[论文解读] From the flat-space S-matrix to the Wavefunction of the Universe

Paolo Benincasa|arXiv (Cornell University)|Nov 6, 2018

Cosmology and Gravitation Theories参考文献 9被引用 18

一句话总结

该论文通过宇宙多面体，在一类玩具模型中建立了宇宙的宇宙学波函数与平坦空间S矩阵元素之间的直接对应关系。它表明，波函数所有奇点——除总能量极点外——均编码了平坦空间散射振幅，且所有波函数留数均可表示为这些振幅的乘积，从而实现了在树图层次上仅通过平坦空间数据与对称性约束即可完全重构波函数。

ABSTRACT

The physical information encoded in the cosmological late-time wavefunction of the universe is tied to its singularity structure and its behaviour as such singularities are approached. One important singularity is identified by the vanishing of the total energy, where the wavefunction reduces to the physics of scattering in flat space. In this paper, we discuss the behaviour of the perturbative wavefunction as its other singularities are approached and the role played by the flat-space scattering, in the simplified context of the class of toy models admitting a first principle definition in terms of cosmological polytopes. The problems then translates into the analysis of the structure of its facets, one of which -- the scattering facet -- beautifully encodes the flat-space S-matrix. We show that all the boundaries of the cosmological polytope encode information about the flat-space physics. In particular, a subset of its facets turns out to have a similar structure as the scattering facet, with the vertices which can be grouped together to form lower dimensional scattering facets. The other facets admit one (and only one) triangulation in terms of products of lower dimensional scattering facets. As a consequence, the whole perturbative wavefunction can be represented as a sum of product of flat-space scattering amplitudes. Finally, we turn the table around and ask whether the knowledge of the flat-space scattering amplitudes suffices to reconstruct the wavefunction of the universe. We show that, at least for our class of toy models, this is indeed the case at tree level if we are also provided with a subset of symmetries that the wavefunction ought to satisfy. Once the tree cosmological polytopes are reconstructed, the loop ones can be obtained as a particular projection of them.

研究动机与目标

• 理解宇宙学波函数的解析结构及其在总能量奇点之外的物理诠释。
• 研究平坦空间散射振幅是否编码了宇宙学模型中的完整微扰波函数。
• 确定波函数是否可从平坦空间数据与对称性（特别是树图层次）中重构。
• 分析宇宙多面体面在编码平坦空间物理（包括散射与非散射边界）中的作用。
• 通过从树图层次多面体投影，将重构程序扩展至环图层次的波函数。

提出的方法

• 将宇宙多面体用作编码波函数奇点结构的第一性原理组合对象。
• 分析宇宙多面体的面，识别与波函数极点对应的面及其物理诠释。
• 证明与总能量极点相关的散射面通过其对偶结构编码了平坦空间切割规则与洛伦兹不变性。
• 表明非散射面可唯一地剖分为低维散射面的乘积，暗示其留数可分解为平坦空间振幅的乘积。
• 对所有极点处的波函数留数应用柯西定理，将其表示为平坦空间振幅与低阶波函数的组合。
• 通过能量空间形变（例如 $x_i \to x_i + \zeta$）提取留数，并验证其与已知递推关系的一致性。

实验结果

研究问题

• RQ1 宇宙学波函数所有超越总能量极点的奇点是否均编码了平坦空间散射物理？
• RQ2 是否可仅通过平坦空间散射振幅与一组对称性完全重构整个微扰波函数？
• RQ3 宇宙多面体面在编码平坦空间振幅（包括非散射边界）中的作用是什么？
• RQ4 当接近极点时，环图层次的波函数如何分解？是否可从树图层次结构投影得到？
• RQ5 波函数的奇点结构是否完全由其留数决定，而每个留数均可表示为平坦空间振幅的形式？

主要发现

• 宇宙多面体的所有边界，包括非散射面，均可剖分为低维散射面的乘积，每个面均编码平坦空间振幅。
• 极点 $y_{12} + x_2 + x_{23} = 0$ 的留数可表示为两个类洛伦兹传播子的乘积，等价于三体点平坦空间振幅 $A_3(x_1, x_3)$。
• 波函数 $\psi_3$ 可表示为平坦空间振幅乘积之和，其中振幅 $A_3(x_1, x_3)$ 显式给出为 $\frac{1}{y_{12}^2 - x_1^2} \cdot \frac{1}{y_{23}^2 - x_3^2}$。
• 对于环图（如蝌蚪图），波函数留数满足柯西恒等式，可表示为平坦空间振幅与低阶波函数的组合。
• 一环三体图可分解为树图层次平坦空间振幅与环波函数留数的乘积，后者通过柯西定理与平坦空间振幅相关联。
• 在树图层次，波函数可完全由平坦空间散射振幅与一组对称性重构，环图波函数可作为树图层次多面体的投影获得。