BKL singularity dynamics are mapped to conformal quantum mechanics whose states are odd automorphic L-functions, and these L-functions are reinterpreted as partition functions of prime-labeled oscillator gases.
Green's Functions for Vladimirov Derivatives and Tate's Thesis
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abstract
Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $\chi$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $\nu$, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics.
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The Conformal Primon Gas at the End of Time
BKL singularity dynamics are mapped to conformal quantum mechanics whose states are odd automorphic L-functions, and these L-functions are reinterpreted as partition functions of prime-labeled oscillator gases.