算子基、$S$ -矩阵及其配分函数
Operator bases, $S$-matrices, and their partition functions
摘要：
相对论量子系统，承认散射实验定量描述的有效场论，其中$S$ -矩阵运动学和对称性的考虑编码在经营者的EFT频谱。在本文中，我们使用的$S$ -矩阵推导出的EFT算子的基础结构，提供互补的描述（i）位置空间利用共形代数和上同调和（ii）动量空间通过代数公式的动量环的运动学实现为一个理想。这些框架系统地处理与运动方程（壳上）和部分积分（动量守恒）相关的冗余。我们引入一个配分函数，称为希尔伯特级数，枚举的运营商的基础-相应的，$S$ -矩阵-并推导出一个矩阵积分表达式计算希尔伯特级数。该表达式是通用的，容易应用于任何时空维度，具有任意场内容和（线性实现的）对称性。除了计算，我们还讨论了基础的构造。简单的算法从动量空间中的代数公式推导出来。我们显式计算涉及多达$n=5$标量场的运算符的基础。这种构造普遍适用于具有自旋的场，因为标量的算符基编码了$n$点振幅的动量依赖性。我们详细讨论了非线性实现的对称性的运营商的基础。在无质量粒子的存在下，可以自由地以软极限的形式在$S$ -矩阵上施加附加结构。无质量标量的最简单的实现导致了π介子的算符基，我们使用非线性实现的标准CCWZ公式来确认。
英文摘要：
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where $S$-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the $S$-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis--correspondingly, the $S$-matrix--and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to $n=5$ scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of $n$-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the $S$-matrix in the form of soft limits. The most naïve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.
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