String Theory Demystified

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CHAPTER 6 BRST Quantization 117 It is assumed that Q= Q † . We say that the BRST operator is nilpotent of degree two. This means that if we square the operator we get zero: Q 2 = 0 (6.6) Notice that Eq. (6.6) can also be expressed as { Q, Q } = 0. We label the BRST operator with a Q to imply that this is a conserved charge of the system, we often call this the BRST charge. A second operator that is composed solely of ghost fi elds is called the ghost number operator U. This is given by U c b i = (6.7) (Again note the Einstein summation convention is being used.) This operator has integer eigenvalues. If the dimension of the Lie algebra is n, then the eigenvalues of U are the integers 0,...,n. A state ψ has ghost number m if U ψ = m ψ . EXAMPLE 6.1 Show that Q raises the ghost number by 1. SOLUTION Using the anticommutation relations for the ghost fi elds [Eq. (6.3)], notice that i r i r i i Uc K = c b c K = c −c b K i ∑ r i ∑ δr r r i i r = cK −ccbK i r r i ( ) i i r i i = cK + cK cb = cK + cKU i ∑ i ∑ Now consider some state ψ with ghost number m, that is U ψ = m ψ . Then ⎛ i 1 k i j U( Q ψ ) = U c Ki − fij cc b ⎞ k ψ ⎝ ⎜ ⎠ ⎟ 2 ⎛ i ⎛ r ⎞ 1 k i j ⎞ = Uc Ki − ∑ c br ⎝ ⎜ ⎠ ⎟ ⎝ ⎜ fij cc bk 2 ⎠ ⎟ ⎛ i i 1 = cKi + cKU i − fij ⎝ ⎜ 2 r ∑ r i r ψ k r i j cbccb r k r ⎞ ⎠ ⎟ ψ i r i i i i i ( Use Uc K = c K + c K U ) i i i
118 <strong>String</strong> <strong>Theory</strong> Demystifi ed ⎛ i i 1 k r i i j ⎞ = + − ( − ) ⎝ ⎜cKi cKU i fij ∑ c δr cbr cbk 2 ⎠ ⎟ ψψ [Apply Eq. (6.3)] ⎛ i i 1 = cKi + cKU− f ⎝ ⎜ 2 = + − 1 ⎛ ⎝ ⎜ r ∑ k i j k i r j i ij ccbk + fij c cbcb r k 2 r 1 ⎞ ⎠ ⎟ ψ i i k i j k i r j j cKi cKU i fij ccbk + fij c∑c ( r −c br) bk 2 2 r 1 i (Use the properties of δr ) ⎞ δ ⎠ ⎟ ψ [Apply Eq. (6.3) again] ⎛ i 1 k i r j j ⎞ = Q+ c KU i + fij c c ( δr −c br) bk ⎝ ⎜ ∑ 2 ⎠ ⎟ ψ [Use Eq. (6.5) to write Q] r ⎛ i 1 k i j 1 k i j r ⎞ = Q+ c KU i + fij cc bk − fij cc∑c bkbr ⎝ ⎜ 2 2 ⎠ ⎟ ψ ⎛ i 1 k i j 1 k i j k r ⎞ = Q + cKU i + fij ccbk − fij cc∑ ( δr −bc k ) br ⎝ ⎜ 2 2 ⎠ ⎟ ψ = Q+ c i ⎛ 1 k i j r ⎞ KU i − fij cc c br ⎝ ⎜ ∑ r ⎠ ⎟ ψ 2 = ( Q+ QU) ψ = ( 1+ m)(Q ψ ) Apply U ψ = m ψ r r ( ) Hence the BRST operator raises the ghost number by 1. BRST-Invariant States (Kroneker delta again) [Yet again use Eq. (6.3)] A state ψ is called BRST invariant if it is annihilated by the BRST operator [Eq. (6.5)]: Q ψ = 0 ⇒ ψ is BRST invariant (6.8) BRST-invariant states are the physical states of the theory. Since Q2 = 0, it follows that any state ψ = Q χ ≠0 is BRST invariant, since 2 Q ψ = Q χ = 0
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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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Page 82 and 83: CHAPTER 3 Symmetries and Worldsheet
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Page 104 and 105: CHAPTER 5 Conformal Field Theory Pa
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Page 142 and 143: CHAPTER 7 RNS Superstrings The real
Page 144 and 145: CHAPTER 7 RNS Superstrings 129 EXAM
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Page 168 and 169: 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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