String Theory Demystified

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CHAPTER 7 RNS Superstrings 129 EXAMPLE 7.1 Show that the Dirac matrices on the worldsheet obey an anticommutation relation known as the Dirac algebra by explicit computation. SOLUTION This result is easy to verify. Since α β αβ { ρ , ρ } =−2 η (7.4) η αβ = − ⎛ 1 0⎞ ⎝ ⎜ 0 1⎠ ⎟ The Dirac algebra will be satisfi ed by the ρ α if the following relations hold: 1 0 and likewise for { ρ , ρ } . Now, 0 0 ⎛ 0 ρρ = ⎝ ⎜ i 0 0 0 0 0 0 0 0 00 { ρ, ρ} = ρρ + ρρ = 2ρρ =− 2η= 2I 1 1 1 1 1 1 1 1 11 { ρ , ρ } = ρ ρ + ρρ = 2ρρ =− 2η=−2I 0 1 0 1 1 0 { ρ , ρ } = ρ ρ + ρ ρ =−2ηη 01 −i⎞ ⎛ 0 0 ⎠ ⎟ ⎝ ⎜ i = 0 − ⎞ 0 0 0 0 ⎠ ⎟ = i ⎛ ⋅ + ( −i)⋅i ⋅( −i) + ( −i)⋅0⎞⎛1 0⎞ ⎜ ⎝ ⋅ + ⋅ ⋅( − )+ ⋅ ⎟ = i 0 0 i i i 0 0 ⎠ ⎝ ⎜ 0 1 ⎠ ⎟ = I 0 0 Hence the fi rst relation { ρ , ρ } = 2I is satisfi ed. We verify that the second 1 1 relation { ρ , ρ } =−2I is also satisfi ed: ρρ = ⎛ ⎝ ⎜ i 1 1 0 Finally, noting that ⎛ ρρ = ⎝ ⎜ i 0 1 0 1 0 ⎛ 0 ρρ = ⎝ ⎜ i ⎞ 0 0⎠ ⎟ ⎛ i ⎝ ⎜ i 0 1 1 0 we see that { ρ , ρ } = { ρ , ρ } = 0. ⎞ 0 0 0 0 0⎠ ⎟ 0 = i ⎛ ⋅ + i⋅i ⋅ i+ i⋅ ⎞ ⎛ − ⎞ ⎝ ⎜ i ⋅ + ⋅ ⋅ + ⋅ ⎠ ⎟ = ⎝ ⎜ − ⎠ ⎟ =− 1 0 I 0 i i i 0 0 0 1 − ⎞ ⎛ 0 0 ⎠ ⎟ ⎝ ⎜ ⎞ 0 0 0⎠ ⎟ 0 = i i i ⎛ ⋅ + ( −i)⋅i ⎜ ⎝ ⋅ + ⋅ ⋅ i+ ( −i)⋅⎞ ⋅ + ⋅ ⎟ ⎠ = i 0 0 i 0 ⎛ 1 i i 0 0 ⎝ ⎜ 0 0⎞ −1⎠ ⎟ i ⎞ ⎛ 0 0⎠ ⎟ ⎝ ⎜ i −i⎞ 0 0 i i 0 ⎠ ⎟ i 0 0 i i 0 = ⎛ ⋅ + ⋅ ⎜ ⎝ ⋅ +0⋅ ⋅( − )+ ⋅ ⎞ −1 ⋅( − )+ 0⋅0 ⎟ = ⎠ 0 0 1 ⎛ i i i ⎝ ⎜ ⎞ ⎠ ⎟
130 <strong>String</strong> <strong>Theory</strong> Demystifi ed MAJORANA SPINORS µ µ The fi elds introduced in the action, ψ = ψ ( σ, τ), are two-component Majorana spinors on the worldsheet. Given that they have two components, they are sometimes written with two indices ψ µ , where A µ = 01 , , …, D −1is the space-time index and A =± is the spinor index. We can write ψ µ as a column vector in the A following way (suppressing the space-time index): ψ ψ = ψ ⎛ ⎞ − ⎝ ⎜ ⎠ ⎟ Under Lorentz transformations, these fi elds transform as vectors in space-time µ [recall that a contravariant vector fi eld V ( x) is one that transforms as µ µ µ ν µ µ µ ν µ V ( x) → V′ ( x′ ) =Λ V ( x) under ν x → x′ =Λ x where Λ is a Lorentz ν ν transformation]. Following the convention used with Dirac spinors in quantum fi eld theory, we have the defi nition: + µ † µ 0 ψ = ( ψ ) ρ Note that the defi nitions used here depend on the basis used to write down the Dirac matrices Eq. (7.3), and that other conventions are possible. We can also introduce a matrix you’re familiar with from studies of third Dirac matrix analogous to the γ 5 the Dirac equation, which in this context we denote by ρ 3 : 3 0 1 ⎛ 1 0⎞ ρ = ρ ρ = ⎝ ⎜ 0 −1⎠ ⎟ It will be of interest to make left movers and right movers manifest. This can be done by recalling the following defi nitions: EXAMPLE 7.2 µ α Show that ψ ρ ∂ ψ = 2( ψ ⋅∂ ψ + ψ ⋅∂ ψ ). α µ ± σ = τ ± σ (7.5) 1 ∂ = ( ∂ ±∂ ) ± 2 τ σ (7.6) ∂ =∂ + ∂ ∂ =∂ −∂ (7.7) τ + − σ + − − + − + − +
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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 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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