String Theory Demystified

More documents

Recommendations

Info

CHAPTER 5 Conformal Field Theory Part I In this chapter we study conformal fi eld theory, an area of quantum fi eld theory that relies heavily on complex variables. In this chapter, we will introduce some of the basic concepts of conformal fi eld theory. The topic will be expanded and utilized in many areas in the rest of the book. In the next chapter, we discuss other aspects of conformal fi eld theory along with BRST quantization. Conformal fi eld theory is an important tool used in the analysis of perturbative string theory, so it plays a central role in our task at hand (understanding the physics of quantized strings). In particular, since the worldsheet can be described using two coordinates ( τ, σ ) two-dimensional conformal fi eld theory is used. The theory of complex variables plays an important role in the study of theoretical physics, and string theory is no exception. If you are not familiar with complex variables you should take time out to study the topic before proceeding any further. You can do so using my book Complex Variables Demystifi ed, also published by McGraw-Hill. Copyright © 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
90 String Theory Demystifi ed In the theory of complex variables, a conformal transformation is one that maps a region of the complex plane to a new more convenient region while preserving angles, but not lengths. For example, you can map the unit disk to the upper-half plane using a conformal transformation. The notion that angles are preserved is a geometric interpretation and leads us to the notion of a conformal transformation of space-time coordinates. Let us consider a transformation of the space-time coordinates such that x→ x′. In general, the metric gµν x ( ) is found to transform in the following way: α β x x g ′ µν x′ = g x µ ν αβ x x ∂ ∂ ( ) ( ) ∂ ′ ∂ ′ (5.1) Now consider a function of the space-time coordinates given by Ω( x ) . If the metric transforms in the following way: g ′ ( x′ ) = Ω ( x) g ( x) (5.2) µν µν Then we have a conformal transformation of the metric. Notice that Ω( x) acts as a scaling factor, hence it will preserve angles but not lengths. If a metric is related to the fl at Minkowski metric as g =Ω( x) η ( x)we say that the metric is conformally fl at. µν µν To see how a conformal transformation preserves angles, consider two tangent vectors u v and . Using the metric, the angle between them is given by cosθ = g( u, v) g( u, u) g( v, v) Now we apply the transformation given by Eq. (5.2) and fi nd cosθ → g′ ( u, v) = g′ ( u, u) g′ ( v, v) Ω( xg ) ( u, v) = Ω( xg ) ( u, u) Ω( xg ) ( v, v) g( u, v) g( u, u) g( v, v) Hence a conformal transformation preserves angles. A conformal fi eld theory is a quantum fi eld theory that is invariant under conformal transformations. These theories are Euclidean quantum fi eld theories, meaning that a Euclidean metric is used. The symmetry group of such a theory will contain Euclidean symmetries (we will review those in a moment) along with local conformal transformations. It turns out that two-dimensional conformal fi eld theories are of particular use. Two-dimensional conformal fi eld theories have an infi nite number of conserved charges.
Page 2 and 3:
Accounting Demystified Advanced Cal
Page 4 and 5:
Copyright © 2009 by The McGraw-Hil
Page 6 and 7:
For more information about this tit
Page 8 and 9:
Contents vii BRST in String Theory-
Page 10 and 11:
Contents ix CHAPTER 14 Black Holes
Page 12 and 13:
PREFACE String theory is the greate
Page 14 and 15:
Preface xiii time as this one to he
Page 16 and 17:
CHAPTER 1 Introduction General rela
Page 18 and 19:
CHAPTER 1 Introduction 3 fl at spac
Page 20 and 21:
CHAPTER 1 Introduction 5 we say tha
Page 22 and 23:
CHAPTER 1 Introduction 7 initial an
Page 24 and 25:
CHAPTER 1 Introduction 9 a spin-2 b
Page 26 and 27:
CHAPTER 1 Introduction 11 So if α
Page 28 and 29:
CHAPTER 1 Introduction 13 Figure 1.
Page 30 and 31:
CHAPTER 1 Introduction 15 • Type
Page 32 and 33:
CHAPTER 1 Introduction 17 Close up,
Page 34 and 35:
