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29 March 2024
 
  » arxiv » gr-qc/0410094

 Article overview


Invariant operator due to F. Klein quantizes H. Poincare's dodecahedral 3-manifold
Peter Kramer ;
Date 19 Oct 2004
Journal J.Phys. A38 (2005) 3517-3540
Subject gr-qc
AbstractThe eigenmodes of the Poincaré dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $pi_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck( ilde{M})$, acting on the universal cover $ ilde{M}$. ($pi_1(M)$, $ ilde{M}$) are known to be the binary icosahedral group ${cal H}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck( ilde{M}) sim {cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${cal H}^r_3$ as normal subgroup and from a second group ${cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein’s fundamental icosahedral ${cal H}_3$-invariant, we construct a novel hermitian ${cal H}_3$-invariant polynomial (generalized Casimir) operator ${cal K}$. Its eigenstates with eigenvalues $kappa$ quantize a complete orthogonal basis on Poincaré’s dodecahedral 3-manifold. The eigenstates of lowest degree $lambda=12$ are 12 partners of Klein’s invariant polynomial. The analysis has applications in cosmic topology cite{LA},cite{LE}. If the Poincaré 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${cal K}$.
Source arXiv, gr-qc/0410094
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