Skip to main content
SpringerLink
Log in

Algebraic Quantum Field Theory and Causal Symmetric Spaces

  • Conference paper
  • First Online:
Geometric Methods in Physics XXXIX (WGMP 2022)

Part of the book series: Trends in Mathematics ((TM))

Included in the following conference series:

Abstract

In this chapter, we review our recent work on the causal structure of symmetric spaces and related geometric aspects of algebraic quantum field theory. Motivated by some general results on modular groups related to nets of von Neumann algebras, we focus on Euler elements of the Lie algebra, i.e., elements whose adjoint action defines a 3-grading. We study the wedge regions they determine in corresponding causal symmetric spaces and describe some methods to construct nets of von Neumann algebras on causal symmetric spaces that satisfy abstract versions of the Reeh–Schlieder and the Bisognano–Wichmann condition.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adamo, M. S., Neeb, K.-H., Schober, J.: Reflection positivity and Hardy spaces on disc, half plane and the strip (2022). In preparation

    Google Scholar 

  2. Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Mathematical Phys. 4, 1343–1362 (1963). https://doi.org/10.1063/1.1703912

    Article  MathSciNet  MATH  Google Scholar 

  3. Araki, H.: Von Neumann algebras of local observables for free scalar field. J. Mathematical Phys. 5, 1–13 (1964). https://doi.org/10.1063/1.1704063

    Article  MathSciNet  MATH  Google Scholar 

  4. Araki, H., Woods, E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Mathematical Phys. 4, 637–662 (1963). https://doi.org/10.1063/1.1704002

    Article  MathSciNet  Google Scholar 

  5. Baumgärtel, H., Jurke, M., Lledó, F.: Twisted duality of the CAR-algebra. J. Math. Phys. 43(8), 4158–4179 (2002). https://doi.org/10.1063/1.1483376

    Article  MathSciNet  MATH  Google Scholar 

  6. Borchers, H.J.: The CPT-theorem in two-dimensional theories of local observables. Comm. Math. Phys. 143(2), 315–332 (1992). http://projecteuclid.org/euclid.cmp/1104248958

    Article  MathSciNet  MATH  Google Scholar 

  7. Bratteli, O., Robinson, D.: Operator Algebras and Quantum Statistical Mechanics 1: C∗- and W∗-Algebras. Symmetry Groups. Decomposition of States. Operator Algebras and Quantum Statistical Mechanics. Springer (1987). https://books.google.pl/books?id=YuR4VQOQQUIC

    Book  MATH  Google Scholar 

  8. Bros, J., Moschella, U.: Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8(3), 327–391 (1996). https://doi.org/10.1142/S0129055X96000123

    Article  MathSciNet  MATH  Google Scholar 

  9. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Comm. Math. Phys. 156(1), 201–219 (1993). http://projecteuclid.org/euclid.cmp/1104253522

    Article  MathSciNet  MATH  Google Scholar 

  10. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14(7–8), 759–785 (2002). https://doi.org/10.1142/S0129055X02001387

    Article  MathSciNet  MATH  Google Scholar 

  11. Buchholz, D., Dreyer, O., Florig, M., Summers, S.J.: Geometric modular action and spacetime symmetry groups. Rev. Math. Phys. 12(4), 475–560 (2000). https://doi.org/10.1142/S0129055X00000174

    Article  MathSciNet  MATH  Google Scholar 

  12. Connes, A., Rovelli, C.: von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories. Classical Quantum Gravity 11(12), 2899–2917 (1994). http://stacks.iop.org/0264-9381/11/2899

  13. Eckmann, J.P., Osterwalder, K.: An application of Tomita’s theory of modular Hilbert algebras: duality for free Bose fields. J. Functional Analysis 13, 1–12 (1973). https://doi.org/10.1016/0022-1236(73)90062-1

    Article  MathSciNet  MATH  Google Scholar 

  14. Frahm, J., Neeb, K.-H., Ólafsson, G.: Nets of standard subspaces on non-compactly causal symmetric spaces (2022). arXiv https://arxiv.org/abs/2303.10065

  15. Gindikin, S., Krötz, B.: Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces. Trans. Amer. Math. Soc. 354(8), 3299–3327 (2002). https://doi.org/10.1090/S0002-9947-02-03012-X

    Article  MathSciNet  MATH  Google Scholar 

  16. Gindikin, S., Krötz, B., Ólafsson, G.: Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum. Math. Ann. 327(1), 25–66 (2003). https://doi.org/10.1007/s00208-003-0409-x

    Article  MathSciNet  MATH  Google Scholar 

  17. Gindikin, S., Krötz, B., Ólafsson, G.: Holomorphic H-spherical distribution vectors in principal series representations. Invent. Math. 158(3), 643–682 (2004). https://doi.org/10.1007/s00222-004-0376-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Haag, R.: Local quantum physics, second edn. Texts and Monographs in Physics. Springer-Verlag, Berlin (1996). https://doi.org/10.1007/978-3-642-61458-3

