String Theory Demystified

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CHAPTER 2 Equations of Motion 49 In going from ordinary classical mechanics to quantum theory, we follow Dirac and use the correspondence between the Poisson brackets and commutators. In string theory, we consider equal τ Poisson brackets as our starting point when we quantize the modes on the strings. Later, we will discuss hamiltonians and the Virasoro algebra, an important concept in string theory, but for now we simply introduce the important Poisson brackets which will allow us to quantize the string. First we note that µ ν { X ( τσ , ), X 1 ( τσ , ′ )} = δσ ( − σ′ ) η (2.67) µν T or, in terms of momentum µ ν { P ( τσ , ), X ( τσ , ′ )} = δσ ( − σ′ ) η (2.68) The Fourier expansion can be used to derive Poisson brackets for the modes. These are µ ν { αm, αn}=− imδm+ n, σ ηµν (2.69) The Poisson brackets will be the starting point to quantize the theory. Quiz 1. Consider the lagrangian given in the Eq. (2.10) which includes the auxiliary fi eld. Use the Euler-Lagrange equations to derive the equation of motion. 2. Start with the Nambu-Goto lagrangian L = XX 2 2 ′ − XX 2 ( ⋅ ′ ) and consider the gauge choice which gives the fl at metric h . Using the constraints αβ 2 2 X + X′ = 0, X ⋅ X′ = 0, fi nd the equations of motion. 3. Consider the Polyakov action. A Weyl transformation is one of the form φ( τ, σ) h → e h and δ αβ αβ µ X = 0. Determine the form of the Polyakov action αβ − φ( τ , σ ) αβ under a Weyl transformation [hint: h → e h ]. 4. Using the Polyakov action, defi ne the energy-momentum tensor on the worldsheet by varying the action with respect to the intrinsic metric T αβ 2 1 δS =− T −h δh P αβ µν Poisson Brackets
50 <strong>String</strong> <strong>Theory</strong> Demystifi ed 1 αβ Using the fact that δ − h =− −hh δh fi nd an expression for the αβ 2 induced metric so that it can be eliminated from the action. This should allow you to recover the Nambu-Goto action. 5. Consider an open string with free endpoints. Find the variation of the center of mass position of the string with τ by calculating 1 π µ σ τ σ π ∫ d X ( , ) 0 6. Find the conserved momentum of the open string with free endpoints by calculating π µ µ P T dσX ( τ, σ) = ∫ 0 where the dot represents differentiation with respect to τ.
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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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Page 68 and 69: CHAPTER 3 Symmetries and Worldsheet
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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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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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