CHAPTER 1 Introduction 19 5. The mi
Page 36 and 37:
CHAPTER 2 The Classical String I: E
Page 38 and 39:
CHAPTER 2 Equations of Motion 23 To
Page 40 and 41:
CHAPTER 2 Equations of Motion 25 No
Page 42 and 43:
CHAPTER 2 Equations of Motion 27 No
Page 44 and 45:
CHAPTER 2 Equations of Motion 29 ct
Page 46 and 47:
CHAPTER 2 Equations of Motion 31 Th
Page 48 and 49:
CHAPTER 2 Equations of Motion 33 is
Page 50 and 51:
CHAPTER 2 Equations of Motion 35 Si
Page 52 and 53:
CHAPTER 2 Equations of Motion 37 ho
Page 54 and 55: CHAPTER 2 Equations of Motion 39 Th
Page 56 and 57: CHAPTER 2 Equations of Motion 41 wh
Page 58 and 59: CHAPTER 2 Equations of Motion 43 In
Page 60 and 61: CHAPTER 2 Equations of Motion 45 No
Page 62 and 63: CHAPTER 2 Equations of Motion 47 We
Page 64 and 65: CHAPTER 2 Equations of Motion 49 In
Page 66 and 67: CHAPTER 3 The Classical String II:
Page 68 and 69: CHAPTER 3 Symmetries and Worldsheet
Page 84 and 85: CHAPTER 4 String Quantization At th
Page 86 and 87: CHAPTER 4 String Quantization Here,
Page 88 and 89: CHAPTER 4 String Quantization That
Page 90 and 91: CHAPTER 4 String Quantization Using
Page 92 and 93: CHAPTER 4 String Quantization expec
Page 94 and 95: CHAPTER 4 String Quantization † A
Page 96 and 97: CHAPTER 4 String Quantization To ob
Page 98 and 99: CHAPTER 4 String Quantization lower
Page 100 and 101: CHAPTER 4 String Quantization this)
Page 102 and 103: CHAPTER 4 String Quantization Norma
Page 106 and 107: CHAPTER 5 Conformal Field Theory Pa
Page 130 and 131: CHAPTER 6 BRST Quantization So far
Page 132 and 133: CHAPTER 6 BRST Quantization 117 It
Page 134 and 135: CHAPTER 6 BRST Quantization 119 Cal
Page 136 and 137: CHAPTER 6 BRST Quantization 121 The
Page 138 and 139: CHAPTER 6 BRST Quantization 123 To
Page 140 and 141: CHAPTER 6 BRST Quantization 125 The
Page 142 and 143: CHAPTER 7 RNS Superstrings The real
Page 144 and 145: CHAPTER 7 RNS Superstrings 129 EXAM
Page 146 and 147: CHAPTER 7 RNS Superstrings 131 SOLU
Page 148 and 149: CHAPTER 7 RNS Superstrings 133 In t
Page 150 and 151: CHAPTER 7 RNS Superstrings 135 Now,
Page 152 and 153: CHAPTER 7 RNS Superstrings 137 We c
Page 154 and 155:
CHAPTER 7 RNS Superstrings 139 We o
Page 156 and 157:
CHAPTER 7 RNS Superstrings 141 OPEN
Page 158 and 159:
CHAPTER 7 RNS Superstrings 143 If w
Page 160 and 161:
CHAPTER 7 RNS Superstrings 145 The
Page 162 and 163:
CHAPTER 7 RNS Superstrings 147 From
Page 164 and 165:
CHAPTER 7 RNS Superstrings 149 The
Page 166 and 167:
CHAPTER 7 RNS Superstrings 151 3. A
Page 168 and 169:
CHAPTER 8 Compactifi cation and T-D
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
CHAPTER 9 Superstring Theory Contin
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
CHAPTER 10 A Summary of Superstring
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
CHAPTER 11 Type II String Theories
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
CHAPTER 12 Heterotic String Theory
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
CHAPTER 13 D-Branes One of the most
Page 238 and 239:
CHAPTER 13 D-Branes 223 The Space-T
Page 240 and 241:
CHAPTER 13 D-Branes 225 Once again
Page 242 and 243:
CHAPTER 13 D-Branes 227 and ( X i i
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
Page 258 and 259:
CHAPTER 14 Black Holes 243 Here, G
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 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
CHAPTER 16 String Theory and Cosmol
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Final Exam 1 1. Consider the lagran
Page 296 and 297:
Final Exam 281 0 0 26. Find a simpl
Page 298 and 299:
Final Exam 283 68. When a single sp
Page 300 and 301:
Chapter 1 1. b 6. a 2. a 7. c 3. c
Page 302 and 303:
Quiz Solutions 287 3. i( − ) 4.
Page 304 and 305:
Quiz Solutions 289 Chapter 11 1. c
Page 306 and 307:
1. P 2. E µ 2 ν 1 ( X′ ) X µ
Page 308 and 309:
Final Exam Solutions 293 27. Supers
Page 310 and 311:
Final Exam Solutions 295 73. Types
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
show all

String Theory Demystified

Create successful ePaper yourself

Delete template?

Save as template?