Abstract
We explore the important fundamental question of how quantum information is localized in quantum gravity, in a perturbative approach. Familiar descriptions of localization of information, such as via tensor factorization of the Hilbert space or a net of commuting subalgebras of operators, conflict with basic gravitational properties -- specifically gauge invariance -- already at leading order in perturbation theory. However, previous work found that information can be classically localized in a region in a way such that measurements, including those of the gravitational field, outside the region are insensitive to that information, and only measure total Poincare charges. This paper shows that, working to leading order in the gravitational coupling, a similar quantum result holds, leading to a definition of a "gravitational splitting" on the Hilbert space for gravity. Such localization of information also argues against a role for "soft hair" in resolving the information problem for black holes. This basic mathematical structure plausibly plays a foundational role in the quantum description of gravity.
Highlights
A long-standing question playing an important role in various approaches to quantum gravity is that of how quantum information is localized
The idea that information can be equivalently represented in a dual “boundary” theory is central to the conjectured AdS/CFT equivalence, and the idea that information is accessible through the gravitational field has been suggested [1,2,3] as a possible resolution to the information problem for black holes
This question of localization of information in a gravitational theory arises because the usual notion of localization from local quantum field theory (LQFT) does not respect the gauge symmetries of gravity, which at least in the geometrical approximation correspond to diffeomorphisms
Summary
A long-standing question playing an important role in various approaches to quantum gravity is that of how quantum information is localized. These operators must have nontrivial support extending to infinity, as shown in [8], so are nonlocal, ifpoffinffiffiffieffiffiffiwffiffiffioffi rks perturbatively in the gravitational coupling κ 1⁄4 32πG the nonlocality is small in κ This raises the important question regarding whether in gravitational theory there is any precise notion of localized quantum information, or whether any information is necessarily delocalized in the collective state of a matter distribution and its gravitational field. A good approach towards understanding quantum gravity is to take a “quantum-first” viewpoint that it should respect basic axioms of quantum mechanics, the existence of a Hilbert space; a large part of the problem of formulating the theory is to find appropriate mathematical structure on that Hilbert space for describing gravity [11,12] This is guided partly by a correspondence principle for weak gravitational fields. It appears to lend weight to arguments [13] that soft hair does not offer a way to determine the information content of a region, which plausibly extend to the case of black hole interiors
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