Crossed Products, Extended Phase Spaces and the Resolution of Entanglement Singularities (2306.09314v4)
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.
- R. Haag and D. Kastler, “An Algebraic approach to quantum field theory,” J. Math. Phys. 5 (1964) 848–861.
- S. Leutheusser and H. Liu, “Emergent times in holographic duality,” arXiv:2112.12156 [hep-th].
- S. Leutheusser and H. Liu, “Subalgebra-subregion duality: emergence of space and time in holography,” arXiv:2212.13266 [hep-th].
- L. Ciambelli and R. G. Leigh, “Isolated Surfaces and Symmetries of Gravity,” Phys. Rev. D 104 (2021) no. 4, 046005, arXiv:2104.07643 [hep-th].
- L. Ciambelli, R. G. Leigh, and P.-C. Pai, “Embeddings and Integrable Charges for Extended Corner Symmetry,” Phys. Rev. Lett. 128 (2022) , arXiv:2111.13181 [hep-th].
- L. Ciambelli, “From Asymptotic Symmetries to the Corner Proposal,” 12, 2022. arXiv:2212.13644 [hep-th].
- L. Freidel, “A Canonical Bracket for Open Gravitational System,” arXiv:2111.14747 [hep-th].
- L. Ciambelli and R. G. Leigh, “Universal Corner Symmetry and the Orbit Method for Gravity,” arXiv:2207.06441 [hep-th].
- M. S. Klinger, R. G. Leigh, and P.-C. Pai, “Extended Phase Space in General Gauge Theories,” arXiv:2303.06786 [hep-th].
- H. Araki, “Type of von Neumann Algebra Associated with Free Field,” Progress of Theoretical Physics 32 (1964) no. 6, 956–965, https://academic.oup.com/ptp/article-pdf/32/6/956/5311286/32-6-956.pdf. https://doi.org/10.1143/PTP.32.956.
- R. Longo, “Algebraic and modular structure of von Neumann algebras of physics,” Commun. Math. Phys. 38 (1982) 551.
- K. Fredenhagen, “On the Modular Structure of Local Algebras of Observables,” Commun. Math. Phys. 97 (1985) 79.
- M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications. Lecture Notes in Mathematics. Springer-Verlag, 1970.
- F. Combes, “Poids associé à une algèbre hilbertienne à gauche,” Compositio Mathematica 23 (1971) no. 1, 49–77. http://eudml.org/doc/89076.
- H. J. Borchers, “On revolutionizing quantum field theory with Tomita’s modular theory,” J. Math. Phys. 41 (2000) 3604–3673.
- E. Witten, “APS medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory,” Reviews of Modern Physics 90 (2018) no. 4, . https://doi.org/10.1103%2Frevmodphys.90.045003.
- M. Nakamura and Z. Takeda, “On some elementary properties of the crossed products of von neumann algebras,” Proceedings of the Japan Academy 34 (1958) no. 8, 489–494.
- T. Turumaru, “Crossed product of operator algebra,” Tohoku Mathematical Journal, Second Series 10 (1958) no. 3, 355–365.
- M. Takesaki, “Periodic and homogeneous states on a von neumann algebra. ii,” Bulletin of The American Mathematical Society - BULL AMER MATH SOC 79 (1973) .
- M. Takesaki, “Duality for crossed products and the structure of von Neumann algebras of type III,” Acta Mathematica 131 (1973) no. none, 249 – 310. https://doi.org/10.1007/BF02392041.
- A. Connes, “Une classification des facteurs de type {iii}iii\{\rm iii\}{ roman_iii },” Annales scientifiques de l’École Normale Supérieure 6 (1973) no. 2, 133–252. http://eudml.org/doc/81916.
- U. HAAGERUP, “On the dual weights for crossed products of von neumann algebras i: Removing separability conditions,” Mathematica Scandinavica 43 (1978) no. 1, 99–118. http://www.jstor.org/stable/24491344.
