Spectral methods in quantum field theory and quantum cosmology (original) (raw)
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Spectral boundary conditions in one-loop quantum cosmology
Phys Rev D, 1991
For fermionic fields on a compact Riemannian manifold with a boundary, one has a choice between local and nonlocal (spectral) boundary conditions. The one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be studied using the generalized Riemann ζ function formed from the squared eigenvalues of the four-dimensional fermionic operators. For a massless Majorana spin-1/2 field, the spectral conditions involve setting to zero half of the fermionic field on the boundary, corresponding to harmonics of the intrinsic three-dimensional Dirac operator on the boundary with positive eigenvalues. Remarkably, a detailed calculation for the case of a flat background bounded by a three-sphere yields the same value ζ(0)=11/360 as was found previously by the authors using local boundary conditions. A similar calculation for a spin-3/2 field, working only with physical degrees of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value ζ(0)=-289/360 equal to that for the natural local boundary conditions.
Journal of High Energy Physics, 2005
Recent work on Euclidean quantum gravity on the four-ball has proved regularity at the origin of the generalized ζ-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 ζfunction 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 threegeometry, which suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory.
New developments in the spectral asymptotics of quantum gravity
Journal of Physics A: Mathematical and General, 2006
A vanishing one-loop wave function of the Universe in the limit of small three-geometry is found, on imposing diffeomorphism-invariant boundary conditions on the Euclidean 4-ball in the de Donder gauge. This result suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory. All of this is made possible by a peculiar spectral cancellation on the Euclidean 4-ball, here derived and discussed.
Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions
Classical and Quantum Gravity, 2005
A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry.
Euclidean Quantum Gravity in Light of Spectral Geometry
2003
A proper understanding of boundary-value problems is essential in the attempt of developing a quantum theory of gravity and of the birth of the universe. The present paper reviews these topics in light of recent developments in spectral geometry, i.e. heat-kernel asymptotics for the Laplacian in the presence of Dirichlet, or Robin, or mixed boundary conditions; completely gauge-invariant boundary conditions in Euclidean quantum gravity; local vs. non-local boundary-value problems in one-loop Euclidean quantum theory via path integrals.
Spectral geometry and quantum gravity
1997
Recent progress in quantum field theory and quantum gravity relies on mixed boundary conditions involving both normal and tangential derivatives of the quantized field. In particular, the occurrence of tangential derivatives in the boundary operator makes it possible to build a large number of new local invariants. The integration of linear combinations of such invariants of the orthogonal group yields the boundary contribution to the asymptotic expansion of the integrated heat-kernel. This can be used, in turn, to study the one-loop semiclassical approximation. The coefficients of linear combination are now being computed for the first time. They are universal functions, in that are functions of position on the boundary not affected by conformal rescalings of the background metric, invariant in form and independent of the dimension of the background Riemannian manifold. In Euclidean quantum gravity, the problem arises of studying infinitely many universal functions.
The impact of quantum cosmology on quantum field theory
1997
The basic problem of quantum cosmology is the definition of the quantum state of the universe, with appropriate boundary conditions on Riemannian three-geometries. This paper describes recent progress in the corresponding analysis of quantum amplitudes for Euclidean Maxwell theory and linearized gravity. Within the framework of Faddeev-Popov formalism and zeta-function regularization, various choices of mixed boundary conditions lead to a deeper understanding of quantized gauge fields and quantum gravity in the presence of boundaries.
A New Spectral Cancellation in Quantum Gravity
Papers in Honor of Krzysztof P Wojciechowski, 2006
21:8 Proceedings Trim Size: 9in x 6in holbaek05 2 A general method exists for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity is affected by the lack of strong ellipticity. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is "confined" to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics on the Euclidean four-ball is also obtained; this remains regular at the origin by virtue of a peculiar spectral identity obtained by the authors. There is therefore encouraging evidence in favour of the ζ(0) value with fully diff-invariant boundary conditions remaining well defined, at least on the four-ball, although severe technical obstructions remain in general. 2000 MSC. Primary 58J35, 83C45; Secondary 81S40, 81T20.
The effect of boundaries in one-loop quantum cosmology
1991
The problem of boundary conditions in a supersymmetric theory of quantum cosmology is studied, with application to the one-loop prefactor in the quantum amplitude. Our background cosmological model is flat Euclidean space bounded by a three-sphere, and our calculations are based on the generalized Riemann ζ-function. One possible set of supersymmetric local boundary conditions involves field strengths for spins 1, 3 2 and 2, the undifferentiated spin-1 2 field, and a mixture of Dirichlet and Neumann conditions for spin 0. In this case the results we can obtain are : ζ(0) = 7 45 for a complex scalar field, ζ(0) = 11 360 for spin 1 2 , ζ(0) = − 77 180 (magnetic) and 13 180 (electric) for spin 1, and ζ(0) = 112 45 for pure gravity when the linearized magnetic curvature is vanishing on S 3 . The ζ(0) values for gauge fields have been obtained by working only with physical degrees The Effect of Boundaries in One-Loop Quantum Cosmology of freedom. An alternative set of boundary conditions can be motivated by studying transformation properties under local supersymmetry; these involve Dirichlet conditions for the spin-2 and spin-1 fields, a mixture of Dirichlet and Neumann conditions for spin-0, and local boundary conditions for the spin-1 2 field and the spin-3 2 potential. For the latter one finds : ζ(0) = − 289 360 . The full ζ(0) does not vanish in extended supergravity theories,
New results in one-loop quantum cosmology
1999
A crucial problem in quantum cosmology is a careful analysis of the one-loop semiclassical approximation for the wave function of the universe, after an appropriate choice of mixed boundary conditions. The results for Euclidean quantum gravity in four dimensions are here presented, when linear covariant gauges are implemented by means of the Faddeev-Popov formalism. On using ζ-function regularization and a mode-by-mode analysis, one finds a result for the one-loop divergence which agrees with the Schwinger-DeWitt method only after taking into account the non-trivial effect of gauge and ghost modes. For the gravitational field, however, the geometric form of heat-kernel asymptotics with boundary conditions involving tangential derivatives of metric perturbations is still unknown. Moreover, boundary effects are found to be responsible for the lack of one-loop finiteness of simple supergravity, when only one bounding three-surface occurs. This work raises deep interpretative issues about the admissible backgrounds and about quantization techniques in quantum cosmology.