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

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.

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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)

    ADS  MathSciNet  MATH  Google Scholar 

  2. 2.

    Arveson, W.B.: An Invitation to C*-Algebras. Graduate Texts in Mathematics, vol. 39, 1st edn. Springer, New York (1976)

    MATH  Google 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)

    ADS  MathSciNet  MATH  Google Scholar 

  5. 5.

    Ashtekar, A., Lewandowski, J.: Projective techniques and functional integration for gauge theories. J. Math. Phys. 36(5), 2170–2191 (1995)

    ADS  MathSciNet  MATH  Google 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)

    ADS  MathSciNet  MATH  Google 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)

    ADS  MathSciNet  MATH  Google Scholar 

  8. 8.

    Baez, J.C.: Spin networks in gauge theory. Adv. Math. 117, 253–272 (1996)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Belk, J.: Thompson’s group F. Ph.D. thesis, Cornell University (2004)

  10. 10.

    Blackadar, B.E.: Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)

    MATH  Google Scholar 

  11. 11.

    Borchers, H.-J.: On revolutionizing quantum field theory with Tomita’s modular theory. J. Math. Phys. 41(6), 3604–3673 (2000)

    ADS  MathSciNet  MATH  Google 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)

    MATH  Google 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)

    MATH  Google 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

    Article  MATH  Google 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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Connes, A.: Noncommutative Geometry, 1st edn. Academic Press, San Diego (1994)

    MATH  Google 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)

    MathSciNet  MATH  Google 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)

    ADS  MathSciNet  MATH  Google Scholar 

  24. 24.

    Evans, D.E., Kawahigashi, Y.: Quantum Symmetries on Operator Algebras, p. 848. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  25. 25.

    Evenbly, G., Vidal, G.: Tensor network renormalization. Phys. Rev. Lett. 115(18), 1–14 (2015)

    MathSciNet  Google Scholar 

  26. 26.

    Evenbly, G., Vidal, G.: Tensor network renormalization yields the multiscale entanglement renormalization ansatz. Phys. Rev. Lett. 115, 200401 (2016)

    Google 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)

    MATH  Google Scholar 

  28. 28.

    Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)

    ADS  MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Grundling, H., Rudolph, G.: QCD on an infinite lattice. Commun. Math. Phys. 318, 717–766 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  31. 31.

    Grundling, H., Rudolph, G.: Dynamics for QCD on an infinite lattice. Commun. Math. Phys. 349, 1163–1202 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  32. 32.

    Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Theoretical and Mathematical Physics, 2nd edn. Springer, Berlin (1996)

    MATH  Google 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)

    ADS  MathSciNet  MATH  Google 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)

    Google 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)

    ADS  MathSciNet  MATH  Google Scholar 

  36. 36.

    Jones, V.F.R.: Scale invariant transfer matrices and Hamiltonians. J. Phys. A: Math. Theor. 51, 104001 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  37. 37.

    Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49(1), 214–224 (1948)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Kogut, J.B.: The lattice gauge theory approach to quantum chromodynamics. Rev. Mod. Phys. 55(3), 775–836 (1983)

    ADS  Google 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)

    ADS  Google Scholar 

  40. 40.

    Landsman, N.P.: Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15, 285–319 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  41. 41.

    Lang, T., Liegener, K., Thiemann, T.: Hamiltonian renormalisation I: derivation from Osterwalder–Schrader reconstruction. Class. Quantum Gravity 35(24), 245011 (2017)

    ADS  MathSciNet  MATH  Google 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)

    ADS  MathSciNet  MATH  Google Scholar 

  43. 43.

    Mal’tsev, A.: Nilpotent semigroups. Uchen. Zap. Ivanovsk. Ped. Inst. 4, 107–111 (1953)

    MathSciNet  Google 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)

  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)

    ADS  MathSciNet  MATH  Google 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)

    ADS  MathSciNet  MATH  Google 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, 6

    MATH  Google 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, 8

    MATH  Google Scholar 

  49. 49.

    Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  50. 50.

    Timmermann, T.: An Invitation to Quantum Groups and Duality. EMS Textbooks in Mathematics. European Mathematical Society, Zürich (2008)

    MATH  Google Scholar 

  51. 51.

    Velhinho, J.M.: Functorial aspects of the space of generalized connections. Mod. Phys. Lett. A 20(17–18), 1299–1303 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  52. 52.

    Vidal, G.: A class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett. 101(11), 110501 (2008)

    ADS  Google 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)

    ADS  MATH  Google 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)

    ADS  MATH  Google Scholar 

  55. 55.

    Witten, E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141(1), 153–209 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  56. 56.

    Wren, K.K.: Quantization of constrained systems with singularities using Rieffel induction. J. Geom. Phys. 24, 173–202 (1998)

    ADS  MathSciNet  MATH  Google Scholar 

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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.

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Correspondence to Arnaud Brothier.

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Both authors were partially supported by European Research Council Advanced Grant 669240 QUEST. AS was supported by Alexander-von-Humboldt Foundation through a Feodor Lynen Research Fellowship. AB is supported by a University of New South Wales Sydney starting Grant.

Communicated by Y. Kawahigashi

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Brothier, A., Stottmeister, A. Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups. Commun. Math. Phys. 376, 841–891 (2020). https://doi.org/10.1007/s00220-019-03603-4

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