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Twisted Standard Model and its Krein structure -- in memoriam Manuele Filaci

Pierre Martinetti

Abstract

We review the contributions of Manuele Filaci - a PhD student from the university of Genova prematurely deceased a little more than a year ago - to the description of the Standard Model in noncommutative geometry. Building on Manuele's discovery that there exist various ways to minimally twist the spectral triple of the Standard Model, we study in a systematic way the inner product induced by the twist. Under loose assumptions, this product turns the Hilbert space of the spectral triple into a Krein space. For the Standard Model, the group of unitary with respect to the twisted product contains the symmetry group of twistors as a subgroup.

Twisted Standard Model and its Krein structure -- in memoriam Manuele Filaci

Abstract

We review the contributions of Manuele Filaci - a PhD student from the university of Genova prematurely deceased a little more than a year ago - to the description of the Standard Model in noncommutative geometry. Building on Manuele's discovery that there exist various ways to minimally twist the spectral triple of the Standard Model, we study in a systematic way the inner product induced by the twist. Under loose assumptions, this product turns the Hilbert space of the spectral triple into a Krein space. For the Standard Model, the group of unitary with respect to the twisted product contains the symmetry group of twistors as a subgroup.
Paper Structure (13 sections, 5 theorems, 105 equations)

This paper contains 13 sections, 5 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Minimal twist of the Standard Model
  3. Spectral description of the Standard Model
  4. Twist by grading
  5. Minimally twisted Standard Model
  6. Torsion as a twisted fluctuation
  7. Signature change
  8. Twisted inner product
  9. Expandable twist
  10. Krein structure
  11. Minimal twisting by an operator
  12. Twisted unitaries
  13. Conclusion

Key Result

Proposition III.3

Any expandable twisting automorphism is spatial, and thus induces a non-degenerate twisted product. The latter is Hermitian iff $\rho$ is spatial for some selfadjoint $R$ in ${\mathcal{B}}({\mathcal{H}})$.

Theorems & Definitions (20)

  • Example II.1
  • Remark II.2
  • Remark II.3
  • Remark II.4
  • Remark III.1
  • Definition III.2
  • Proposition III.3
  • proof
  • Corollary III.4
  • proof
  • ...and 10 more