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Crossed product algebras and generalized entropy for subregions
by Shadi Ali Ahmad, Ro Jefferson
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Submission summary
Authors (as registered SciPost users):  Shadi Ali Ahmad · Ro Jefferson 
Submission information  

Preprint Link:  scipost_202308_00005v2 (pdf) 
Date submitted:  20240210 20:06 
Submitted by:  Jefferson, Ro 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type $III_1$, in which entropy cannot be welldefined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type $II_\infty$ factor, in which traces and hence von Neumann entropy can be welldefined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag's assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type $II$ factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
Author comments upon resubmission
List of changes
See the detailed reply to referee suggestions in the attached pdf.
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Reports on this Submission
Anonymous Report 2 on 2024218 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202308_00005v2, delivered 20240218, doi: 10.21468/SciPost.Report.8577
Strengths
The discussion is very natural and the claim of a rigor UVfinite entanglement entropy is intriguing.
Weaknesses
Currently, I have not understood why the construction leads to a regularizationindependent UVfinite entanglement measure.
Report
Dear editor,
I thank the authors for the detailed response, but unfortunately, I still fail to understand equation 3.8. Defining an operator of this type is crucial for the crossedproduct construction, but I do not see it can be regularization scheme independent. I am worried that the crossedproduct construction ends up being regularizationdependent, in which case, we are not "paving the way to the study of entanglement entropy without UV divergences" as the authors claim.
The authors say the operator in 3.8 "... is just the standard vacuum subtraction scheme in any quantum field theory, regardless if there is a large coupling parameter or not."
Can they clarify what they mean by the "standard vacuum subtraction" for a halfboost operator?
I do not know how to "rigorously" define this operator without a regularization scheme. That is precisely why we do not have entanglement entropies in type III1 algebras of QFT. If we are willing to let go of rigor, in what sense, is this different from the conventional discussion of entanglement entropy in QFT.
It is crucial to realize that, in holography, in a sense, large N provides a natural regularization scheme. Here, we do not have that.
Author: Ro Jefferson on 20240304 [id 4334]
(in reply to Report 2 on 20240218)
We thank the referee again for their extremely careful reading of the manuscript. We believe there are two distinct issues being conflated here: the issue of regularization (e.g., by $1/N$) and the issue of renormalization (e.g., by vacuum subtraction).
The factor of $1/N$ arises in holography because the thermofield double state belongs to the canonical ensemble. There, thermal fluctuations scale like $N^2$, hence the need to work perturbatively with $1/N$ as a regularizer. In contrast, thermal fluctuations in Rindler space die out at infinity, so there is no need for this factor (see the penultimate paragraph on page 21 and surrounding discussion). Extensive discussion of this issue can be found in 2209.10454 (ref. [34] of this version), where the authors considered a microcanonical version of the TFD. There the fluctuations are $O(1)$, so the regularization by $1/N$ is similarly not required.
The vacuum subtraction (by which we meant the standard normal ordering prescription in our previous reply) is a separate issue, and is infinite even at finite $N$ (cf. eq. (2.2) of 2112.12828 (ref. [27] in this version), where the Hamiltonian has a factor of $N$ outside a divergent integral). In analogy with a more pedagogical example, the Hamiltonian VEV here suffers from a divergence with $N$ given that it scales as $N^2$, and the usual divergence arising from the integral of the Hamiltonian density that resembles the familiar divergence for a free scalar field. This makes the vacuum subtraction used in previous approaches purely formal, since  as the referee correctly points out  there is no known way to rigorously define this without resorting to some regularization scheme. (We emphasize again that here we refer to some hypothetical regularization scheme for the infinite vacuum divergence, not the thermal fluctuations regularized by $1/N$ above.)
In our approach, we do *not* use this subtraction scheme, for reasons discussed in the two paragraphs immediately below eq. (3.8). We also point out the possible shortcoming of this regularization in large$N$ gauge theory under eq. (4.6). In contrast, our choice has the advantage of avoiding the question of regularization/renormalization raised by the referee, though it is still necessary to work formally. In either case, the (stateindependent) shift in the vacuum energy gets absorbed into the constant $c$, cf. the penultimate paragraph on page 11.
To more accurately reflect the need to work formally, we propose to slightly soften the claim highlighted by the referee to "*formally* paving the way to the study of entanglement entropy..." We would also like to emphasize that our construction (as other works do) relies on the fact that the crossed product of a Type III$_1$ factor by its modular automorphism group generated by the modular Hamiltonian $H_\mathrm{mod}$, which is a welldefined operator in the Type III theory unlike its halfsided counterparts, is a semifinite factor. Generally, the crossed product is a way to incorporate, on the same Hilbert space, the representation of the original algebra and the unitary action of the modular group on this algebra, neither of which makes reference to the halfsided Hamiltonians. Our central claim is true regardless of the concrete way of physically implementing this crossed product as the proposal is to replace the infinite Type III factors with their associated semifinite Type II factors obtainable via the crossed product with the modular group.
Ro Jefferson on 20240210 [id 4305]
Our detailed pointbypoint response to the comments made by the referees can be found in the attached pdf.
Attachment:
Referee_response.pdf