Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 104, 12 pages      arXiv:1111.2672
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

The Space of Connections as the Arena for (Quantum) Gravity

Steffen Gielen
Albert Einstein Institute, Am Mühlenberg 1, 14476 Golm, Germany

Received August 30, 2011, in final form November 09, 2011; Published online November 11, 2011; Reference [9] is added November 17, 2011

We review some properties of the space of connections as the natural arena for canonical (quantum) gravity, and compare to the case of the superspace of 3-metrics. We detail how a 1-parameter family of metrics on the space of connections arises from the canonical analysis for general relativity which has a natural interpretation in terms of invariant tensors on the algebra of the gauge group. We also review the description of canonical GR as a geodesic principle on the space of connections, and comment on the existence of a time variable which could be used in the interpretation of the quantum theory.

Key words: canonical quantum gravity; gravitational connection; semisimple Lie algebras; infinite-dimensional manifolds.

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