- U. HAAGERUP, “On the dual weights for crossed products of von neumann algebras ii: Application of operator valued weights,” Mathematica Scandinavica 43 (1978) no. 1, 119–140. http://www.jstor.org/stable/24491345.
- W. Donnelly, “Decomposition of Entanglement Entropy in Lattice Gauge Theory,” Phys. Rev. D 85 (2012) 085004, arXiv:1109.0036 [hep-th].
- W. Donnelly and L. Freidel, “Local Subsystems in Gauge Theory and Gravity,” JHEP 09 (2016) 102, arXiv:1601.04744 [hep-th].
- J. R. Fliss, X. Wen, O. Parrikar, C.-T. Hsieh, B. Han, T. L. Hughes, and R. G. Leigh, “Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory,” JHEP 09 (2017) 056, arXiv:1705.09611 [cond-mat.str-el].
- J. R. Fliss and R. G. Leigh, “Interfaces and the extended Hilbert space of Chern-Simons theory,” JHEP 07 (2020) 009, arXiv:2004.05123 [hep-th].
- E. Noether, “Invariant Variation Problems,” Transport theory and statistical physics 1 (1971) no. 3, 186–207.
- L. Freidel, M. Geiller, and W. Wieland, “Corner Symmetry and Quantum Geometry,” arXiv:2302.12799 [hep-th].
- A. P. Balachandran, L. Chandar, and A. Momen, “Edge States in Gravity and Black Hole Physics,” Nucl. Phys. B 461 (1996) 581–596, arXiv:gr-qc/9412019.
- S. Carlip, “The Statistical Mechanics of the (2+1)-dimensional Black Hole,” Phys. Rev. D 51 (1995) 632–637, arXiv:gr-qc/9409052.
- S. Carlip, “The Statistical Mechanics of the Three-dimensional Euclidean Black Hole,” Phys. Rev. D 55 (1997) 878–882, arXiv:gr-qc/9606043.
- A. P. Balachandran, L. Chandar, and A. Momen, “Edge States in Canonical Gravity,” in 17th Annual MRST (Montreal-Rochester-Syracuse-Toronto) Meeting on High-energy Physics. 5, 1995. arXiv:gr-qc/9506006.
- T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,” Annals of Physics 88 (1974) no. 1, 286–318. https://www.sciencedirect.com/science/article/pii/0003491674904047.
- W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, “Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics,” JHEP 09 (2021) 008, arXiv:2012.10367 [hep-th].
- M. Geiller, “Edge Modes and Corner Ambiguities in 3d Chern–Simons Theory and Gravity,” Nucl. Phys. B 924 (2017) 312–365, arXiv:1703.04748 [gr-qc].
- L. Freidel, M. Geiller, and D. Pranzetti, “Edge Modes of Gravity. Part I. Corner Potentials and Charges,” JHEP 11 (2020) 026, arXiv:2006.12527 [hep-th].
- L. Freidel, M. Geiller, and D. Pranzetti, “Edge Modes of Gravity. Part II. Corner Metric and Lorentz Charges,” JHEP 11 (2020) 027, arXiv:2007.03563 [hep-th].
- L. Freidel, M. Geiller, and D. Pranzetti, “Edge Modes of Gravity. Part III. Corner Simplicity Constraints,” JHEP 01 (2021) 100, arXiv:2007.12635 [hep-th].
- V. Chandrasekaran, E. E. Flanagan, I. Shehzad, and A. J. Speranza, “Brown-York Charges at Null Boundaries,” JHEP 01 (2022) 029, arXiv:2109.11567 [hep-th].
- W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, “Matrix Quantization of Gravitational Edge Modes,” arXiv:2212.09120 [hep-th].
- L. Freidel, R. Oliveri, D. Pranzetti, and S. Speziale, “Extended Corner Symmetry, Charge Bracket and Einstein’s Equations,” JHEP 09 (2021) 083, arXiv:2104.12881 [hep-th].
