# Talk:Euclidean quantum gravity

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I'm looking through my notes, and Euclidean Quantum Gravity does relate back to ADM formalism used in canonical quantum gravity and recovers the Wheeler–DeWitt equation under various circumstances. If we have some matter field ${\displaystyle \phi }$, then the path integral reads

${\displaystyle Z=\int {\mathcal {D}}{\mathbf {g}}\,{\mathcal {D}}\phi \,\exp \left(-\int d^{4}x{\sqrt {|{\mathbf {g}}|}}(R+{\mathcal {L}}_{\mathrm {matter} })\right)}$

where integration over ${\displaystyle {\mathcal {D}}{\mathbf {g}}}$ includes an integration over the three-metric, the lapse function ${\displaystyle N}$, and shift vector ${\displaystyle N^{a}}$. But we demand that ${\displaystyle Z}$ be independent of the lapse function and shift vector at the boundaries, so we obtain

${\displaystyle {\frac {\delta Z}{\delta N}}=0=\int {\mathcal {D}}{\mathbf {g}}\,{\mathcal {D}}\phi \,\left.{\frac {\delta S}{\delta N}}\right|_{\Sigma }\exp \left(-\int d^{4}x{\sqrt {|{\mathbf {g}}|}}(R+{\mathcal {L}}_{\mathrm {matter} })\right)}$

where ${\displaystyle \Sigma }$ is the three-dimensional boundary. Observe that this expression vanishes implies the functional derivative vanishes, giving us the Wheeler-DeWitt equation. A similar statement may be made for the Diffeomorphism constraint.

There are also some problems with the Euclidean Quantum Gravity programme:

1. Wick Rotation is not a diffeomorphism invariant procedure, i.e., not every Riemannian metric posses a "Lorentzian section". In other words: taking ${\displaystyle x^{0}\to ix^{0}}$ doesn't always give us a Lorentzian metric.
2. We cannot classify four-dimensional manifolds, so it is impossible to sum over all of them.
3. The Euclidean gravitational action is unbounded from below. We can see this by taking some metric, the a "bad" conformal transformation to obtain another metric. The new conformally-transformed metric "blows up". This leads to the conformal-factor problem...see Dasgupta (arXiv:0801.4770) for some discussion.

Consequently, the current approach appears to be using discrete versions of general relativity using Regge calculus. The promising programme of Causal dynamical triangulations appears to be Euclidean quantum gravity's successor...

There are also a few references which may be useful:

• Arundhati Dasgupta, "The Measure in Euclidean Quantum Gravity" (arXiv:1106.1679)
• Arundhati Dasgupta, "The gravitational path integral and trace of the diffeomorphisms" (arXiv:0801.4770) Gen.Rel.Grav.43 (2011) 2237-2255
• Bryce S. DeWitt, Giampiero Esposito, "An introduction to quantum gravity" (arXiv:0711.2445)
• Claus Kiefer, Quantum Gravity. Oxford University Press, second ed.
• Emil Mottola, "Functional Integration Over Geometries" (arXiv:hep-th/9502109) J.Math.Phys. 36 (1995) 2470-2511

Just my notes on the subject... —Pqnelson (talk) 17:44, 2 March 2012 (UTC)

Also note that S.Yu. Alexandrov and D.V. Vassilevich have considered Euclidean quantum gravity using Loop quantum gravity variables, in their paper "Path integral for the Hilbert-Palatini and Ashtekar gravity" (arXiv:gr-qc/9806001)
Pqnelson (talk) 18:03, 2 March 2012 (UTC)

## Erroneous Definition of Euclidean Path Integral

This article states that "It is also assumed that the manifolds are compact, connected and boundaryless (i.e. no singularities)." The first assumption is incorrect; in fact, it is assumed that the manifolds are noncompact. The metrics over which one integrates are assumed to the Asymptotically Euclidean, meaning that in each asymptotic region, there is a coordinate chart such that the metric takes the form (1+4I/(3\pi r^2) (dx^2+dy^2+dz^2+dt^2) for n>0, where r^2=x^2+y^2+z^2+t^2, and I is the value of the action. See Shoen & Yau, Phys. Rev. Lett. 42 (1979) 547. For confirmation and further discussion, see, e.g., Gibbons & Pope, Comm.Math.Phys. 66 (1979) 267. The manifolds can have boundaries, so long as they remain noncompact and so long as you modify the action integral with an appropriate boundary term, as prescribed by Hawking. See, e.g., Gibbons & Hawking, Phys. Rev. D15 (1976) 2752. Whether the region must be connected, I'm not sure, but I think not. This gets into questions of "baby universes". In fact, I don't think one need speak of the manifold at all; it is just an integral over a function space defined by certain constraints on the functions. I hope whoever wrote this article will rewrite carefully. — Preceding unsigned comment added by MidwestGeek (talkcontribs) 20:15, 10 May 2014 (UTC)