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I've been studying Quantum Field Theory in Curved Spacetimes through Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. On Section 4.7, he discusses how to generalize the theory built so far for a real scalar field to other quantum fields. He mentions that a straightforward generalization will work for any real, bosonic, linear field provided that

  1. it has a well-posed initial value problem;
  2. it is derivable from a Lagrangian.

He then mentions that well-posedness is not trivial to satisfy, since the straightforward generalizations to curved spacetime of fields of spin $s > 1$ are not well-posed (this is shown in pages 374-375 of Wald's General Relativity). Quoting page 375 of General Relaivity, "for $s > 1$ there is no natural generalization to curved spacetime of the notion of a 'pure' massless spin $s$ field".

What I find particularly surprising in these remarks is that linearized gravity is described by a spin $s = 2$ field. Hence, if I've read these statements correctly, they imply that one cannot describe the "propagation of free gravitons" on curved backgrounds (where I use quotation marks because I do not mean to imply a particle interpretation, but rather a quantized perturbation).

In summary, is it possible to describe quantized linear perturbations of the gravitational field in a curved background?

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If you are happy to treat GR as the low energy approximation of a full quantum theory of gravity, then the procedure is conceptually straightforward (but can quickly become technically very difficult).

  1. Split the metric into a background plus perturbation, $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}.$
  2. Expand the Einstein-Hilbert action to quadratic order in $h_{\mu\nu}$ (or higher order if you want to look at more complicated diagrams).
  3. Quantize the theory for $h_{\mu\nu}$ in your favorite formulation of quantum mechanics.

This approach can be used, for example, to compute Hawking radiation into gravitons by looking at metric fluctuations around a Schwarzschild background, or the power spectrum of primordial gravitational waves by looking at quantized metric fluctuations around a quasi-de Sitter background during inflation. (For completeness, I'll add that a very important subtlety when doing calculations with quantum theory on curved spacetime -- including both Hawing radiation and inflationary perturbations -- is that there is no obvious privileged vacuum state (unlike in flat spacetime where there is a Poincaire invariant ground state), and so you need to choose an appropriate state to call the vacuum state; however this is not specific to spin-2 fields)

If we expand the action to second order in $h$, then the Lagrangian has the form $\frac{1}{2} h_{\mu\nu} \bar{\mathcal{E}}^{\mu\nu\rho\sigma} h_{\rho\sigma}$, where $\bar{\mathcal{E}}^{\mu\nu\rho\sigma}$ is a second-order differential operator depending on the background metric $\bar{g}_{\mu\nu}$. The propagator (in some gauge) is the inverse of $\bar{\mathcal{E}}$. In the geometric optics limit (when the wavelength of the perturbation is small compared to the curvature, you will find that the solutions to the classical equations of motion follow null geodesics.

The tricky part is going in the opposite direction. In other words, if you don't already know the Einstein-Hilbert action, and you want to construct a theory of a spin-2 field, where would you start? In flat spacetime, we find unitary representations of the Poincaire group which are labeled by mass and spin. Once you do that, it's possible to look for consistent interacting theories of different representations, and you find the only self-consistent interacting theory of a massless-spin 2 particles (at low energies) is GR. It's much harder to define what this procedure means, when you don't have Poincaire invariance in the background spacetime.

One approach that I like is Einstein-Cartan gravity (i.e., using the vielbein instead of the metric as the basic variable), where you can define a local Lorentz frame, and then the spin-2 field falls in the spin-2 representation of the Poincaire group in each local frame. In this approach, you can view gravity as being analogous to Yang-Mills, where the gauge group is the Poincare group, instead of $SU(N)$. (However, I am sure Wald was well aware of the Einstein-Cartan formulation at the time he wrote the passage you quoted).

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    $\begingroup$ @NíckolasAlves I think that's accurate. But even if it isn't straightforward, the "right answer" is still that it has to be GR. There are a lot of papers by Deser about this, here is an example: authors.library.caltech.edu/7595/1/DEScqg07.pdf $\endgroup$
    – Andrew
    Aug 8, 2021 at 10:24
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    $\begingroup$ @NíckolasAlves Another key point is that the background should be on shell. If you don't have matter present, then this means $G_{\mu\nu}=0$, which implies $R_{\mu\nu}=0$ (for the background). However you always can add matter if you want to consider more general backgrounds (for inflation you need to add an inflaton field to source the background expansion). $\endgroup$
    – Andrew
    Aug 8, 2021 at 10:27
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    $\begingroup$ "the only self-consistent interacting theory of a massless-spin 2 particles (at low energies) is GR" Why do generalizations such as $f(R)$, non-zero torsion etc. fail? Or are they precluded by our wanting mass/spin labels for Poincaré representations? $\endgroup$
    – J.G.
    Aug 8, 2021 at 12:56
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    $\begingroup$ @J.G. $f(R)$ can be rewritten by a change of variables in the "Einstein frame", where the gravity sector takes the form of the Einstein-Hilbert action and a scalar field with a normal kinetic term and potential. (The "cost" is that the matter is not minimally coupled to the Einstein-frame metric, but to an effective metric built out of a combination of the scalar field and the Einstein-frame metric). (...) $\endgroup$
    – Andrew
    Aug 8, 2021 at 17:11
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    $\begingroup$ So one way of expressing the difference is that $f(R)$ is not just a theory of a massless spin-2 particle, but contains another scalar degree of freedom. $\endgroup$
    – Andrew
    Aug 8, 2021 at 17:14

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