We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU(2) quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering fermionic cycles through the foam we couple this SU(2) quantum group with the same deformation of SU(3), so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups SU(N), G2 and E8 is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network − in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.
Quantum Gravity at the Fifth Root of Unity
Marcelo Amaral, Raymond Aschheim, Klee Irwin
Quantum Gravity Research
Los Angeles, CA
E-mail: Marcelo@QuantumGravityResearch.org,
Raymond@QuantumGravityResearch.org and Klee@QuantumGravityResearch.org
16 April 2019
Abstract.
We consider quantum transition amplitudes, partition functions and observables
for 3D spin foam models within SU(2) quantum group deformation symmetry,
where the deformation parameter is a complex fifth root of unity. By considering
"fermionic" cycles through the foam we couple this SU(2) quantum group with the
same deformation of SU(3), so that we have quantum numbers linked with spacetime
symmetry and charge gauge symmetry in the computation of observables. The
generalization to higher-dimensional Lie groups SU(N), G2 and E8 is suggested. On
this basis we discuss a unifying framework for quantum gravity. Inside the transition
amplitude or partition function for geometries, we have the quantum numbers of
particles and fields interacting in the form of a spin foam network in the framework of
state sum models, we have a sum over quantum computations driven by the interplay
between aperiodic order and topological order.
Keywords: Quantum Gravity, Spin Foam, Unification Physics, Aperiodic Order,
Topological Order
1. Introduction
Quantum gravity and unification physics programs, in the absence of more concrete
experimental results, rely much on rigorous mathematical results to make progress. The
old and well established quantum gravity programs of loop quantum gravity (LQG) [1]
and string theory [2, 3] are good examples of this new paradigm to find clues about
the underlying physics at the Planck scale. For a review of the motivations and main
results of LQG and spin foam models we recommend the book [1]. We also recommend
a review of the main elements of the higher dimensional Lie algebra unification program
[4, 5, 6, 7, 8], which is closely related to string theory. With those results as guides, this
paper will construct a model that does not have a p