  19. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1978)

    Google Scholar 

  20. Hilgert, J., Ólafsson, G.: Causal symmetric spaces, Perspectives in Mathematics, vol. 18. Academic Press, Inc., San Diego, CA (1997)

    MATH  Google Scholar 

  21. Kaneyuki, S.: The Sylvester’s law of inertia in simple graded Lie algebras. J. Math. Soc. Japan 50(3), 593–614 (1998). https://doi.org/10.2969/jmsj/05030593

    Article  MathSciNet  MATH  Google Scholar 

  22. Kaneyuki, S.: Graded Lie Algebras, Related Geometric Structures and Pseudo-Hermitian Symmetric Spaces, pp. 103–182. Birkhäuser Boston, Boston, MA (2000). https://doi.org/10.1007/978-1-4612-1366-6_8

  23. Karl-Hermann Neeb and Gestur Ólafsson: Wedge domains in compactly causal symmetric spaces. International Mathematics Research Notices (2022). https://doi.org/10.1093/imrn/rnac131. 104pp; arXiv:2107.13288

  24. Krötz, B., Schlichtkrull, H.: Holomorphic extension of eigenfunctions. Math. Ann. 345(4), 835–841 (2009). https://doi.org/10.1007/s00208-009-0379-8

    Article  MathSciNet  MATH  Google Scholar 

  25. Krötz, B., Stanton, R.J.: Holomorphic extensions of representations. I. Automorphic functions. Ann. of Math. (2) 159(2), 641–724 (2004). https://doi.org/10.4007/annals.2004.159.641

  26. Krötz, B., Stanton, R.J.: Holomorphic extensions of representations. II. Geometry and harmonic analysis. Geom. Funct. Anal. 15(1), 190–245 (2005). https://doi.org/10.1007/s00039-005-0504-0

    Article  MATH  Google Scholar 

  27. Krötz, B., Neeb, K.-H.: On hyperbolic cones and mixed symmetric spaces. J. Lie Theory 6(1), 69–146 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Lechner, G.: Algebraic constructive quantum field theory: Integrable models and deformation techniques. In: R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds.) Advances in Algebraic Quantum Field Theory, pp. 397–448. Springer International Publishing, Cham (2015). https://doi.org/10.1007/978-3-319-21353-8_10

    Chapter  MATH  Google Scholar 

  29. Lechner, G., Longo, R.: Localization in nets of standard spaces. Comm. Math. Phys. 336(1), 27–61 (2015). https://doi.org/10.1007/s00220-014-2199-2

    Article  MathSciNet  MATH  Google Scholar 

  30. Loos, O.: Symmetric spaces. I: General theory. W. A. Benjamin, Inc., New York-Amsterdam (1969)

    Google Scholar 

  31. Mack, G., de Riese, M.: Simple space-time symmetries: generalizing conformal field theory. J. Math. Phys. 48(5), 052304, 21 (2007). https://doi.org/10.1063/1.2713999

  32. Morinelli, V., Neeb, K.H.: A family of non-modular covariant AQFTs. Analysis and Mathematical Physics 12(5), 124 (2022). https://doi.org/10.1007/s13324-022-00727-0

    Article  MathSciNet  MATH  Google Scholar 

  33. Morinelli, V., Neeb, K.H., Ólafsson, G.: From Euler elements and 3-gradings to non-compactly causal symmetric spaces (2022). https://doi.org/10.48550/ARXIV.2207.14034. To appear in: Journal of Lie Theory 23(1), 377–432 (2023)

  34. Morinelli, V., Neeb, K.-H.: Covariant homogeneous nets of standard subspaces. Comm. Math. Phys. 386(1), 305–358 (2021). https://doi.org/10.1007/s00220-021-04046-6. ArXiv:math-ph.2010.07128

  35. Morinelli, V., Neeb, K.-H.: From local nets to Euler elements (2022). In preparation

    Google Scholar 

  36. Morinelli, V., Neeb, K.-H., Ólafsson, G.: Modular geodesics and wedge domains in general non-compactly causal symmetric spaces (2022). In preparation

    Google Scholar 

  37. Neeb, K.H.: Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces. Kyoto Journal of Mathematics 62(3), 577–613 (2022). https://doi.org/10.1215/21562261-2022-0017

    Article  MathSciNet  MATH  Google Scholar 

  38. Neeb, K.-H.: Conal orders on homogeneous spaces. Invent. Math. 104(3), 467–496 (1991). https://doi.org/10.1007/BF01245086

    Article  MathSciNet  MATH  Google Scholar 

  39. Neeb, K.-H.: Finite dimensional semigroups of unitary endomorphisms of standard subspaces. Represent. Theory 25, 300–343 (2021). https://doi.org/10.1090/ert/566

