AI 中文总结
研究1+1维中基于链的因果集传播子对曲率的响应，发现其体积依赖性会产生主导曲率修正，大距离下等价于有效曲率耦合为ξR的连续标量传播子，该耦合由微观结构涌现，数值模拟支持预测。
AI 中文摘要
我们研究一种新近引入的、基于链定义的因果集传播子对曲率的响应。在嵌入平直1+1维闵可夫斯基空间的撒点上，该传播子已知会在大尺度上重现无推迟连续格林函数。在具有常曲率$\boldsymbol{\textit{R}}$的共形平直嵌入中，因果序不变，但链关系对所张因果钻石的物理体积敏感。我们证明，这种体积依赖性会对传播子产生主导曲率修正。在大距离下，该修正等价于具有形式为$\boldsymbol{\textit{\u03BE R}}$的有效曲率耦合的连续标量传播子，其中耦合常数$\boldsymbol{\textit{\u03BE_{\rm eff}}}=1/6$，对应有效质量平方参数$\boldsymbol{\textit{m_{\rm eff}^2}}=\boldsymbol{\textit{\u03BE_{\rm eff} R}}$。该耦合并非人为引入，而是从基于链的路径和本身涌现，反映了因果集的微观结构。对嵌入$\boldsymbol{\textit{AdS_{1+1}}}$和$\boldsymbol{\textit{dS_{1+1}}}$的撒点进行的数值模拟支持所预测的曲率响应，表明有效耦合会随撒点密度增加而持续存在。
英文摘要
We study how a recently introduced causal-set propagator defined in terms of links responds to curvature. On sprinklings embedded in a flat 1+1-dimensional Minkowski space, this propagator is known to reproduce the massless retarded continuum Greens function on large scales. In conformally flat embeddings with constant curvature $\mathcal R$, the causal order is unchanged, but the link relationship is sensitive to the physical volume of the spanned causal diamond. We show that this volume dependence generates a leading curvature correction to the propagator. On large distances, this correction is equivalent to a continuum scalar propagator with an effective curvature coupling of the form $ξ\mathcal R$ with the coupling constant $ξ_{\rm eff}=1/6$, equivalently corresponding to an effective mass-squared parameter $m_{\rm eff}^2=ξ_{\rm eff}\mathcal R$. This coupling is not inserted by hand but emerges from the link-based path sum itself, reflecting the microscopic structure of the causal set. Numerical simulations on sprinklings embedded in $\mathrm{AdS}_{1+1}$ and $\mathrm{dS}_{1+1}$ support the predicted curvature response and indicate that the effective coupling persists as the sprinkling density is increased.
Comments17 pages, 6 figures
