Originally posted by Mentat
I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.
Odd? You mean you find it odd that michio kaku, string theorist and full professor at CCNY who graduated at the top of his class at harvard knows less then marcus? The best people from whom to learn string theory is from string theorists, not marcus or LQG people. Now, I recently addressed this issue in detail when selfAdjoint specifically asked me too. I repost it here for your convenience. If you have any questions, don't hesitate to ask.
Firstly, any correct theory of quantum gravity must in the appropriate limits give rise to helicity-2 excitations of the gravitational field because GR reduces in it's weak-field approximation to the theory of a helicity-2 excitation that couples to itself and everything else in a way that respects the general covariance of GR. In the other direction, such a self-interacting theory implies GR. Now for the hard part:
Transition amplitudes in ST are defined in a 1st quantized formalism based on the world-sheet action
S
G = - (1/4πα′) ∫ dμ
γγ
abG
μν(X)∂
aX
μ∂
bX
ν
in which the basic fields X
μ of the theory embed the world-sheet with metric γ
ab and measure dμ
γ in a background spacetime with metric G
μν. Recall that in QFT the tree level feynman diagram for an interaction consists of a vertex where legs representing incoming and outgoing particles meet. Analogously, for closed strings we have a sphere with punctures to which are glued the ends of "world-tubes" representing incoming or outgoing strings. The invariance, known as
weyl-invariance, of S
G under rescalings γ
ab → e
φγ
ab of the world-sheet metric allows the projection (continuous deformation) of world-tubes onto the punctures, effectively sealing each one by insertion of a point sitting at which is a
vertex operator defined in terms of X
μ and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as the
state-operator correspondence, an example of which is given at the end of this post. Higher order interactions are obtained as compact oriented boundaryless surfaces of genus g with a vertex operator insertion V
i(k
i) for each incoming/outgoing closed string of momentum k
i. Hence, amplitudes for n external string states have the form of a sum of path-integrals with insertions
<V
1(k
1)⋅⋅⋅V
n(k
n)> ~ ∑
g=0,1,2,... ∫
g Dγ
abDX
μ V
1(k
1)⋅⋅⋅V
n(k
n)e
-SG.
Now, take
G
μν(X) = η
μν + ε
μν(X)
with
ε
μν(X) = ∫ d
26k ε
μν(k)e
ik⋅X
everywhere small compared to η
μν. Then
e
-SG = e
-(Sη + Sε) = e
-Sη ∑
n=0,1,...(- 4πα′)
-n(1/n!) ∫ d
26k
1⋅⋅⋅d
26k
n V(k
1)⋅⋅⋅V(k
n)
in which
V(k) ≡ ε
μν(k)V
μν(k) ≡ ε
μν(k) ∫ dμ
γ γ
ab∂
aX
μ∂
bX
νe
ik⋅X
is a vertex operator coupling strings to fluctuations in the background metric G
μν. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point. Next, observe that ε
μν picks out the symmetric part of V
μν, so V is the vertex operator of a spin-2 state. Also, since the state-operator correspondence (see the example at the end of this post) requires that vertex operators transform like the string state vectors they represent, they must include the factor e
ik⋅X to transform properly under spacetime translations X
μ → X
μ + a
μ. Now, any insertion must respect the
local weyl symmetry of the theory. In particular, demanding that V be weyl-invariant requires (see polchinski I Chap 3.6)
k
2 = k
2ε
μν(k) = 0 ↔ ⇑ε
μν(X) = ⇑G
μν(X) = 0
k
με
μν(k) = 0 ↔ ∂
με
μν(X) = ∂
μG
μν(X) = 0,
ε
μμ(k) = 0 ↔ ε
μμ(X) = 0.
In addition to showing that the spin-2 excitations are massless, because the ricci tensor R
μν satisfies
R
μν ∝ ∂
μ∂
νε
λλ - 2∂
λ∂
(με
μ)λ + ⇑ε
μν + O(ε
2),
this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G
μν satisfy the vacuum einstein equations R
μν = 0.
Because massless states are transversally polarized, V must be invariant under the shift
ε
μν(k) → ε
μν(k) + k
μξ
ν + k
νξ
μ
by longitudinal polarizations. In terms of the metric, this gauge-invariance
ε
μν(X) → ε
μν(X) + k
μξ
ν(X) + k
νξ
μ(X)
is an infinitesimal diffeomorphism generated by the vector field ξ
μ(X) in the approximation where O(ε
2) terms are neglected and under which R
μν = 0 is invariant. In fact R
μν = 0 is the only spacetime diffeo-invariant equation that reduces to ⇑G
μν(X) = 0 in the linearized limit.
In sum, weyl-invariance requires spin-2 excitations be massless and couple in a gauge-invariant way, that is, it requires the general covariance of GR, justifying the interpretation of helicity-2 excitations as gravitons.
State-operator correspondence for the graviton vertex operator:
Define world-sheet coordinates
z = e
-iσ + τ , z* = e
iσ + τ
with σ = σ + 2π the periodic coordinate along the string and τ the time coordinate on the world-sheet. We then have
V ∝ ε
μν∫d
2z ∂
zX
μ(z)∂
z*X
ν(z*)e
ik⋅X(z,z*)
in which we've taken the world-sheet metric in "conformal gauge" so that it effectively drops out. Then (up to proportionality) the state-operator correspondence is
∂
zX
μ(0) ↔ α
-1μ , ∂
z*X
μ(0) ↔ (α
-1μ)* , e
ik⋅X(0,0) ↔ |0;0;k>
where α
-1μ and (α
-1μ)* excite left- and right-moving n = 1 modes.
Putting these together gives
V ↔ ε
μνα
-1μ(α
-1ν)*|0;0;k>.
). However, you have left me torn. On the one hand, I have been interested in and awed by string theory since I first heard of it. On the other hand, I strive to remain open-minded, and so I want to learn as much as I can about LQG as well, since it seems to be the "competition" for TOE. 
