Quantization of axions on dS_D yields Hilbert space H = L^2(S^1) ⊗ F with zero-mode U(1) charge, producing non-dS-invariant charged sectors and Hadamard Wightman functions that become asymptotically invariant.
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We show that the late time Hartle-Hawking wave function for a free massless scalar in a fixed de Sitter background encodes a sharp ultrametric structure for the standard Euclidean distance on the space of field configurations. This implies a hierarchical, tree-like organization of the state space, reflecting its genesis as a branched diffusion process. An equivalent mathematical structure organizes the state space of the Sherrington-Kirkpatrick model of a spin glass.
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Axions on de Sitter space
Quantization of axions on dS_D yields Hilbert space H = L^2(S^1) ⊗ F with zero-mode U(1) charge, producing non-dS-invariant charged sectors and Hadamard Wightman functions that become asymptotically invariant.