Symmetry

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.

We discuss the problem of defining rest frames for mixed particles, showing that Quantum Reference Frames are necessary to incorporate mass superpositions. This approach reveals a strictly frame-dependent nature of entanglement. We investigate the phenomenological impact on neutrinos and neutral mesons, demonstrating that the transition to the rest frame generates entanglement.

Geophysical forward modeling serves as a fundamental theoretical approach for characterizing subsurface structures and material properties, essentially involving the computation of gravity responses at surface or spatial observation points based on a predefined density distribution. With the rapid development of data-driven techniques such as deep learning in geophysical inversion, forward algorithms are facing increasing demands in terms of computational scale, observable types, and efficiency. To address these challenges, this study develops an efficient forward modeling method based on voxel discretization, the enabling rapid calculation of gravity anomalies and radial gravity gradients on multiple observational surfaces. Leveraging the parallel computing capabilities of graphics processing units (GPU), together with tensor acceleration, Compute Unified Device Architecture (CUDA) execution, and Just-in-time (JIT) compilation strategies, the method achieves high efficiency and automation in the forward computation process. Numerical experiments conducted on several typical theoretical models demonstrate the convergence and stability of the calculated results, indicating that the proposed method significantly reduces computation time while maintaining accuracy, thus being well-suited for large-scale 3D modeling and fast batch simulation tasks. This research can efficiently generate forward datasets with multi-view and multi-metric characteristics, providing solid data support and a scalable computational platform for deep-learning-based geophysical inversion studies.