有限固有时间标量场论作为谱算子演算

中文摘要 AI 辅助

我们为欧几里得标量λφ⁴理论构建了有限固有时间构造，其中保留一个端点s₀以完成内部历史，而非独立分配给各个施温格段。该理论通过配对的开、闭谱函数及其弗雷歇/杜哈梅尔层级定义。泛函微分沿现有固有时间历史插入算子并划分其总长度，而相互作用顶点则缝合各自完整的历史。我们引入对应的完整历史图解演算，并推导得到单圈和双圈结构。我们将保留的端点与辅助正则化子或重整化群标度区分开来，测试其效应是否在普通参数匹配和容许场重定义下存续。对于单圈四点函数，固定重整化质量、场归一化及四次耦合后，仍会留下有限的动量依赖余项。我们在该阶建立了保留端点相对于有限可重整参数重定义及定域S矩阵守恒场重定义的微扰非冗余描述。低能理论可表示为有效场论，但其高阶导数系数并非独立，这为保留有限固有时间标度与任意截断方案提供了具体区分。

英文摘要

We formulate a finite-proper-time (FPT) construction for real scalar $λϕ^4$ theory in which a non-zero lower endpoint $s_0$ is retained as physical spectral data for complete virtual histories. Functional differentiation partitions a pre-existing history, while interaction sewing creates closed momentum circulations. Local momentum conservation identifies the primitive closed circulations with graph circuits; applying the retained endpoint condition gives \[ s_e\ge0, \qquad \sum_{e\in c}s_e\ge s_0 \quad(c\in\mathcal{C}(G)). \] An individual edge or bridge may be arbitrarily short, but no complete closed circulation may collapse below $s_0$. This routing-independent prescription differs from damping every propagator. We prove that it gives a positive loop quadratic form and ultraviolet-finite massive amplitudes, including overlapping short-distance regions. Under a physical cut, precisely the circuit conditions crossed by the cut disappear, leaving the independently constructed daughter domains together with the ordinary pole residues and positive scalar phase space. At fixed $s_0$ we give an all-order local construction; for $λ\ge0$, $\mathcal{K}_{\barϕ}\ge-\partial^2+m^2$ gives a background-uniform heat-kernel bound. The open Euclidean history kernel is virtual spectral machinery rather than the physical propagator, and physical positivity is imposed on complete boundary amplitudes. After local matching, the first non-constant one-loop on-shell correction is proportional to $s_0^2(s^2+t^2+u^2)$, providing a correlated observable test of the retained scale.

发表机构

• University of Oxford（牛津大学）
• National University of Singapore（新加坡国立大学）

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