The Conformal Universe II: Conformal Symmetry, its Spontaneous Breakdown and Higgs Fields in Conformally Flat Spacetime

TL;DR

This paper explores the structure of conformal symmetry in spacetime, its spontaneous breaking, and the role of Higgs fields within conformally flat spacetime, providing multiple geometric representations and linking to cosmology.

Contribution

It introduces the conformal group as an invariance group in spacetime, discusses symmetry breaking mechanisms, and compares different geometric frameworks for conformal general relativity.

Findings

Detailed description of conformal group structure and symmetries

Analysis of spontaneous conformal symmetry breaking involving scalar fields

Comparison of three geometric representations of conformal general relativity

Abstract

This is the second of three papers on Conformal General Relativity (CGR). The conformal group is introduced here as the invariance group of the partial order of causal events in $n$D spacetime. Its general structure, discrete symmetries and field representations are described in detail. The spontaneous breakdown of conformal symmetry is then discussed and the role played by a ghost scalar field and a physical scalar field in 4D spacetime are evidenced. Kinematic--, conformal-- and proper--time hyperbolic coordinates are introduced in a negatively curved Milne spacetime for the purpose of providing three different but equivalent representations of CGR. The first of these is grounded in a Riemannian manifold and is manifestly conformal invariant, the second is grounded in a conformally connected Cartan manifold but its conformal invariance is hidden, the third is grounded in the Riemannian manifold of the Milne spacetime and has the formal structure of General Relativity (GR). The relation between CGR and standard inflationary cosmology is also briefly discussed. Lastly, in view of the detailed study of Higgs--field dynamics carried out in the third paper, the action integrals, motion equations and total energy--momentum tensors of the Higgs field interacting with the dilation field are described in the three representations mentioned above.