Exploring the Wheeler-DeWitt Equation: A Comprehensive Introduction

[YZer]

I'm looking for a good introduction source on the Wheeler-DeWitt equation. Anybody know any literature (book or article) that will help me wrap my mind around it.

Thanks

 

[marcus]

I'm looking for a good introduction source on the Wheeler-DeWitt equation. Anybody know any literature (book or article) that will help me wrap my mind around it.

Thanks

YZer, see what you think of John Norbury's
http://arxiv.org/physics/980604
From Newton's Laws to the Wheeler-DeWitt Equation

it is a pedagogical essay that is intent on crashing thru
in about 10 pages to a rough idea of the WdW
without ever doing General Relativity (!)

this may seem odd

he expects you to know Schroedinger equation
and he lowers his head and charges
through a Lagrangian and Euler-Lagrange thicket
and in the end something appears that looks
reassuringly like the Schroedinger equation but with an intricate
new-fangled potential
and that is the WdW for the wavefunction describing the cosmos.
doo-dah doo-dah

I think that he has a good pedagogical idea
a good teacher doesn't always play by the rules
It might just happen to be the right approach for you.

and if not, trash it and don't worry.

 

[YZer]

thanks, I am actually looking for a general relativistic formulation, if that is even possible. I would like to understand it and its relation in terms of general relativty.

 

[marcus]

thanks, I am actually looking for a general relativistic formulation, if that is even possible. I would like to understand it and its relation in terms of general relativty.

that would be the normal way to approach wheeler-dewitt, which is known as the "Quantum Einstein Equation"
So that guy John Norbury was taking a shortcut through the bushes.

I believe the normal way would be to go through General Relativity and the ADM (arnowitt deser misner) formulation and quantize that

the problem is all that stuff is so old that it is not online. It is in books.
Because it came before the internet.

do you know that book titled "Gravitation" where John Wheeler is one of the authors? Misner, Thorne, Wheeler. I don't have it but I remember using it a long time ago and I have the idea (could be wrong) that it could have WdW, and show how to develop it from Gen Rel.

this is something selfAdjoint could answer in a jiffy.

You want to see the standard development of WdW starting from classical Gen Rel. there could be an online source, but i don't know one right off

 

[marcus]

I'm looking for a good introduction source on the Wheeler-DeWitt equation. Anybody know any literature (book or article) that will help me wrap my mind around it.

Thanks

I have another idea for you (but actually I like the 10-page Norbury shortcut approach---it's so basic)

My suggestion is to read pages 293 and 294 of Rovelli's book Quantum Gravity, the free downloadable draft version.

Anybody interested in QG should have this book----either the storebought final version or the free draft version.

These 2 pages give a thumbnail historical sketch of how the WdW emerged.
(knowing how humans stumble onto something is often a good approach to essential understanding)

download the book PDF, put it on yr desktop as an electronic copy of the book. If you stay interested in QG you will be referring to it.
And for starters look at page 293 and 294

anyway such is my second suggestion (anybody have other online resource for this?)

How to get the book? Just google the single word "rovelli"
that will get you his website and the link to the PDF draft is on the first page you see

 

[selfAdjoint]

Just coming in with that reference Marcus promised.
MTW (as Misner, Taylor, and Wheelers's Gravitation is usually referred to) has the development of the Wheeler-Dewitt equation in section #43.3, "The Einstein-Hamilton-Jacobi Equation". Interestingly, you won't find "Wheeler-DeWitt" in the index; Wheeler here modestly calls it the DeWitt equation.

 

[marcus]

Just coming in with that reference Marcus promised.
MTW (as Misner, Taylor, and Wheelers's Gravitation is usually referred to) has the development of the Wheeler-Dewitt equation in section #43.3, "The Einstein-Hamilton-Jacobi Equation". Interestingly, you won't find "Wheeler-DeWitt" in the index; Wheeler here modestly calls it the DeWitt equation.

selfAdjoint thanks!
you obviously have the book in hand and can verify that it is
Kip Thorne who supplies the T in MTW
(unless it is our man Tuesday in the first three days of the week)

did either of you have any reactions to Baez latest TWF?

 

[YZer]

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?

 

[marcus]

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

 

[selfAdjoint]

Sorry about typing Taylor for Thorne, I must have been thinking of Wheeler and Taylor's Spacetime Physics.

The Wheeler-Dewitt equation is what you get when you set out to do Hamiltonian physics on GR spacetime. The Hamiltonian itself boils down to just an expression that's identically zero, the "Hamiltonian Constraint". String physicists might conceivably use it in the math of the worldsheet, where among several other math approaches they treat it as a two-dimensional Einstein manifold (the proper name is Lorentzian manifold, but Lorentz never had anything to do with them, while Einstein had EVERYTHING to do with them; he was their creator!).