Affine Group formulation of General Relativity
Chou Ching-Yi1*
1National Center for Theoretical Sciences (South), National Cheng Kung University, Tainan, Taiwan
* presenting author:Ching-Yi Chou, email:ccmaple014@gmail.com
The Hamiltonian constraint of four-dimensional gravity with Lorentzian signature is reformulated as an affine algebra relation. It is remarkable that the affine quantization program of Klauder is applicable to not just pure gravity but also when fermions, Yang-Mills and Higgs fields of the whole Standard Model are taken into account, provided the cosmological constant is non-vanishing.

Techniques of affine group representations are used to construct exact solutions from a fiducial state, and algebraic affine group methodology and results are employed to study them. The crucial observation that just the imaginary part, rather than the full Chern-Simons functional, is needed in the reformulation of the Hamiltonian constraint of pure gravity theory as an affine algebra ensures that the generators of the algebra are all Hermitian. The extension to full Standard Model incorporating Weyl fermions (with Hermitian action) with Higgs and Yang-Mills fields is also carried out. Within the context of this work, all physical states of quantum gravity must come from representations of the affine algebra. It is intriguing the formulation of the Hamiltonian constraint as an affine algebra is predicated upon a non-vanishing cosmological constant.

Ashtekar variables are introduced and the associated reality conditions are studied. With Standard Model fermions, the necessity of using complex (anti-)self-dual Ashtekar connection is pointed out. A key observation is that \mathcal{CPT} maps the Ashtekar variable to its Hermitian adjoint. Moreover, with Lorentz invariance and spin-statistics rule, the CPT theorem ensures that the action is mapped to its Hermitian adjoint under \mathcal{CPT}. Thus despite the complications which arise from fermions, the generic reality condition for the Ashtekar connection can be formulated as a pseudo-Hermiticity condition with respect to \mathcal{CPT} even in the presence of fermions and other Standard Model fields.


Keywords: Affine commutation relation, General Relativity