Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space (1712.02803v1)

Abstract: We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how Radon transforms can be used to convert this data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson's "entanglement equilibrium" that the geometry should obey Einstein's equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.

• The paper presents a framework that derives Einstein’s linearized gravitational dynamics from quantum entanglement patterns in Hilbert space.
• It employs multi-dimensional scaling and the Radon transform to reconstruct spatial geometry from mutual information networks.
• The study links quantum error correction principles and modified entanglement equilibrium to explain the emergence of spacetime dynamics.

Insights into Bulk Entanglement Gravity: Deriving Spacetime from Quantum States

The paper "Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space" by ChunJun Cao and Sean M. Carroll presents a conceptual framework for understanding spacetime geometries and gravitational dynamics as emergent properties from quantum states in Hilbert spaces. The authors explore the potential for deriving Einstein's equation in the weak-field limit solely from quantum mechanics principles, without relying on boundary conditions or holographic dualities such as AdS/CFT.

Emergence of Spatial Geometry from Quantum States

Cao and Carroll begin by investigating how spatial geometries could emerge from the entanglement patterns of quantum states. This idea revolves around defining an "information graph" whose vertices represent tensor factors of a Hilbert space, connected by edges weighted according to mutual information. The phenomenon of redundancy-constrained (RC) states is pivotal, where entanglement entropy adheres to an area-law approximation, akin to quantum many-body systems and holographic codes.

To translate this entanglement into a tangible spatial geometry, the authors employ multi-dimensional scaling (MDS) to generate an embedded metric space, leading to a manifold that represents emergent spatial geometry. However, they propose utilizing the Radon transform as an advanced tool to refine the association between the mutual information (reflective of area measures) and the spatial metric tensor, emphasizing its utility in reconstructing the geometry, especially from perturbations of a quantum state.

Gravitational Dynamics via Entanglement Equilibrium

Following the geometric emergence, the focus shifts to the dynamics of gravity originating from entanglement equilibrium in quantum states. The paper argues that the Hamiltonian constraint, a crucial element of general relativity, can be expressed in terms of perturbations derived from quantum entanglement. The modified entanglement equilibrium condition (MEEC), akin to Jacobson's proposals, becomes a central motif, linking variations in geometric and field-theory entropies across spatial cuts to gravitational dynamics.

The considerations result in a conceptual equivalence between the tensor Radon transform of spatial curvature perturbations and the entanglement entropy differences in Hilbert space. The authors suggest that the quantum state dynamically evolves over time slices, forming a Lorentzian spacetime, with the ensuing gravitational field equations following Einstein's linearized forms.

Quantum Error Correction and the Generalized RT Formula

An intriguing extension of their work includes a discussion on quantum error correction codes (QECC) and their resemblance to holography. The RT-like entropy relations derived within error correction frameworks highlight a separation between entanglement contributions: one identifying geometric terms and the other matter-field entropies. Thus, spatial geometry in these quantum systems can be seen as analogous to holographic codes, providing a complementary way to explore spacetime emergence.

The inquiry concludes with the proposition for future developments, including addressing the emergence of holography without boundary constraints, supplementing existing research avenues in quantum gravity. Evaluations such as emergent Lorentz invariance, tensor network models, and distinctions between UV/IR entanglement contributions exemplify the broader prospects for the presented framework.

Implications and Future Work

The paper identifies multiple conjectural assumptions yet offers vital insights into bridging quantum mechanics with general relativity principles beyond conventional quantization processes. The Bulk Entanglement Gravity model holds promise for a comprehensive understanding of gravity's inception from quantum mechanics, advocating deeper explorations into emergent spacetime from purely quantum states.

Future work should aim to solidify these theoretical constructs, examine the feasibility of dynamics in different background geometries, and investigate novel applications aligning with condensed matter paradigms and emergent gauge theories. The path delineated in this research amplifies the appeal of questioning gravity's quintessential role within quantum frameworks, proposing that entanglement could indeed encode the foundational blueprint of spacetime itself.