- A. J. Speranza, “Local Phase Space and Edge Modes for Diffeomorphism-invariant Theories,” JHEP 02 (2018) 021, arXiv:1706.05061 [hep-th].
- H. Casini and J. M. Magan, “On completeness and generalized symmetries in quantum field theory,” Mod. Phys. Lett. A 36 (2021) no. 36, 2130025, arXiv:2110.11358 [hep-th].
- E. Witten, “Algebras, Regions, and Observers,” arXiv:2303.02837 [hep-th].
- R. M. Wald, “Black Hole Entropy is the Noether Charge,” Physical Review D 48 (1993) no. 8, R3427.
- V. Iyer and R. M. Wald, “Some Properties of the Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical review D 50 (1994) no. 2, 846.
- M. Banados, C. Teitelboim, and J. Zanelli, “The Black Hole in Three-dimensional Space-time,” Phys. Rev. Lett. 69 (1992) 1849–1851, arXiv:hep-th/9204099.
- A. Strominger, “Black Hole Entropy from Near Horizon Microstates,” JHEP 02 (1998) 009, arXiv:hep-th/9712251.
- W. Donnelly and A. C. Wall, “Entanglement Entropy of Electromagnetic Edge Modes,” Phys. Rev. Lett. 114 (2015) no. 11, 111603, arXiv:1412.1895 [hep-th].
- W. Donnelly, “Entanglement Entropy and Nonabelian Gauge Symmetry,” Class. Quant. Grav. 31 (2014) no. 21, 214003, arXiv:1406.7304 [hep-th].
- W. Donnelly and A. C. Wall, “Geometric Entropy and Edge Modes of the Electromagnetic Field,” Phys. Rev. D 94 (2016) no. 10, 104053, arXiv:1506.05792 [hep-th].
- D. Das and S. Datta, “Universal Features of Left-right Entanglement Entropy,” Phys. Rev. Lett. 115 (2015) no. 13, 131602, arXiv:1504.02475 [hep-th].
- X. Wen, S. Matsuura, and S. Ryu, “Edge Theory Approach to Topological Entanglement Entropy, Mutual Information and Entanglement Negativity in Chern-Simons Theories,” Phys. Rev. B 93 (2016) no. 24, 245140, arXiv:1603.08534 [cond-mat.mes-hall].
- S. Carlip, “Black Hole Entropy from Bondi-Metzner-Sachs Symmetry at the Horizon,” Phys. Rev. Lett. 120 (2018) no. 10, 101301, arXiv:1702.04439 [gr-qc].
- L.-Q. Chen, W. Z. Chua, S. Liu, A. J. Speranza, and B. d. S. L. Torres, “Virasoro Hair and Entropy for Axisymmetric Killing Horizons,” Phys. Rev. Lett. 125 (2020) 241302, arXiv:2006.02430 [hep-th].
- M. Geiller and P. Jai-akson, “Extended Actions, Dynamics of Edge Modes, and Entanglement Entropy,” JHEP 09 (2020) 134, arXiv:1912.06025 [hep-th].
- T. Faulkner, R. G. Leigh, O. Parrikar, and H. Wang, “Modular Hamiltonians for Deformed Half-spaces and the Averaged Null Energy Condition,” Journal of High Energy Physics 2016 (2016) no. 9, 1–35.
- T. Faulkner, R. G. Leigh, and O. Parrikar, “Shape Dependence of Entanglement Entropy in Conformal Field Theories,” Journal of High Energy Physics 2016 (2016) no. 4, 1–39.
- V. Balasubramanian, J. R. Fliss, R. G. Leigh, and O. Parrikar, “Multi-boundary Entanglement in Chern-Simons Theory and Link Invariants,” Journal of High Energy Physics 2017 (2017) no. 4, 1–34.
