Cycle expansions for higher-dimensional spatiotemporal zeta functions

Determine whether the spatiotemporal zeta function 1/ζ[β,z] for two- and higher-dimensional lattice field theories, together with the expectation-value formula ⟨a⟩ = ⟨A⟩_ζ/⟨N⟩_ζ, can be reorganized into cycle expansions dominated by prime periodic states of small spacetime volume, analogous to the established one-dimensional temporal case.

Background

The paper introduces a spatiotemporal zeta function for deterministic lattice field theories that sums contributions from prime spatiotemporal periodic states with weights built from stability exponents. In one temporal dimension, conventional dynamical systems theory benefits from cycle expansions that accelerate convergence by exploiting shadowing.

Extending cycle expansions to two and higher spacetime dimensions could substantially improve convergence and practical computation of observables within the spatiotemporal framework, but whether and how such expansions can be organized in higher dimensions remains unresolved.