Recovering the Standard Model Gauge Group from Loop Quantum Gravity: a Unified Relative-Entropy, Flux-Gauge Network, and Geometric-Engineering Construction

Abstract

Mohammad Hannan

We give a single, self-contained construction in which the Standard Model gauge group SU(3)×SU(2)×U(1) arises alongside gravity from an enlarged connection built on the Ashtekar variables of loop quantum gravity (LQG). The paper unifies, into one logical development, three previously separate lines of work by the author: (I) a relative-entropy renormalization-group (RG) toolkit, built from the data-processing inequality, that is unconditionally monotone under lattice and spin-network coarse-graining; (II) a flux-gauge network model in which a confining gauge connection is the sole microscopic substrate, geometry and the graviton are coarse-grained derived quantities, and SU(N) matter content is generated at ADE-orbifold degenerations of an internal fiber; and (III) an explicit application of both toolkits to the Ashtekar connection, in which the matter sector is fixed to su(5) by a rank-4 A4 singularity and broken to the Standard Model by the network’s own adjoint vertex-Higgs mechanism. Every group-theoretic step of part (III) is proved as a theorem with complete proof and verified constants: a conservativity theorem showing the enlargement leaves the LQG kinematical Hilbert space an exact tensor product with the unchanged geometric sector; the Borel–de Siebenthal classification identifying su(3) ⊕su(2) ⊕u(1) as the unique maximal-rank subalgebra of su(5) with the Standard Model rank profile; the exact adjoint branching with hypercharges from root data and the resulting sin2θW = 3/8 matching relation; the Li vacuum-alignment theorem fixing the open region of quartic couplings for which the network’s vertex scalar breaks SU(5) to the Standard Model rather than to SU(4)×U(1); the resulting X/Y boson mass; and the global form of the unbroken group, S(U(3)×U(2)) ∼= (SU(3)×SU(2)×U(1))/Z6, from which Standard Model hypercharge and electric-charge quantization follow as corollaries rather than inputs. We report the numerical tests carried out for parts (I)–(II) — exact confirmation of the relative-entropy monotonicity under block-summing coarse-graining, an independently measured percolation threshold consistent with the known three-dimensional value, and a suggestive (not conclusive) correlation between the growth of the half-dissipation scale and the percolation crossover — and we state precisely, in a unified status ledger, what the whole construction proves, what it imports from established results elsewhere in the literature, and what remains open: the dynamical selection of the singularity rank and the sign of the alignment coupling, and, entirely outside this paper’s scope, fermion content, chirality, generation number, Yukawa structure, and unification phenomenology.

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