Casimir 物理中的光滑截断与解析延拓
Smooth cutoffs and analytic continuation in Casimir physics
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中文总结 AI 辅助
本文基于 Tao 的光滑求和理论，统一了 Casimir 能量计算中的正则化减除与解析延拓方法，通过 Mellin 变换揭示发散与有限项的极点结构，并分解为普适与截断依赖部分，类比重整化。
中文摘要 AI 辅助
在 Casimir 能量计算中出现的发散级数有两种看似不同的处理方法：一种是基于物理动机的正则化与减除程序，另一种是形式化的解析延拓方法，后者直接将有限值赋予发散表达式。对于平行板之间的 Casimir 能量，这两种方法的一致性众所周知，但其根源往往被隐晦地略过。在本工作中，我们提供了一个统一的框架，使这种等价性变得透明。基于 Tao 的光滑求和理论，我们证明引入光滑截断会导致一个渐近展开，其有限的、与正则化无关的部分由相关的 Dirichlet 级数的解析延拓普遍决定。这一识别源于 Mellin 变换表示，其中发散和有限贡献被编码在被积函数的极点结构中。我们将此方法扩展到与 Casimir 问题相关的一类级数。在此背景下，我们证明对数发散源于极点重合，并得到结果自然分解为普适项和截断依赖项，这反映了量子场论中重整化的结构。
英文摘要
Divergent series arising in the computation of Casimir energies admit two seemingly different treatments: a physically motivated regularization and subtraction procedure, and formal analytic continuation methods that assign finite values directly to divergent expressions. The agreement between these approaches for the Casimir energy between parallel plates is well known, but its origin is often left implicit. In this work, we provide a unified framework that makes this equivalence transparent. Building on Tao's theory of smoothed sums, we show that the introduction of a smooth cutoff leads to an asymptotic expansion whose finite, regulator-independent part is universally determined by the analytic continuation of the associated Dirichlet series. This identification follows from a Mellin-transform representation, in which divergences and finite contributions are encoded in the pole structure of the integrand. We extend this approach to a class of series relevant to Casimir problems. In this setting, we show that logarithmic divergences arise from pole coincidences, and we obtain a natural decomposition of the result into universal and cutoff-dependent terms, mirroring the structure of renormalization in quantum field theory.
发表机构
- Instituto Balseiro, Centro Atómico Bariloche（巴尔塞罗研究所，巴里洛切原子中心）
- Instituto de Física Teórica, Universidade Estadual Paulista（理论物理研究所，圣保罗州立大学）
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