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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
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