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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_00005v3  (pdf)
Date accepted: 2024-03-22
Date submitted: 2024-03-13 18:03
Submitted by: Jefferson, Ro
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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 well-defined 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 well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally 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.

List of changes

Slight modification of claim in the abstract as discussed in previous exchange.

Published as SciPost Phys. Core 7, 020 (2024)

Reports on this Submission

Anonymous Report 1 on 2024-3-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202308_00005v3, delivered 2024-03-15, doi: 10.21468/SciPost.Report.8715


After carefully reviewing the manuscript, I concur with the assessment provided by Referee 2.
Indeed, although I think what the authors are trying to accomplish is valuable, and their line of thinking is on the right track, the idea that the cross product of a type III1 algebra with its modular flow gives a type $II_\infty$ algebra was already known in the literature, and many have speculated that this might pave the way to a rigorous notion of entropy for local algebra of quantum field theories.
Regrettably, I find that the authors have not made significant progress in advancing this idea.
Furthermore, the response provided by the authors does not adequately address the concerns raised by Referee 2.
Overall, the manuscript raises numerous unresolved questions, and while it may have merit for publication, it falls short of the standards expected for Scipost Physics.
Consequently, I recommend the publication in Scipost Physics Core instead.

  • validity: low
  • significance: ok
  • originality: low
  • clarity: good
  • formatting: perfect
  • grammar: perfect

Author:  Ro Jefferson  on 2024-03-18  [id 4374]

(in reply to Report 1 on 2024-03-15)

We thank the referee for their very prompt response. While it is true that the following statement "the crossed product of a Type III$_1$ factor with its modular automorphism group gives a semifinite, Type II$_\infty$ factor" was known in the mathematical literature and indeed was proven by Takesaki in 1973, to the best of our knowledge, we have not come across any article which relates this result to the local algebras of a generic quantum field theory, nor the regularization of their von Neumann entropies until our work which appeared in June 2023. The first paper we know which uses this result concretely is Witten's "Gravity and the Crossed Product" (which indeed was the impetus for our investigation), but this was claimed to only apply to theories with gravity and moreover did not address the issue of subregions. Indeed, generalizations, again only in the context of gravitational theories, have been made to attempt to eliminate the issue of subregions, cf. 2306.01837. In the context of the current activity in the field, the emphasis that this result is independent of gravity is important, as was recognized by referee 1, as this is clearly not appreciated by most workers in high-energy theory. In this sense, our paper does contribute something novel and important to the discussion of entanglement entropy in QFT.

We proposed the crossed product as a way to enhance Haag's net of observable algebras, basically by replacing each local Type III$_1$ with its better behaved Type II$_\infty$ factor. This accomplishes two things that are significant in the study of entanglement entropy: (1) the crossed product can serve as a regulator for generic quantum field theories and not just those with gravity, and (2) implementing this procedure for subregions is immediate in our approach. Since our result appeared, two papers have made similar claims. The first is 2306.09314, which relates the crossed product construction for a generic QFT to imposing ​"​gauge" constraints in the presence of spatial subregions, and again claims divergence resolution in entanglement entropy for general QFTs independently of gravity. The second is 2312.07646, which proposes the crossed product as a regulator for generic QFTs, once again independently of gravity, but they analyze entropy differences instead of entropies and choose an explicit embedding into the extended Hilbert space of the crossed product. Both works were posted *after* ours.

Frankly, the report we received does not go into any specifics about our work and makes general vague claims about ​"speculations" in the community that are unsubstantiated. The referee has not provided any sources for the claims that they are making regarding the novelty of our work, nor specified which "unresolved questions" they would like answered, so we are unable to reply in greater detail.

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