arXiv subjects

Pierre-Philippe Dechant

Publications and source records attributed to Pierre-Philippe Dechant.

21 records · Page 2Linked to original sources

arXiv2012

Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional structures, which our study sheds new light on.

math-ph↗

arXiv2010

Cracking the Taub-NUT

We present further analysis of an anisotropic, non-singular early universe model that leads to the viable cosmology presented in Dechant et al (arXiv:0809.4335). Although this model (the DLH model) contains scalar field matter, it is reminiscent of the Taub-NUT vacuum solution in that it has biaxial Bianchi IX geometry and its evolution exhibits a dimensionality reduction at a quasi-regular singularity that one can identify with the big-bang. We show that the DLH and Taub-NUT metrics are related by a coordinate transformation, in which the DLH time coordinate plays the role of conformal time for Taub-NUT. Since both models continue through the big-bang, the coordinate transformation can become multivalued. In particular, in mapping from DLH to Taub-NUT, the Taub-NUT time can take only positive values. We present explicit maps between the DLH and Taub-NUT models, with and without a scalar field. In the vacuum DLH model, we find a periodic solution expressible in terms of elliptic integrals. Mapping the vacuum solution over to Taub-NUT coordinates, recovers the standard (non-periodic) Taub-NUT solution in the Taub region, where Taub-NUT time takes positive values, but does not exhibit the two NUT regions known in the standard Taub-NUT solution. Conversely, mapping the complete Taub-NUT solution to the DLH case reveals that the NUT regions correspond to imaginary time and space in DLH coordinates. We show that many of the well-known `pathologies' of the Taub-NUT solution arise because the traditional coordinates are connected by a multivalued transformation to the physically more meaningful DLH coordinates. In particular, the `open-to-closed-to-open' transition and the Taub and NUT regions of the (Lorentzian) Taub-NUT model are replaced by a closed pancaking universe with spacelike homogeneous sections at all times.

gr-qc↗

arXiv2009

An anisotropic, non-singular early universe model leading to a realistic cosmology

We present a novel cosmological model in which scalar field matter in a biaxial Bianchi IX geometry leads to a non-singular `pancaking' solution: the hypersurface volume goes to zero instantaneously at the `Big Bang', but all physical quantities, such as curvature invariants and the matter energy density remain finite, and continue smoothly through the Big Bang. We demonstrate that there exist geodesics extending through the Big Bang, but that there are also incomplete geodesics that spiral infinitely around a topologically closed spatial dimension at the Big Bang, rendering it, at worst, a quasi-regular singularity. The model is thus reminiscent of the Taub-NUT vacuum solution in that it has biaxial Bianchi IX geometry and its evolution exhibits a dimensionality reduction at a quasi-regular singularity; the two models are, however, rather different, as we will show in a future work. Here we concentrate on the cosmological implications of our model and show how the scalar field drives both isotropisation and inflation, thus raising the question of whether structure on the largest scales was laid down at a time when the universe was still oblate (as also suggested by Pitrou et al and Contaldi et al). We also discuss the stability of our model to small perturbations around biaxiality and draw an analogy with cosmological perturbations. We conclude by presenting a separate, bouncing solution, which generalises the known bouncing solution in closed FRW universes.

gr-qc↗