A chaotic lattice field theory in two dimensions

Area of Science:

• Physics
• Statistical Mechanics
• Field Theory

Background:

• Chaos theory traditionally focuses on low-dimensional temporal dynamics.
• Existing models often lack a unified framework for spatiotemporal chaos.
• Field theories are typically formulated without explicit time evolution.

Purpose of the Study:

• To recast chaos theory in the language of statistical mechanics, field theory, and solid-state physics.
• To develop a framework for analyzing spatiotemporally chaotic field theories.
• To introduce a spatiotemporal zeta-function formulation for these theories.

Main Methods:

• Discretizing d-dimensional lattices for spatiotemporally chaotic field theories.
• Summing over multi-periodic orbits within the field-theoretical formulation.
• Treating temporal and spatial directions equally to identify spatiotemporally periodic orbits.
• Utilizing the Bloch theorem to evaluate orbit weights via the Jacobian operator spectrum.

Main Results:

• A reformulation of chaos theory applicable to d-dimensional lattices.
• Identification of spatiotemporally periodic orbits contributing to the partition sum.
• Development of a spatiotemporal zeta-function formulation based on prime orbits.
• Demonstration of multiplicative weights for spacetime orbit repeats.

Conclusions:

• The study provides a novel perspective on spatiotemporal chaos in field theories.
• The proposed zeta-function formulation offers a new tool for analyzing complex chaotic systems.
• This work bridges concepts from dynamical systems theory and advanced physics.