Quantum Gravity

284 STRING THEORYD − 2 = 24 degrees of freedom. Since for N = 1 one has M 2 = 0, this statedescribes a massless vector boson (a ‘photon’). In fact, it turns out that hadwe chosen D ≠ 26, we would have encountered a breakdown of Lorentz invariance.Excited states for N>1 correspond to massive particles. They are usuallyneglected because their masses are assumed to be of the order of the Planckmass—this is the mass scale of unification where string theory is of relevance(since we expect apriorithat l s ∼ l P ).We emphasize that here we are dealing with higher-dimensional representationsof the Poincaré group, which do not necessarily have analogues in D =4.Therefore, the usual terminology of speaking about photons, etc., should not betaken literally.For the closed string one has the additional restriction L 0 = ˜L 0 ,leadingto∞∑∞∑α −n α n = ˜α −n ˜α n .n=1The ground state is again a tachyon, with mass squared M 2 = −4/α ′ . The firstexcited state is massless, M 2 = 0, and described by|e, p ν 〉 = e µν α µ −1 ˜αν −1 |0,pν 〉 , (9.25)where e µν is a transverse polarization tensor, p µ e µν = 0. The state (9.25) can bedecomposed into its irreducible parts. One thereby obtains a symmetric tracelesstensor, a scalar, and an antisymmetric tensor. The symmetric tensor describesa spin-2 particle in D = 4 and can therefore—in view of the uniqueness featuresdiscussed in Chapter 2—be identified with the graviton. It is at this stage thatstring theory makes its first contact with quantum gravity. The perturbationtheory discussed in Chapter 2 will thus be implemented in string theory. But aswe shall see in the next section, string theory can go beyond it.The scalar is usually referred to as the dilaton,Φ.InD = 4, the antisymmetrictensor has also spin zero and is in this case called the axion. The fact that masslessfields appear in the open- and the closed-string spectrum is very interesting. Boththe massless vector boson as well as the graviton couple to conserved currentsand thereby introduce the principle of gauge invariance into string theory. Higherexcited states also lead for closed strings to massive (‘heavy’) particles.Up to now we have discussed oriented strings, that is, strings whose quantumstates have no invariance under σ →−σ. We note that one can also havenon-oriented strings by demanding this invariance to hold. For closed stringsthis invariance would correspond to an exchange between right- and left-movingmodes. It turns out that the graviton and the dilaton are also present for nonorientedstrings, but not the axion.9.2 Quantum-gravitational aspects9.2.1 The Polyakov path integralWe have seen in the last section that the graviton appears in a natural way inthe spectrum of closed strings. Linearized quantum gravity is, therefore, auto-n=1