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Lorentz signature and twisted spectral triples

  • A. Devastato
  • S. Farnsworth
  • F. Lizzi
  • P. Martinetti
Open Access
Regular Article - Theoretical Physics

Abstract

We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twists and Wick rotation.

Keywords

Non-Commutative Geometry Differential and Algebraic Geometry SpaceTime Symmetries 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    A. Connes, Noncommutative geometry, Academic Press, U.S.A., (1994) [INSPIRE].
  2. [2]
    A.H. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186 (1997) 731 [hep-th/9606001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A.H. Chamseddine, A. Connes and M. Marcolli, Gravity and the Standard Model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007) 991 [hep-th/0610241] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    W.D. van Suijlekom, Noncommutative geometry and particle physics, Springer, Dordrecht The Netherlands, (2015) [INSPIRE].
  5. [5]
    A.H. Chamseddine and A. Connes, Why the Standard Model, J. Geom. Phys. 58 (2008) 38 [arXiv:0706.3688] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    C. Brouder, N. Bizi and F. Besnard, The Standard Model as an extension of the noncommutative algebra of forms, arXiv:1504.03890 [INSPIRE].
  7. [7]
    C. Brouder, N. Bizi and F. Besnard, The Standard Model as an extension of the noncommutative algebra of forms, arXiv:1504.03890 [INSPIRE].
  8. [8]
    A. Devastato, Spectral action and gravitational effects at the Planck scale, Phys. Lett. B 730 (2014) 36 [arXiv:1309.5973] [INSPIRE].
  9. [9]
    ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
  10. [10]
    CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
  11. [11]
    D. Buttazzo et al., Investigating the near-criticality of the Higgs boson, JHEP 12 (2013) 089 [arXiv:1307.3536] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    C.-S. Chen and Y. Tang, Vacuum stability, neutrinos and dark matter, JHEP 04 (2012) 019 [arXiv:1202.5717] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J. Elias-Miro, J.R. Espinosa, G.F. Giudice, H.M. Lee and A. Strumia, Stabilization of the electroweak vacuum by a scalar threshold effect, JHEP 06 (2012) 031 [arXiv:1203.0237] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A.H. Chamseddine and A. Connes, Resilience of the spectral Standard Model, JHEP 09 (2012) 104 [arXiv:1208.1030] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    C.A. Stephan, New scalar fields in noncommutative geometry, Phys. Rev. D 79 (2009) 065013 [arXiv:0901.4676] [INSPIRE].
  16. [16]
    C.A. Stephan, Noncommutative geometry in the LHC-era, in Quantum mathematical physics, Birkhauser, (2013) [arXiv:1305.3066] [INSPIRE].
  17. [17]
    A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Inner fluctuations in noncommutative geometry without the first order condition, J. Geom. Phys. 73 (2013) 222 [arXiv:1304.7583] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Beyond the spectral Standard Model: emergence of Pati-Salam unification, JHEP 11 (2013) 132 [arXiv:1304.8050] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    S. Farnsworth and L. Boyle, Rethinking Connes’ approach to the Standard Model of particle physics via non-commutative geometry, New J. Phys. 17 (2015) 023021 [arXiv:1408.5367] [INSPIRE].
  20. [20]
    F. D’Andrea and L. Dabrowski, The Standard Model in noncommutative geometry and Morita equivalence, J. Noncommut. Geom. 10 (2016) 551 [arXiv:1501.00156] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    L. Dabrowski, F. D’Andrea and A. Sitarz, The Standard Model in noncommutative geometry: fundamental fermions as internal forms, Lett. Math. Phys. (2017) 1 [arXiv:1703.05279] [INSPIRE].
  22. [22]
    T. Brzezinski, N. Ciccoli, L. Dabrowski and A. Sitarz, Twisted reality condition for Dirac operators, Math. Phys. Anal. Geom. 19 (2016) 16 [arXiv:1601.07404] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Devastato, F. Lizzi and P. Martinetti, Grand symmetry, spectral action and the Higgs mass, JHEP 01 (2014) 042 [arXiv:1304.0415] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    A. Devastato, F. Lizzi and P. Martinetti, Higgs mass in noncommutative geometry, Fortsch. Phys. 62 (2014) 863 [arXiv:1403.7567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Devastato and P. Martinetti, Twisted spectral triple for the Standard Model and spontaneous breaking of the grand symmetry, Math. Phys. Anal. Geom. 20 (2017) 2 [arXiv:1411.1320] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  26. [26]
