发表机构
Leinweber Institute for Theoretical Physics and Department of Physics, University of California, Berkeley; RIKEN iTHEMS Center for Interdisciplinary Theoretical and Mathematical Sciences; Joseph Henry Laboratories and Leinweber Forum for Theoretical Physics, Princeton University; Department of Physics, Harvard University（加州大学伯克利分校莱因韦伯理论物理研究所与物理系; 理化学所iTHEMS跨学科理论与数学科学中心; 普林斯顿大学约瑟夫·亨利实验室与莱因韦伯理论物理论坛; 哈佛大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出引力路径积分作为内积或统计平均的充要条件，并证明更强条件对虫洞振幅施加额外约束，通过玩具模型和自举研究展示其诊断与加强界限的作用。
AI 中文摘要
我们找到了引力路径积分（GPI）能够被解释为（i）开放/闭合宇宙希尔伯特空间中的内积，或（ii）边界观测量的统计平均的必要且充分条件。满足条件（i）所需的条件是满足条件（ii）的必要条件，但并非充分条件。要求更强的条件（ii）对所有虫洞振幅施加了额外的约束，从而为GPI计算的是平均值还是仅仅是伪平均值提供了具体的诊断标准。为了说明这两种约束之间的差异，我们提出了一类满足条件（i）但不满足更强条件（ii）的引力玩具模型；为了强调更强条件的力量，我们基于条件（ii）启动了统计系综的自举研究，发现由此产生的约束显著加强了通过施加条件（i）所获得的系综矩的界限。更广泛地说，这两个正性条件都对虫洞振幅施加了非平凡的约束，从而限制了哪些引力有效场论可以被嵌入到一致的量子引力理论中。
英文摘要
We find the necessary and sufficient conditions for the gravitational path integral (GPI) to admit an interpretation in terms of (i) an inner product in the Hilbert space of open/closed universes or (ii) a statistical average of boundary observables. The conditions needed for (i) are necessary but not sufficient for (ii). Requiring the stronger condition (ii) places additional constraints on all wormhole amplitudes, giving a concrete diagnostic of whether the GPI computes an average or merely a pseudo-average. To exemplify the difference between the two constraints, we present a class of gravitational toy models that satisfy (i) but fail the stronger condition (ii); to emphasize the power of the stronger condition, we initiate a bootstrap study of statistical ensembles based on condition (ii), finding that the resulting constraints substantially sharpen the bounds on the moments of the ensemble obtained by imposing (i). More broadly, both of these positivity conditions impose nontrivial constraints on wormhole amplitudes and thereby restrict which gravitational effective field theories can be embedded in consistent theories of quantum gravity.
Comments10 pages plus appendices. 33 pages total, 10 figures
