YZer said:
Thanks a lot, very helpful.
Is the Wheeler-Dewitt equation used much in modern research? It seems like LGQ uses it as a foundation, what about string theory?
Hi YZer, I will have to let someone else answer as regards string theory. I see WdW used rather much because as you say it is part of the foundations of LQG----in the sense that WdW is the "semiclassical limit" which Loop agrees with a few orders of magnitude above Planck scale.
So it serves as a check and in Loop cosmology papers (Bojowald and others) what they say, in effect, is:
"our own loop version of the equation is a difference equation, not a differential equation like WdW, and right around the classical singularity our equation works and doesn't blow up, but after about 100 Planck time units have elapsed our difference equation is showing results which match what the WdW shows----the two equations merge---so our theory has the correct semiclassical limit"
They actually do
computer runs calculating with their difference equation and calculating, for comparison, with WdW! So they are referring to, and calculating with, the WdW a lot for comparison sake. they are showing that their approach gives very similar results (for cosmology) to the vintage approach, away from singularities, but that it fixes a place where the WdW went bad and blew up.
Cosmologists use a simplified version of Einstein equation called the Friedmann equation (Friedmann made this in early 1920s IIRC). It is simpler because it assumes uniformity so that just a few numbers suffice. The Loop cosmology (difference) equation is one version of the Quantum Friedmann equation and the WdW equation provides another version.
Loop is not the only approach that is referring back to WdW as something to improve on but stay in touch with.
There is work by Viqar Husain (U new brunswick, Canada) and Reuters (IIRC Univ. Bremen) and several others, where they show that they can get simila results to Loop---like remove the classical big-bang singularity---if they just keep the old vintage 1970 "Geometrodynamics" and the old WdW but do the analysis differently. they tweak things to get the old car to run like the new car. This seems to me to be more on an ad hoc or problem by problem basis, instead of part of a comprehensive theory. But it is reassuring because it shows that major results, like removing the black hole and big bang singularities, are not just dependent on something special in Loop formalism but are sort of generally true robust results.
So there are a few people that are so-to-speak resuscitating old WdW.
This is all in the non-String department (the "Loop-and-allied" approaches)
I do not know of anything in String where they use WdW or variations on it, and where they, like, do computer runs to crank out the evolution of the universe around the big bang and compare numerical results. My impression is they are not into that kind of thing.
here are a couple recent Bojowald, to give the flavor
http://arxiv.org/gr-qc/0402053
http://arxiv.org/gr-qc/0408094
The first one is a general overview paper with a lot of links to other papers so it is kind of an up-to-date Loop cosmology "hub". It is called
Loop Quantum Cosmology: Recent Progress
he gave it at this year's ICGC conference
But the other paper, being more specialized, has more graphs
of results of computation and comparisons and stuff like that.
I guess this is more than you asked for, basically a long way to say yes you are absolutely right. Loop researchers use the WdW a whole lot
