3,212 research outputs found
Spectral geometry of spacetime
Spacetime, understood as a globally hyperbolic manifold, may be characterized
by spectral data using a 3+1 splitting into space and time, a description of
space by spectral triples and by employing causal relationships, as proposed
earlier. Here, it is proposed to use the Hadamard condition of quantum field
theory as a smoothness principle.Comment: AMS-LaTeX, 6 pages,Talk presented at the Euroconference on
"Non-Commutative Geometry And Hopf Algebras In Field Theory And Particle
Physics" Torino, Villa Gualino, September 20 - 30, 199
Spectral Geometry and Causality
For a physical interpretation of a theory of quantum gravity, it is necessary
to recover classical spacetime, at least approximately. However, quantum
gravity may eventually provide classical spacetimes by giving spectral data
similar to those appearing in noncommutative geometry, rather than by giving
directly a spacetime manifold. It is shown that a globally hyperbolic
Lorentzian manifold can be given by spectral data. A new phenomenon in the
context of spectral geometry is observed: causal relationships. The employment
of the causal relationships of spectral data is shown to lead to a highly
efficient description of Lorentzian manifolds, indicating the possible
usefulness of this approach. Connections to free quantum field theory are
discussed for both motivation and physical interpretation. It is conjectured
that the necessary spectral data can be generically obtained from an effective
field theory having the fundamental structures of generalized quantum
mechanics: a decoherence functional and a choice of histories.Comment: AMS-Latex, 14 pages, 3 figures, using epsf macr
Spectral Geometry and Quantum Gravity
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.Comment: 6 pages, Latex, invited talk given at the Tomsk Conference: Quantum
Field Theory and Gravity (July-August 1997
Vacuum Energy as Spectral Geometry
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical
audience as a topic in spectral theory. Then some one-dimensional systems are
solved exactly, in terms of closed classical paths and periodic orbits. The
relations among local spectral densities, energy densities, global eigenvalue
densities, and total energies are demonstrated. This material provides
background and motivation for the treatment of higher-dimensional systems
(self-adjoint second-order partial differential operators) by semiclassical
approximation and other methods.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Spectral Geometry of Heterotic Compactifications
The structure of heterotic string target space compactifications is studied
using the formalism of the noncommutative geometry associated with lattice
vertex operator algebras. The spectral triples of the noncommutative spacetimes
are constructed and used to show that the intrinsic gauge field degrees of
freedom disappear in the low-energy sectors of these spacetimes. The quantum
geometry is thereby determined in much the same way as for ordinary superstring
target spaces. In this setting, non-abelian gauge theories on the classical
spacetimes arise from the K-theory of the effective target spaces.Comment: 14 pages LaTe
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