arXiv 2310.014842023-10hep-thhep-phmath.AGmath.LOquant-ph

驯良量子理论中的复杂性

Complexity in Tame Quantum Theories

Thomas W. Grimm, Lorenz Schlechter, Mick van Vliet

备注：: 48 pages, 2 figures
发表信息：: JHEP 05 (2024) 001

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AI总结：

本文提出利用o-minimal结构量化物理理论中函数与集合的逻辑信息复杂性，并建议以sharp o-minimal结构作为度量量子理论复杂性的统一框架。

摘要：

受物理系统只能包含有限信息量或复杂性这一观念的启发，我们引入了一个框架，用于量化指定一个函数或集合所需的逻辑信息量。随后，我们将这一方法应用于多种物理系统，并推导出有效拉格朗日量中出现的依赖参数的物理可观测量和耦合函数的复杂性。为了实现这些想法，必须考虑能够在o-minimal结构中定义的物理理论。O-minimality是数理逻辑中的一个概念，它体现了一种驯良性原理。最近有人论证，这一性质内在于许多已知的量子场论，并与理论的紫外完备化相关联。为了给这些理论中的每个命题赋予复杂性，必须进一步约束所允许的o-minimal结构。为举例说明这一点，我们展示了许多物理系统可以使用Pfaffian o-minimal结构来表述，这些结构具有明确的复杂性概念。更一般地，我们提议采用Binyamini和Novikov最近引入的sharp o-minimal结构，作为度量量子理论中复杂性的总体框架。

英文摘要：

Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.

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