标题： 混合几何动力学：经典引力与量子物质耦合的哈密顿描述 显示英文标题

标题： Hybrid Geometrodynamics: A Hamiltonian description of classical gravity coupled to quantum matter

摘要： 我们推广了已知的几何动力学——广义相对论与经典物质耦合的哈密顿图像——到这种物质由弯曲时空中的量子场论描述，而引力仍由空间超曲面上的经典度规张量场及其相伴的动量来描述的情况。因此，在我们的方法中，除了事件流形之外，没有非动力学的背景结构，并且引力和量子自由度的动力学是不可分割地耦合在一起的。鉴于框架的哈密顿性质，我们使用了流形上量子态的超曲面变形生成元。该构造大量依赖于量子态集在引力变量集上的纤维丛的微分几何。本工作的一个重要特征是利用物质场空间上的高斯测度和Hida分布来定义一个共同的超空间，适用于所有具有不同测度的不同希尔伯特空间，以正确表征弯曲时空中QFT的薛定谔波泛函图像。这使我们能够在真空态或依赖引力自由度的测度的情况下（例如Ashtekar复结构相关的那些）关联不同希尔伯特空间中的态。这是通过引入纤维丛的量子厄米联络实现的，它将有深远的物理意义。该构造最显著的物理特性包括：量子态的范数守恒（即使总动力学是非幺正的）、明确识别混合守恒量以及描述量子物质对几何的动力反作用及其相反方向的作用，这些都将修改在无反作用情况下引力场所具有的物理属性。 显示英文摘要

摘要： We generalize the Hamiltonian picture of General Relativity coupled to classical matter, known as geometrodynamics, to the case where such matter is described by a Quantum Field Theory in Curved Spacetime, but gravity is still described by a classical metric tensor field over a spatial hypersurface and its associated momentum. Thus, in our approach there is no non-dynamic background structure, apart from the manifold of events, and the gravitational and quantum degrees of freedom have their dynamics inextricably coupled. Given the Hamiltonian natureof the framework, we work with the generators of hypersurface deformations over the manifold of quantum states. The construction relies heavily on the differential geometry of a fibration of the set of quantum states over the set of gravitational variables. An important feature of this work is the use of Gaussian measures over the space of matter fields and of Hida distributions to define a common superspace to all possible Hilbert spaces with different measures, to properly characterize the Schrodinger wave functional picture of QFT in curved spacetime. This allows us to relate states within different Hilbert spaces in the case of vacuum states or measures that depend on the gravitational degrees of freedom, as the ones associated to Ashtekar's complex structure. This is achieved through the inclusion of a quantum Hermitian connection for the fibration, which will have profound physical implications. The most remarkable physical features of the construction are norm conservation of the quantum state (even if the total dynamics are non-unitary), the clear identification of the hybrid conserved quantities and the description of a dynamical backreaction of quantum matter on geometry and vice versa, which shall modify the physical properties the gravitational field would have in the absence of backreaction.