Physical states in N=1N = 1 supergravity

By solving the supersymmetry constraints for physical wave-functions, it is shown that the only two allowed bosonic states in $N = 1$ supergravity are of the form const.~exp~$(\pm I / \hbar)$, where $I$ is an action functional of the three-metric. States containing a finite number of fermions are forbidden. In the case that the spatial topology is $S^3$, the state const.~exp~$(- I / \hbar)$ is the wormhole ground state, and the state const.~exp~$(I / \hbar)$ is the Hartle--Hawking state. $N = 1$ supergravity has no quantum ultraviolet divergences, and no quantum corrections.Comment: 9 pages, DAMTP Preprint DAMTP-R-93-

Scalar--Field Amplitudes in Black--Hole Evaporation

We study the quantum-mechanical decay of a Schwarzschild-like black hole into almost-flat space and weak radiation at a very late time, evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. No information is lost. The model contains gravity and a massless scalar field. The quantum amplitude to go from given initial to final bosonic data in a slightly complexified time-interval $T={\tau}{\exp}(-i{\theta})$ at infinity is approximately $\exp(-I)$, where $I$ is the (complex) Euclidean action of the classical solution filling in between the boundary data. And in a locally supersymmetric (supergravity) theory, the amplitude const. exp(-I) is exact. Dirichlet boundary data for gravity and the scalar field are posed on an initial spacelike hypersurface extending to spatial infinity, just prior to collapse, and on a corresponding final spacelike surface, sufficiently far to the future of the initial surface to catch all the Hawking radiation. In an averaged sense this radiation has an approximately spherically-symmetric distribution. If the time-interval $T$ were exactly real, the resulting `hyperbolic Dirichlet boundary-value problem' would not be well posed. If instead (`Euclidean strategy'), one takes $T$ complex, as above ($0<\theta{\leq}{\pi}/2$), the field equations become strongly elliptic, with a unique solution to the classical boundary-value problem. Expanding the bosonic part of the action to quadratic order in perturbations about the classical solution gives the quantum amplitude for weak-field final configurations, up to normalization. Such amplitudes are calculated for weak final scalar fields.Comment: Submitted to Physics Letters B, Friday 23rd July 200

Classical Boundary-value Problem in Riemannian Quantum Gravity and Self-dual Taub-NUT-(anti)de Sitter Geometries

The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact ($S^{3}$) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii $(a,b).$ For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti)de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii $(a,b),$ we find that there are three solutions for the infilling geometry of this type. We obtain exact solutions for them and for their Euclidean actions. The case of negative cosmological constant is investigated further. For reasonable squashing of the three-sphere, all three infilling solutions have real-valued actions which possess a ``cusp catastrophe'' structure with a non-self-intersecting ``catastrophe manifold'' implying that the dominant contribution comes from the unique real positive-definite solution on the ball. The positive-definite solution exists even for larger deformations of the three-sphere, as long as a certain inequality between $a$ and $b$ holds. The action of this solution is proportional to $-a^{3}$ for large $a (\sim b)$ and hence larger radii are favoured. The same boundary-value problem with more complicated interior topology containing a ``bolt'' is investigated in a forthcoming paper.Comment: 20 pages, 11 figures; Latex; Revised version with important new results on real infilling solutions and corrections. To appear in Nuclear Physics B, issue 648 (1,2), pp. 397-41

Coherent and squeezed states in black-hole evaporation

In earlier Letters, we adopted a complex approach to quantum processes in the formation and evaporation of black holes. Taking Feynman's $+i\epsilon$ prescription, rather than than one of the more usual approaches, we calculated the quantum amplitude (not just the probability density) for final weak-field configurations following gravitational collapse to a black hole with subsequent evaporation. What we have done is to find quantum amplitudes relating to a pure state at late times following black-hole matter collapse. Such pure states are then shown to be susceptible to a description in terms of coherent and squeezed states - in practice, this description is not very different from that for the well-known highly-squeezed final state of the relic radiation background in inflationary cosmology. The simplest such collapse model involves Einstein gravity with a massless scalar field. The Feynman approach involves making the boundary-value problem for gravity and a massless scalar field well-posed. To define this, let T be the proper-time separation, measured at spatial infinity, between two space-like hypersurfaces on which initial (collapse) and final (evaporation) data are posed. Then, in this approach, one rotates T by a complex phase exp(-i\delta) into the lower half-plane. In an adiabatic approximation, the resulting quantum amplitude may be expressed in terms of generalised coherent states of the quantum oscillator, and a physical interpretation is given. A squeezed-state representation, as above, then follows

