Euclidean Quantum Gravity

The following sections are included:

• CONTENTS
• INTRODUCTION

All one-loop divergencies of pure gravity and all those of gravitation interacting with a scalar particle are calculated. In the case of pure gravity, no physically relevant divergencies remain; they can all be absorbed in a field renormalization. In case of gravitation interacting with scalar particles, divergencies in physical quantities remain, even when employing the socalled improved energy-momentum tensor.

The following sections are included:

• Introduction
• The action
• Complex spacetime
• The indefiniteness of the gravitational action
• The stationary-phase approximation
• Zeta function regularization
• The background fields
• Gravitational thermodynamics
• Beyond one loop
• Spacetime foam

The following sections are included:

• INTRODUCTION
• THERMAL PROPERTIES
• THE GRAVITATIONAL VACUUM
• ASYMPTOTICALLY LOCALLY EUCLIDEAN METRICS
• SPACETIME FOAM
• REFERENCES

The Euclidean action for gravity is not positive definite unlike those of scalar and Yang-Mills fields. Indefiniteness arises because conformal transformations can make the action arbitrarily negative. In order to make the path integral converge one has to take the contour of integration for the conformal factor to be parallel to the imaginary axis. The path integral will then converge at least in the one-loop approximation if a certain positive action conjecture holds. We perform a zeta function regularization of the one-loop term for gravity and obtain a non-trivial scaling behaviour in cases in which the background metric has non-zero curvature tensor, and hence non-trivial topologies.

We extend our previous method of proving the positive-mass conjecture to prove the positive-action conjecture of Hawking for asymptotically Euclidean metric. This result is crucial in proving the path integral convergent in the Euclidean quantum gravity theory.

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises to n dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.

The classical Euclidean action for general relativity is unbounded below; therefore Euclidean functional integrals weighted by this action are manifestly divergent. However, as a consequence of the positive-energy theorem, physical amplitudes for asymptotically flat spacetimes can indeed be expressed as manifestly convergent Euclidean functional integrals formed in terms of the physical degrees of freedom. From these integrals, we derive expressions for these same physical quantities as Euclidean integrals over the full set of variables for gravity computed as metric perturbations off a flat background. These parametrized Euclidean functional integrals are weighted by manifestly positive actions with rotated conformal factors. They are similar in form to Euclidean functional integrals obtained by the Gibbons-Hawking-Perry prescription of contour rotation.

In the path-integral approach to the decay of a metastable state by quantum tunneling, the tunneling process is dominated by a solution to the imaginary-time equations of motion, called the bounce. In all known cases, the second variational derivative of the euclidean action at the bounce has one and only one negative eigenvalue. This note explains this phenomenon by showing it is an inevitable feature of the bounce for a wide class of systems. This class includes a set of particles interacting through potentials obeying some mild technical restrictions, and also theories of interacting scalar and gauge fields. There may exist solutions in other ways like bounces and which have more than one negative eigenvalue, but, even if they do exist, they have nothing to do with tunneling.

Let M be a compact Riemannian manifold having volume V, mean curvature, bounded below by B, and diameter D. Let V₁ be the volume of a ball of radius D in the simply-connected Riemannian manifold having constant sectional curvature, mean curvature B, and the same dimension as M. Then V ≤ V₁ Equality holds only if M has constant curvature. The proof consists in getting bounds on the Jacobians of an exponential map of M, using the minimizing properties of Jacobi fields with respect to 2nd variation, and integrating the bound on the ball of radius D in the tangent space, the domain of exp. The details for a similar result on Kaehler manifolds will appear in a paper by the author and S. I. Goldberg. Myers’ theorem that when $B>0,D\le \pi /\sqrt{B/\left(n-1\right)}$ is a corollary to the proof.

It is suggested that the apparent cosmological constant is not necessarily zero but that zero is by far the most probable value. One requires some mechanism like a three-index antisymmetric tensor field or topological fluctuations of the metric which can give rise to an effective cosmological constant of arbitrary magnitude. The action of solutions of the euclidean field equations is most negative, and the probability is therefore highest, when this effective cosmological constant is very small.

In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature $\frac{hk}{2\pi k}\approx {10}^{-6}\left(\frac{M_{\odot }}{M}\right){}^{\circ }\text{K}$ where k is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 10¹⁵ g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law: $S+\frac{1}{4}A$ never decreases never decreases where S is the entropy of matter outside black holes and A is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.

