Euclidean quantum gravity

Short description: Approach to quantum gravity utilizing Wick rotations

In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics.

## Contents

* 1 Introduction in layperson's terms
* 1.1 The Wick rotation
* 1.2 Application in quantum mechanics
* 2 More rigorous treatment
* 3 Relation to ADM formalism
* 4 References

##  Introduction in layperson's terms

### The Wick rotation

Main article: Physics:Wick rotation

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. More precisely, it substitutes a mathematical problem in Minkowski space into 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 complex numbers are represented as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of π/2 radians 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 model the bath volume with the different gradients of temperature we would have to subdivide this volume into infinitesimal volumes and see how they interact. We know such infinitesimal volumes are in fact water molecules. If we represent all molecules in the bath by only one molecule in an attempt to simplify the problem, this unique molecule should walk along all possible paths that the real molecules might follow. The path integral formulation is the conceptual tool used to describe the movements of this unique molecule, and Wick rotation is one of the mathematical tools that are very useful to analyse a path integral problem.

### Application in quantum mechanics

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 have different speeds. It differs clearly to the movement of a classical object (e.g. a billiard ball) for which a single path with precise position and speed can be described. A quantum object does not move from A to B with a single path, but moves from A to B by all ways possible at the same time. According to the Feynman path-integral formulation of quantum mechanics, 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 appealing in this new approach is its lack of singularities when they are unavoidable in general relativity.

Another operational problem with general relativity is the computational difficulty, because of the complexity of the mathematical tools used. Path integrals in contrast have been used in mechanics since the end of the nineteenth century and are well known. In addition, the path-integral formalism is used both in classical and quantum physics so it might be a good starting point for unifying general relativity and quantum theories. For example, the quantum-mechanical Schrödinger equation and the classical heat equation are related by Wick rotation. So the Wick relation is a good tool to relate a classical phenomenon to a quantum phenomenon. The ambition of Euclidean quantum gravity is to use the Wick rotation to find connections between a macroscopic phenomenon, gravity, and 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.

∫𝒟𝐠𝒟ϕexp⁡(∫d4x|𝐠|(R+ℒmatter))

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 ϕ, then the path integral reads

Z=∫𝒟𝐠𝒟ϕexp⁡(∫d4x|𝐠|(R+ℒmatter))

where integration over 𝒟𝐠 includes an integration over the three-metric, the lapse function N, and shift vector Na. But we demand that Z be independent of the lapse function and shift vector at the boundaries, so we obtain

δZδN=0=∫𝒟𝐠𝒟ϕδSδN|Σexp⁡(∫d4x|𝐠|(R+ℒmatter))

where Σ 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

* DeWitt, Bryce S. (1967-10-25). "Quantum Theory of Gravity. II. The Manifestly Covariant Theory" (in en). Physical Review (American Physical Society (APS)) 162 (5): 1195–1239. doi:10.1103/PhysRev.162.1195. ISSN 0031-899X. Bibcode: 1967PhRv..162.1195D. https://link.aps.org/doi/10.1103/PhysRev.162.1195.
* Dewitt, Bryce S.; Esposito, Giampiero (February 2008). "An introduction to quantum gravity" (in en). International Journal of Geometric Methods in Modern Physics 05 (1): 101–156. doi:10.1142/S0219887808002679. ISSN 0219-8878. Bibcode: 2008IJGMM..05..101D. https://www.worldscientific.com/doi/abs/10.1142/S0219887808002679.
* Feynman, Richard P.; Morinigo, Fernando B.; Wagner, William G. (1995) (in en). Feynman lectures on gravitation. Boulder, Colo. : Westview. ISBN 978-0-8133-4038-8. https://archive.org/details/feynmanlectureso0000feyn_p1z8.
* Gibbons, G W; Hawking, S W (1993). Gibbons, Gary W.. ed (in en). Euclidean quantum gravity. Singapore River Edge, NJ: World Scientific. doi:10.1142/1301. ISBN 978-981-02-0516-4. https://www.worldscientific.com/worldscibooks/10.1142/1301.
* Hamber, H. W. (2009). Quantum gravitation: the Feynman path integral approach. Berlin: Springer. ISBN 978-3-540-85293-3.
* Hawking, S. W. (May 1993). "15. The path-integral approach to quantum gravity" (in en). Euclidean Quantum Gravity. WORLD SCIENTIFIC. pp. 29–72. doi:10.1142/9789814539395_0002. ISBN 978-981-02-0515-7. https://www.worldscientific.com/doi/10.1142/9789814539395_0002.
* Hartle, J. B.; Hawking, S. W. (1983-12-15). "Wave function of the Universe" (in en). Physical Review D (American Physical Society (APS)) 28 (12): 2960–2975. doi:10.1103/PhysRevD.28.2960. ISSN 0556-2821. Bibcode: 1983PhRvD..28.2960H. https://link.aps.org/doi/10.1103/PhysRevD.28.2960. Formally relates Euclidean quantum gravity to ADM formalism.
* Kiefer, Claus (2012). Quantum gravity. International series of monographs on physics (3rd ed.). Oxford: Oxford University Press. ISBN 978-0-19-958520-5.
* Mottola, Emil (1995). "Functional integration over geometries". Journal of Mathematical Physics 36 (5): 2470–2511. doi:10.1063/1.531359. ISSN 0022-2488. Bibcode: 1995JMP....36.2470M.
* Veltman, Martin J.G. (March 1993). "Quantum theory of gravitation" (in en). Methods in Field Theory. North Holland Publishing Company. pp. 265–328. doi:10.1142/9789814412674_0007. ISBN 978-9971-83-078-6. https://dspace.library.uu.nl/handle/1874/4837.