    Article  MathSciNet  MATH  Google Scholar 

  40. Neeb, Karl-Hermann, Ólafsson, Gestur: Antiunitary representations and modular theory. In: A. Fialowski, K. Kowalczewska-Grabowska, J. Grabowski, K.H. Neeb (eds.) 50th Seminar “Sophus Lie”, Banach Center Publ., vol. 113, pp. 291–362. Polish Acad. Sci. Inst. Math., Warsaw (2017)

    Google Scholar 

  41. Neeb, Karl-Hermann, Ólafsson, Gestur: KMS conditions, standard real subspaces and reflection positivity on the circle group. Pacific J. Math. 299(1), 117–169 (2019). https://doi.org/10.2140/pjm.2019.299.117

    Article  MathSciNet  MATH  Google Scholar 

  42. Neeb, Karl-Hermann, Ólafsson, Gestur: Reflection positivity on spheres. Anal. Math. Phys. 10(1), Paper No. 9, 59 (2020). https://doi.org/10.1007/s13324-019-00353-3

  43. Neeb, Karl-Hermann, Ólafsson, Gestur: Nets of standard subspaces on Lie groups. Adv. Math. 384, Paper No. 107715, 69 (2021). https://doi.org/10.1016/j.aim.2021.107715. ArXiv:2006.09832

  44. Neeb, Karl-Hermann, Ólafsson, Gestur: Wedge domains in non-compactly causal symmetric spaces (2022). https://doi.org/10.48550/ARXIV.2205.07685

  45. Neeb, Karl-Hermann, Ørsted, Bent, Ólafsson, Gestur: Standard subspaces of Hilbert spaces of holomorphic functions on tube domains. Comm. Math. Phys. 386(3), 1437–1487 (2021). https://doi.org/10.1007/s00220-021-04144-5

    Google Scholar 

  46. Neumann, A., Ólafsson, G.: Minimal and maximal semigroups related to causal symmetric spaces. Semigroup Forum 61(1), 57–85 (2000). https://doi.org/10.1007/PL00006015

    Article  MathSciNet  MATH  Google Scholar 

  47. Oeh, D.: Nets of standard subspaces induced by antiunitary representations of admissible Lie groups I. J. Lie Theory 32(1), 29–74 (2022). ArXiv:2104.02465

    Google Scholar 

  48. Ólafsson, G.: Symmetric spaces of Hermitian type. Differential Geom. Appl. 1(3), 195–233 (1991). https://doi.org/10.1016/0926-2245(91)90001-P

    Article  MathSciNet  MATH  Google Scholar 

  49. Ólafsson, G., Ørsted, B.: Causal compactification and Hardy spaces. Trans. Amer. Math. Soc. 351(9), 3771–3792 (1999). https://doi.org/10.1090/S0002-9947-99-02101-7

    Article  MathSciNet  MATH  Google Scholar 

  50. Schroer, B.: Wigner representation theory of the Poincaré group, localization, statistics and the S-matrix. Nuclear Phys. B 499(3), 519–546 (1997). https://doi.org/10.1016/S0550-3213(97)00358-1

    Article  MathSciNet  MATH  Google Scholar 

  51. Simon, B.: The P(ϕ)2 Euclidean (quantum) field theory. Princeton Series in Physics. Princeton University Press, Princeton, N.J. (1974)

    Google Scholar 

  52. Summers, S.J.: Tomita-Takesaki Modular Theory (2005). https://doi.org/10.48550/ARXIV.MATH-PH/0511034

  53. Wiesbrock, H.W.: Half-sided modular inclusions of von-Neumann-algebras. Comm. Math. Phys. 157(1), 83–92 (1993). http://projecteuclid.org/euclid.cmp/1104253848

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of K.-H. Neeb was partially supported by DFG-grant NE 413/10-1. The research of G. Ólafsson was partially supported by Simons grant 586106.

Author information

Authors and Affiliations

  1. Department Mathematik, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Erlangen, Germany

    Karl-Hermann Neeb

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA

    Gestur Ólafsson

Authors
  1. Karl-Hermann Neeb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gestur Ólafsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gestur Ólafsson .

Editor information

Editors and Affiliations

  1. Departamento de Física, CINVESTAV, Ciudad de México, Mexico

    Piotr Kielanowski

  2. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Alina Dobrogowska

  3. Departments of Mathematics, Physics, Learning & Teaching, Rutgers University, New Brunswick, NJ, USA

    Gerald A. Goldin

  4. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Tomasz Goliński

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Neeb, KH., Ólafsson, G. (2023). Algebraic Quantum Field Theory and Causal Symmetric Spaces. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_20

Download citation

Publish with us

Policies and ethics

Search

Navigation