Quantum Gravity in Four DimensionsNova Publishers, 2001 - 153 pagine Main section headings: Ideas and Problems in Quantum Gravity; On Ellipticity and Quantum Gravity; Non-Local Boundary Data in Quantum Gravity; Non-Locality and Ellipticity for Gauge Theories; New Kernels in Quantum Gravity; Quantum Gravity from First Principles; Quantum Gravity and Spectral Geometry; Bibliography; Index. |
Sommario
IDEAS AND PROBLEMS IN QUANTUM GRAVITY | xiii |
12 Manifestly covariant form of field theory | 1 |
122 Infinitesimal disturbances and background fields | 2 |
123 The Green theorem | 3 |
124 Groupinvariance properties | 4 |
125 Supplementary conditions and Green functions | 6 |
126 The Peierls bracket | 9 |
13 On Feynman quantization of general relativity | 17 |
43 Ghost and gaugefield operators | 78 |
44 Ellipticity | 80 |
45 Concluding remarks | 87 |
Ghost boundary conditions | 88 |
NEW KERNELS IN QUANTUM GRAVITY | 91 |
52 A new set of nonlocal boundary conditions | 94 |
53 Ghost boundary operator and consistency conditions | 95 |
54 Euclidean quantum gravity | 97 |
131 Euclidean approach to quantization | 21 |
14 Path integrals for gauge theories | 23 |
15 Lack of perturbative renorrealization | 25 |
151 Asymptotic expansions | 26 |
152 Summability methods | 27 |
On the meaning of Wick rotation | 30 |
ON ELLIPTICITY AND QUANTUM GRAVITY | 33 |
212 Weak solutions of elliptic equations | 35 |
214 Elementary spectral theory | 39 |
22 Boundary conditions in quantum gravity | 43 |
24 Open problems | 50 |
NONLOCAL BOUNDARY DATA IN QUANTUM GRAVITY | 53 |
32 The adjoint with nonlocal boundary conditions | 58 |
33 Application to the gravitational field | 62 |
34 Modebymode equations on the Euclidean 4baIl | 64 |
35 Concluding remarks | 69 |
Pseudodifferential operators | 71 |
Local boundary conditions | 72 |
NONLOCALITY AND ELLIPTICITY FOR GAUGE THEORIES | 75 |
nonlocality and gauge invariance | 77 |
55 Quantized Maxwell theory | 100 |
56 All boundary operators which are projectors | 102 |
57 A new boundaryvalue problem | 105 |
58 Concluding remarks | 107 |
Operators with pseudohomogeneous kernel | 110 |
QUANTUM GRAVITY FROM FIRST PRINCIPLES | 113 |
62 Strong ellipticity | 116 |
621 Ellipticity in the interior | 119 |
623 Strong ellipticity in the presence of boundaries | 120 |
63 Application of the strong ellipticity criterion | 122 |
64 Further applications | 125 |
65 Concluding remarks | 127 |
QUANTUM GRAVITY AND SPECTRAL GEOMETRY | 129 |
72 Conformal variations and heatkernel coefficients | 135 |
73 The A₅₂ coefficient | 137 |
74 Concluding remarks | 140 |
BIBLIOGRAPHY | 143 |
| 149 | |
Parole e frasi comuni
action analysis application approach asymptotic Avramidi and Esposito boundary operator boundary-value problem bundle chapter coefficients complete consider corresponding defined definition denoted derivatives DeWitt differential operator Dirichlet equation Euclidean quantum gravity example exist expressed Feynman finds formulation framework functions fundamental gauge gauge-field gauge-invariant geometry ghost operator Gilkey given Green Grubb hand hence holds homogeneous implies infinitesimal integral invariance involves kernel Laplace type leading symbol linear manifold mathematical metric perturbations Moreover nature non-local boundary conditions notation obtained occurring one-loop particular Peierls bracket physical positive possible presented principles projector properties prove pseudo-differential operators quantization quantum field theory quantum mechanics quantum theory reads relativity remains requirement respect restriction resulting Riemannian satisfied scheme singularity smooth solution space-time strong ellipticity studied sufficient tensor term transformations universal vanishes vector virtue әм
Brani popolari
Pagina xii - One then finds two families of eigenfunctions of the Hamiltonian: surface states which decrease exponentially as one moves away from the boundary, and bulk states which remain instead smooth and non-vanishing. The generalization to an Abelian gauge theory such as Maxwell theory can fulfill non-locality, ellipticity and complete gauge invariance of boundary conditions providing one learns to work with pseudo-differential operators in one-loop quantum theory.
Pagina xi - General relativity might be therefore viewed as a low-energy limit of a richer theory, which achieves the synthesis of both the basic principles of modern physics and the fundamental interactions in the form presently known. Within the framework just outlined it remains however true that the various approaches to quantum gravity developed so far suffer from mathematical inconsistencies, or incompleteness in their ability of accounting for some basic features of the laws of nature.
Pagina xi - The aim of theoretical physics is to provide a clear conceptual framework for the wide variety of natural phenomena, so that not only are we able to make accurate predictions to be checked against observations, but the underlying mathematical structures of the world we live in can also become sufficiently well understood by the scientific community.
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Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds ... Bernhelm Booss,Gerd Grubb,Krzysztof P. Wojciechowski Anteprima non disponibile - 2005 |