Marin said:
...I know that QM is a probability theory, whereas GR is a deterministic one, but after all - both of them include our reality as a separate 'boundary' case, don't they?
Then what splits them up and makes them so different to be even incompatible (at least for now)?
Hi Marin,
I see you are new here---welcome to PhysicsForums!
I have a different point of view from Gianni's which I will lay out briefly for you.
GR is our current theory of geometry. It does not presuppose any fixed geometrical background. The metric (the distance function describing geometrical relationships) is the dynamical variable. The theory determines geometry. On the other hand, much of the rest of physics is developed by assuming at the start a fixed geometrical background, thus fixing the metric. It assumes the geometry is already determined!
I do not consider this an incompatibility. I consider it a problem, and one should appreciate the scale of the problem. In sum, while the rest of physics is based on provisional fixed-geometry spacetimes, GR is about the nature of spacetime itself: geometry and how matter interacts with it.
To get a clearer idea of what a quantum geometry (QG) or quantum general relativity (QGR) should look like, please have a look at the July Scientific American article by Ambjorn and Loll. This article gives an idea of what the main goals of QG are.
http://www.scribd.com/doc/3366486/SelfOrganizing-Quantum-Universe-SCIAM-June-08
(I also have a link to this article in my signature.)
The goal is to produce a quantum continuum, where there are fluctuations and uncertainty in the local geometry especially at small scale, but where overall, on average, we see a nice smooth classical geometry emerge, like the macroscopic real world.
As you can see, Ambjorn Loll and their co-workers have made remarkable progress towards this goal recently. They generate random quantum universes in the computer and when they average them up, they get a classical spacetime. The individual quantum universes can be captured and explored--they show interesting non-classical properties at small scale. In work like this a
new idea of the continuum is emerging.
This is what one should expect from QG as it matures. Their model also requires a positive cosmological constant in order to work---so it begins to suggest what the meaning of the cosmological constant is, where it comes from so to speak. No clear conclusions, but this is another thing that a mature QG should do: explain the acceleration in expansion.
A singularity is a place where a theory breaks down, in the past singularities have been eliminated by improving the theory. This indicates another thing that QG should do. It should say what is really going on at the center of a black hole and describe conditions around the time of the big bang. A number of recent QG papers pursue these two goals.
A good way to sample the current literature on this is to do a keyword search at slac.stanford spires database using the keyword "quantum cosmology". Here is a search of recent papers ranked by citation count. The most highly cited ones use the Loop approach and are largely based on computer models of the big bang. I set the date so as to select highly cited papers published since 2005.
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=DK+quantum+cosmology+and+date+%3E+2005&FORMAT=WWW&SEQUENCE=citecount%28d%29
If you do the same spires search but change the date to be before, say, 2003, it will turn up a lot of highly cited string-based papers.
The shift in researcher's interest seems to follow from the fact that loop QG models resolve the big bang singularity and allow one to go back earlier in time beyond the singularity in the classical theory. There have recently been a number of papers using the same methodology to resolve the black hole singularity. I just saw one by Boehmer and Vandersloot that you could find on arxiv if you want a sample of how that is going.
So resolving classical singularities is another obvious goal and progress is currently being made on that.
So my perspective is I don't see an incompatibility. That is just how some people talk. Instead of an incompatibility, I see it in terms of goals.
The most important goal is to get a quantum geometry----which means a new mathematical model of the continuum. To see what I mean, look at the SciAm article. They aren't there yet but that's the direction. The quantum spacetime continuum should reproduce GR dynamical geometry at large scale and should eliminate the classic GR singularities so we can go back in time before the big bang and also get an idea of what happens in black holes. It should also clarify the way that matter and geometry interact, perhaps leading to a new understanding of what matter is. These are major challenges, not incompatibilities.
