Euclidean quantum gravity and stochastic approach: Physical reality of complex-valued instantons
Dong-han Yeom
备注：
4 pages, 1 figure; A proceeding for the Multiverse and Fundamental Cosmology Conference (Multicosmofun'12). Talk on September 10, 2012, Szczecin, Poland
In this talk, we compare two states: the stationary state in stochastic inflation and the ground state wave function of the universe. We already know that, for the potential with a static field, two pictures give the same probability distribution. Here, we go beyond this limit and assert that two pictures indeed have deeper relations. We illustrate a simple example so that there is a corresponding instanton if a certain field value has a non-zero probability in the statistical side. This instanton should be complex-valued. Furthermore, the compact manifold in the Euclidean side can be interpreted as a coarse-graining grid size in the stochastic universe. Finally, we summarize the recent status and possible applications.
We compute the on-shell Euclidean action of Schwarzschild-de Sitter black holes, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. Although Euclidean de Sitter black hole geometries have conical singularities for generic masses, their on-shell action is finite and is shown to be independent of the Euclidean time periodicity and equal to minus the sum of the black hole and cosmological horizon entropy. We apply this result to compute the probability for a nonrotating, neutral arbitrary mass black hole to nucleate spontane…
We study the emergence of de Sitter space in Euclidean dynamical triangulations (EDT). Working within the semi-classical approximation, it is possible to relate the lattice parameters entering the simulations to the partition function of Euclidean quantum gravity. We verify that the EDT geometries behave semi-classically, and by making contact with the Hawking-Moss instanton solution for the Euclidean partition function, we show how to extract a value of the renormalized Newton's constant from the simulations. This value is consistent with that of our previous determination coming from the int…
We study the geometry of Euclidean instantons in loop quantum cosmology (LQC) such as those relevant for the no-boundary proposal. Confining ourselves to the simplest case of a cosmological constant in minisuperspace cosmologies, we analyze solutions of the semiclassical (Euclidean) path integral in LQC. We find that the geometry of LQC instantons have the peculiar feature of an infinite tail which distinguishes them from Einstein gravity. Moreover, due to quantum-geometry corrections, the small-a behaviour of these instantons seem to naturally favor a closing-off of the geometry in a regular …
I present an informal overview of several recent results about Euclidean saddle points sourced by axion fields in quantum gravity (AdS/CFT), such as wormholes, their extremal "D-instanton" limits and their under-extremal singular counterparts. Concerning wormholes we argue they cannot contribute to the path integral because a stability analysis suggests they fragment like other super-extremal objects. For concrete AdS/CFT embeddings the Euclidean saddle point solutions are neatly described by geodesic curves living inside moduli spaces and can typically be solved for using group theory. Our wo…
The Euclidean path integral is well approximated by instantons. If instantons are dynamical, then instantons are necessarily complexified. These fuzzy instantons can have various physical applications. In slow-roll inflation models, fuzzy instantons can explain the probability distribution of the initial conditions of the universe. Even though the potential shape does not satisfy the slow-roll conditions following the swampland criteria, still the fuzzy instantons can explain the origin of the universe. If we extend the Euclidean path integral beyond the no-boundary proposal, we may study fuzz…
Nucleating an Inflationary Universe: Euclidean Wormholes and their No-Boundary Limit
Department of Theoretical Physics, A.Razmadze Mathematical Institute at I.Javakhishvili Tbilisi State University（伊万·贾瓦希什维利第比利斯国立大学A.拉兹马泽数学研究所理论物理系）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））
The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well known Eguchi-Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically {\it{globally}} Euclidean, i.e., the boundary of the space as $r\longrightarrow\infty$ is $S^3=SU(2)$.
$α'$ corrections to self-dual gravitational instantons
Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学自由艺术学院科学系）
This paper is devoted to an investigation of Euclidean wormholes made by fuzzy instantons. We investigate the Euclidean path integral in anti-de Sitter space. In Einstein gravity, we introduce a scalar field with a potential. Because of the analyticity, there is a contribution of complex-valued instantons, so-called fuzzy instantons. If we have a massless scalar field, then we obtain Euclidean wormholes, where the probabilities become smaller and smaller as the size of the throat becomes larger and larger. If we introduce a non-trivial potential, then in order to obtain a non-zero tunneling ra…
In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained aft…
We study supergravity instantons sourced by axion (and saxion) fields in the Euclidean $AdS_3\times S^3 \times CY_2$ vacua of IIB supergravity. Such instantons are described by geodesic curves on the moduli space; the timelike geodesics can describe Euclidean wormholes, the lightlike geodesics describe (generalisations of) D-instantons and spacelike geodesics are sub-extremal versions thereof. We perform a concrete classification of such geodesics and find that, despite earlier claims, the wormholes fail to be regular. A subclass of the lightlike geodesics is supersymmetric and, up to dualitie…
The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\mathbb{R}^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a …
We present novel regular Euclidean solutions to General Relativity in presence of Maxwell and conformally coupled scalar fields. In particular, we consider metrics of the Eguchi-Hanson and Taub-NUT families to solve the field equations analytically. The solutions have nontrivial topology labeled by the Hirzebruch signature and Euler characteristic that we compute explicitly. We find that, although the solutions are locally inequivalent with the original (anti-)self-dual Eguchi-Hanson metric, their asymptotically locally Euclidean limit leads to the same global properties. We revisit the Taub-N…
Instanton is known to exist in Euclidean spacetime only. Their role in real time dynamics is usually understood as tunneling effect by Wick rotation. We illustrate other effects of instanton in holography by investigating 5d effective gravity theory of the black D3-brane-D-instanton system. The supergravity description of the D3-brane-D-instanton system is dual to the super Yang-Mills theory with topological excitations of the vacuum. We obtain Euclidean correlators in the presence of instantons by analyzing of the fluctuations of the bulk fields in the 5d effective theory. Furthermore, analyt…
We investigate supergravity instantons in Euclidean $\rm AdS_5\times S^5/\mathbb{Z}_k$. These solutions are expected to be dual to instantons of $\mathcal{N}=2$ quiver gauge theories. On the supergravity side the (extremal) instanton solutions are neatly described by the (lightlike) geodesics on the AdS moduli space for which we find the explicit expression and compute the on-shell actions in terms of the quantised charges. The lightlike geodesics fall into two categories depending on the degree of nilpotency of the Noether charge matrix carried by the geodesic: For degree 2 the instantons pre…
In this paper, we investigate complex-valued Euclidean wormholes in the Starobinsky inflation. Due to the properties of the concave inflaton potential, the classicality condition at both ends of the wormhole can be satisfied, as long as the initial condition of the inflaton field is such that it is located sufficiently close to the hilltop. We compare the probabilities of classicalized wormholes with the Hartle-Hawking compact instantons and conclude that the Euclidean wormholes are probabilistically preferred than compact instantons, if the inflation lasts more than 50 e-foldings. Our result …
