Entropy and Topology for Gravitational Instantons

In this work a relation between topology and thermodynamical features of gravitational instantons is shown. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form that makes the relation between them self-evident. A new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by $S=\chi A/8$, is obtained. This formula provides the correct results for a wide class of gravitational instantons described by both spherically and axially symmetric metrics.Comment: 25 pages, RevTeX, accepted for publication in Phys. Rev.

Twistors and Spin 3/2 Potentials in Quantum Gravity

Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-lowering and spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains. Alternative boundary conditions for supergravity involving the spinor-valued 1-forms for gravitinos and the normal to the boundary are also studied.Comment: 20 pages, plain-tex, recently appearing in: Twistor Theory, edited by Stephen Huggett (Marcel Dekker, New York, 1994). The authors apologize for the delay in circulating the paper, which was due to technical problems now fixe

Boundary Terms for Massless Fermionic Fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal to the boundary and a pair of independent spinor fields. This paper studies the corresponding classical properties, i.e. the classical boundary-value problem and boundary terms in the variational problem. Interestingly, a link is found with the classical boundary-value problem when spectral boundary conditions are imposed on a 3-sphere in the massless case. Moreover, the boundary term in the action functional is derived.Comment: 8 pages, plain-tex, recently appearing in Foundations of Physics Letters, volume 7, pages 303-308, year 199

Euclidean Quantum Gravity

This chapter studies the linearized gravitational field in the presence of boundaries. For this purpose, zeta-function regularization is used to perform the mode-by-mode evaluation of Faddeev-Popov amplitudes in the case of flat Euclidean four-space bounded by two concentric three-spheres, or just one three-sphere. On choosing the de Donder gauge-averaging term, the resulting ζ(0) value is found to agree with the space-time covariant calculation of the same amplitudes, which relies on the recently corrected geometric formulae for the asymptotic heat kernel in the case of mixed boundary conditions. Two sets of mixed boundary conditions for Euclidean quantum gravity are then compared in detail. The analysis proves that one cannot restrict the path-integral measure to transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are only obtained on considering from the beginning all perturbative modes of the gravitational field, jointly with ghost modes. Unlike the mixed boundary conditions involving (complementary) projectors, one knows from chapter six that boundary conditions completely invariant under infinitesimal diffeomorphisms involve both normal and tangential derivatives of metric perturbations. The corresponding ζ(0) value is obtained, and the proof of symmetry of the Laplace operator in such a case is obtained. Mixed boundary conditions are also considered which lead to Robin conditions on spatial metric perturbations, and Dirichlet conditions on normal metric perturbations. Last, a review of Hawking's proposal to consider smooth simply connected four-manifolds as the building blocks of Euclidean quantum gravity is presented. This makes it necessary to study physical processes in S 2 x S 2 , K3 and CP 2 geometries. Yet another open problem is a consistent formulation of quantum supergravity on manifolds with boundary

Euclidean Maxwell Theory in the Presence of Boundaries. II

Zeta-function regularization is applied to complete a recent analysis of the quantized electromagnetic field in the presence of boundaries. The quantum theory is studied by setting to zero on the boundary the magnetic field, the gauge-averaging functional and hence the Faddeev-Popov ghost field. Electric boundary conditions are also studied. On considering two gauge functionals which involve covariant derivatives of the 4-vector potential, a series of detailed calculations shows that, in the case of flat Euclidean 4-space bounded by two concentric 3-spheres, one-loop quantum amplitudes are gauge independent and their mode-by-mode evaluation agrees with the covariant formulae for such amplitudes and coincides for magnetic or electric boundary conditions. By contrast, if a single 3-sphere boundary is studied, one finds some inconsistencies, i.e. gauge dependence of the amplitudes.Comment: 24 pages, plain-tex, recently appearing in Classical and Quantum Gravity, volume 11, pages 2939-2950, December 1994. The authors apologize for the delay in circulating the file, due to technical problems now fixe

Spin-Raising Operators and Spin-3/2 Potentials in Quantum Cosmology

Local boundary conditions involving field strengths and the normal to the boundary, originally studied in anti-de Sitter space-time, have been recently considered in one-loop quantum cosmology. This paper derives the conditions under which spin-raising operators preserve these local boundary conditions on a 3-sphere for fields of spin 0,1/2,1,3/2 and 2. Moreover, the two-component spinor analysis of the four potentials of the totally symmetric and independent field strengths for spin 3/2 is applied to the case of a 3-sphere boundary. It is shown that such boundary conditions can only be imposed in a flat Euclidean background, for which the gauge freedom in the choice of the potentials remains.Comment: 13 pages, plain-tex, recently appearing in Classical and Quantum Gravity, volume 11, April 1994, pages 897-903. Apologies for the delay in circulating the file, due to technical problems now fixe

One-Loop Effective Action on the Four-Ball

This paper applies $\zeta$-function regularization to evaluate the 1-loop effective action for scalar field theories and Euclidean Maxwell theory in the presence of boundaries. After a comparison of two techniques developed in the recent literature, vacuum Maxwell theory is studied and the contribution of all perturbative modes to $\zeta'(0)$ is derived: transverse, longitudinal and normal modes of the electromagnetic potential, jointly with ghost modes. The analysis is performed on imposing magnetic boundary conditions, when the Faddeev-Popov Euclidean action contains the particular gauge-averaging term which leads to a complete decoupling of all perturbative modes. It is shown that there is no cancellation of the contributions to $\zeta'(0)$ resulting from longitudinal, normal and ghost modes.Comment: 25 pages, plain Te

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

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