* v
* t
* e

Theories of gravitation

Standard|

| Newtonian gravity (NG)|

* Newton's law of universal gravitation
* Gauss's law for gravity
* Poisson's equation for gravity
* History of gravitational theory

|

General relativity (GR)|

* Introduction
* History
* Mathematics
* Exact solutions
* Resources
* Tests
* Post-Newtonian formalism
* Linearized gravity
* ADM formalism
* Gibbons–Hawking–York boundary term

Alternatives to
general relativity|

| Paradigms|

* Classical theories of gravitation
* Quantum gravity
* Theory of everything

|

Classical|

* Einstein–Cartan
* Bimetric theories
* Gauge theory gravity
* Teleparallelism
* Composite gravity
* f(R) gravity
* Infinite derivative gravity
* Massive gravity
* Modified Newtonian dynamics, MOND
* AQUAL
* Tensor–vector–scalar
* Nonsymmetric gravitation
* Scalar–tensor theories
* Brans–Dicke
* Scalar–tensor–vector
* Conformal gravity
* Scalar theories
* Nordström
* Whitehead
* Geometrodynamics
* Induced gravity
* Chameleon
* Pressuron
* Degenerate Higher-Order Scalar-Tensor theories

Quantum-mechanical|

* Euclidean quantum gravity
* Canonical quantum gravity
* Wheeler–DeWitt equation
* Loop quantum gravity
* Spin foam
* Causal dynamical triangulation
* Asymptotic safety in quantum gravity
* Causal sets
* DGP model
* Rainbow gravity theory

Unified-field-theoric|

* Kaluza–Klein theory
* Dilaton
* Supergravity

Unified-field-theoric and
quantum-mechanical|

* Noncommutative geometry
* Semiclassical gravity
* Superfluid vacuum theory
* Logarithmic BEC vacuum
* String theory
* M-theory
* F-theory
* Heterotic string theory
* Type I string theory
* Type 0 string theory
* Bosonic string theory
* Type II string theory
* Little string theory
* Twistor theory
* Twistor string theory

Generalisations /
extensions of GR|

* Scale relativity
* Liouville gravity
* Lovelock theory
* (2+1)-dimensional topological gravity
* Gauss–Bonnet gravity
* Jackiw–Teitelboim gravity

Pre-Newtonian
theories and
toy models|

* Aristotelian physics
* CGHS model
* RST model
* Mechanical explanations
* Fatio–Le Sage
* Entropic gravity
* Gravitational interaction of antimatter
* Physics in the medieval Islamic world
* Theory of impetus

Related topics|

* Graviton

* v
* t
* e

Quantum gravity

Central concepts|

* AdS/CFT correspondence
* Causal patch
* Gravitational anomaly
* Graviton
* Holographic principle
* IR/UV mixing
* Planck scale
* Quantum foam
* Trans-Planckian problem
* Weinberg–Witten theorem

Toy models|

* 2+1D topological gravity
* CGHS model
* Jackiw-Teitelboim gravity
* Liouville gravity
* RST model
* Topological quantum field theory

Quantum field theory
in curved spacetime|

* Bunch–Davies vacuum
* Hawking radiation
* Semiclassical gravity
* Unruh effect

Black holes|

* Black hole complementarity
* Black hole information paradox
* Black-hole thermodynamics
* Bousso's holographic bound
* ER=EPR
* Firewall (physics)
* Gravitational singularity

Approaches|

| String theory|

* Bosonic string theory
* M-theory
* Supergravity
* Superstring theory

|

Canonical quantum gravity|

* Loop quantum gravity
* Wheeler-DeWitt equation

Euclidean quantum gravity|

* Hartle-Hawking state

Others|

* Causal dynamical triangulation
* Causal sets
* Noncommutative geometry
* Spin foam
* Group field theory
* Superfluid vacuum theory
* Twistor theory
* Dual graviton

Applications|

* Quantum cosmology
* Eternal inflation
* Multiverse
* FRW/CFT duality

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