydrogen atom, have frequencies that stand in simple ratios with one another. For example, the phase of one quantum state might cycle nine times while another cycles four times. That's a lot like the four-against-three cross rhythm of the Fantasie-Impromptu. But the frequencies in quantum mechanics are usually a lot faster. For example, in a hydrogen atom, characteristic frequencies are on the scale of 1015 oscillations or cycles per second. That's indeed a lot faster than the Fantasie-Impromptu, in which the right hand plays about 12 notes per second.

The rhythmic fascination of the Fantasie-Impromptu is hardly its greatest charm—at least, not when it's played rather better than I ever could. Its melody floats above a melancholy bass. The notes run together in a chromatic blur. The harmonies shift slowly, contrasting with the almost desultory flitting of the main theme. The subtle four-against-three rhythm provides just the backdrop for one of Chopin's more memorable compositions. Quantum mechanics is like this. Its underlying graininess, with quantum states at definite frequencies, blurs at larger scales into the colorful, intricate world of our experience. Those quantum frequencies leave an indelible mark on that world: for example, the orange light from a street lamp has a definite frequency, associated with a particular cross rhythm in sodium atoms. The frequency of the light is what makes it orange.

In the rest of this chapter, I'm going to focus on three aspects of quantum mechanics: the uncertainty principle, the hydrogen atom, and the photon. Along the way, we'll encounter energy in its new quantum mechanical guise, closely related to frequency. Analogies with music are apt for those aspects of quantum mechanics having to do with frequency. But as we'll see in the next section, quantum physics incorporates some other key ideas that are less readily compared with everyday experience.

(Continues...)

Excerpted from the LITTLE BOOK of STRING THEORYby Steven S. Gubser Copyright © 2010 by Steven S. Gubser. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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