String Theory Demystified

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CHAPTER 4 String Quantization Normal ordering leads to 1 2 D−2 ∞ i= 1 n=−∞ D−2 ∞ i i 1 i i D − 2 α−nαn= : α−nαn: + 2 2 ∑∑∑∑ ∑ i= 1 n=−∞ The regularization trick can be applied to make the second term fi nite. We fi nd D − 2 2 Again taking a = 1, one fi nds D = 26. ∞ ∑ n= 1 D − 2 n =− 24 In the previous two chapters, we constructed a relativistic theory of the string, the classical theory. In this chapter we have introduced the simplest possible quantum extension of the classical theory. This is a theory that consists only of bosons. While the theory is not realistic since it does not include fermionic states, it is easier to deal with and introduces important concepts and methods that will play a role in the full quantum theory. The classical theory was quantized using two different methods. The fi rst method, called covariant quantization, is a straightforward approach that imposes commutation relations on the X µ and their conjugate momenta. This leads to negative norm states. The Virasoro constraints are imposed to rid the theory of these states. When this is done, we fi nd that the theory must have 26 space-time dimensions. We concluded the chapter with a different approach, known as light-cone quantization. Quiz µ ν 1. Explicitly calculate the commutators [ ∂ + X ( σ, τ), ∂ −′ X ( σ′ , τ )] and µ ν [ ∂ − X ( σ, τ), ∂ ′ + X ( σ′ , τ )] . µ ν 2. Consider the closed string and explicitly calculate [ x , p ] . µ ν 3. Consider the fi rst excited state of the closed string εµν ( k) α−1α−1 0 , k . Using the condition satisfi ed by physical states ψ , in particular L1 ψ = L1 ψ = 0, µ fi nd ε µνk . ∞ n= 1 n 87 Summary
88 String Theory Demystifi ed i 4. Let α−1 0, k be the state of an open string and suppose that the normal ordering constant a is undetermined. What is the mass of this state? i j ij i j ij 5. In the light-cone gauge, use [ x , p ] = iδ ,[ X ( σ), P ( σ′ )] = iδ δ( σ − σ ′ ) i j to fi nd a commutation relation for the transverse modes, [ α , α ] . 6. Consider light-cone quantization for the closed string case. What additional commutation relation do you think should be imposed for the modes? m n
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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 27 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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Page 66 and 67: CHAPTER 3 The Classical String II:
Page 68 and 69: CHAPTER 3 Symmetries and Worldsheet
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Page 104 and 105: CHAPTER 5 Conformal Field Theory Pa
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Page 142 and 143: CHAPTER 7 RNS Superstrings The real
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Page 146 and 147: CHAPTER 7 RNS Superstrings 131 SOLU
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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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CHAPTER 13 D-Branes One of the most
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CHAPTER 13 D-Branes 223 The Space-T
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CHAPTER 13 D-Branes 225 Once again
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CHAPTER 13 D-Branes 227 and ( X i i
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CHAPTER 13 D-Branes 229 fi rst exci
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CHAPTER 13 D-Branes 231 a set of D-
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CHAPTER 13 D-Branes 233 D-brane 1 a
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CHAPTER 13 D-Branes 235 Tachyons ca
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CHAPTER 13 D-Branes 237 We consider
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CHAPTER 14 Black Holes Black holes,
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CHAPTER 14 Black Holes 241 This equ
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CHAPTER 14 Black Holes 243 Here, G
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CHAPTER 14 Black Holes 245 When the
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CHAPTER 14 Black Holes 247 hole squ
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CHAPTER 14 Black Holes 249 Entropy
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CHAPTER 14 Black Holes 251 Recallin
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CHAPTER 14 Black Holes 253 Now, the
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CHAPTER 15 The Holographic Principl
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CHAPTER 16 String Theory and Cosmol
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Final Exam 1 1. Consider the lagran
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Final Exam 281 0 0 26. Find a simpl
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Final Exam 283 68. When a single sp
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Chapter 1 1. b 6. a 2. a 7. c 3. c
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Quiz Solutions 287 3. i( − ) 4.
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Quiz Solutions 289 Chapter 11 1. c
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1. P 2. E µ 2 ν 1 ( X′ ) X µ
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Final Exam Solutions 293 27. Supers
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Final Exam Solutions 295 73. Types
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Books References Becker, K., M. Bec
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|NS〉 ⊕ |NS〉 Sector, 203 |NS
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INDEX 301 Cyclic model, 277 Cylinde
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INDEX 303 Mechanics, quantum. See Q
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INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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