- V. Balasubramanian, M. DeCross, J. Fliss, A. Kar, R. G. Leigh, and O. Parrikar, “Entanglement Entropy and the Colored Jones Polynomial,” Journal of High Energy Physics 2018 (2018) no. 5, 1–41.
- V. Chandrasekaran and A. J. Speranza, “Anomalies in Gravitational Charge Algebras of Null Boundaries and Black Hole Entropy,” JHEP 01 (2021) 137, arXiv:2009.10739 [hep-th].
- S. Doplicher, D. Kastler, and D. W. Robinson, “Covariance algebras in field theory and statistical mechanics,” Communications in Mathematical Physics 3 (1966) no. 1, 1–28.
- F. Hiai, “Concise lectures on selected topics of von neumann algebras,” arXiv:2004.02383 [math.OA].
- Springer, 2003.
- E. Witten, “Gravity and the crossed product,” JHEP 10 (2022) 008, arXiv:2112.12828 [hep-th].
- T. Digernes, “Poids dual sur un produit croisé,” tech. rep., Aix-Marseille 2. Cent. Phys. Part., Marseille, 1973. http://cds.cern.ch/record/414178.
- 1979.
- J. D. Bekenstein, “Black holes and the second law,” Lett. Nuovo Cim. 4 (1972) 737–740.
- J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7 (1973) 2333–2346.
- J. D. Bekenstein, “Generalized second law of thermodynamics in black hole physics,” Phys. Rev. D 9 (1974) 3292–3300.
- V. Chandrasekaran, G. Penington, and E. Witten, “Large N algebras and generalized entropy,” arXiv:2209.10454 [hep-th].
- K. Jensen, J. Sorce, and A. Speranza, “Generalized entropy for general subregions in quantum gravity,” arXiv:2306.01837 [hep-th].
- S. Ali Ahmad and R. Jefferson, “Crossed product algebras and generalized entropy for subregions,” arXiv:2306.07323 [hep-th].
- K. C. Mackenzie and K. C. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids. No. 213. Cambridge University Press, 2005.
- M. Crainic and R. L. Fernandes, “Integrability of Lie Brackets,” Annals of Mathematics (2003) 575–620.
- W. Jia, M. S. Klinger, and R. G. Leigh, “BRST cohomology is Lie algebroid cohomology,” Nucl. Phys. B 994 (2023) 116317, arXiv:2303.05540 [hep-th].
- C. Fournel, S. Lazzarini, and T. Masson, “Formulation of Gauge Theories on Transitive Lie Algebroids,” J. Geom. Phys. 64 (2013) 174–191, arXiv:1205.6725 [math-ph].
- L. Ciambelli and R. G. Leigh, “Lie Algebroids and the Geometry of Off-shell BRST,” Nucl. Phys. B 972 (2021) 115553, arXiv:2101.03974 [hep-th].
- C. Blohmann, M. C. B. Fernandes, and A. Weinstein, “Groupoid Symmetry and Constraints in General Relativity,” Commun. Contemp. Math. 15 (2013) no. 01, 1250061, arXiv:1003.2857 [math.DG].
- S. Lazzarini and T. Masson, “Connections on Lie Algebroids and on Derivation-based Noncommutative Geometry,” Journal of Geometry and Physics 62 (2012) no. 2, 387–402. https://www.sciencedirect.com/science/article/pii/S0393044011002506.
- U. Carow-Watamura, M. A. Heller, N. Ikeda, T. Kaneko, and S. Watamura, “Off-shell Covariantization of Algebroid Gauge Theories,” PTEP 2017 (2017) no. 8, 083B01, arXiv:1612.02612 [hep-th].
- A. Kotov and T. Strobl, “Lie Algebroids, Gauge Theories, and Compatible Geometrical Structures,” Rev. Math. Phys. 31 (2018) no. 04, 1950015, arXiv:1603.04490 [math.DG].
