String Theory Demystified

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CHAPTER 13 D-Branes 225 Once again we apply the quantization procedure, considering the bosonic string theory case. The fi rst step is to write down the modal expansions. These will be different depending on which coordinates we look at because now we have NN coordinates and DD coordinates. The fi rst step is to write out the modal expansions. Now let’s recall the open string modal expansion, which can be written in the following way: µ µ µ µ αn −inτ X ( στ , ) = x + 2α′ p τ+ i 2 α′ ∑ e cos( nσ) (13.9) 0 0 n≠ n Taking the derivative of this expression with respect to σ we fi nd Clearly this expression satisfi es ∑ µ µ −inτ ∂ X ( στ , ) =− i 2α σ ′ α e sin( nσ) n n≠0 µ ∂ X 0 = 0 σ σ= , π which are the Neumann boundary conditions. So, we take the modal expansion for the X i to be i i i i αn −inτ X ( στ , ) = x + 2α′ p τ+ i 2 α′ ∑ e cos( nσ) (13.10) 0 0 n≠ n For the DD coordinates, we really have two requirements that have to be satisfi ed. We need the summation over the modes ∑ in the expansion to vanish at σ = 0, π. n≠0 This indicates that we should use sin( nσ ) instead of cos( nσ ) which is in the a a a usual open string expansion. However, we also need X ( 0, τ) = X ( π, τ) = x . Looking at the usual modal expansion, this tells us that we must take x → x 0 a p → 0 a a 0 0 0 Quantization
226 <strong>String</strong> <strong>Theory</strong> Demystifi ed So, the presence of a momentum term in the modal expansion would mean that Dirichlet boundary conditions could not be satisfi ed. Putting everything together, the modal expansion for the DD coordinates is i a a αn −inτ X ( στ , ) = x + i 2α′ ∑ e sin( nσ) n≠0 n Quantization will involve imposing the usual commutators: a b ab ⎡ ⎣X ( στ , ), X ( σ′ , τ) ⎤ ⎦ = iδδσ ( − σ′ ) a b ⎡ ⎣α , α ⎤ m n ⎦ = m + ab δ δ m n,0 (13.11) (13.12) Now, for a moment we consider the general light-cone expansion of the string (so + for a moment we let i = 2, ..., 25). We gauge fi x by choosingα n = 0 for all n ≠ 0 and so The momentum p + is defi ned as The light-cone gauge condition is Now, X − It follows that is an NN coordinate, so + + + X ( στ , ) = x + 2α′ p τ p 0 1 = α (13.13) 2α ′ + + 0 p + 1 = 2α ′ (13.14) − − − − αn −inτ X ( στ , ) = x + 2α′ p τ+ i 2α′∑e cos( nσ) n n≠0 X − − − − ± X′ = 2α′ p + 2α′∑αne n≠0 − in( τ± σ) (13.15)
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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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Page 190 and 191: CHAPTER 9 Superstring Theory Contin
Page 202 and 203: CHAPTER 10 A Summary of Superstring
Page 210 and 211: CHAPTER 11 Type II String Theories
Page 222 and 223: CHAPTER 12 Heterotic String Theory
Page 236 and 237: CHAPTER 13 D-Branes One of the most
Page 238 and 239: CHAPTER 13 D-Branes 223 The Space-T
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Page 244 and 245: CHAPTER 13 D-Branes 229 fi rst exci
Page 246 and 247: CHAPTER 13 D-Branes 231 a set of D-
Page 248 and 249: CHAPTER 13 D-Branes 233 D-brane 1 a
Page 250 and 251: CHAPTER 13 D-Branes 235 Tachyons ca
Page 252 and 253: CHAPTER 13 D-Branes 237 We consider
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 260 and 261: CHAPTER 14 Black Holes 245 When the
Page 262 and 263: CHAPTER 14 Black Holes 247 hole squ
Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
Page 266 and 267: CHAPTER 14 Black Holes 251 Recallin
Page 268 and 269: CHAPTER 14 Black Holes 253 Now, the
Page 270 and 271: CHAPTER 15 The Holographic Principl
Page 280 and 281: CHAPTER 16 String Theory and Cosmol
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CHAPTER 16 String Theory and Cosmol
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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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