Difference between the role of a non-flat metric and the graviton in string theory

Question

To preface, I am not after a highly involved answer, this is definitely just a conceptual point of confusion.

In string theory the target spacetime into which the string is embedded can have a non-flat metric and this non-flat background metric describes the background gravitational field. So to my thinking, any particle placed in this background metric simply follows the geodesics determined by the metric. However, we also have a particle that governs the gravitational interaction, the graviton.

My problem is reconciling these two points, if the graviton mediates the gravitational interaction, what is the purpose of having a non-flat background metric, and vice versa, if the background metric determines the behaviour of particles under gravity, why would you also have a new particle that mediates the gravitational interaction?

A non-technical explanation is definitely sufficient here.

Answer 1

The bottom line here is that the background metric in the target spacetime and the gravitons arising from the spectrum of the (let's say, bosonic) string are not independent concepts. Any sigma model into a (pseudo-)Riemannian manifold (like string theory) will obviously have a background metric that can be varied (but don't conflate this with being dynamical). On the other hand, through the standard analysis of the Polyakov action in flat space, you can show that the bosonic string contains within its spectrum a massless spin-2 particle which must be the graviton and these, perturbatively, build up the metric deviations. This is not unique to string theory: gravitons will arise out of any perturbative treatment of quantum gravity, at least at low energy scales, and should reduce to the "field picture" of general relativity. In fact, this is not unique to gravity per se: the electromagnetic interaction can be described by virtual photon exchange and alternatively, as a "classical" approximation, by the interaction with the electromagnetic field (which is "built up" by the photons).

Here's how it works: when you expand the background metric as a small perturbation about (Euclidean) flat space: to first order, $G_{\mu\nu} = \delta_{\mu\nu} + h_{\mu\nu}$, the Polyakov action in the partition function splits as:

$$ Z = \int\mathcal{D}X\ \mathcal{D}g \ \exp\{-S_\text{curved}\}=\int\mathcal{D}X\ \mathcal{D}g \ \exp\left\{-\frac{1}{4\pi\alpha'}\int\mathrm{d}^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu G_{\mu\nu}\right\} \\=\int\mathcal{D}X\ \mathcal{D}g \ \exp\left\{-\frac{1}{4\pi\alpha'}\int\mathrm{d}^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu(\delta_{\mu\nu}+h_{\mu\nu}(X))\right\} \\=\int\mathcal{D}X\ \mathcal{D}g \ \exp\left\{-\frac{1}{4\pi\alpha'}\int\mathrm{d}^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu\delta_{\mu\nu}\right\}\times\\ \exp\left\{-\frac{1}{4\pi\alpha'}\int\mathrm{d}^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu h_{\mu\nu}(X)\right\} \\=\int\mathcal{D}X\ \mathcal{D}g \ \exp\{-S_\text{flat}\}\exp\{-V\} $$ where $V$ is the vertex operator corresponding to the graviton state of the bosonic closed string. If you don't know what a vertex operator is, think of it (at least in the context of scattering amplitudes) as an insertion of an external state of the scattering process that you want to calculate - e.g. you might insert vertex operators for the tachyon if you wanted to compute tachyon scattering amplitudes. In this case, the presence of $\exp\{-V\}$ for the graviton dictates that, in addition to the existing external in/out states, we should insert a (coherent, due to the exponential) superposition of plane-wave like graviton modes as an external state. Accordingly, when these graviton vertex operators are integrated over the worldsheet, it modifies the scattering amplitudes that are to be calculated in the same way that a perturbed background metric would. So altering the background metric is equivalent to adding external (on-shell) graviton states, which interact with the other strings at finite time.

A more authoritative (but more involved) reference is Luboš' answer here. Another good picture is offered in this answer.

Answer 2

To give a slightly different perspective: there is nothing in this question that is particularly about string theory. We can consider, for example, that on the one hand that photons are elementary excitations of the electromagnetic field, and on the other hand that any negatively charged particle carries with it a background electric field in the form of the Coulomb potential. At first glance, these appear to be two incompatible ways of understanding the electromagnetic field.

In fact, we can understand the Coulomb potential as being a coherent state of photons that surround the electron.

In single-particle quantum mechanics, a coherent state $|\alpha\rangle$ is an eigenstate of the annihilation operator \begin{equation} a |\alpha\rangle = \alpha | \alpha \rangle \end{equation} It corresponds to a state that is "very classical" in the following ways:

• The state contains many-particle excitations; indeed it is a superposition of particle states involving states with arbitrarily large numbers of particles \begin{equation} |\alpha\rangle = e^{\alpha a^\dagger-\alpha^\star a} | 0 \rangle = e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle \end{equation}

• The mean position and momentum in a coherent state oscillate just as in the classical solution of the equations of motion. Writing $\alpha=|\alpha|e^{i\varphi}$, we have \begin{eqnarray} \langle \alpha | x | \alpha \rangle &=& |\alpha|\sqrt{\frac{2 \hbar }{m \omega}}\cos(\omega t - \varphi) \\ \langle \alpha | p | \alpha \rangle &=& -|\alpha|\sqrt{2 \hbar \omega m}\sin(\omega t - \varphi) \\ \end{eqnarray}

The idea of a coherent state is generalized to field theory by promoting the creation and annihilation operators as well as $\alpha$ to be functions of the photon momentum and spin, and by replacing the position and momentum of a particle with the field operator and its conjugate momentum.

In this way, the background Coulomb electromagnetic field surrounding a charged particle can be understood as a coherent state of photons. Indeed, any background electromagnetic field is really a coherent state.

Similarly, in gravity, the space-time metric should be thought of as being a coherent state built up out of graviton excitations.

However, in gravity, this is a more conceptually tricky than for your average field, because the metric also has to provide a background spacetime. In other words: we can start off by writing the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ as a background Minkowski spacetime and then a graviton $h_{\mu\nu}$, and we can think of a non-trivial geometry as being a coherent state of gravitons. (This is more or less what the other answer in terms of vertex operators is doing in a different language). However, since GR doesn't depend on any prior geometry, there should be some way to formulate this idea without needing to start with a background Minkowski spacetime $\eta_{\mu\nu}$; the background spacetime should itself be a coherent superposition of gravitons. In other words, a full quantum theory of gravity should tell us how spacetime emerges from a deeper picture, in terms of gravitons or something else; there are many ideas for how this might work, but at the moment we don't know the full story.