The Feynman path-integral method is applied to the quantum mechanics of a scalar particle moving in the background geometry of a Schwarzschild black hole. The amplitude for the black hole to emit a scalar particle in a particular mode is expressed as a sum over paths connecting the future singularity and infinity. By analytic continuation in the complexified Schwarzschild space this amplitude is related to that for a particle to propagate from the past singularity to infinity and hence by time reversal to the amplitude for the black hole to absorb a particle in the same mode. The form of the connection between the emission and absorption probabilities shows that a Schwarzschild black hole will emit scalar particles with a thermal spectrum characterized by a temperature which is related to its mass, M, by T = ħ c³/8π GMk. Thereby a conceptually simple derivation of black-hole radiance is obtained. The extension of this result to other spin fields and other black-hole geometries is discussed.

This paper concerns itself with the possibility of thermal equilibrium between a black hole and a heat bath implied by Hawking’s discovery of black hole emission. We argue that in an isolated box of radiation, for sufficiently high energy density a black hole will condense out. We introduce thermal Green functions to discuss this equilibrium and are able to extend the original arguments, that the equilibrium is possible based on fields interacting solely with the external gravitational field, to the case when mutual and self interactions are included.

One can evaluate the action for a gravitational field on a section of the complexified spacetime which avoids the singularities. In this manner we obtain finite, purely imaginary values for the actions of the Kerr-Newman solutions and de Sitter space. One interpretation of these values is that they give the probabilities for finding such metrics in the vacuum state. Another interpretation is that they give the contribution of that metric to the partition function for a grand canonical ensemble at a certain temperature, angular momentum, and charge. We use this approach to evaluate the entropy of these metrics and find that it is always equal to one quarter the area of the event horizon in fundamental units. This agrees with previous derivations by completely different methods. In the case of a stationary system such as a star with no event horizon, the gravitational field has no entropy.

The instabilities of quantum gravity are investigated using the path-integral formulation of Einstein’s theory. A brief review is given of the classical gravitational instabilities, as well as the stability of flat space. The Euclidean path-integral representation of the partition function is employed to discuss the instability of flat space at finite temperature. Semiclassical, or saddle-point, approximations are utilized. We show how the Jeans instability arises as a tachyon in the graviton propagator when small perturbations about hot flat space are considered. The effect due to the Schwarzschild instanton is studied. The small fluctuations about this instanton are analyzed and a negative mode is discovered. This produces, in the semiclassical approximation, an imaginary part of the free energy. This is interpreted as being due to the metastability of hot flat space to nucleate black holes. These then evolve by evaporation or by accretion of thermal gravitons, leading to the instability of hot flat space. The nucleation rate of black holes is calculated as a function of temperature.

The Bekenstein-Parker Gaussian path-integral approximation is used to evaluate the thermal propagator for a conformally invariant scalar field in an ultrastatic metric. If the ultrastatic metric is conformal to a static Einstein metric, the trace anomaly vanishes and the Gaussian approximation is especially good. One then gets the ordinary flat-space expressions for the renormalized mean-square field and stress-energy tensor in the ultrastatic metric. Explicit formulas for the changes in 〈ϕ²〉 and 〈 T_µv〉 resulting from a conformal transformation of an arbitrary metric are found and used to take the Gaussian approximations for these quantities in the ultrastatic metric over to the Einstein metric. The result for 〈ϕ²〉 is exact for de Sitter space and agrees closely with the numerical calculations of Fawcett and Whiting in the Schwarzschild metric. The result for 〈 T_µv〉 is exact in de Sitter space and the Nariai metric and is close to Candelas’s values on the bifurcation two-sphere in the Schwarzschild metric. Thus one gets a good closed-form approximation for the energy density and stresses of a conformal scalar field in the Hartle-Hawking state everywhere outside a static black hole.

The vacuum expectation value of the stress-energy tensor for the Hartle-Hawking state in Schwarzschild space-time has been calculated for the conformal scalar field. $〈T_{\mu }^v〉$ separates naturally into the sum of two terms. The first coincides with an approximate expression suggested by Page. The second term is a “remainder” that may be evaluated numerically. The total expression is in good qualitative agreement with Page’s approximation. These results are at variance with earlier results given by Fawcett whose error is explained.

