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Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

  • Arnaud BrothierEmail author
  • Alexander Stottmeister
Article
  • 35 Downloads

Abstract

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct \(1+1\)-dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson’s group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson’s group and semi-finite von Neumann algebras. For \(G={\mathbf {S}}\) the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita–Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson’s group T, for geometrically motivated choices of families of heat-kernel states.

Notes

Acknowledgements

Part of this project was done when both authors were working at the University of Rome, Tor Vergata thanks to the very generous support of Roberto Longo. We are very grateful to Roberto for giving us this great opportunity besides his constant encouragement and support during various stages of the project. AS thanks the Alexander-von-Humboldt Foundation for generous financial support during his stay at the University of Rome, Tor Vergata. Moreover, AS acknowledges financial support and kind hospitality by the Isaac Newton Institute and the Banff International Research Station where parts of this work were developed. Furthermore, we are grateful to Thomas Thiemann, Yoh Tanimoto, Luca Giorgetti and Vincenzo Morinelli for comments and discussions during various stages of this work. Finally, we are grateful to the referee for constructive comments which improved the clarity of this manuscript.

References

  1. 1.
    Arici, F., Stienstra, R., van Suijlekom, W.: Quantum lattice gauge fields and groupoid C*-algebras. Annales Henri Poincaré 19(11), 3241–3266 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arveson, W.B.: An Invitation to C*-Algebras. Graduate Texts in Mathematics, vol. 39, 1st edn. Springer, New York (1976)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomy C*-algebras. In: Baez, J.C. (ed.) Knots and Quantum Gravity, pp. 21–62. Oxford University Press, Oxford (1994)Google Scholar
  4. 4.
    Ashtekar, A., Lewandowski, J.: Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys. 17(3), 191–230 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ashtekar, A., Lewandowski, J.: Projective techniques and functional integration for gauge theories. J. Math. Phys. 36(5), 2170–2191 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J.M.C., Thiemann, T.: Quantisation of diffeomorphism invariant theories of connections with local degrees of freedom. J. Math. Phys. 36, 6456–6493 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., Thiemann, T.: SU(N) quantum Yang–Mills theory in two dimensions: a complete solution. J. Math. Phys. 38(11), 5453–5482 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Baez, J.C.: Spin networks in gauge theory. Adv. Math. 117, 253–272 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Belk, J.: Thompson’s group F. Ph.D. thesis, Cornell University (2004)Google Scholar
  10. 10.
    Blackadar, B.E.: Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)zbMATHCrossRefGoogle Scholar
  11. 11.
    Borchers, H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41(6), 3604–3673 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1: \({C}^{*}\)- and \({W}^{*}\)-Algebras, Symmetry Groups, Decomposition of States. Theoretical and Mathematical Physics, 2nd edn. Springer, Berlin (1987)zbMATHCrossRefGoogle Scholar
  13. 13.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States, Models in Quantum Statistical Mechanics. Theoretical and Mathematical Physics, 2nd edn. Springer, Berlin (1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    Brothier, A.: Haagerup property for wreath products constructed with Thompson’s groups. Preprint, arXiv:1906.03789 (2019)
  15. 15.
    Brothier, A., Jones, V.F.R.: On the Haagerup and Kazhdan property of R. Thompson’s groups. J. Group Theory 22(5), 795–807 (2019).  https://doi.org/10.1515/jgth-2018-0114 CrossRefzbMATHGoogle Scholar
  16. 16.
    Brothier, A., Jones, V.F.R.: Pythagorean representations of Thomspon’s groups. J. Funct. Anal. 277(7), 2442–2469 (2019).  https://doi.org/10.1016/j.jfa.2019.02.009 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brothier, A., Stottmeister, A.: Canonical quantization of \(1+1\)-dimensional yang-mills theory: an operator-algebraic approach. Preprint, arXiv:1907.05549 (2019)
  18. 18.
    Cannon, J., Floyd, W., Parry, W.: Introductory notes on Richard Thompson’s groups. Enseign. Math. 42, 215–256 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Choksi, J.R., Kakutani, S.: Residuality of ergodic measurable transformations and of ergodic transformations which preserve an infinite measure. Indiana Univ. Math. J. 28(3), 453–469 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Connes, A.: Noncommutative Geometry, 1st edn. Academic Press, San Diego (1994)zbMATHGoogle Scholar
  21. 21.
    Creutz, M.: Quarks, Gluons and Lattices. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1985)Google Scholar
  22. 22.
    Dimock, J.: Canonical quantization of Yang–Mills on a circle. Rev. Math. Phys. 8, 85–102 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Driver, B.K., Hall, B.C.: Yang–Mills theory and the Segal–Bargmann transform. Commun. Math. Phys. 201(2), 249–290 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Evans, D.E., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras, p. 848. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  25. 25.
