纠缠、杨-米尔斯理论与作为SU(N)-等变核的散射矩阵
Entanglement, Yang-Mills, and the Scattering Matrix as an SU(N)-equivariant Kernel
AI总结：
通过SU(N)-等变映射研究两体散射，揭示颜色空间纠缠作为有效算子的层析探针，并证明杨-米尔斯四次耦合由最大纠缠保持唯一确定。
摘要：
我们将两体散射研究为作用在张量积表示空间上的SU(N)-等变映射，并分析S矩阵产生的纠缠。这种表示论视角将群结构与动力学分离：R⊗R'的分解固定了不变算子代数，从而确定了过程的定性纠缠能力。对于基础表示中的粒子，End_{SU(N)}(N⊗N)=Span{I,S}，因此只有恒等和交换方向保持可分离性，而一般组合产生纠缠。伴随-伴随散射涉及包含d-张量的更大不变代数，本质上是纠缠的。在杨-米尔斯理论中，可以利用颜色-运动学对偶性证明颜色核位于该算子空间的固定射线上，从而得到直角散射时出射纠缠的普适最大值，对于SU(2)为E_*^{(2)}=3/4，对于SU(3)为E_*^{(3)}≈0.91，与运动学无关。维度六算子保持这种普适性，而维度八形变填充新的颜色扇区并改变E_*^{(N)}，表明颜色空间中的纠缠作为有效算子的层析探针。在螺旋度空间中，要求最大纠缠输入散射为最大纠缠输出唯一地选择杨-米尔斯四次耦合并强制颜色雅可比恒等式，将壳上沃德约束重述为纠缠保持条件。我们的结果表明，信息论观点统一了散射的代数、几何和动力学方面。
英文摘要：
We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the $S$-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of $R\!\otimes\!R'$ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, $\mathrm{End}_{\mathrm{SU}(N)}(N\!\otimes\!N)=\mathrm{Span}\{\mathbb{I},\mathbb{S}\}$, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving $d$-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, $E_\star^{(2)}=\tfrac{3}{4}$ for $SU(2)$ and $E_\star^{(3)}\simeq0.91$, independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift $E_\star^{(N)}$, suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.
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