Euclidean quantum gravity

Euclidean quantum gravity
Name	Euclidean quantum gravity
Field	Theoretical physics
Introduced	1970s
Key people	Stephen Hawking, G. W. Gibbons, Bryce DeWitt, James Hartle, Gerald 't Hooft, John Wheeler, Roger Penrose, Paul Dirac, Sidney Coleman, Yakov Borisovich Zel'dovich, Bryce S. DeWitt

Euclidean quantum gravity Euclidean quantum gravity is an approach in theoretical physics that formulates a quantum theory of gravitation by analytically continuing time coordinates to imaginary values and studying a Riemannian metric path integral. It draws on techniques developed in Paul Dirac's canonical quantization program, Richard Feynman's path integral formalism, and semiclassical methods associated with Stephen Hawking and G. W. Gibbons, aiming to connect results from General relativity with frameworks from Quantum field theory and Statistical mechanics. The approach has influenced research programs connected to Cosmology, Black hole thermodynamics, and the quest for a consistent ultraviolet completion such as String theory and Loop quantum gravity.

Introduction

Euclidean quantum gravity emerged from attempts by Stephen Hawking, G. W. Gibbons, and Bryce DeWitt to apply Feynman's path integral to gravitational fields by rotating to imaginary time, leveraging ideas from Paul Dirac and semiclassical analysis by John Wheeler and Roger Penrose. Proponents used analytic continuation to convert a Lorentzian Lorentzian manifold with signature (−,+,+,+) into a Riemannian manifold with signature (+,+,+,+), invoking techniques familiar in Statistical mechanics, Thermodynamics, and Quantum field theory in curved spacetime developed by N. D. Birrell and P. C. W. Davies. This introduction set the stage for work on instantons, tunnelling, and entropy calculations associated with Black holes and Cosmology by figures like James Hartle and Gary Gibbons.

Mathematical formulation

The mathematical formulation uses a Euclidean path integral Z = ∫ Dg Dφ exp(−SE[g,φ]/ħ) where the Euclidean action SE is constructed from the Einstein–Hilbert action augmented by boundary terms such as the Gibbons–Hawking–York boundary term introduced by G. W. Gibbons and James York Jr.. Fields include metrics g and matter fields φ drawn from representations familiar in Dirac and Weyl formulations. The formal measure Dg raises issues related to the Diff(M) gauge group, Fadeev–Popov ghosts introduced by L. D. Faddeev and V. N. Popov, and anomalies analogous to those studied by Gerard 't Hooft and Stephen Adler. Rigorous treatments invoke tools from Differential geometry, Spectral theory associated with the Laplace–Beltrami operator, and index theorems such as the Atiyah–Singer index theorem.

Path integral and semiclassical approximation

Semiclassical evaluation employs saddle-point expansions around classical solutions to the Euclidean field equations, connecting to the stationary-phase methods developed by Ludwig Faddeev and the WKB techniques used by Kramer and Jeffreys. Dominant contributions come from classical stationary points including Euclidean instantons found by Stephen Hawking, G. W. Gibbons, Sidney Coleman, and Alexander Polyakov. One computes one-loop determinants using heat-kernel methods explored by B. S. DeWitt and renormalization inputs from Gerard 't Hooft and Martinus Veltman. These approximations underpin derivations of entropy formulas for Schwarzschild black holes and thermodynamic relations first analyzed by Jacob Bekenstein and Stephen Hawking.

Instantons and gravitational tunnelling

Instantons in Euclidean quantum gravity are finite-action solutions to the Euclidean Einstein equations; notable examples include the Schwarzschild instanton, the Taub–NUT solution, and cosmological instantons studied by James Hartle and Stephen Hawking in the no-boundary proposal. Instanton methods, adapted from Sidney Coleman's work on false-vacuum decay and A. D. Linde's studies of inflationary tunnelling, explain nucleation processes in Inflation and black hole pair production in backgrounds like de Sitter space and anti-de Sitter space examined by Edward Witten and Gary Gibbons. Gravitational instantons also connect to topological transitions considered in Euclidean topology and by Michael Atiyah and Isadore Singer.

Renormalization and ultraviolet issues

Renormalization in Euclidean quantum gravity confronts the nonrenormalizability of the perturbative expansion around a fixed background as shown by Gerard 't Hooft and Martinus Veltman and later by Goroff and Sagnotti. Counterterm structures involving higher-derivative invariants were analyzed by K. S. Stelle and motivated investigations into asymptotic safety proposed by Steven Weinberg and pursued by Martin Reuter and collaborators. Ultraviolet completions studied include String theory by Michael Green, John Schwarz, and Edward Witten, and nonperturbative frameworks like Loop quantum gravity by Carlo Rovelli and Lee Smolin, as well as functional renormalization group approaches by Reuter and Christof Wetterich.

Relation to Lorentzian quantum gravity

Relating Euclidean formulations to Lorentzian quantum gravity requires analytic continuation prescriptions such as the Wick rotation used in Richard Feynman's work and more geometric approaches explored by Bryce DeWitt and John Hartle. The connection intersects with canonical quantization programs initiated by Paul Dirac and developed by Bryce DeWitt and Abhay Ashtekar in loop approaches. Debates over contour choices, Picard–Lefschetz theory invoked by C. B. M. H. Thacker and developed further in modern applications, and the role of causality highlighted by Roger Penrose and Stephen Hawking remain active research themes connecting Euclidean methods to real-time propagation and the S-matrix frameworks of Gerard 't Hooft.

Applications and implications in cosmology and black holes

Euclidean techniques underpin proposals like the Hartle–Hawking no-boundary wavefunction by James Hartle and Stephen Hawking and semiclassical derivations of black hole entropy and Hawking radiation studied by Stephen Hawking and Jacob Bekenstein. They inform analyses of primordial perturbations in inflationary models by Alan Guth, Andrei Linde, and Andrei Starobinsky, and calculations of pair creation of black holes in de Sitter backgrounds by Gary Gibbons and Steven Carlip. Euclidean methods also connect to holographic principles through the AdS/CFT correspondence by Juan Maldacena and thermodynamic phase structures akin to the Hawking–Page phase transition examined by Stephen Hawking and Don Page, influencing ongoing work in Quantum cosmology and quantum aspects of singularities.

Category:Quantum gravity