Lectures on String Theory

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Exercise 6. Show that the Polyakov action is invariant under reparametrizations
δX µ = ξ α ∂ α X µ
δh αβ = ∇ α ξ β + ∇ β ξ α
δ( √ h) = ∂ α (ξ α√ h)
Exercise 7. Show that the Weyl invariance implies the tracelessness of the stressenergy
tensor T αβ .
Exercise 8. Show that the Gauss-Bonnet term
χ = 1 ∫
d 2 σ √ hR
4π
is topological, i.e. it vanishes under smooth variations of the world-sheet metric h αβ .
Take into account that in 2dim the Ricci tensor is proportional to Ricci scalar and
also
δ( √ (
hR) ∼ R αβ − 1 )
2 h αβR δh αβ .
Exercise 9. Let S(q, t; q 0 , t 0 ) be the action of the classical path between (q 0 , t 0 )
and (q, t). Show that
∂S
∂q = p(t) ,
where p(t) is the conjugate momentum of q at time t. Show that
∂S
∂t
= −H(q,
∂S
∂q ) ,
where H is the Hamiltonian. Suppose that H(q, p) = p2
+ V (q) and define
2m
ψ(q, t) = e i ~ S(q,t;q 0,t 0 ) .
Show that the schrödinger equation approximately holds for ψ,
i ∂ψ (q,
∂t = H −i ∂ )
ψ + O() .
∂q
This is of course related to Dirac’s idea that the phase of the wave function is proportional
to the classical action.