String Theory Demystified

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CHAPTER 7 RNS Superstrings The real world as we know it described by the standard model of particle physics contains two general classes of particles. These are defi ned in terms of their spin angular momentum as follows. Those particles with whole integer spin are called bosons, while those with half-integer spin are called fermions. At the level of fundamental particles—electrons, neutrinos, quarks, photons, gauge bosons, and gluons—matter particles are fermions while force-mediating particles are bosons. A symmetry which relates fermions and bosons is called a supersymmetry. The string theory we have described so far in the book consists only of bosons. Obviously, this cannot be a realistic theory that describes our universe since we see fermions in the everyday world. This tells us that the theory we have developed so far, starting with the Polyakov action, is not the whole story. The theory must be extended to include fermions. In addition, recall that when we quantized the theory, the ground state was a tachyon—a particle that travels faster than the speed of light. These states with negative mass squared are physically unrealistic. More importantly, a quantum theory with a tachyon has an unstable vacuum. The remedy to this situation is to introduce supersymmetry into the theory. This will allow us to develop string theory so that fermions are included in our description of nature. We will also see that the unwanted tachyon state goes away. An interesting Copyright © 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
128 String Theory Demystifi ed side effect of this effort will be that the critical dimension drops from 26 to 10. If you aren’t familiar with the description of fermionic fi elds (and supersymmetry) in quantum theory, try reviewing your favorite quantum fi eld theory book before tackling this chapter (Quantum Field Theory Demystifi ed provides a relatively painless introduction). The Superstring Action We can proceed with a straightforward modifi cation of the theory to include fermions using an approach called the Ramond-Neveu-Schwarz (RNS) formalism. This approach is supersymmetric on the worldsheet. Later we consider the Green- Schwarz formalism, which is supersymmetric in space-time. When the number of space-time dimensions is 10, these two approaches are equivalent. The program we will follow can be done using basically the same which was applied in the bosonic case: introduce an action, fi nd the equations of motion, and quantize the theory. However, this time we are going to include fermionic fi elds on the worldsheet. We start with the Polyakov action, fi rst described in Eq. (2.27) and reproduced here in the conformal gauge: T 2 µ α S =− ∫ d σ ∂ X ∂ X α µ 2 (7.1) To include free fermions in the theory using the RNS formalism, we add a kinetic energy term for a Dirac fi eld to the lagrangian. That is, we include D free fermionic fi elds ψ µ to the action, so that it assumes the form ( α µ ) T 2 µ α µ α S =− ∫ d σ ∂ X ∂ X −iψ ρ ∂ ψ α µ 2 (7.2) Again, if you are not familiar with Dirac fi elds, consult Quantum Field Theory Demystifi ed or your own favorite quantum fi eld theory text. The ρ α are Dirac matrices on the worldsheet. Since the worldsheet has 1 1 dimensions, the ρ α are Dirac matrices in 1 1 dimensions. Hence there are two such 2× 2 matrices, which can be written in the form using an appropriate choice of basis. 0 ⎛ 0 − ⎞ 1 0 ρ = ρ ⎝ ⎜ 0 ⎠ ⎟ = ⎛ i i ⎝ ⎜ i i ⎞ 0⎠ ⎟ (7.3)
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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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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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