It is shown that the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant. An observer in these models will have an event horizon whose area can be interpreted as the entropy or lack of information of the observer about the regions which he cannot see. Associated with the event horizon is a surface gravity k which enters a classical “first law of event horizons” in a manner similar to that in which temperature occurs in the first law of thermodynamics. It is shown that this similarity is more than an analogy: An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon. If the observer absorbs some of this radiation, he will gain energy and entropy at the expense of the region beyond his ken and the event horizon will shrink. The derivation of these results involves abandoning the idea that particles should be defined in an observer-independent manner. They also suggest that one has to use something like the Everett-Wheeler interpretation of quantum mechanics because the back reaction and hence the spacetime metric itself appear to be observer-dependent, if one assumes, as seems reasonable, that the detection of a particle is accompanied by a change in the gravitational field.

It is possible for a classical field theory to have two stable homogeneous ground states, only one of which is an absolute energy minimum. In the quantum version of the theory, the ground state of higher energy is a false vacuum, rendered unstable by barrier penetration. There exists a well-established semiclassical theory of the decay of such false vacuums. In this paper, we extend this theory to include the effects of gravitation. Contrary to naive expectation, these are not always negligible, and may sometimes be of critical importance, especially in the late stages of the decay process.

The universe might have had a prolonged exponentially expanding phase caused by its being stuck in a metastable state of the grand unified phase transition. The only way that it could exit from this exponential expansion without introducing too much inhomogeneity or spatial curvature would be through a homogeneous “bubble” solution in which quantum tunnelling occured everywhere at the same time. This would produce more baryons than the conventional scenarios.

The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the “ground state” or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and Λ > 0. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.

The quantum state of the universe is determined by the specification of the class of metrics and matter field configurations that are summed over in the path integral. The only natural choice of this class seems to be compact euclidean (i.e. positive definite) metrics and matter fields that are regular on them. This choice incorporates the idea that the universe is completely self-contained and has no boundary or asymptotic region. I show that in a simple “minisuperspace” model this boundary condition leads to a wave function which can be interpreted as a superposition of quantum states which are peaked around a family of classical solutions of the field equations. These solutions are non-singular and represent oscillating universes with a long inflationary period. They could be a good description of the observed universe. I also show that the features of the minisuperspace model that give rise to such a wave function are also present in models that contain all the degrees of freedom of the gravitational and matter fields.

It is assumed that the Universe is in the quantum state defined, by a path integral over compact four-metrics. This can be regarded as a boundary condition for the wave function of the Universe on superspace, the space of all three-metrics and matter field configurations on a three-surface. We extend previous work on finite-dimensional approximations to superspace to the full infinite-dimensional space. We treat the two homogeneous and isotropic degrees of freedom exactly and the others to second order. We justify this approximation by showing that the inhomogeneous or anisotropic modes start off in their ground state. We derive time-dependent Schrödinger equations for each mode. The modes remain in their ground state until their wavelength exceeds the horizon size in the period of exponential expansion. The ground-state fluctuations are then amplified by the subsequent expansion and the modes reenter the horizon in the matter- or radiation-dominated era in a highly excited state. We obtain a scale-free spectrum of density perturbations which could account for the origin of galaxies and all other structure in the Universe. The fluctuations would be compatible with observations of the microwave background if the mass of the scalar field that drives the inflation is 10¹⁴ GeV or less.

Any reasonable theory of quantum gravity will allow closed universes to branch off from our nearly fiat region of spacetime. I describe the possible quantum states of these closed universes. They correspond to wormholes which connect two asymptotically Euclidean regions, or two parts of the same asymptotically Euclidean region. I calculate the influence of these wormholes on ordinary quantum fields at low energies in the asymptotic region. This can be represented by adding effective interactions in flat spacetime which create or annihilate closed universes containing certain numbers of particles. The effective interactions are small except for closed universes containing scalar particles in the spatially homogeneous mode. If these scalar interactions are not reduced by sypersymmetry, it may be that any scalar particles we observe would have to be bound states of particles of higher spin, such as the pion. An observer in the asymptotically flat region would not be able to measure the quantum state of closed universes that branched off. He would therefore have to sum over all possibilities for the closed universes. This would mean that the final state would appear to be a mixed quantum state, rather than a pure quantum state.

We consider a system comprised of an axion (described by a rank-three antisymmetric tensor field strength) coupled to gravity. Instantons are found which describe the nucleation of a Planck-sized baby Robertson-Walker universe. Information loss to the baby universes can lead to an effective loss of quantum coherence. An estimate of the magnitude of this effect on particle propagation is made in the semi-classical approximation. This magnitude depends on the parameters of the theory (which includes a cutoff since the theory is non-renormalizable) and on the quantum state of the many-universe system. In contrast to the naive expectation that Planck-scale dynamics should lead to very small effects at low energies, the effects of these instantons can be large. The case of string theory is considered in some detail, and it is found that a massless dilaton can suppress the tunneling.

