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Covariant information-density cutoff in curved space-time
1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
Physical Review Letters
|July 13, 2004
Summary
Sampling theory, crucial for information theory, is extended to pseudo-Riemannian manifolds. This generalization offers new mathematical tools for studying Planck-scale spacetime and quantum gravity theories.
Area of Science:
- Theoretical Physics
- Information Theory
- Differential Geometry
Background:
- Sampling theorems link continuous and discrete information, enabling perfect signal reconstruction.
- Existing theories often treat spacetime as continuous, posing challenges for quantum gravity.
Purpose of the Study:
- Generalize sampling theory to pseudo-Riemannian manifolds.
- Develop new mathematical tools for Planck-scale spacetime analysis.
- Explore connections between spacetime theories and lattice theories.
Main Methods:
- Extension of classical sampling theorems to the domain of pseudo-Riemannian manifolds.
- Mathematical framework for analyzing differentiable spacetime manifolds.
- Investigation of equivalences between continuous field theories and discrete lattice theories.
Main Results:
- A generalized sampling theory applicable to pseudo-Riemannian manifolds is established.
- Demonstrated that theories on differentiable spacetime manifolds can be equivalent to lattice theories.
- Revealed a connection to generalized uncertainty relations in quantum gravity.
Conclusions:
- The generalized sampling theory provides novel mathematical tools for quantum gravity research.
- This work bridges continuous and discrete descriptions of spacetime at the Planck scale.
- The findings suggest potential new avenues for unifying string theory and other quantum gravity approaches.