Quantum Renormalization Group and Holography

Quantum renormalization group scheme provides a microscopic understanding of holography through a general mapping between the beta functions of underlying quantum field theories and the holographic actions in the bulk. We show that the Einstein gravity emerges as a long wavelength holographic description for a matrix field theory which has no other operator with finite scaling dimension except for the energy-momentum tensor. We also point out that holographic actions for general large N matrix field theories respect the inversion symmetry along the radial direction in the bulk if the beta functions of single-trace operators are gradient flows with respect to the target space metric set by the beta functions of double-trace operators.Comment: 5 pages; 1 figure; v2) references adde

Emergent gravity from relatively local Hamiltonians and a possible resolution of the black hole information puzzle

In this paper, we study a possibility where gravity and time emerge from quantum matter. Within the Hilbert space of matter fields defined on a spatial manifold, we consider a sub-Hilbert space spanned by states which are parameterized by spatial metric. In those states, metric is introduced as a collective variable that controls local structures of entanglement. The underlying matter fields endow the states labeled by metric with an unambiguous inner product. Then we construct a Hamiltonian for the matter fields that is an endomorphism of the sub-Hilbert space, thereby inducing a quantum Hamiltonian of the metric. It is shown that there exists a matter Hamiltonian that induces the general relativity in the semi-classical field theory limit. Although the Hamiltonian is not local in the absolute sense, it has a weaker notion of locality, called relative locality : the range of interactions is set by the entanglement present in target states on which the Hamiltonian acts. In general, normalizable states are not invariant under the transformations generated by the Hamiltonian. As a result, a physical state spontaneously breaks the Hamiltonian constraint, and picks a moment of time. The subsequent flow of time can be understood as a Goldstone mode associated with the broken symmetry. The construction allows one to study dynamics of gravity from the perspective of matter fields. The Hawking radiation corresponds to a unitary evolution where entanglement across horizon is gradually transferred from color degrees of freedom to singlet degrees of freedom. The underlying quantum states remain pure as evaporating black holes keep entanglement with early Hawking radiations in the singlet sector which is not captured by the Bekenstein-Hawking entropy.Comment: 66 pages, 19 figures; v4) minor typos correcte