Abstract
We point out that the MERA network for the ground state of a 1+1-dimensional conformal field theory has the same structural features as kinematic space---the geometry of CFT intervals. In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry. We argue that in these settings MERA is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself. As a test of this kinematic proposal, we compare the MERA representation of the thermofield-double state with the space of geodesics in the two-sided BTZ geometry, obtaining a detailed agreement which includes the entwinement sector. We discuss how the kinematic proposal can be extended to excited states by generalizing MERA to a broader class of compression networks.
Highlights
Gravitational physics presents us with a paradox
In holographic theories kinematic space becomes identified with the space of bulk geodesics studied in integral geometry
We argue that in these settings Multi-Scale Entanglement Renormalization Ansatz (MERA) is best viewed as a discretization of the space of bulk geodesics rather than of the bulk geometry itself
Summary
Gravitational physics presents us with a paradox. On the one hand, its most successful formulation to date — the general theory of relativity — relies on differential geometry, which emphasizes local dynamics. [5], we outlined a semantically evident answer to this question: to complement Einstein’s apparatus of differential geometry, we need an approach based on integral geometry [6] This beautiful field of mathematics is concerned with translating between local and global properties of geometric spaces. Our key insight is to recognize that this de Sitter space is the vacuum kinematic space, which carries an information metric determined by entanglement This allows us to propose a generalization of MERA to excited states [30]. We emphasize that the relation between MERA and holographic duality is primarily qualitative; its main purpose is to stimulate progress by offering a compelling analogy This modus operandi has been extremely fruitful far, but we believe that further MERA-inspired progress hinges on other aspects of Vidal’s ansatz, which are only clarified with reference to kinematic space.
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