A chaotic lattice field theory in two dimensions

We describe spatiotemporally chaotic (or turbulent) field theories discretized over d-dimensional lattices in terms of sums over their multi-periodic orbits. `Chaos theory' is here recast in the language of statistical mechanics, field theory, and solid state physics, with the traditional periodic orbits theory of low-dimensional, temporally chaotic dynamics a special, one-dimensional case. In the field-theoretical formulation, there is no time evolution. Instead, treating the temporal and spatial directions on equal footing, one determines the spatiotemporally periodic orbits that contribute to the partition sum of the theory, each a solution of the system's defining deterministic equations, with sums over time-periodic orbits of dynamical systems theory replaced here by sums of d-periodic orbits over d-dimensional spacetime geometries, the weight of each orbit given by the Jacobian of its spatiotemporal orbit Jacobian operator. The weights, evaluated by application of the Bloch theorem to the spectrum of periodic orbit's Jacobian operator, are multiplicative for spacetime orbit repeats, leading to a spatiotemporal zeta function formulation of the theory in terms of prime orbits.