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Semifinite von Neumann algebras in gauge theory and gravity

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arxiv 2407.01695 v2 pith:QS35RLIY submitted 2024-07-01 hep-th gr-qcmath-phmath.MPmath.OA

Shadi Ali Ahmad , Marc S. Klinger , Simon Lin This is my paper · ORCID
classification hep-thgr-qcmath-phmath.MPmath.OA
keywords crossedgaugegroupproductalgebrasconditiongravitymodular
verification ladder T0 review T1 audit T2 compute T3 formal
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von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a useful tool for constructing gauge invariant algebras. The crossed product of a Type III algebra with its modular automorphism group is semifinite, which means that the crossed product regulates divergences in local quantum field theories. In this letter, we find a sufficient condition for the semifiniteness of the crossed product of a type III algebra with any locally compact group containing the modular automorphism group. Our condition surprisingly implies the centrality of the modular flow in the symmetry group, and we provide evidence for the necessity of this condition. Under these conditions, we construct an associated trace which computes physical expectation values. We comment on the importance of this result and and its implications for subregion physics in gauge theory and gravity.

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Cited by 4 Pith papers

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  1. Quantization of Gravity on Null Hypersurfaces

    hep-th 2026-07 conditional novelty 7.0 of 10

    An operator-algebraic quantization of the characteristic initial-value problem yields a candidate on-shell algebra for a gravitational subregion bounded by two null hypersurfaces.

  2. Algebras for generalized entanglement wedges

    hep-th 2025-11 conditional novelty 7.0 of 10

    Generalized (Bousso–Penington) entanglement wedges are conjectured to carry von Neumann algebras such that S_gen(W) = S(ω|A_W) − log Ind(E) + K_Ω (eq. 2.7), making BP's monotonicity and strong subadditivity consequenc...

  3. Entanglement Entropy of Quantum Corners

    hep-th 2025-07 conditional novelty 6.0 of 10

    For a two-dimensional corner symmetry algebra, coherent corner states give an entanglement entropy that scales with the area when mapped to near-extremal Reissner-Nordström black holes.

  4. The Crossed Product, Modular (Tomita) Dynamics and its Role in the Transition of Type $III$ to Type $II_{\infty}$ v.Neumann Algebras and Connections to Quantum Gravity

    gr-qc 2025-07 reject novelty 3.0 of 10

    A review of Takesaki crossed product duality plus an unproven conjecture that modular Hamiltonian evolution of vacuum fluctuations is the microscopic mechanism of gravitational time dilation.

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