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arXiv · 2306.09314

Crossed Products, Extended Phase Spaces and the Resolution of Entanglement Singularities

Marc S. Klinger·Robert G. Leigh
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Abstract

We identify a direct correspondence between the crossed product construction which plays a crucial role in the theory of Type III von Neumann algebras, and the extended phase space construction which restores the integrability of non-zero charges generated by gauge symmetries in the presence of spatial substructures. This correspondence provides a blue-print for resolving singularities which are encountered in the computation of entanglement entropy for subregions in quantum field theories. The extended phase space encodes quantities that would be regarded as `pure gauge' from the perspective of the full theory, but are nevertheless necessary for gluing together, in a path integral sense, physics in different subregions. These quantities are required in order to maintain gauge covariance under such gluings. The crossed product provides a consistent method for incorporating these necessary degrees of freedom into the operator algebra associated with a given subregion. In this way, the extended phase space completes the subregion algebra and subsequently allows for the assignment of a meaningful, finite entropy to states therein.

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Marc S. Klinger, Robert G. Leigh. 2023-06-15. Crossed Products, Extended Phase Spaces and the Resolution of Entanglement Singularities. https://arxiv.org/abs/2306.09314

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