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WheelerdeWitt equation 
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#1
Dec3003, 03:56 AM

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What has WheelerdeWitt equation got to do with gravity and the problem of time?



#2
Dec3003, 07:15 AM

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The WDW equation may be viewed as a kind of schrodinger equation for space, giving it's quantum "evolution" with respect to some "time" parameter. However, unlike in the classical case, different choices of time parameter produce inequivalent theories. The issue then of how a time parameter should be chosen is known as the problem of time and afflicts all attempts (LQG is a good example) to quantize gravity that require a separation of spacetime into space and time.



#3
Dec3003, 08:40 AM

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What do you mean by different choices of time parameter producing inequivalent theories? Is it compatible with GR? If it is afflicting, why then quantizing gravity? Gravity shouldn't be quantized then. A separation of spacetime? Doesn't it violate GR? 


#4
Dec3003, 04:59 PM

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PF Gold
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WheelerdeWitt equation
The December 30 draft (probably already sent to publisher Cambridge U. P.) is at Rovelli's website. Alexok just gave the link to it in the string/brane/loop forum He has a 16 page History of Quantum Gravity appendix at the end (Appendix B) and it has a timeline. The story of their collaboration on this is in the timeline around 1965 or 1967. 


#5
Dec3003, 05:11 PM

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PF Gold
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Yeah, it was alexok, in the thread "S particles and LQG", replying to mentat. He said:
"Also, in case you're connived by the theory, you could give Rovelli's latest book (still not finished, but a draft is pretty much complete  as of December 30th) a spin :) http://www.cpt.univmrs.fr/~rovelli/book.pdf Enjoy! :)" Also a propos of time, Rovelli discusses time in GR and time in quantum GR early in the book around section 1.3.1 beginning page 20 and again in more detail around section 2.4.4 beginning page 58 


#6
Dec3003, 07:41 PM

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I saw on page 308 a diagram on the development of the quantum theory of gravitational field. In one part, it shows that WheelerdeWitt equation leads to loop quantum gravity. Does it mean that the WDW equation has been solved by loop quantum? No offence but it I didn't realise this has extended to quantum. :P 


#7
Dec3003, 09:26 PM

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PF Gold
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1967Bryce DeWitt publishes the "EinsteinSchroedinger equation [later called the WheelerDeWitt, this is page 312(or 294) and it has the story of how they found it] 1980...discussion focuses on understanding the disappearance of the time coordinate from the WheelerDeWitt theory. The problem [once called 'the problem of time'] has actually nothing to do with quantum gravity, since the time coordinate disappears in the classical HamiltonJacobi form of GR as well, and, in any case, physical observables are coordinate independent, and thus, in particular independent from the time coordinate, in whatever correct formulation of GR... 1988Ted Jacobson and Lee Smolin find looplike solutions to the WheelerDeWitt equation...in the connection formulation, opening the way to LQG. end of quotes The disappearance of the time coordinate (seen in one formulation of classical 1915 GR and also in the first quantum GR equation, WDW) at one time puzzled people and they called it "the problem of time". As far as I know, Rovelli doesnt mention "the problem of time" in his book because from his more contemporary perspective there is no problem. He talks a lot about time in classical GR. The role of time in 1915 GR largely carries over to LQG because LQG is a quantization of GR. The fact that the time coordinate disappears in the main equations comes with the territory and is not special to LQG. Indeed people were focussing on "the problem of time" in the 1980s before there even was any LQG. In GR time is not physically meaningful on its own apart from matter and the gravitational field. One can declare a particular material objectsome mechanical or electronic deviceis "the clock", but no one can certify that it will keep good time consistently and forever. In cosmology papers the increasing size or scale factor of the universe is sometimes used to clock other processes, since no absolute time coordinate is defineable. Einstein said: "...the requirement of general covariance takes away from space and time the last remnant of physical objectivity... All our spacetime verifications invariably amount to a determination of spacetime coincidences...coincidences between the hands of a clock and points on the clock dial...The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences..."[the basic 1916 paper 'Grundlage der allgemeinen Relativitaetstheorie'] Two of Rovelli's section headings in the table of contents are significant: 3.1 "Nonrelativistic mechanics is about time evolution." 3.2.4 "[Relativistic]mechanics is about relations between observables." So the Hamiltonian equation becomes a "Hamiltonian constraint" free of any time variable and of the form H Ψ = 0 this type of equation is common to the (HamiltonJacobi form of) GR and to the 1967 WheelerDeWitt equation and to LQG as well. [[[this is not immediately relevant to the question you raised but I should mention that in LQG case the theory is still being worked on and several forms of the Hamiltonian are being studied. One can be certain that it will be a Hamiltonian constraintH Ψ = 0something set equal to zero instead of some more Schroedingerlooking time evolution thingbut the state space, the inner product, the H operator still appear to have room for modification as the theory develops]]] hope this strikes a good medium between being oversimple and too wordy. more people should read Rovelli, the book explains a lot 


#8
Dec3103, 08:42 AM

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PF Gold
P: 8,147

This is off topic, but I was astonished to learn that DeWitt had published in 1964 a result equivalent to FadeevPopov's, published four years later, although DeWitt's method was different and less convenient. Next they'll be telling me YangMills was secretly renormalized before t'Hooft and Veltzmann!



