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String Theory Demysti ed
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that bound a higher-dimensional bulk. This idea was extended in the ekpyrotic model, which allows the branes to move and collide, explaining the big bang and providing a string theory alternative to in ation. The ekpyrotic scenario does not say the big bang never happened, rather it explains the big bang without evoking a singularity. Once the brane collision has occurred, the universe evolves according to standard big-bang theory on the branes.
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1. Let g = det g , wherge g is the Kasner metric. Using the rst Kasner condition, nd an expression for g . 1 2. Suppose that p j = for all j in the Kasner metric. Is the second Kasner D 1 condition satis ed 3. Consider the metric derived in the text: t ds 2 = dt 2 + a 1 t
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2 p
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t D 1 2 3 dxi2 + b 1 t 4 dxm m= i=1
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Show that the Kasner conditions are satis ed.
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1 2 X X 2 ( X X )2 . 1. Consider the lagrangian for a classical string L = 2 Write down the canonical momentum. 2. Consider a classical string which is in a con guration described by a rigid rod rotating about the origin with angular velocity . Find the energy of the l 1 1 string from E = l d 1 2 2 . 2 3. Consider the action for a p-brane with cosmological constant T S = d p +1 h h X X + d p +1 h . Find the classical equations 2 of motion for the metric. 4. Using the action of the previous problem, nd a constraint on the cosmological constant 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 at 6. Using the action of Prob. 5, what is the canonical momentum
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String Theory Demysti ed
1 7. The mass of a classical open string is M 2 = n n . How is the mass n =1 of a closed string different 8. Consider the classical string. What is the algebra satis ed by the Virasoro generators 1 9. Using the normal ordering prescription, Lm = : m n n : nd L0 . 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 .
12. Using the result of Prob. 11, nd p0 , J . 13. Find [ J , J ] . 14. For the Virasoro operator Lm , nd [ Lm , J ]. 15. Consider the rst excited state of the open bosonic string. Let be a polarization vector and consider the action of L1 on the state 1 0, k . What condition on the polarization and momentum follows from the Virasoro constraint L1 = 0 for physical states 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 ( ) and dx . Use the usual variational action S = m d x x , where x = d procedure to 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 21. Consider your solution to Prob. 19. Take the static gauge, where x 0 = t . What is the action in this case if we use the usual de nition of the particle dx velocity v = dt 22. What is the momentum in this case (using the action from Prob. 21) 23. How do the spinors in the RNS formalism transform under Lorentz transformations
In problems 24 26, let be a gamma matrix in D = 10 space-time dimensions and following the text let 11 = 0 1 9. 24. Calculate { 11 , }. 25. Calculate 11 0 .
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