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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:  https://arxiv.org/abs/2006.17000v2 (pdf) 
Date accepted:  20201001 
Date submitted:  20200916 14:11 
Submitted by:  Anous, Tarek 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study density matrices in quantum gravity, focusing on topology change. We argue that the inclusion of braket wormholes in the gravity path integral is not a free choice, but is dictated by the specification of a global state in the multiuniverse Hilbert space. Specifically, the HartleHawking (HH) state does not contain braket wormholes. It has recently been pointed out that braket wormholes are needed to avoid potential bagsofgold and strong subadditivity paradoxes, suggesting a problem with the HH state. Nevertheless, in regimes with a single large connected universe, approximate braket wormholes can emerge by tracing over the unobserved universes. More drastic possibilities are that the HH state is nonperturbatively gauge equivalent to a state with braket wormholes, or that the thirdquantized Hilbert space is onedimensional. Along the way we draw some helpful lessons from the wellknown relation between worldline gravity and KleinGordon theory. In particular, the commutativity of boundarycreating 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 KleinGordon field operators do not commute at timelike separation.
Author comments upon resubmission
List of changes
1. To give an early disclaimer that expectation values in the HartleHawking 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 HartleHawking state."
2. To clarify the points about commutativity we have added to the introduction:
"For example, the KleinGordon 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 boundarycreating
operators to commute."
as well as footnote 4:
"Superspace has infinitely many negativesignature directions and this analytic continuation might be subtle."
3. We have added footnote 14:
"Note that this density matrix is illdefined 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)