Noether定理在平直时空中的一阶与二阶场论
Noether's theorems for first and second order field theories in flat spacetimes
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中文总结 AI 辅助
针对高阶张量场和高阶导数场论，完整推导Noether恒等式及第一、第二定理，并给出物理应用实例与对称性起源说明。
中文摘要 AI 辅助
Noether恒等式（即使用变分法展示并解释所有步骤）以及Noether第一和第二定理的技术推导，在数学文献中通常针对作用量中具有一阶导数的标量函数以显式细节呈现。这种呈现方式对于希望学习这些技术以应用于具有高阶张量场（势）或作用量中高阶导数的物理场论的物理学家而言存在缺陷。受此动机驱动，并深受Gelfand和Fomin的变分法表述的影响，我们详细给出了在平直（如闵可夫斯基）时空中具有任意秩张量场的一阶和二阶场论的Noether恒等式和Noether定理的完整推导，显式展示所有步骤。随后，我们给出一些将第一和第二定理应用于物理学中场论的物理实例，并详细说明常规时空对称性源自Killing和共形Killing方程。最后，我们简要回顾了这些定理已被讨论和应用的物理学诸多领域。
英文摘要
Technical derivations of the Noether identity (that is, showing and explaining all steps using methods of the calculus of variations), as well as of Noether's first and second theorems, are typically presented in explicit detail in the mathematics literature for scalar functions with first order derivatives in the action. This presentation has drawbacks for physicists wishing to learn these techniques to apply to physical field theories that have higher rank tensor fields (potentials) or higher orders of derivatives in the action. Motivated by this, and heavily influenced by the calculus of variations presentation of Gelfand and Fomin, we detail complete derivations of the Noether identity and Noether's theorems for first and second order field theories with tensor fields of arbitrary rank in flat (e.g., Minkowski) spacetimes, showing all steps explicitly. We then give some physical examples of how to apply the first and second theorems to field theories in physics, and detail the origin of conventional spacetime symmetries from the Killing and conformal Killing equations. We conclude by briefly reviewing the many areas of physics where the theorems have been discussed and applied.
