String Theory Demystified

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CHAPTER 16 String Theory and Cosmology 267 and cools with its dynamics evolving according to Einstein’s equations. Interestingly, the universe exhibits a great deal of uniformity on large scales that the standard bigbang model is hard pressed to explain. To understand the type of uniformity we are talking about, we can think of everyday life. Imagine heating a cup of tea in the microwave and then taking it out and setting it on the counter. Over time, the cup of tea will cool and if we leave it there long enough, it will reach an equilibrium point where it is the same temperature as its surroundings. The same kind of behavior has occurred on the largest scales of the universe. If we examine the universe on large scales where we divide it up into cubes that have sides which are on the order of hundreds of millions of light years across, we fi nd • Homogeneity: On large scales on the average the universe is the same everywhere. That is each cube has the same galaxy density, the same mass density, and the same luminosity. • Isotropy: We have already mentioned that standard cosmology assumes the universe is isotropic, or the same in every direction. Observation bears this out to an incredibly high degree. The problem with standard big-bang theory and these observations is that the universe evolved too quickly for equilibrium in the sense we described with the cup of tea, could have occurred. There would not have been enough time for light signals to connect different spatial regions, so how could they have “communicated” so as to end up in exactly the same confi guration? Another problem with standard big-bang cosmology is known as the fl atness problem. The universe is fl at and the mass density of the early universe was apparently so exactly fi ne-tuned to give the observed fl atness that it is hard to imagine how this could be coincidence. The critical mass density is defi ned in terms of the Hubble constant: H ρ = c πG 3 2 8 where G is Newton’s gravitational constant. Now defi ne (16.3) Ω= ρ (16.4) ρc where r is the actual mass density in the universe. Now let ∆ be the cosmological constant. If Ω Λ + 3 2 H (16.5)
268 String Theory Demystifi ed exceeds 1, then the universe is a closed space like a sphere. If it is less than one, then it is an open space with negative curvature (like a saddle). If it is exactly 1, then the universe is a fl at open space. Observation indicates that the universe is a fl at open space, so the fl atness problem boils down to the equation of why early conditions in the universe fi xed the mass density so close to the critical mass density. The issues which cannot be explained by classical physics-homogeneity, isotropy, and the fl atness problem can be explained by a theory known as infl ation. This is a theory that proposes that the early universe went through a brief phase of exponential expansion. Just prior to the phase of exponential expansion, all regions of the universe were causally connected. This explains the homogeneity and isotropic problems. The expansion is driven by a scalar fi eld f (a quantum fi eld called the infl aton) which has negative pressure. This acts like a repulsive gravitational fi eld causing different regions of the universe to repel one another and to expand outward. The infl aton fi eld is believed to have a false vacuum, which is a metastable point that is higher in energy than the true vacuum (the lowest energy state). For a brief period, the infl aton was at the false vacuum and could cause infl ation, then it “rolled down the hill” to the true vacuum or lowest energy state. During the expansion, the total energy of the universe remains constant (as it must). During infl ation, the energy of matter, which is positive, is increasing exponentially. Energy from the infl aton fi eld can be used to actually create matter through Einstein’s equation E = mc As matter is added to the universe, the gravitational fi eld gets larger as well. The gravitational fi eld has negative energy density. So the increasing negative energy of the gravitational fi eld balances out the increasing positive energy of matter keeping the total energy of the universe constant. Quantum fl uctuations in the infl aton fi eld when the universe was very small are believed to have magnifi ed during the exponential expansion providing seedlike structures for the universe as a whole. These seeds led to the formation of the galaxies. This is an amazing connection between quantum theory and the large-scale structure of the universe. Infl ation theory makes several predictions that are consistent with observation to date. The Kasner Metric The Kasner metric is a solution to the Einstein fi eld equations that has an interesting property that makes it useful from a string theory perspective. We can characterize the Kasner metric by considering the notion of isotropy. If space is isotropic, then it is the same in all directions. This is a reasonable assumption that is used routinely in cosmology when considering the 3 + 1 dimensional space-time we appear to 2 .
