String Theory Demystified

280 String Theory 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