String Theory Demystified

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Final Exam 1 1. Consider the lagrangian for a classical string L = X 2 µ 2 µ 2 X′ − ( X X µ µ ′ ) . Write down the canonical momentum. 2πα ′ 2. Consider a classical string which is in a confi guration described by a rigid rod rotating about the origin with angular velocity ω . Find the energy of the 1 l 1 string from E = d ′ ∫ σ 2 −l 2 2 πα 1− ωσ . 3. Consider the action for a p-brane with cosmological constant T p+ 1 αβ p+ 1 S =− ∫d σ −hh ∂ X⋅∂ X + Λ ∫d σ −h. α β Find the classical equations 2 of motion for the metric. 4. Using the action of the previous problem, fi nd a constraint on the µν cosmological constant Λ using h h = p+1. 5. Consider the classical string, describing its dynamics using the Polyakov action. What form does the action take if the worldsheet metric is taken to be fl at? 6. Using the action of Prob. 5, what is the canonical momentum? Copyright © 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use. µν
280 <strong>String</strong> <strong>Theory</strong> Demystifi ed ∞ 2 7. The mass of a classical open string is M = ∑ ⋅ −n n α ′ n= 1 of a closed string different? 1 α α . How is the mass 8. Consider the classical string. What is the algebra satisfi ed by the Virasoro generators? ∞ 1 9. Using the normal ordering prescription, L = m ∑ : α ⋅α : fi nd L m−n n 0. 2 n=−∞ 10. What is the mass-shell condition for states of the bosonic string, written in terms of Virasoro operators? In Probs. 11–14, consider the bosonic string. 11. Find the angular momentum operators J µν . µ ρλ 12. Using the result of Prob. 11, fi nd ⎡⎣ p0, J ⎤⎦ . µν ρλ 13. Find [ J , J ] . µν 14. For the Virasoro operator Lm , fi nd [ L , J ] . 15. Consider the fi rst excited state of the open bosonic string. Let ξ µ be a polarization vector and consider the action of L1 on the state ξ⋅α 0, . −1 k What condition on the polarization and momentum follows from the Virasoro constraint L1 ψ = 0 for physical states ψ ? 16. What condition cancels the conformal anomaly for the bosonic string? 17. Consider the Polyakov action in the conformal gauge. State the constraints on the components of the energy momentum tensor. 18. State the Neumann boundary conditions for classical, open bosonic strings. 19. Consider a relativistic point particle with space-time coordinates x µ ( τ) and action S =−m∫d −x x τ µ µ µ dx , where x = . Use the usual variational µ dτ procedure to fi nd the equations of motion, and write down these equations in terms of the conjugate momentum. 20. Consider your solution to Prob. 19. What is the condition on p µ ? 0 21. Consider your solution to Prob. 19. Take the static gauge, where x = t. What is the action in this case if we use the usual defi nition of the particle velocity dx v = ? dt 22. What is the momentum in this case (using the action from Prob. 21)? 23. How do the spinors ψ µ in the RNS formalism transform under Lorentz transformations? In problems 24–26, let Γ µ be a gamma matrix in D = 10 space-time dimensions and following the text let Γ = Γ Γ Γ . 11 0 1 9 µ 24. Calculate { Γ , Γ } . 11 0 25. Calculate Γ11Γ . m
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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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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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Page 254 and 255: CHAPTER 14 Black Holes Black holes,
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Page 264 and 265: CHAPTER 14 Black Holes 249 Entropy
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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
Page 320 and 321: INDEX 305 Schwarzschild metric, 242
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String Theory Demystified

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