Combinatorial quantisation of the Euclidean torus universe

Details Combinatorial quantisation of the Euclidean torus universe Combinatorial quantisation of the Euclidean torus universe

2010-07-27

arxiv.org/abs/1007.4615v2

Nuclear Physics B, Volume 841, Issue 3 (2010) 463-505

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

47 pages, 2 eps figures, minor changes, typos corrected

Scientific paper

10.1016/j.nuclphysb.2010.08.014

We quantise the Euclidean torus universe via a combinatorial quantisation formalism based on its formulation as a Chern-Simons gauge theory and on the representation theory of the Drinfel'd double DSU(2). The resulting quantum algebra of observables is given by two commuting copies of the Heisenberg algebra, and the associated Hilbert space can be identified with the space of square integrable functions on the torus. We show that this Hilbert space carries a unitary representation of the modular group and discuss the role of modular invariance in the theory. We derive the classical limit of the theory and relate the quantum observables to the geometry of the torus universe.

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