Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2008-07-07
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
27 pages
Scientific paper
This paper is the first of a series of three which construct coherent states for spin networks with planar symmetry. Constructing such states is not straightforward. After gauge-fixing, the full SU(2) symmetry is broken to U(1), but one cannot simply use the U(1) limit of SU(2) coherent states, because the planar states exhibit an unexpected O(3) symmetry arising from the closed loop character of the transverse directions. This paper uses an intuitive, rather than mathematically rigorous approach to construct a candidate set of O(3) symmetric coherent states. However, paper 2 of the series then demonstrates explicitly that the proposed coherent states are approximate eigenvectors of the holonomy and momentum operators (as expected for coherent states), up to small correction terms. These coherent states are superpositions of holonomies which obey the residual U(1) symmetry only on average; that is, some holonomies in the superposition violate the symmetry, although the U(1) quanntum numbers of these holonomies are peaked at values which obey the symmetry. An appendix discusses the closely related case of cylindrical symmetry.
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