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 nature of 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 mathematical feature of this work is the use of Minlos’s theorem to characterize 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 Schrödinger 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.

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混合几何动力学：经典引力与量子物质耦合的哈密顿描述

我们将广义相对论与经典物质（称为几何动力学）耦合的哈密顿图推广到这样的情况，即这种物质由弯曲时空中的量子场论描述，但引力仍然由空间超曲面上的经典度量张量场描述，并且其相关动量。因此，在我们的方法中，除了多种事件之外，不存在非动态背景结构，并且引力和量子自由度的动力学不可避免地耦合在一起。考虑到该框架的哈密顿性质，我们使用多种量子态上的超表面变形生成器。该结构在很大程度上依赖于一组量子态在一组引力变量上的纤维化的微分几何。这项工作的一个重要数学特征是使用 Minlos 定理来表征物质场和 Hida 分布空间上的高斯测度，以定义具有不同测度的所有可能希尔伯特空间的公共超空间，以正确表征弯曲时空中的 QFT。这使我们能够在真空状态或取决于重力自由度的测量的情况下将不同希尔伯特空间内的状态联系起来，就像与阿什特卡复杂结构相关的情况一样。这是通过包含纤维的量子厄米特联系来实现的，这将具有深远的物理意义。该结构最显着的物理特征是量子态的范数守恒（即使总动力学是非单一的）、混合守恒量的清晰识别以及量子物质在几何上的动力学逆反应的描述，反之亦然，这将改变引力场在没有反反应的情况下所具有的物理特性。