    Evenbly, G., Vidal, G.: Tensor network renormalization. Phys. Rev. Lett. 115(18), 1–14 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Evenbly, G., Vidal, G.: Tensor network renormalization yields the multiscale entanglement renormalization ansatz. Phys. Rev. Lett. 115, 200401 (2016)CrossRefGoogle Scholar
  27. 27.
    Fernández, R., Fröhlich, J., Sokal, A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Texts and Monographs in Physics. Springer, Berlin (1992)zbMATHCrossRefGoogle Scholar
  28. 28.
    Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Grundling, H.: A group algebra for inductive limit groups. Continuity problems of the canonical commutation relations. Acta Appl. Math. 46, 107–145 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Grundling, H., Rudolph, G.: QCD on an infinite lattice. Commun. Math. Phys. 318, 717–766 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Grundling, H., Rudolph, G.: Dynamics for QCD on an infinite lattice. Commun. Math. Phys. 349, 1163–1202 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics, 2nd edn. Springer, Berlin (1996)zbMATHCrossRefGoogle Scholar
  33. 33.
    Huebschmann, J., Rudolph, G., Schmidt, M.: A gauge model for quantum mechanics on a stratified space. Commun. Math. Phys. 286, 459–494 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Jones, V.F.R.: Some unitary representations of Tompson’s groups F and T. J. Combin. Algebra 1(1), 1–44 (2017)CrossRefGoogle Scholar
  35. 35.
    Jones, V.F.R.: A no-go theorem for the continuum limit of a periodic quantum spin chain. Commun. Math. Phys. 357(1), 295–317 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Jones, V.F.R.: Scale invariant transfer matrices and Hamiltonians. J. Phys. A: Math. Theor. 51, 104001 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49(1), 214–224 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kogut, J.B.: The lattice gauge theory approach to quantum chromodynamics. Rev. Mod. Phys. 55(3), 775–836 (1983)ADSCrossRefGoogle Scholar
  39. 39.
    Kogut, J.B., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D: Part. Fields 11, 395–408 (1975)ADSCrossRefGoogle Scholar
  40. 40.
    Landsman, N.P.: Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15, 285–319 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Lang, T., Liegener, K., Thiemann, T.: Hamiltonian renormalisation I: derivation from Osterwalder–Schrader reconstruction. Class. Quantum Gravity 35(24), 245011 (2017)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Löffelholz, J., Morchio, G., Strocchi, F.: Mathematical structure of the temporal gauge in quantum electrodynamics. J. Math. Phys. 44, 5095–5107 (2003)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Mal’tsev, A.: Nilpotent semigroups. Uchen. Zap. Ivanovsk. Ped. Inst. 4, 107–111 (1953)MathSciNetGoogle Scholar
  44. 44.
    Sengupta, A.N.: The Yang–Mills measure and symplectic structure over spaces of connections. In: Quantization of Singular Symplectic Quotients. Progress in Mathematics, vol. 198, pp. 329–355. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  45. 45.
    Stottmeister, A., Thiemann, T.: Coherent states, quantum gravity, and the Born–Oppenheimer approximation. II. Compact Lie groups. J. Math. Phys. 57, 073501 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Stottmeister, A., Thiemann, T.: Coherent states, quantum gravity, and the Born–Oppenheimer approximation. III. Applications to loop quantum gravity. J. Math. Phys. 57, 083509 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Takesaki, M.: Theory of Operator Algebras. II. Encyclopaedia of Mathematical Sciences, vol. 125. Springer, Berlin (2003). Operator Algebras and Non-commutative Geometry, 6zbMATHCrossRefGoogle Scholar
  48. 48.
    Takesaki, M.: Theory of Operator Algebras. III. Encyclopaedia of Mathematical Sciences, vol. 127. Springer, Berlin (2003). Operator Algebras and Non-commutative Geometry, 8zbMATHCrossRefGoogle Scholar
  49. 49.
    Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2007)zbMATHCrossRefGoogle Scholar
  50. 50.
    Timmermann, T.: An Invitation to Quantum Groups and Duality. EMS Textbooks in Mathematics. European Mathematical Society, Zürich (2008)zbMATHCrossRefGoogle Scholar
  51. 51.
    Velhinho, J.M.: Functorial aspects of the space of generalized connections. Mod. Phys. Lett. A 20(17–18), 1299–1303 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Vidal, G.: A class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101(11), 110501 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    Wilson, K.G.: Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B: Condens. Matter Mater. Phys. 4(9), 3174–3183 (1971)ADSzbMATHCrossRefGoogle Scholar
  54. 54.
    Wilson, K.G.: Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior. Phys. Rev. B: Condens. Matter Mater. Phys. 4(9), 3184–3205 (1971)ADSzbMATHCrossRefGoogle Scholar
  55. 55.
    Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141(1), 153–209 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Wren, K.K.: Quantization of constrained systems with singularities using Rieffel induction. J. Geom. Phys. 24, 173–202 (1998)ADSMathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsUniversity of Rome Tor VergataRomeItaly

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