A chaotic lattice field theory in one dimension (2201.11325v2)

Abstract: Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is noLyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations. The reformulation aligns chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic $d$-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective,time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

• The paper introduces a deterministic framework that replaces time-periodic orbits with Bravais cell tilings to analyze one-dimensional chaotic lattice field theories.
• It employs discrete lattice recurrence relations on models like the Bernoulli and cat maps to derive numerical partition sums and stability measures.
• The study bridges chaos theory, solid-state physics, and statistical mechanics, setting a pathway for future higher-dimensional generalizations.

Analyzing One-Dimensional Chaotic Lattice Field Theory

The paper presents a compelling exploration into the deterministic underpinnings of chaotic lattice field theory, particularly focused on one-dimensional systems. It builds on the historical foundations of periodic orbit theory, integrating novel ideas about deterministic chaos within the framework of field theories discretized over lattice structures. The discourse inverts the traditional paradigm by treating chaotic systems through the lens of global spatiotemporal solutions.

Gutzwiller's semiclassical quantization is a crucial motivator for the analysis conducted herein, where periodic orbits serve as the so-called WKB skeleton for a chaotic quantum mechanics context. The authors extend this analogy into infinite-dimensional lattice-discretized scalar field theories and propose a framework that brings together deterministic skeletons and chaos theory. This reformulates our understanding of infinite-dimensional chaotic field theory, aligning it with elements of solid-state physics, field theory, and statistical mechanics.

Key Theoretical Contributions

The paper demonstrates how time-periodic orbits in the dynamical systems theory can be replaced by d-dimensional Bravais cell tilings in spacetime. These are weighted according to their instability; this rearrangement is pivotal for deriving certain predictions of the theory. Noteworthy is the emphasis on the role of space group symmetries and its connection to the reciprocal lattice in defining a field theory's partition function.

Determined through the application of discrete lattice recurrence relations for elementary chaotic models such as the Bernoulli and cat map, the lattice field theories presented seek a tangible realization of spatiotemporal chaos in explicit scalar models. The introduction of ‘temporal Bernoulli’ and ‘temporal cat’ is significant, paving the way for d-dimensional generalizations while retaining a focus on one-dimensional demonstrations for clarity.

Numerical and Analytical Insights

Strong numerical results are evidenced via the construction of partition sums, the determination of Hill determinants, and the symbolic dynamics formatted through group-theoretic zeta functions. Theoretical rigor is complemented by practical illustrations of these results, such as the probabilistic density returns orbits to the state space points, reinforcing the theoretical assertions regarding spatiotemporal chaos.

Appendices consisting of historical context and foundational mathematics lend deeper insight into the exploration, providing a rich substrate of works ranging from lattice theory to symbolic dynamics and periodic orbit analyses. Particular attention is drawn to Hill's formula, employed here as a means to connect time evolution stability with orbit stability in high-dimensional space.

Theoretical Implications and Future Directions

The implications of this work stretch over several intertwined domains. By situating chaotic lattice field theory within a recognizable structure, it provides fresh insights into many theoretical subjects, including turbulence in strongly nonlinear deterministic field theories akin to Navier-Stokes or Kuramoto-Sivashinsky equations, and even quantum, chaotic, or stochastic field theories.

As the research evolves, extension into higher dimensions and more complex lattice structures is anticipated. Exploration into computational algorithms and frameworks that can efficiently tackle these expanded models will also be paramount. The theoretical edges touched by this paper could set the stage for new paradigms in understanding the chaotic behaviors of infinitely complex systems.

In summary, this work bridges several theoretical domains by recalibrating chaotic field theory through deterministic lattice frameworks. Its alignment with notions drawn from established physics offers a comprehensive pathway towards understanding and predicting complex spatiotemporal behaviors.