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jby
What has Wheeler-de-Witt equation got to do with gravity and the problem of time?
Originally posted by jeff
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.
Originally posted by jby
How did they develop the WDW equation in the first place?
Originally posted by marcus
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
Originally posted by jby
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?
...
Originally posted by jby
How did they develop the WDW equation in the first place?
Originally posted by jby
What do you mean by different choices of time parameter producing inequivalent theories?
Originally posted by jby
If it is afflicting, why then quantizing gravity? Gravity shouldn't be quantized then. A separation of spacetime? Doesn't it violate GR?
Originally posted by jeff
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.
Originally posted by selfAdjoint
But I agree with you they should, and probably have, consider more general kinds of manifold.
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(hij, ðij)Ø = 0
in which Ø is the "wavefunction of the universe". However, the connection to reparametrization invariance is lost...
Originally posted by lethe
why are these the only topologies allowed by classical GR?
Originally posted by lethe
...why do we restrict ourselves to spacetimes [M4xK6]?
Originally posted by maddy
why and how reparametrization invariance is lost when the Hamiltonian constraint is imposed upon hij and nij?
Originally posted by jeff
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(hij, ðij)Ø = 0
in which Ø is the "wavefunction of the universe". However, the connection to reparametrization invariance is lost...
Originally posted by maddy
May I ask why and how reparametrization invariance is lost when the Hamiltonian constraint is imposed upon hij and nij?
Originally posted by jeff
Canonical quantization requires one choose a time parameter before quantizing, with different choices yielding physically inequivalent theories.
Originally posted by marcus
interesting conversation, Maddy and Jeff!
Originally posted by jeff
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.
Originally posted by lethe
ok, but there are many noncompact spaces that are not Cartesian products with the real line.
Originally posted by lethe
...we have no idea what the extra dimensions might look like...
Originally posted by lethe
...how M4 would be embedded among them...
Canonical quantization requires one choose a time parameter before quantizing, with different choices yielding physically inequivalent theories.
Originally posted by maddy
But will the reparametrization invariance be lost when the Hamiltonian constraint is imposed upon hij and nij?
Originally posted by jeff
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.
Originally posted by maddy
"Canonical Quantum Gravity and the Problem of Time" by Chris Isham at gr-qc/9210011
Originally posted by maddy
I'm trying to get a hand on a preprint by Barvinsky, A. on gauge invariance at xxx.lanl.gov
Originally posted by jeff
What we've been discussing is reviewed in Sections 4.1.1-2 of Isham's paper. In it the hamiltonian constraint is denoted [itex]\mathcal{H}_\bot = 0[/itex]. The three bulleted items near the bottom of p44 are the three possible ways to fix a time variable. If [itex]\mathcal{H}_\bot = 0[/itex] could be solved, we could write out the physical hamiltonian, called [itex]\hat{H}_{true}[/itex] in isham, containing only the true dynamical degrees of freedom. The resulting schrodinger equation is given in isham as equation (4.1.1).
Would you mind supplying a link to this paper?
Originally posted by maddy
In Isham's paper, it is said that the Robertson-Walker model allows the usage of different types of internal time (pg.s 51 & 56) and that the 3 Hamiltonians correspond with 3 different quantum theories of the same classical system. Are these different quantum theories 'physically inequivalent' even though they are derived from the same classical system?
The Wheeler-de-Witt equation is a mathematical equation that describes the evolution of the wave function of a quantum system in the absence of time. It is a crucial component of quantum gravity theories, particularly in the study of the universe as a whole.
The Wheeler-de-Witt equation is named after physicist John Wheeler, who proposed it in 1967. It is significant because it attempts to reconcile the principles of quantum mechanics and general relativity, two fundamental theories of physics that have been difficult to unify.
Unlike other equations in quantum mechanics, the Wheeler-de-Witt equation does not contain a time variable. This is because it is derived from the principle of timelessness in general relativity, which posits that time is an emergent property of the universe rather than a fundamental aspect of it.
The Wheeler-de-Witt equation has been used in various quantum gravity theories, such as loop quantum gravity and string theory, to study the early universe and the nature of black holes. It has also been applied to other areas of physics, such as cosmology and particle physics.
No, the Wheeler-de-Witt equation is not considered a complete theory of quantum gravity. It is a theoretical framework that provides insight into the nature of the universe and has been used to make predictions, but it is still an area of active research and development.