Fall factor

Introduction in layman terms

In physics, a Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to dynamics problems in n dimensions, by transposing their descriptions in n+1 dimensions, by trading one dimension of space for one dimension of time. For the more mathematically inclined people it substitutes a mathematical problem in Minkowski space to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.
It is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of pi/2 about the origin.
For example, a Wick rotation could be used to relate a macroscopic event temperature diffusion (like in a bath) to the underlying thermal movements of molecules. If we attempt to modelize the bath volume with the different gradients of temperature we would have to subdivize this volume in infinetesimal volumes and see how they interact. We know such infenitesimal volumes are in fact water molecules. If we represent all molecules in the bath by only one molecule in an attempt to simplify and manage easily this problem, this unique molecule should walk along all possible paths that every real molecules will follow. Path integral is the conceptual tool used to describe the movements of this unique molecule, and Wick rotation is one of the mathematical tool that are very useful to analyse an integral path problem.

In a somewhat similar manner, the motion of a quantum object as described by quantum mechanics implies that it can exist simultaneously in different positions and different speeds. It differs clearly to the movement of a classical object (e.g., a billiard ball), since in this case a single path with precise position and speed can be described. An quantum object does not move from A to B with a single path, but it moves from A to B by all ways possible at the same time. According to the principle of superposition (Richard Feynman's integral of path in 1963), the path of the quantum object is described mathematically as a weighted average of all those possible paths. In 1966 an explicitly gauge invariant functional-integral algorithm was found by DeWitt, which extended Feynman’s new rules to all orders. What is appealling in this new approach is its lack of singularities when they are un-avoidable in General Relativity.
Another operational problem with General Relativity is the difficulty to do calculations, because of the complexity of the mathematical tools used. Integral of path in contrast is used in mechanics since the end of the 19th century and is well known. In addition Path integral is a formalism used both in mechanics and quantum theories so it might be a good starting point for unifying General Relativity and Quantum theories. Some quantum features like the Schrödinger equation and the heat equation are also related by Wick rotation. So the Wick relation is a good tool to relate a classical phenomena to a quantum phenomena. The ambition of Euclidian quantum gravity is to use the Wick rotation to find connections between a macroscopic phenomena, the gravity, to something more microscopic.

More rigorous treatment

Euclidean quantum gravity refers to a Wick rotated version of quantum gravity, formulated as a quantum field theory. The manifolds that are used in this formulation are 4 dimensional Riemannian manifolds instead of pseudo Riemannian manifolds. It is also assumed that the manifolds are compact, connected and boundaryless (i.e. no singularities). Following the usual quantum field-theoretic formulation, the vacuum to vacuum amplitude is written as a functional integral over the metric tensor, which is now the quantum field under consideration.

$\int \mathcal{𝒟}\mathbf{g}\mathcal{𝒟}\varphi \exp (\int d^{4}x\sqrt{\left|\mathbf{g}\right|}(R+{\mathcal{ℒ}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}))$

where φ denotes all the matter fields. See Einstein-Hilbert action.

Relation to ADM Formalism

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 $\varphi$, then the path integral reads

$Z=\int \mathcal{𝒟}\mathbf{g}\mathcal{𝒟}\varphi \exp (\int d^{4}x\sqrt{\left|\mathbf{g}\right|}(R+{\mathcal{ℒ}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}))$

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

$\frac{\delta Z}{\delta N}=0=\int \mathcal{𝒟}\mathbf{g}\mathcal{𝒟}\varphi {\frac{\delta S}{\delta N}|}_{\mathrm{\Sigma }}\exp (\int d^{4}x\sqrt{\left|\mathbf{g}\right|}(R+{\mathcal{ℒ}}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}}))$

where $\mathrm{\Sigma }$ 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 (take functional derivative with respect to the shift functions instead).

References

• Arundhati Dasgupta, "The Measure in Euclidean Quantum Gravity." Eprint arXiv:1106.1679.
• Arundhati Dasgupta, "The gravitational path integral and trace of the diffeomorphisms." Gen.Rel.Grav. 43 (2011) 2237–2255. Eprint arXiv:0801.4770.
• Bryce S. DeWitt, Giampiero Esposito, "An introduction to quantum gravity." Int.J.Geom.Meth.Mod.Phys. 5 (2008) 101–156. Eprint arXiv:0711.2445.
• G. W. Gibbons and S. W. Hawking (eds.), Euclidean quantum gravity, World Scientific (1993)
• J. B. Hartle and S. W. Hawking, "Wave function of the Universe." Phys. Rev. D 28 (1983) 2960–2975, eprint. Formally relates Euclidean quantum gravity to ADM formalism.
• Claus Kiefer, Quantum Gravity. Oxford University Press, second ed.
• Emil Mottola, "Functional Integration Over Geometries." J.Math.Phys. 36 (1995) 2470–2511. Eprint arXiv:hep-th/9502109.