String Theory Demystified

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CHAPTER 15 The Holographic Principle and AdS/CFT Correspondence In this chapter we will touch on one of the most interesting ideas to come out of the study of quantum gravity and string theory in particular: the holographic principle. This is an idea closely related to entropy, so we present it here after we have completed our discussion of black holes and entropy in the last chapter. The holographic principle appears to be a quite general feature of quantum gravity, but we discuss it in the context of string theory. Our discussion largely follows that of Susskind and Witten. 1 Our focus will be on showing how the holographic principle leads to an entropy bound of the type we found for black holes. 1 The topics discussed here are quite advanced, so our discussion will be more qualitative and heuristic. For a detailed exposition you may consult L. Susskind and E. Witten, “The Holographic Bound in Antide Sitter Space,” http://xxx.lanl.gove/abs/hep-th/9805114; and J. M. Maldacena, “The Large N Limit of Superconformal Field Theories and Super Gravity,” Adv. Theor. Math. Phys. 2:231–252, 1998. Copyright © 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
256 <strong>String</strong> <strong>Theory</strong> Demystifi ed A Statement of the Holographic Principle The holographic principle was fi rst proposed by Gerard t’Hooft in 1993 and has been worked on extensively by Leonard Susskind. It can be asserted using two postulates: • The total information content in a volume of space is equivalent to a theory that lives only on the surface area that encloses the region. • The boundary of a region of space-time contains at most a single degree of freedom per Planck area. The holographic principle really applies to gravity and we have already seen it in action when talking about black holes. Information content, which is another way of saying entropy, is about counting the number of states in a system and so is proportional to area. We have already seen that in the case of a black hole that entropy is proportional to the area of the event horizon: S A = 4G where G is Newton’s gravitational constant. The area A is measured in Planck units. This is a surprising result because we would intuitively expect that the number of states is proportional to the volume of the enclosed region. Following Susskind, we illustrate that this is in fact the case when gravity is not involved. Imagine that a volume V contains a set of spins on a lattice. We take the lattice spacing to be a, and imagine that the lattice fi lls the entire volume. Then the total number of spins contained in V is # spins = V a 3 The total number of states the system can have is N V a = 2 3 / Using thermodynamics, we arrive at a relationship between the number of states and entropy S: N ∝ exp S
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Copyright © 2009 by The McGraw-Hil
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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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CHAPTER 1 Introduction General rela
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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 31 Th
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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 43 In
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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 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 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 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 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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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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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
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INDEX 305 Schwarzschild metric, 242
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