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Density matrices in quantum gravity

by Tarek Anous, Jorrit Kruthoff, Raghu Mahajan

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Submission summary

Authors (as registered SciPost users): Tarek Anous · Jorrit Kruthoff · Raghu Mahajan
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-10-01
Date submitted: 2020-09-16 14:11
Submitted by: Anous, Tarek
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


We study density matrices in quantum gravity, focusing on topology change. We argue that the inclusion of bra-ket wormholes in the gravity path integral is not a free choice, but is dictated by the specification of a global state in the multi-universe Hilbert space. Specifically, the Hartle-Hawking (HH) state does not contain bra-ket wormholes. It has recently been pointed out that bra-ket wormholes are needed to avoid potential bags-of-gold and strong subadditivity paradoxes, suggesting a problem with the HH state. Nevertheless, in regimes with a single large connected universe, approximate bra-ket wormholes can emerge by tracing over the unobserved universes. More drastic possibilities are that the HH state is non-perturbatively gauge equivalent to a state with bra-ket wormholes, or that the third-quantized Hilbert space is one-dimensional. Along the way we draw some helpful lessons from the well-known relation between worldline gravity and Klein-Gordon theory. In particular, the commutativity of boundary-creating operators, which is necessary for constructing the alpha states and having a dual ensemble interpretation, is subtle. For instance, in the worldline gravity example, the Klein-Gordon field operators do not commute at timelike separation.

Author comments upon resubmission

We would like to thank both reviewers for their encouraging and detailed comments. We hope to have addressed the reviewer's with the changes listed below.

List of changes

1. To give an early disclaimer that expectation values in the Hartle-Hawking state do not have wormholes connecting the sandwiched |HH>'s, but that other wormholes may exist when acting on the third quantized Hilbert space, we have added the following sentence at the end of the introduction:

"In section 4.2 we discuss how boundary entropy computations in AdS are conceptually different, and emphasize that the replica wormholes in that case are simply the wormholes between the various ket boundaries in the bulk Hartle-Hawking state."

2. To clarify the points about commutativity we have added to the introduction:

"For example, the Klein-Gordon field operators do not commute at timelike separation. However, as usual, if the target space (which, in general, would be “superspace”) is analytically continued to Euclidean signature, we do expect the boundary-creating
operators to commute."

as well as footnote 4:

"Superspace has infinitely many negative-signature directions and this analytic continuation might be subtle."

3. We have added footnote 14:

"Note that this density matrix is ill-defined if the Hilbert space is infinite dimensional"

when discussing the page density matrix being proportional to the identity matrix. This is done in order to address the referee's concern that such a density matrix does not exist in most relevant cases.

Published as SciPost Phys. 9, 045 (2020)

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