Wormholes are topology-changing configurations in euclidean quantum gravity whose importance has recently been advocated by several authors. I argue here that if wormholes exist, they have the effect of making the cosmological constant vanish. The argument involves approximations in dealing with physics at the wormhole energy scale (assumed to be somewhat below the Planck mass) but is exact in all interactions at all lower energies.

We review Coleman’s wormhole mechanism for the vanishing of the cosmological constant. We show that in a minisuperspace model wormhole-connected universes dominate the path integral. We also provide evidence that the euclidean path integral over geometries with spherical topology is unstable with respect to formation of infinitely many wormhole-connected 4-spheres. Consistency is restored by summing over all topologies, which leads to Coleman’s result. Coleman’s argument for determination of other parameters is reviewed and applied to the mass of the pion. A discouraging result is found that the pion mass is driven to zero. We also consider qualitatively the implications of the wormhole theory for cosmology. We argue that a small number of universes containing matter and energy may exist in contact with infinitely many cold and empty universes. Contact with the cold universes insures that the cosmological constant in the warm ones is zero.

The proposal that wormholes in spacetime cause the cosmological constant Λ to vanish is reviewed, and its implications are studied. Wormholes also drive Newton’s constant G to the lowest possible value. The requirement that G is at its minimum determines, in principle, all other constants of Nature. In practice, the values of fundamental constants other than Λ cannot be predicted without a detailed knowledge of Planck-scale physics.

We calculate in chiral perturbation theory the dependence of Newton’s gravitational constant G on the θ parameter of quantum chromodynamics, and we find that G, as a function of θ, is minimized at θ≃ π. This calculation suggests that quantum fluctuations in the topology of spacetime would cause θ to assume a value very near π, contrary to the phenomenological evidence indicating that θ is actually near 0.

In an attempt to find gravitational analogs of Yang–Mills pseudoparticles, we obtain two classes of self-dual solutions to the euclidean Einstein equations. These metrics are free from singularities and approach a flat metric at infinity.

We present a new family of self-dual positive definite metrics which are asymptotic to Euclidean space modulo identifications under discrete subgroups of O(4). these solutions contain 3_τ − 3 parameters where τ is the signature. We show that a fully general self-dual solution with these boundary conditions should have this number of parameters.

The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics on R⁴ and a large class of more complicated topologies and for self-dual metrics. We show that if $R_{\mu }^{\mu }$ ≧ 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under an SU(2) or SO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric on K3 - the only simply connected compact manifold which admits a self-dual metric.

1. There has been much interest recently in ‘instantons’. These correspond, in differential geometric terms, to connexions in principal bundles over Euclidean 4-space whose curvature satisfies the Yang-Mills equations. The connexion is also required to approach the trivial connexion at infinity. The self-dual solutions have been classified by converting the problem into one of algebraic geometry using the ideas of R. Penrose and R. Ward.

The following sections are included:

• Introduction
• A family of hyper-Kähler manifolds
• Properties of the manifolds
• The period map
• References

A compact rotating gravitational instanton (a positive-definite metric solution of the Einstein equations with Λ term) is presented. The manifold is the nontrivial S² fibre bundle over S² and has X = 4, τ = 0, but no spinor structure. The metric can be obtained from a special limit of the positive-definite analytic extension of the Kerr-de Sitter metric or alternatively from the Taub-NUT metric with Λ term. The action is about $4\frac{1}{2}\%$ less negative than that of the Einstein metric on the trivial bundle S² × S².

We classify the action of one parameter isometry groups of Gravitational Instantons, complete non singular positive definite solutions of the Einstein equations with or without Λ term. The fixed points of the action are of 2-types, isolated points which we call “nuts” and 2-surfaces which we call “bolts”. We describe all known gravitational instantons and relate the numbers and types of the nuts and bolts occurring in them to their topological invariants. We perform a 3 + 1 decomposition of the field equations with respect to orbits of the isometry group and exhibit a certain duality between “electric” and “magnetic” aspects of gravity. We also obtain a formula for the gravitational action of the instantons in terms of the areas of the bolts and certain nut charges and potentials that we define. This formula can be interpreted thermodynamically in several ways.

The interaction of slowly moving BPS-monopoles is described and it is shown that monopoles can get converted into dyons.