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.
How did they develop the WDW equation in the first place? 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?
Rovelli has an interesting account of Wheeler and DeWitt coming up with the equation, in his book Quantum Gravity 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.
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.univ-mrs.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
Thx for the website n the pdf. I saw on page 308 a diagram on the development of the quantum theory of gravitational field. In one part, it shows that Wheeler-de-Witt 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
I see there are two sets of page numbers, page 308 of the file is page 290 of the hardcopy. That diagram on page 308(or 290)refers to the year by year listing of developments. So I will quote some of those, for example: 1967--Bryce DeWitt publishes the "Einstein-Schroedinger equation [later called the Wheeler-DeWitt, 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 Wheeler-DeWitt 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 Hamilton-Jacobi 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... 1988---Ted Jacobson and Lee Smolin find loop-like solutions to the Wheeler-DeWitt 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 object---some mechanical or electronic device---is "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 space-time verifications invariably amount to a determination of space-time 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 (Hamilton-Jacobi form of) GR and to the 1967 Wheeler-DeWitt 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 constraint---H Ψ = 0-----something set equal to zero instead of some more Schroedinger-looking time evolution thing---but 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
This is off topic, but I was astonished to learn that DeWitt had published in 1964 a result equivalent to Fadeev-Popov's, published four years later, although DeWitt's method was different and less convenient. Next they'll be telling me Yang-Mills was secretly renormalized before t'Hooft and Veltzmann!
They began with the canonical or ADM formulation of GR due to arnowitt, deser and misner. In this approach, any spacetime M is viewed as a union U_{t}Σ_{t} of nonintersecting spacelike hypersurfaces Σ_{t} indexed by some time parameter t. In standard canonical formulations of any theory, the basic objects are always the phase space coordinates consisting of the configuration variables and their conjugate momenta. Here these are respectively, the spatial part h_{ij}(t) of the spacetime metric giving the geometry that is intrinsic to each Σ_{t}, and π_{ij}(t) which roughly speaking encodes how each Σ_{t} is embedded in M. 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 4-geometry of M that's observer-independent, 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... It's a famous fact that quantum theories unrelated by unitary transformations are physically inequivalent. The problem here is that the aforementioned canonical transformations between different phase space variables in the classical theory can't in general be implemented by unitary operators in the quantum theory. Hence different choices of canonical pairs - or equivalently, of time parameters - despite representing physically equivalent classical theories, will in general produce physically inequivalent quantum theories. Only canonical approaches to QG suffer from the "problem of time". Such treatments, though not necessarily incompatible with classical GR, restrict the topology of spacetime to be the product of the real line with some 3D manifold, these being the only types of spacetimes allowed in classical GR. But the thing is that one would expect that QG would allow all possible topologies. In fact it's precisely these other topologies that seem to give the most interesting effects, and hence the most likely to shed light on a number of problems associated with quantum cosmology, as research has already shown. 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.
why are these the only topologies allowed by classical GR? 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?
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.
I agree, see for instance Daniele Oriti's thesis, page 41 "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 4-manifolds having the given fixed boundaries, giving rise to a foam-like structure of spacetime [59], with quantum fluctuation from one 4-metric 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 professional-grade articles to that effect etc. Could be though. Would seem odd to have topology subject to quantum fluctuations, a "fuzzy" topology that is a foam-work of alternative topologies is what Oriti seems to be describing. We could write Oriti about it, I guess.
May I ask why and how reparametrization invariance is lost when the Hamiltonian constraint is imposed upon hij and nij?
By "allowed" I meant physically allowed. For example, only noncompact spacetimes can satisfy the chronology condition or be globally hyperbolic. We choose to work on this particular background because it's the vacuum state. Canonical quantization requires one choose a time parameter before quantizing, with different choices yielding physically inequivalent theories.
ok, but there are many noncompact spaces that are not Cartesian products with the real line. i am thinking here of something like a Möbius strip, of infinite width. the time direction this infinite line. 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? all we know about what the vacuum looks like is the M4 part, we have no idea what the extra dimensions might look like, or how M4 would be embedded among them, unless i am mistaken.
By "noncompact" I meant - as is customary in GR - of the product form. For example, to be predictable, i.e. globally hyperbolic, spacetimes must be of the product form, more general noncompact spacetimes being disallowed. We don't know what the specific compactification would be (recent work on large extra dimensions notwithstanding) our most important guide in this being the resulting low energy phenomenology. We have to perturb around a sensible classical solution, and the kind of state your thinking of probably wouldn't be, as we've discussed.
But will the reparametrization invariance be lost when the Hamiltonian constraint is imposed upon hij and nij?
If the classical hamiltonian constraint could be solved, then of course, reparametrizaton invariance would be broken since an explicit choice of gauge has been made. However, the hamiltonian constraint's quadratic dependence on the conjugate momentum makes this practically impossible so that in this forumulation of GR, it seems phase space can't be paired down to only the true dynamical degrees of freedom. But as I pointed out, the classical meaning of this gauge freedom is lost in canonical quantization since different parameterizations produces distinct quantum theories, and in fact solving the quantum hamiltonian constraint remains the central unsolved problem facing canonical approaches to quantizing gravity.