Microscopic Black Hole Production in TeV-Scale Gravity

Models with extra spatial dimensions and TeV-scale gravity offer the first opportunity to test the conjecture of black hole formation in trans-Planckian energy scattering with small impact parameters. After a brief review of gravitational scattering at ultrahigh energies and scenarios of TeV-scale gravity, search strategies at the LHC, at the Pierre Auger (cosmic ray) Observatory and at the neutrino telescopes AMANDA/IceCube are illustrated with the simplest but nevertheless representative example: production of Schwarzschild black holes and their observation via Hawking radiation in the large extra dimension scenario. Some more general features of the production of higher-dimensional black holes and/or uncertainties in the estimates are also outlined.Comment: 18 pages, 5 figures; Talk presented at XXX ITEP Winter School of Physics, Moscow, Russia, February 2002, references adde

Relativistic Gauge Conditions in Quantum Cosmology

This paper studies the quantization of the electromagnetic field on a flat Euclidean background with boundaries. One-loop scaling factors are evaluated for the one-boundary and two-boundary backgrounds. The mode-by-mode analysis of Faddeev-Popov quantum amplitudes is performed by using zeta-function regularization, and is compared with the space-time covariant evaluation of the same amplitudes. It is shown that a particular gauge condition exists for which the corresponding operator matrix acting on gauge modes is in diagonal form from the beginning. Moreover, various relativistic gauge conditions are studied in detail, to investigate the gauge invariance of the perturbative quantum theory.Comment: 26 pages, plain TeX, no figure

Gravitons in One-Loop Quantum Cosmology: Correspondence Between Covariant and Non-Covariant Formalisms

The discrepancy between the results of covariant and non-covariant one-loop calculations for higher-spin fields in quantum cosmology is analyzed. A detailed mode-by-mode study of perturbative quantum gravity about a flat Euclidean background bounded by two concentric 3-spheres, including non-physical degrees of freedom and ghost modes, leads to one-loop amplitudes in agreement with the covariant Schwinger-DeWitt method. This calculation provides the generalization of a previous analysis of fermionic fields and electromagnetic fields at one-loop about flat Euclidean backgrounds admitting a well-defined 3+1 decomposition.Comment: 29 pages, latex, recently appearing in Physical Review D, volume 50, pages 6329-6337, November 1994. The authors apologize for the delay in circulating the paper, due to technical problems now fixe

Missing energy in black hole production and decay at the Large Hadron Collider

Black holes could be produced at the Large Hadron Collider in TeV-scale gravity scenarios. We discuss missing energy mechanisms in black hole production and decay in large extra-dimensional models. In particular, we examine how graviton emission into the bulk could give the black hole enough recoil to leave the brane. Such a perturbation would cause an abrupt termination in Hawking emission and result in large missing-energy signatures.Comment: addressed reviewer comments and updated reference

A Non-Singular One-Loop Wave Function of the Universe From a New Eigenvalue Asymptotics in Quantum Gravity

Recent work on Euclidean quantum gravity on the four-ball has proved regularity at the origin of the generalized zeta-function built from eigenvalues for metric and ghost modes, when diffeomorphism-invariant boundary conditions are imposed in the de Donder gauge. The hardest part of the analysis involves one of the four sectors for scalar-type perturbations, the eigenvalues of which are obtained by squaring up roots of a linear combination of Bessel functions of integer adjacent orders, with a coefficient of linear combination depending on the unknown roots. This paper obtains, first, approximate analytic formulae for such roots for all values of the order of Bessel functions. For this purpose, both the descending series for Bessel functions and their uniform asymptotic expansion at large order are used. The resulting generalized zeta-function is also built, and another check of regularity at the origin is obtained. For the first time in the literature on quantum gravity on manifolds with boundary, a vanishing one-loop wave function of the Universe is found in the limit of small three-geometry, which suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory.Comment: 21 Pages, Latex and .eps files with JHEP3 style. The discussion in Section 5 has been improved, and Ref. 26 has been adde

Zeta function determinant of the Laplace operator on the DD-dimensional ball

We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, $D$, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions $D=2,3,4,5$ and $6$.Comment: 22 pages, one figure appended as uuencoded postscript fil