物理学报2025，Vol.74Issue(8)：29-40,12.DOI:10.7498/aps.74.20241751
芝诺悖论与黑洞信息丢失问题
Zeno's paradox and black hole information loss problem
摘要
Abstract
The black hole information paradox—a fundamental conflict between quantum unitarity and semi-classical gravity—remains unresolved within the framework of quantum gravity.This work builds a conceptual bridge between this modern dilemma and Zeno's ancient paradox of motion,emphasizing that they all rely on limit processes to reconcile infinite divisibility with finite physical outcomes.Although previous researches have established information conservation in black hole evaporation by copying wormholes,this study transcends mere unitarity verification to answer a deeper question:How do we quantify the statistical evolution and inherent irreversibility of Hawking radiation? Methods: ⅰ)Gravitational path integral formalism:We compute Rényi entropies Sn=1/1-n ln Tr(ρn)by analytic continuation(n → 1),incorporating both disconnected geometries(Hawking saddles)and connected replica wormhole saddles. ⅱ)JT gravity model:A two-dimensional Jackiw-Teitelboim(JT)gravity coupled to a conformal bath is analyzed,with End-of-the-World(EOW)branes modeling early Hawking radiation states. ⅲ)Modular thermodynamics:Modular entropy Sm and entanglement capacity Cn are defined as:Sm=log Tr(ρn)-n∂n log Tr(ρn),Cn=n2∂2n log Tr(ρn),mirroring thermodynamic entropy and heat capacity. Key results: ⅰ)Page curve restoration:The von Neumann entropy transitions from linear growth(Hawking phase)to zero(wormhole-dominated phase)at the Page time:tPage∝SBH/(nkc),SBH=A/(4G N),indicating faster entropy reduction for higher n. ⅱ)Replica wormhole mechanism:The gravitational path integral reveals a topological transition:Tr(ρn)≈kZn1+knZn/(kZ1)n⇒Dominance shift at k～es0.Quantum non-cloning constraints:A reversibility paradox arises:while n-copies-copies of a known state allow full reconstruction,the non-cloning theorem forbids replication of unknown states.This necessitates the a priori existence of multiple replicas to ensure consistency. ⅲ)Connection to Zeno's paradox:The resolution of Zeno's paradox through Newtonian limits(v=limΔt→0Δx/Δt)parall)parallels the replica trick's role in black hole physics:ⅳ)Infinite-to-finite mapping:Both employ limit operations to compress divergent processes(infinite steps or entropy growth)into finite observables. Discrete vs.continuous:Challenge classical spacetime continuity,hinting at holographic or discrete quantum spacetime. Irreversibility and generalized second law:The relative entropy S(ρ‖σ),defined as:S(ρ‖σ)=Tr(ρlogρ)-Tr(ρlogσ),monotonically increases during the Hawking-to-wormhole transition,establishing statistical irreversibility in modular space.This generalized second law persists despite microscopic unitarity,analogous to thermodynamic irreversibility in closed systems with the formula:(Sn-Sρ)-n(<H>n-<H>ρ)≥0. Conclusion: Our results address the black hole information paradox and the irreversible problem through three pillars: ⅰ)Unitarity:Replica wormholes enforce unitarity via non-perturbative spacetime topology changes. ⅱ)Modular thermodynamics:A framework linking entanglement entropy,relative entropy,and irreversibility. ⅲ)Replica priority principle:Multiple radiation replicas exist a priori,circumventing quantum cloning constraints. This work transcends mere information conser-vation,advancing into the second layer of black hole physics:Characterizing the irreversible evolution of Hawking radiation in modular space.关键词
引力/规范对偶/冯·诺伊曼熵/Rényi熵/副本技巧/量子不可克隆定理/相对熵第二定律Key words
gravity/gauge duality/von Neumann entropy/Rényi entropy/replica trick/quantum non-cloning theorem/generalized second law of thermodynamics via relative entropy引用本文复制引用
葛先辉..芝诺悖论与黑洞信息丢失问题[J].物理学报,2025,74(8):29-40,12.基金项目
国家自然科学基金(批准号:12275166,12311540141)资助的课题. Project supported by the National Natural Science Foundation of China(Grant Nos.12275166,12311540141). （批准号:12275166,12311540141）