- J. Attard, J. François, S. Lazzarini, and T. Masson, “Cartan Connections and Atiyah Lie Algebroids,” Journal of Geometry and Physics 148 (2020) 103541. https://www.sciencedirect.com/science/article/pii/S0393044019302220.
- T. Strobl, “Algebroid Yang-Mills Theories,” Phys. Rev. Lett. 93 (2004) 211601, arXiv:hep-th/0406215.
- M. Bojowald, A. Kotov, and T. Strobl, “Lie Algebroid Morphisms, Poisson Sigma Models, and Off-shell Closed Gauge Symmetries,” Journal of Geometry and Physics 54 (2005) no. 4, 400–426. https://www.sciencedirect.com/science/article/pii/S0393044004001731.
- C. Mayer and T. Strobl, “Lie Algebroid Yang Mills with Matter Fields,” J. Geom. Phys. 59 (2009) 1613–1623, arXiv:0908.3161 [hep-th].
- N. P. Landsman, “Lie groupoid c*-algebras and weyl quantization,” Communications in mathematical physics 206 (1999) no. 2, 367–381.
- N. P. Landsman and B. Ramazan, “Quantization of poisson algebras associated to lie algebroids,” arXiv:math-ph/0001005 [math-ph].
- B. Kostant, “On Certain Unitary Representations which Arise from a Quantization Theory,” Conf. Proc. C 690722 (1969) 237–253.
- B. Kostant, “Quantization and Unitary Representations,” in Lectures in Modern Analysis and Applications III, pp. 87–208. Springer, 1970.
- V. Guillemin and S. Sternberg, “Geometric Quantization and Multiplicities of Group Representations,” Inventiones mathematicae 67 (1982) no. 3, 515–538.
- V. Guillemin and S. Sternberg, “A Normal Form for the Moment Map,” Differential geometric methods in mathematical physics 6 (1984) 161–175.
- V. Guillemin and S. Sternberg, Symplectic Techniques in Physics. Cambridge university press, 1990.
- B. Kostant and S. Sternberg, “Symplectic Reduction, BRS Cohomology, and Infinite Dimensional Clifford Algebras,” Annals Phys. 176 (1987) 49.
- L. Ciambelli, L. Freidel, and R. G. Leigh, “Toward Quantum Raychaudhuri,” to appear .
- A. Connes, “On the spatial theory of von neumann algebras,” Journal of Functional Analysis 35 (1980) no. 2, 153–164. https://www.sciencedirect.com/science/article/pii/0022123680900026.
- J. J. Bisognano and E. H. Wichmann, “On the Duality Condition for Quantum Fields,” J. Math. Phys. 17 (1976) 303–321.
- S. Banerjee, M. Dorband, J. Erdmenger, and A.-L. Weigel, “Geometric Phases Characterise Operator Algebras and Missing Information,” arXiv:2306.00055 [hep-th].
- R. Kubo, “Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems,” Journal of the Physical Society of Japan 12 (1957) no. 6, 570–586, https://doi.org/10.1143/JPSJ.12.570. https://doi.org/10.1143/JPSJ.12.570.
- P. C. Martin and J. S. Schwinger, “Theory of many particle systems. 1.,” Phys. Rev. 115 (1959) 1342–1373.
- R. Haag, N. M. Hugenholtz, and M. Winnink, “On the Equilibrium states in quantum statistical mechanics,” Commun. Math. Phys. 5 (1967) 215–236.
- A. Connes, Sur le theoreme de radon nikodym pour les poids normaux fideles semi-finis. Centre de Physique Théorique, 1973.
- Springer, 1982.
- Springer Science & Business Media, 2012.
- M. Khoshkam and G. Skandalis, “Crossed products of c*-algebras by groupoids and inverse semigroups,” Journal of Operator Theory (2004) 255–279.
- C. Anantharaman-Delaroche, “Some remarks about the weak containment property for groupoids and semigroups,” arXiv:1604.01724 [math.OA].