#9
Jan104, 08:48 AM

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Now, different choices of time parameter, say t and t' for example, describe different "slicings" Σ_{t} and Σ_{t'} of a spacetime into spacelike hypersurfaces, each slicing simply representing the definitions of simultaneity held by different observers. It's only the 4geometry of M that's observerindependent, that is M = U_{t} Σ_{t} = U_{t'} Σ_{t'}. The canonical pairs (h_{ij}(t), π_{ij}(t)) for different slicings are related by the usual canonical transformations one learns about in classical mechanics. The equation H(h_{ij}, π_{ij}) = 0 most closely connected with the physical equivalence among the different canonical pairs (or equivalently, reparametrizations by different time variables) is known as the hamiltonian constraint. The WDW equation is simply the quantum version of this and is obtained in more or less the usual way by canonical quantization as an operator equation H(h_{ij}, π_{ij})Ψ = 0 in which Ψ is the "wavefunction of the universe". However, the connection to reparametrization invariance is lost... One QG research program oft discussed in these forums that takes the canonical approach and thus suffers from the problem of time is LQG, though it uses a different choice of phase space variables then those discussed above and partially as a result of this is rather different than previous canonical approaches, but is also widely believed to be inconsistent with GR and therefore wrong. On the other hand, string theory is spacetime covariant so string theorists don't worry much about this issue. String theory is consistent with and in fact automatically contains GR, and remains our only known bonafide  though not necessarily correct  quantum theory of gravity. 


#10
Jan104, 07:30 PM

P: 657

also, a related question: you know how in string theory, one assumes that spacetime is M4xCY, where M4 is minkowski space, and CY is some compact manifold? well, it seems that there are lots of manifolds that would contain minkowski space as a submanifold. so why do we restrict ourselves to spacetimes that are Cartesian products like this? 


#11
Jan104, 07:33 PM

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PF Gold
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I think it's because you have to hide the extra six dimensions but keep the observed four, so it's easier if you use the cartesian product. But I agree with you they should, and probably have, consider more general kinds of manifold.



#12
Jan104, 09:38 PM

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"Topology itself is also allowed to change, and the path integral can be completed by a sum over different topologies, i.e. over all possible 4manifolds having the given fixed boundaries, giving rise to a foamlike structure of spacetime [59], with quantum fluctuation from one 4metric to another for a given topology, but also from one topology to another..." There is a link to Oriti's thesis in the "Rovelli's program" thread in string/loop forum. He did the work while at Cambridge under one of the few women LQG people (Ruth Williams) and a fair amount of the findings in the thesis is from collaborations with E. Livine (whose thesis also came out 2003 as well). Oriti is one of the people Rovelli puts in the acknowledgements page of his book. I would guess he is postdoc now either at Marseille or at PI, but havent followed him and dont know. Anyway topology change is certainly something in the context of loop/spin foam gravity that people are ready and willing to consider. I have not checked into it since I dont see the immediate point or urgency when so much else to consider, but I have never seen any informed opinion that the theory was limited in that sense and couldnt handle topology change. No links to professionalgrade articles to that effect etc. Could be though. Would seem odd to have topology subject to quantum fluctuations, a "fuzzy" topology that is a foamwork of alternative topologies is what Oriti seems to be describing. We could write Oriti about it, I guess. 


#13
Jan204, 09:49 PM

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#14
Jan504, 12:15 PM

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#15
Jan504, 12:27 PM

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#16
Jan504, 04:35 PM

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#17
Jan604, 06:35 PM

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ok, this isn t very physical, i suppose we want a physical spacetime to be orientable, among other things? but the point is, there are lots of ways that a space can include a copy of the real line, and thus be noncompact. like the tangent bundle of a sphere: it is noncompact, 4 dimensional, orientable, and includes a copy of the real line, that we could call the time dimension. but it is not a Cartesian product with the real line since it is a nontrivial bundle. so is this not a physical spacetime? why not? 


#18
Jan704, 09:43 AM

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