Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: paros on 27/01/2014 09:40:32
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I have been studying canonical gravity theories lately and I have encountered the Wheeler-Dewitt equation. I was wondering if anyone here knows physics well enough to answer some questions I have about that equation. To the best of my knowledge, I will attempt to translate what the equation says into plain english. My translation is probably wrong. I would welcome any corrections and the harshest criticism.
We are going to be looking at integrals taken over something called a phase space. This is different from the so-called "path-integral formulation" of QFT. Those are usually called "flat space quantization". The integral we are taking is the integral of the gravitational field over all possible (static) wave functions of all possible universes. This integral will have an associated functional that will associate the integral with a scalar. Call this scalar the "Hamiltonian Constraint." The Wheeler-DeWitt equation in english goes:
H(x)|psi> = 0
"The functional over all possible universe phase states, given as a wave function, has a Hamiltonian Constraint equal to zero."
I don't disagree with the validity of that statement. But I want to really understand what that means. What sorts of realistic changes would see in a universe where the Hamiltonian Constraint was non-zero?
The only situation in which this equation had any application, that I know of, is in the Hartle-Hawking State. This might mean it is a toy model for cosmology, rather than a description of the universe as it exists now chock full of galaxies. If I am wrong, correct away as you feel fit!
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Deep at the heart of cosmology there is this unproved and unprovable belief that the whole bulk, that is, everything that there is not just our universe but the sum total of all universes, is a zero sum game. That is all the disturbances wave functions etc that there are, balance themselves out to result in the concept that if there was not something (which there is) there would be nothing.
This is in effect the full extension of the well known and accepted law of the conservation of energy writ as large as possible.
It is possibly the nearest thing to the modern cosmologists view of the concept of "god" which in religions is seen as something that acts on everything to produce things.
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Thanks for the reply.
I noticed you didn't mention anything about time in your response. All time vanishes in the Wheeler-DeWitt, which is usually the big, hot topic involved in this equation.
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It is possibly the nearest thing to the modern cosmologists view of the concept of "god" which in religions is seen as something that acts on everything to produce things.
Would that not make more sense as: ".....something that acts on nothing to produce things."
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Seems real tricky doesn't it? If you are stating that there is no way to introduce 'time' in it I'm not sure though. Take a look at http://cds.cern.ch/record/324729/files/9704061.pdf
When it comes to this 'zero sum game' that SoulSurfer refers to I tend to agree. It's a sort of belief of mine too, doesn't matter what your constrains or limits are for it. Either it is a zero sum game and then we don't lose anything, it only transforms, or it isn't. To that you then need to decide if you need to count in time in such a description, or if it is time (local arrow) that describes the outcomes/transformations. If you count time into such a 'zero sum game' you should come to a 'static description' of a 'universe' I suspect. And, if you decide that time is what makes outcomes? Then you need to find a way to describe it both ways, to keep a zero sum game, as you otherwise 'lose' the past irrecoverably.
Einstein thought of time as being ultimately a illusion, I don't agree there. Instead I prefer a local definition in where your arrow connects to 'c', valid wherever you go, at whatever speed. From that aspect the arrow has a definite direction and are valid as a definer of a past, a 'now', and a future, and it defines outcomes.
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The point is that if it isn't a zero sum game, then what drives it? What I mean is that if it isn't, all conservation laws becomes questionable. And it becomes a very different 'universe' in where most physics should need to be redefined, probably this equation too. When Einstein defined time as a illusion I think he went from a archetypal idea of a universe we all expect us to exist in, in a way a 'container universe', although defined through observer dependencies. Dimensions and constants setting its limits. If you instead use 'degrees of freedom' to define it, adding local constants as the 'common ground' we observe from, you can get to a different definition in where dimensions becomes our 'illusion', not time.
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Paros I did not mention specifically. All wave functions are integrals over all of space AND time. Quantum theory implies that every particle in our universe exists in all of space and time. It may however be extremely improbable to find it in very unexpected places but the probability is never absolutely zero in a mathematical sense.