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Old 08-11-2003, 08:11 AM
SentientMeat SentientMeat is offline
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The Wheeler DeWitt equation

I am reading "The End of Time" by Julian Barbour. He makes a compelling argument for a timeless universe with reference to the Wheeler DeWitt equation, wherein the time dependent Schrodinger equation of the universe is solved for two space coordinates with respect to a third.

Barbour contends that it shows that the quantum universe is static. However, as with any popular science book one has little sense of how this notion goes down in mainstream science.

So, what does the Wheeler DeWitt equation mean?
Old 08-11-2003, 10:41 AM
Ring Ring is offline
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I have no idea but here's what the eminent theoretical physicist John Baez has to say about it:

One can try to use spin foams to find solutions of the Wheeler-DeWitt equations - Rovelli and Reisenberger wrote some papers on this. On the other, it's equally plausible that the whole idea of trying to find solutions of the Wheeler-DeWitt equation is ultimately misguided. In loop quantum gravity, spatial diffeomorphisms don't have any infinitesimal generator - so why should diffeomorphisms that push time forward? These days, I've sort of given up on the Wheeler-DeWitt equation. Maybe sometime later I'll decide this was overly pessimistic.
Who knows?

In the quantum theory of gravity, which doesn't quite exist yet, one expects
all physical states to be solutions of the Wheeler-DeWitt equation
H psi = 0, i.e., they're all annihilated by the Hamiltonian constraint
(which for pure gravity and no matter is simply another way of stating the Einstein equation G_{00} = 0).

The problem with quantum gravity
for many years was that the perturbative theory a la Feynman diagrams wasn't renormalizable, and it seemed hopeless to find *exact* -- nonperturbative -- solutions of the Wheeler-DeWitt equation, since it was so bloody complicated and nonlinear (leading also to nasty "factor-ordering problems" in trying to make sense of it). The Ashtekar"new variables" revolution seems to have changed all that.
Old 08-11-2003, 11:04 AM
SentientMeat SentientMeat is offline
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Thanks Ring - I understand you were trying to be helpful but pasting Google results out of context does not necessarily lead to enlightenment!
Old 08-11-2003, 11:28 AM
Ring Ring is offline
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So sorry. I'll make very sure it doesn't happen again.
Old 08-11-2003, 12:02 PM
MC Master of Ceremonies MC Master of Ceremonies is offline
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It's the major equation in canonical quantum gravity (the attempt to quantize general relativity) thought by some to describe the universe, it's time invariant, in otherwords it doesn't change with time.
Old 08-11-2003, 12:22 PM
Brad Amundsen Brad Amundsen is offline
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oh poppycock!
Old 08-11-2003, 08:35 PM
bonzer bonzer is offline
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You're not the first to have asked this question. At a guess, the literature on Wheeler-DeWitt must split about 50:50 between those who try to solve it and those who argue about what it means.
While I presume that Barbour explains this in the book (which I haven't read, though I may get around to it), the equation arises when you try to talk about a wavefunction for the entire universe. Superficially, this looks simple. Quantum mechanics tells us how to find the wavefunction for any particular classical system and general relativity tells us what the classical system corresponding to a universe is. So you do the maths and out falls:

H psi = 0.

What does this remind you of? Since H is the Hamiltonian, this looks like a time-dependent Schrodinger equation in which, umm, there's no time-dependence.
We'll come back to the alleged philosophical difficulties here below, but a few technical remarks are in order. It's actually better to think of it as a constraint equation. Even in simpler cases, that's a subtle subject in its own right. The classic introduction is in Chapter XII of Dirac's Principles (Dirac worried about the issue a lot, though not particularly in this context) and I can't think of anything more elementary than that. One of those nasty bits of field theory that never gets popularised.
Though the notation is liable to make it look trivial, solving it is not. That benign H hides all sorts of nastiness in realistic cases. Still, courtesy of the work of Jacobson, Rovelli and Smolin - which I suspect John Baez is alluding to in the quote - one can actually write down an infinite number of solutions. Is this useful? Maybe, maybe not. After all, go back to thinking of the equation as an eigenvalue equation for a stationary state with E=0. One can invent an infinity of solutions to that problem. The difficulty is that only a handful of them correspond to any problem that'd be interesting otherwise. Still, big claims are made for existing exact solutions to the Wheeler-DeWitt equation.

So back to the philosophical issues. The big one is: where's time gone? We've got a wavefunction that doesn't evolve. It doesn't do anything, it just is. To answer the question about whether Barbour's views are mainstream, in arguing that there's something to be explained here, that's the concensus. One may take the Baez position that this is something that will ultimately be explained away, and so isn't a real physics problem, but there's still that notion that there's something odd going on. Maybe not profound, urgent or requiring an exact answer now, but kind of odd.
One view is that this is just what we'd expect. In relativity, you're already used to thinking of space-time as a whole. Time isn't something that's happening to some priviledged observer and so there's no universal now. There's just stuff in space-time which have relations between them. In GR, picking out one coordinate and saying that the universe evolves in that direction may be intuitive, but it doesn't quite obviously follow from the maths.
The other pole is to identify a specific clock within the solutions to the equation. You pick something like the scale factor of the solution and call that t. Clearly, this tends to nudge up against arguments about the arrow of time.

Both poles boil down to sort of the same attitude. A solution to the Wheeler-DeWitt equation tells us that is the universe is that big in that direction, then it's this big in this direction. One group says you can label one of these as a clock, the other says forget about clocks, are things actually related in this way.
To me, the second seems more profound, but I haven't had to worry about such issues in any day-to-day manner.
Old 08-12-2003, 07:14 AM
SentientMeat SentientMeat is offline
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Fascinating, bonzer, and well presented indeed. Many thanks.


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