# How do you derive a quantum field theory from a spacetime metric?

What are the first steps in converting a metric into a quantum field theory? I know roughly what to do once I have a pair of non-commuting operators, but how do I get to that point?

Specifically, I'd like to start with the weak-field ($$r \gg r_s$$) low-speed ($$v = \frac{dr}{dt} \ll c$$) limit of the Schwarzschild metric:

$${ds}^2 = -c^2{dt}^2 + {dr}^2 + r^2{d\Omega}^2 + 2\frac{GM}{r}c^2{dt}^2$$

which is flat Minkowski spacetime plus time dilation, and gives Newtonian gravity. (It is amusing to note that Misner-Thorne-Wheeler Gravitation states (in Box 12.2 on p.296) that a "spacetime metric cannot be defined" for Newtonian spacetime. Apparently this metric violates one or more of their assumptions (see exercise 12.10), such as playing nicely with the covariant derivative. This metric is also almost completely ignored in the physics literature; one exception is that Sean Carroll discusses it briefly in Chapter 4 of his online lecture notes on GR, specifically around equations 4.10 to 4.22 on pages 105-106.)

Since all the curvature is in the time dimension, is this likely to run into the "there is no time operator in quantum mechanics" problem?

• Start from perturbing the flat metric $g_{\mu\nu}=\delta_{\mu\nu}+h_{\mu\nu}$. Use the perturbation as the field. – Gradient137 Nov 29 '18 at 19:28
• Also here's a good source: weylmann.com/gravity_waves.pdf It talks about gravitational waves but it's a good starting point. – Gradient137 Nov 29 '18 at 19:30
• Well, in this case the flat metric is the first 3 terms on the right side, which give flat Minkowski spacetime (in spherical coordinates). So the perturbation is just the last term (the time dilation field). – Howard A. Landman Nov 30 '18 at 22:59

## 2 Answers

The first question is: do we assume that there is a single valid QFT from a given metric? This question might be studied by looking at the dynamic process that leads to the metric in question, but also how you match the local metric with the far-away asymptotic Minkowski or De-Sitter vacuum.

Assuming you have figured that out already, you might need to start with the regular scalar Klein-Gordon field and write Bogoliubov transformations that connect the quantum modes in the asymptotic theory you are familiar with (either the Minkowski or De-Sitter far-away spacetime) with the modes in the vicinity of your nontrivial metric features

In principle, you would have to do the same with the Dirac/vector boson fields, but this is typically a complex mathematical procedure, and I've never seen it performed in its entirety

The following paper is a work on this subject of constructing such theories given the background geometry: https://arxiv.org/abs/1407.3612

The standard (QFT) way of quantizing gravity is by applying what is known as the background field method. Here, you write the metric tensor $$g_{\mu \nu}$$ as a sum of some classical background spacetime and a quantum perturbation:

$$g_{\mu \nu} = \bar{g}_{\mu \nu} + h_{\mu \nu}$$

An action (containing $$g_{\mu \nu}$$) is then expanded in a Taylor series of perturbations in $$h_{\mu \nu}$$ around the background. From this expansion, you can read off the free and interaction parts of the gravity theory. You can then use the path integral approach to compute scattering amplitudes. This is how 't Hooft and Veltman showed in their seminal paper, One-loop divergencies in the theory of gravitation, general relativity's renormalizability (or lack, thereof).

This is just perturbation theory and one can do the above metric split in classical GR too, as is done in cosmological perturbation theory. This is what Carroll does; where some constraints are imposed on the metric perturbations to ensure one stays in the low speed, weak gravity regime.

Although the problem of time in quantum gravity is fundamental, it is not related to the issue you mention. Carroll chose to impose specific conditions on metric perturbations in order to stay within the regime mentioned above. One can let go of some (or all) constraints and construct a general theory of perturbations. For example, in Carroll's notes, eq. (6.29) shows the line element that describes another weak field approximation to flat space, but allowing particle velocities to be relativistic (which was not the case before). In this case, we see that there are spatial components to the metric perturbation, in addition to the timelike component we had earlier.