Title: Covariant phase space and the semi-classical Einstein equation Show Chinese title

Title: 协变相空间和半经典爱因斯坦方程

Abstract: The covariant phase space formalism in general relativity is a covariant method for constructing the symplectic two-form, Hamiltonian and other conserved charges on the phase space of solutions to the Einstein equation with classical matter. In this note, we consider a generalization of this formalism to the semi-classical Einstein equation coupled to quantum matter. Given a family of solutions in semi-classical gravity, we define the semi-classical symplectic two-form -- a natural generalization of the classical sympelctic two-form -- as the sum of the gravitational symplectic form and the Berry curvature associated to the quantum state of matter. We show that the semi-classical symplectic two-form is independent of the Cauchy slice, and satisfies the quantum generalization of the classical Hollands-Iyer-Wald identity. For small perturbations, we also extend our discussion to gauge-invariantly defined subregions of spacetime, where the quantum contribution is replaced by the Berry curvature of certain special purifications involving the Connes cocycle. In the AdS/CFT context, the semi-classical symplectic form defined here is naturally dual to the Berry curvature in the boundary CFT. Show Chinese abstract

Abstract: 广义相对论中的协变相空间形式是一种协变方法，用于在爱因斯坦方程与经典物质耦合的解的相空间上构造辛两形式、哈密顿量和其他守恒电荷。 在本注释中，我们考虑将这种形式推广到与量子物质耦合的半经典爱因斯坦方程。 给定半经典引力中的一族解，我们将半经典辛两形式定义为引力辛形式和与物质量子态相关的贝里曲率之和——这是经典辛两形式的一个自然推广。 我们证明半经典辛两形式与柯西片无关，并满足经典Hollands-Iyer-Wald恒等式的量子推广。 对于小扰动，我们还将讨论扩展到规范不变定义的时空子区域，在这些子区域中，量子贡献被涉及康尼斯余换的特定纯化所对应的贝里曲率所取代。 在AdS/CFT背景下，此处定义的半经典辛形式自然地与边界CFT中的贝里曲率对偶。