Talk:Wheeler–DeWitt equation

Missing definitions of used units

What does N, beta, g, R stand for? What kind of derivative is delta-derivative? It may be obvious, but this needs to be defined! — Preceding unsigned comment added by 151.236.226.75 (talk) 23:06, 5 October 2016 (UTC)

Simplify?

1) Should 'mini-superspace' be linked to the Superspace article?

No: ""Superspace" has had two meanings in physics. The word was first used by John Wheeler to describe the configuration space of general relativity;", but next parts of the article "Superspace" are only about second sense of the term superspace.

2) Can someone add a simplifying summary to the beginning of this article?

Simplify? What?

It starts with, "In theoretical physics, the Wheeler-deWitt equation is a functional differential equation."

I think that sentence should continue with, "describing ...", instead of so quickly going on to say "It is ill-defined in the general case". —Preceding unsigned comment added by 121.210.170.141 (talk) 12:39, 12 October 2009 (UTC)

This page does seem quite...technical. I would like to think a rewrite is in order. The Wheeler-DeWitt equation should be derived from quantizing the Hamiltonian constraint in the ADM formalism (since...that's the logical approach). Another section could be added deriving it from the Euclidean quantum gravity approach (c.f., the scratch work done on the talk page for Euclidean quantum gravity).

Steve Carlip's "Quantum Gravity: a Progress Report" (arXiv:gr-qc/0108040) discusses the derivation quite well.

Perhaps something could be said of the semiclassical approximation which "solves" the Wheeler-DeWitt equation (well, approximates it). Or the Wheeler-DeWitt equation in Ashtekar variables. Or... --Pqnelson (talk) 22:01, 14 September 2012 (UTC)

If no one objects, I am going to push the current "overview" into a "relation with the Euclidean quantum gravity approach" section; then rewrite the overview using the canonical gravity Hamiltonian constraint, considered using Dirac quantization of constraints.

I will also use notation consistent with the Euclidean quantum gravity page, so not to confuse the gentle reader! --Pqnelson (talk) 02:49, 16 September 2012 (UTC)

3) the entire paragraph "Quantum Gravity" is so poorly worded that I cannot parse it (I have taught quantum gravity)

Energy Functional

What is the meaning of the last paragraph, which sets up an energy functional? The n's are not defined anywhere, and the paragraph does not really add anything to the article. —Preceding unsigned comment added by 134.76.215.201 (talk) 13:31, 14 November 2007 (UTC)

It basically says that the expectation value of the Hamiltonian is the energy eigenstates for the Wheeler-DeWitt equation... —Preceding unsigned comment added by 67.166.144.11 (talk) 23:25, 14 December 2007 (UTC)

Useful References

Here is a couple of references which may be of use...

Jacobson, Ted (4 April 1988). "Nonperturbative quantum geometries". Nuclear Physics B. 299 (4): 295–345. doi:10.1016/0550-3213(88)90286-6. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Dirac, P. A. M. (19 August 1958). "The Theory of Gravitation in Hamiltonian Form". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 246 (1246): 333–343.

DeWitt, Bryce S. (August 1967). "Quantum Theory of Gravity. I. The Canonical Theory". Phys. Rev. 160 (5). American Physical Society: 1113–1148. doi:10.1103/PhysRev.160.1113.

Kiefer, Claus (2007). Quantum Gravity. 2nd ed. International Series of Monographs on Physics. Vol. 136. Oxford University Press.

The opening paragraph should be more readable.

It should explain the significance of the equation more clearly. 67.243.1.21 (talk) 21:47, 16 January 2010 (UTC)

I've made a stab at it, do you think it's going in the right direction? Sonarjetlens (talk) 18:21, 2 February 2012 (UTC)

Hilbert space and Bra Ket Notation

I think the use of the bra ket notation from quantum mechanics is somewhat misleading in this article. It usually implies the existence of an inner product. However, in quantum geometrodynamics, i.e. for the spaces concideres here, such an inner product is nearly never known. In particular there is no canonical choice like the Lebesgue integral for square integrable funtions, because this construction fails in infinite dimensional spaces. To my knowledge the only cases, in which a suitable inner product is known, are those in which the theory is reduced to a theory on a discrete set, i.e. a mini-superspace. (In the case of cosmological models, the symmetry reduction with respect to the assumed homogeneity and isotropy of the universe effectively reduces the universe to one point.)

Likewise the term "Hilbert space" is in general wrong, because a Hilbert space is a Banach space with an inner product. As argued above, there usually is no inner product. Moreover, in general the spaces considered in quantum geometrodynamics are not even normed spaces and hence no Banach spaces. This happenes, because the function spaces over each point are in general not at all connected to each other. — Preceding unsigned comment added by Doenermaster (talk • contribs) 23:38, 11 January 2011 (UTC)

This is an interesting couple points, but as I understand it...really we are working with "wave functionals". A good reference geometrodynamic-specific may be

Claus Kiefer, "Quantum geometrodynamics: whence, whither?" Gen.Rel.Grav. 41 (2009) 877–901. Eprint arXiv:0812.0295 [gr-qc].

Although, for the sake of completeness, it does date back to John Wheeler's work in the '60s, see his contribution to Relativity, Groups, and Topology (eds. C. DeWitt and B. DeWitt, Gordon & Breach, 1964).

At any rate, it is as kosher as the functional integral. From the strictly mathematical point of view, you are correct...but physicists seldom care about rigor, and work with the generality of algebra instead. --Pqnelson (talk) 15:37, 20 September 2012 (UTC)

Wikipedia in Finland

According to Wikipedian Finnish version (written by me): [1] Michio Kaku says that the equation works in all possible universes. Kartasto (talk) 17:42, 2 February 2015 (UTC)

References

equation

Forgive me if this is dumb, but the equation shown in This section is, well, I wouldn't say to simple, but I find versions where its derived from the Hamilton-Jacobi equation (via substituting the squared derivatives with second derivatives ) or made via Wheeler-DeWitt metrics to be more common. In fact, I have never seen this form of the equation.--Jacob851215.64 (talk) 16:59, 19 March 2022 (UTC)

Lee Smolin

According to Lee Smolin (page 62 in Finnish version) the string theory and quantum loop gravity may be different sides of the same theory.

This information is relevant when dealing with Wheeler-deWitt Equation because; 'it is said that its solutions are determining certain quantum states which describe the whole universum.' Kartasto (talk) 18:18, 22 October 2023 (UTC)