    A. Connes and H. Moscovici, Type III and spectral triples, in Traces in number theory, geometry and quantum fields, Aspects Math. Friedt. Vieweg E 38, Wiesbaden Germany, (2008), pg. 57.Google Scholar
  27. [27]
    G. Landi and P. Martinetti, Gauge transformations for twisted spectral triples, arXiv:1704.06121.
  28. [28]
    A. Connes and M. Marcolli, Noncommutative geometry, quantum fields and motives, AMS, U.S.A., (2008) [INSPIRE].
  29. [29]
    A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155 [hep-th/9603053] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    G. Landi and P. Martinetti, On twisting real spectral triples by algebra automorphisms, Lett. Math. Phys. 106 (2016) 1499 [arXiv:1601.00219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Farnsworth, The graded product of real spectral triples, J. Math. Phys. 58 (2017) 023507 [arXiv:1605.07035] [INSPIRE].
  32. [32]
    A.H. Chamseddine and A. Connes, Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I, Fortsch. Phys. 58 (2010) 553 [arXiv:1004.0464] [INSPIRE].
  33. [33]
    A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Grand unification in the spectral Pati-Salam model, JHEP 11 (2015) 011 [arXiv:1507.08161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    U. Aydemir, D. Minic, C. Sun and T. Takeuchi, Pati-Salam unification from noncommutative geometry and the TeV-scale W R boson, Int. J. Mod. Phys. A 31 (2016) 1550223 [arXiv:1509.01606] [INSPIRE].
  35. [35]
    K.v.d. Dungen, Krein spectral triples and the fermionic action, Math. Phys. Anal. Geom. 19 (2016) 4 [arXiv:1505.01939] [INSPIRE].
  36. [36]
    N. Bizi, C. Brouder and F. Besnard, Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics, arXiv:1611.07062 [INSPIRE].
  37. [37]
    N. Franco and M. Eckstein, An algebraic formulation of causality for noncommutative geometry, Class. Quant. Grav. 30 (2013) 135007 [arXiv:1212.5171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N. Franco and M. Eckstein, Exploring the causal structures of almost commutative geometries, SIGMA 10 (2014) 010 [arXiv:1310.8225] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  39. [39]
    N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26 (2014) 1430007 [arXiv:1210.6575] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    F. D’Andrea, M.A. Kurkov and F. Lizzi, Wick rotation and fermion doubling in noncommutative geometry, Phys. Rev. D 94 (2016) 025030 [arXiv:1605.03231] [INSPIRE].
  41. [41]
    M.A. Kurkov and F. Lizzi, Clifford structures in noncommutative geometry and the extended scalar sector, arXiv:1801.00260 [INSPIRE].
  42. [42]
    F. Lizzi, G. Mangano, G. Miele and G. Sparano, Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories, Phys. Rev. D 55 (1997) 6357 [hep-th/9610035] [INSPIRE].
  43. [43]
    K. Osterwalder and R. Schrader, Euclidean Fermi fields and a Feynman-Kac formula for boson-fermion models, Helv. Phys. Acta 46 (1973) 277 [INSPIRE].MathSciNetGoogle Scholar
  44. [44]
    P. van Nieuwenhuizen and A. Waldron, On Euclidean spinors and Wick rotations, Phys. Lett. B 389 (1996) 29 [hep-th/9608174] [INSPIRE].
  45. [45]
    J.W. Barrett, A Lorentzian version of the non-commutative geometry of the Standard Model of particle physics, J. Math. Phys. 48 (2007) 012303 [hep-th/0608221] [INSPIRE].
  46. [46]
    K. Dungen and A. Rennie, Indefinite Kasparov modules and pseudo-Riemannian manifolds, Annales Henri Poincaré 17 (2016) 3255 [arXiv:1503.06916] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.INFN sezione di NapoliNapoliItaly
  2. 2.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Potsdam-GolmGermany
  3. 3.Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico IINapoliItaly
  4. 4.Institut de Cíencies del Cosmos (ICCUB), Universitat de BarcelonaBarcelonaSpain
  5. 5.Dipartimento di MatematicaUniversità di GenovaGenovaItaly

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