AI 中文总结
研究量子场论与引力中离壳等价性问题，基于维尔科夫斯基 - 德维特有效作用量建立操作标准，通过标量可观测量测试等价性，以度规 $f(R)$ 引力等为例展示如何区分不同等价性，明确了相关等价性概念。
AI 中文摘要
场重定义连接了同一物理的多种表述，但标准等价定理是壳上结果，无法用于判定两个量子描述在壳外是否等价。本文依据维尔科夫斯基 - 德维特有效作用量为此问题建立了一个操作标准。核心思想是通过将构型空间张量与可允许探针配对构建的标量可观测量来测试等价性。这使该标准对所考虑的可观测量类别敏感，并区分了通常的壳上等价概念与引力、宇宙学和非平衡量子场论中所需的更强的壳外等价概念。以度规 $f(R)$ 引力和标量场理论为例，展示了形式标准如何区分局部、分支和真正的全局等价性。特别地，当在路径积分中施加辅助约束时，度规 $f(R)$ 引力及其辅助场重新表述产生相同的量子理论。这与将度规和标量视为独立积分变量对相应标量 - 张量作用量进行量子化不同，后者定义了一个不同的量子理论。明显的量子不等价性可追溯到不同量子对象之间的比较，而非实际等价性的失败。这导致了在场重定义和辅助变量扩展下局部和全局等价性的一般且精确的概念。
英文摘要
Field redefinitions connect many formulations of the same physics, but the standard equivalence theorem is an on-shell result and cannot be used to decide when two quantum descriptions are equivalent off shell. This paper develops an operational criterion for that problem in terms of the Vilkovisky--DeWitt effective action. The central idea is that equivalence should be tested on scalar observables built by pairing configuration-space tensors with admissible probes. This makes the criterion sensitive to the observable class under consideration and separates the usual on-shell notion of equivalence from the stronger off-shell notions needed in gravity, cosmology and non-equilibrium quantum field theory. Metric $f(R)$ gravity and a scalar field theory example serve as prototypes, showing how the formal criterion distinguishes local, branchwise and genuinely global equivalence. In particular, we show that metric $f(R)$ gravity and its auxiliary-field reformulation yield the same quantum theory when the auxiliary constraint is enforced in the path integral. This is distinct from quantizing the corresponding scalar--tensor action with the metric and scalar treated as independent integration variables, which defines a different quantum theory. Apparent quantum inequivalences can then be traced to comparisons between different quantum objects, rather than to a failure of actual equivalences. This leads to general and precise notions of local and global equivalence under both field redefinitions and auxiliary-variable extensions.
