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arXiv 2201.11325nlin.CDhep-thmath-phmath.MP
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一维混沌格点场论

A chaotic lattice field theory in one dimension

Han Liang, Predrag Cvitanović

备注：
70 pages, 40 figures. Contribution to S. Gnutzmann et al "Special issue in honour of the life and work of Fritz Haake", J. Phys. A 54 (2021). Ver.2 differs from ver. 1 only in several sections being demoted to appendices. For additional material -online talks, related drafts- see https://ChaosBook.org/overheads/spatiotemporal/
发表信息：
J. Phys. A 55, 304002 (2022)
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摘要：

Gutzwiller的半经典量子化，其中低维确定性动力学的不稳定周期轨道作为一个WKB的“骨架”的混沌量子力学，我们构建了相应的确定性骨架的无限维格点离散标量场理论。在场论的表述中，没有时间上的演化，也没有“李雅普诺夫视界”;只有对理论的分区和有贡献的晶格态的枚举，每个晶格态都是系统确定性欧拉-拉格朗日方程的全局时空解。这种重新表述使“混沌理论”与标准的固态、场论和统计力学相一致。在一个时空的晶体学公式中，动力系统理论中的时间周期轨道被时空的周期性$d$维布拉维细胞镶嵌所取代，每个细胞镶嵌都由其不稳定性的倒数（希尔行列式）加权。大细胞的双曲阴影较小的确保理论的预测是由最小的布拉维细胞。一个给定的场论的配分函数的形式由它的时空对称群决定，也就是说，由它的格离散化的空间群决定，在它的倒易格上研究得最好。已经1维格离散化是足够的兴趣是本文的重点。特别是，从时空场论的角度来看，“时间”反转是一个纯粹的晶体学概念，一个反射点群，导致一个新的，对称性的时间可逆理论和相关的拓扑zeta函数的角度。

英文摘要：

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 no `Lyapunov 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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