勞侖茲反德西特空間中之自旋傳播子

Abstract

在本論文中，我們將歐幾里得情景下的鑲嵌形式(embedding formalism)拓展至勞倫茲情景以描述勞倫茲反德西特空間下的對稱無跡張量。利用此形式，我們藉由兩由塊到邊界傳播子來建立諧和函數。經由此方式所建構的諧和函數可被表示成能被陰影轉換(shadow transform)與自旋陰影轉換(spin-shadow transform)所聯繫的由塊至塊傳播子的線性組合。此外，我們也建構了在勞倫茲情景下傳遞帶質量自旋場的由塊至塊傳播子的分裂表示(split representation)。在這表示中，該傳播子被表示成諧和函數的線性組合。作為初步應用，我們運用該傳播子計算描述任意純量一次運算子(primary operator)間的四點樹狀維騰圖(Witten diagram)。該計算結果可表示成共形部分波的形式，意指其可表示成兩個三點相關函數的內積積分。

In this work, we extend the embedding formalism developed in Euclidean signature to Lorentzian case to describe the symmetric traceless tensors in Lorentzian Anti-de sitter space. Using this formalism, we construct the harmonic function by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. The harmonic functions constructed in this way are shown to be the linear combination of bulk-to-bulk propagators which are related to each other by the so-called shadow transform and spin-shadow transform. Furthermore, We developed the split representation of the bulk-to-bulk propagators of massive spinning fields in Lorentzian signature, the propagators are expressed as a linear combination of harmonic functions. As a application, we computed the tree-level four-point Witten diagram describing spin J exchange between scalar primaries of arbitrary dimension. The Witten diagrams eventually could be related to the conformal partial wave, which is defined as the integral of the product of two three-point functions.