I have just posted on arXiv yet another preprint about Newtonian semi-classical gravity. It is not really an improvement from previous work but rather an easy introduction to the idea and the concepts, using a mathematically simpler basis.
Let me recall how I understand the problem of semi-classical gravity. If a semi-classical theory of gravity is to be fundamental, it has to specify two things:
- How quantum matter moves in a given classical curved space-time.
- How quantum matter curves this classical space-time.
The first point is I think pretty standard. In the general relativistic context it is answered by quantum field theory in curved space time but in the Newtonian limit, being so fancy is not even necessary. To specify the influence of space-time on quantum matter, one just puts the gravitational field as an external field in the Schrödinger equation. The tricky part is the second aspect: how quantum matter back-reacts on space-time. This problem is basically open. In the Newtonian limit, it means finding a good candidate for the gravitating mass density.
To construct a (classical) mass density field from quantum matter, a tempting thing is to do is just to project down to real space some properties of the wave function (that lives in configuration space). This is the historical idea but this is a bad idea. Taking for example 
(where 
is the mass density operator) as a candidate for the mass density gives a gravitational field depending quadratically on the wave function. So in the end, the Schrödinger equation contains linear but also cubic terms in the wave function. And with that, all hell breaks loose and the quantum mechanical toolbox no longer works.
Basically, all the ways to construct a gravitational field from the wave function directly will add a deterministic non-linearity in the Schrödinger equation. So what classical object can one construct in quantum mechanics that does not create problems? Within the orthodox formalism, there is one clear candidate: measurement outcomes. An experimentalist can measure a quantum system and then apply some control on the system depending on this outcome. One can see the classical outcome as acting on the quantum system. This is an example, albeit quite trivial, of consistent quantum to classical to quantum influence. To source the gravitational field, one could just measure some operator corresponding to the mass density and use the outcome as the source of the gravitational field. By construction, this will yield dynamics consistent with the quantum mechanical toolbox because it is formally just measurement + feedback. Obviously, one should not imagine that this measurement is done by a real physical observer. The new unorthodox dynamics should be “as if” someone where doing that within orthodox quantum theory. Interestingly, this is just what collapse models do (without the feedback step).
For discrete collapse models like the Ghirardi-Rimini-Weber (GRW) model, the equivalent of the “measurement outcome” is just the space-time event corresponding to the center of the collapse or the “flash”. For continuous collapse models, an equivalent can also be constructed and corresponds to what people call the “signal” in continuous measurement theory. However, for continuous collapse models this quantity was not historically introduced so it is a bit harder to explain how one should source gravity in one sentence. This brings us to a pedagogical difficulty that ultimately motivated the writing of the preprint. With Lajos Diosi, we have discussed the continuous models only because the resulting theory of gravity looks more appealing in this form. But again, in this context, the good object (this “signal”) to source gravity had not been previously introduced in the context of collapse models. As a result, in all my presentations, lacking time, I had to say that the idea was basically to source gravity from “the continuous equivalent of the discrete GRW flashes”. This is arguably unclear.
In July, I was attending a very nice school on philosophy and physics in Saig, Germany. I presented the work I did together with Lajos on semi-classical gravity. As usual I said that the main idea is to source gravity from the “continuous equivalent of the flashes of discrete models”. In the discussion, the philosophers (mostly David Albert if I remember correctly) told me it would be much easier to just do everything with the discrete model that people know. So this is what I did. I took the GRW model instead of the continuous ones and used the flashes as the sources of the gravitational field. I thought I would run into divergence problems but I did not. The resulting model is quite simple and the main equations are straightforward to obtain (although the final master equation is perhaps less transparent than the one obtained for continuous models). I do not think it provides a better theory of semi-classical gravity than the earlier ones but it gives a working theory of semi-classical gravity that has the same basic features and that is easier to obtain.
From right to left: Barry Lowers, David Albert, Nick Hugget and myself
At the conference, after the talk, Shelly Goldstein asked me if I really believed in these semi-classical gravity proposals (that is, either the continuous ones or the [yet to be written at the time] discrete one). To be honest, I would only put a very low probability on them. What I think they achieve is prove that it is not impossible to construct a theory in which gravity is fundamentally semi-classical (in the Newtonian limit) and in which there is no measurement problem. Probably nothing more.
update 21/09/2017: Anil Ananthaswamy has written a nice article on this preprint for Newscientist. I totally agree with Klaus Hornberger’s assessment reported in the article: the approach is interesting but has severe limitations so far.
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