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Accounting Demystified Advanced Cal
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Copyright © 2009 by The McGraw-Hil
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For more information about this tit
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Contents vii BRST in String Theory-
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Contents ix CHAPTER 14 Black Holes
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PREFACE String theory is the greate
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Preface xiii time as this one to he
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CHAPTER 1 Introduction General rela
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CHAPTER 1 Introduction 3 fl at spac
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CHAPTER 1 Introduction 5 we say tha
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CHAPTER 1 Introduction 7 initial an
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CHAPTER 1 Introduction 9 a spin-2 b
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CHAPTER 1 Introduction 11 So if α
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CHAPTER 1 Introduction 13 Figure 1.
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CHAPTER 1 Introduction 15 • Type
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CHAPTER 1 Introduction 17 Close up,
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CHAPTER 1 Introduction 19 5. The mi
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CHAPTER 2 The Classical String I: E
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CHAPTER 2 Equations of Motion 23 To
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CHAPTER 2 Equations of Motion 25 No
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CHAPTER 2 Equations of Motion 29 ct
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CHAPTER 2 Equations of Motion 31 Th
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CHAPTER 2 Equations of Motion 33 is
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CHAPTER 2 Equations of Motion 35 Si
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CHAPTER 2 Equations of Motion 37 ho
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CHAPTER 2 Equations of Motion 39 Th
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CHAPTER 2 Equations of Motion 41 wh
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CHAPTER 2 Equations of Motion 43 In
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CHAPTER 2 Equations of Motion 47 We
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CHAPTER 2 Equations of Motion 49 In
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CHAPTER 3 The Classical String II:
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CHAPTER 3 Symmetries and Worldsheet
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CHAPTER 4 String Quantization At th
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CHAPTER 4 String Quantization Here,
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CHAPTER 4 String Quantization That
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CHAPTER 4 String Quantization Using
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CHAPTER 4 String Quantization expec
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CHAPTER 4 String Quantization † A
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CHAPTER 4 String Quantization To ob
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CHAPTER 4 String Quantization lower
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CHAPTER 4 String Quantization this)
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CHAPTER 4 String Quantization Norma
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CHAPTER 5 Conformal Field Theory Pa
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CHAPTER 6 BRST Quantization So far
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CHAPTER 6 BRST Quantization 117 It
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CHAPTER 6 BRST Quantization 119 Cal
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CHAPTER 6 BRST Quantization 121 The
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CHAPTER 6 BRST Quantization 123 To
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CHAPTER 6 BRST Quantization 125 The
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CHAPTER 7 RNS Superstrings The real
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CHAPTER 7 RNS Superstrings 129 EXAM
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CHAPTER 7 RNS Superstrings 131 SOLU
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CHAPTER 7 RNS Superstrings 133 In t
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CHAPTER 7 RNS Superstrings 135 Now,
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CHAPTER 7 RNS Superstrings 137 We c
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CHAPTER 7 RNS Superstrings 139 We o
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CHAPTER 7 RNS Superstrings 141 OPEN
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CHAPTER 7 RNS Superstrings 143 If w
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CHAPTER 7 RNS Superstrings 145 The
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CHAPTER 7 RNS Superstrings 147 From
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CHAPTER 7 RNS Superstrings 149 The
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CHAPTER 7 RNS Superstrings 151 3. A
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CHAPTER 8 Compactifi cation and T-D
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CHAPTER 9 Superstring Theory Contin
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CHAPTER 10 A Summary of Superstring
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CHAPTER 11 Type II String Theories
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CHAPTER 12 Heterotic String Theory
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Page 236 and 237: CHAPTER 13 D-Branes One of the most
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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
Page 256 and 257: CHAPTER 14 Black Holes 241 This equ
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Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
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Page 308 and 309: Final Exam Solutions 293 27. Supers
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Page 312 and 313: Books References Becker, K., M. Bec
Page 314 and 315: |NS〉 ⊕ |NS〉 Sector, 203 |NS
Page 316 and 317: INDEX 301 Cyclic model, 277 Cylinde
Page 318 and 319: INDEX 303 Mechanics, quantum. See Q
Page 320 and 321: INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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