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Non[com,mut]ative geometry

                                        &
                            Supersymmetry
                            yrtemmysrepuS


             Is | Thijs van den Broek

      From | Radboud University Nijmegen / Nikhef
      With | Walter van Suijlekom & Wim Beenakker
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time
   General relativity (Einstein, 1916):




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time
   General relativity (Einstein, 1916):
                     Curved spacetime




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time
   General relativity (Einstein, 1916):
                     Curved spacetime
                     Metric




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time
   General relativity (Einstein, 1916):
                     Curved spacetime
                     Metric
         Einstein equations:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
Geometry & physics
   Classical physics: flat space, time
   General relativity (Einstein, 1916):
                     Curved spacetime
                     Metric
         Einstein equations:


         Or: Einstein - Hilbert action



Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Wednesday, May 30, 2012
?           Does this set-up allow for
                      generalisations to
                      noncommutative spaces
Wednesday, May 30, 2012
                                                   3
Noncommutative geometry | Basics
      Analogy:
      QM:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12   4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:




                          Noncommutative space (...)

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:




                          Noncommutative space (...)

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:




                             Determines metric
                          Noncommutative space (...)

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:

                                      Fermions / spinors

                             Determines metric
                          Noncommutative space (...)

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:

                                      Fermions / spinors

                             Determines metric
                          Noncommutative space (...)

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
Noncommutative geometry | Basics
      Analogy:
      QM:                                            acting on wave functions.


     Extend geometry to noncommutative geometry:

                                      Fermions / spinors

                             Determines metric
                          Noncommutative space (...)
     GRT is a special example.
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               4
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Details
       To be a bit more precise, we’re working with in the case
       of GRT:




                                                                  5
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12   6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action


                                 Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry
                                 Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:               An action
                          That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman      Cross
                                Lagrangian             rules     sec#ons  


      ‘Spectral action’:

       Analogy:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                           6
Wednesday, May 30, 2012
NCG | Action
      Wish:                An action
                           That is fixed by the noncommutative geometry

      NCG                        Ac#on  /            Feynman          Cross
                                Lagrangian             rules         sec#ons  


      ‘Spectral action’:

       Analogy:



       GRT:               gives Einstein - Hilbert action + more (               )
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                               6
Wednesday, May 30, 2012
?           How different are the internal
                      and external degrees of
                      freedom of a particle?
Wednesday, May 30, 2012
Standard Model | Set up
              The geometry:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8
Wednesday, May 30, 2012
Standard Model | Set up
              The geometry:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8
Wednesday, May 30, 2012
Standard Model | Set up
              The geometry:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12               8
Wednesday, May 30, 2012
Standard Model | Set up
                                                            “Gauge group”
              The geometry:                          (internal degrees of freedom)

                                                           Curved spacetime
                                                     (external degrees of freedom)




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   8
Wednesday, May 30, 2012
Standard Model | Set up
                                                            “Gauge group”
              The geometry:                          (internal degrees of freedom)

                                                           Curved spacetime
                                                     (external degrees of freedom)



           Choice for F automatically gives fermions (spinors)
           ‘inner structure’ (i.e. color etc).




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   8
Wednesday, May 30, 2012
Standard Model | Set up
                                                            “Gauge group”
              The geometry:                          (internal degrees of freedom)

                                                           Curved spacetime
                                                     (external degrees of freedom)



           Choice for F automatically gives fermions (spinors)
           ‘inner structure’ (i.e. color etc).

               Can naturally coincide with the SM particle content.




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   8
Wednesday, May 30, 2012
Standard Model | Set up
                                                            “Gauge group”
              The geometry:                          (internal degrees of freedom)

                                                           Curved spacetime
                                                     (external degrees of freedom)



           Choice for F automatically gives fermions (spinors)
           ‘inner structure’ (i.e. color etc).

               Can naturally coincide with the SM particle content.

           Spectral action gives:
               General Relativity + Standard Model + more (              )


Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   8
Wednesday, May 30, 2012
Standard Model | Set up
                                                            “Gauge group”
              The geometry:                          (internal degrees of freedom)

                                                           Curved spacetime
                                                     (external degrees of freedom)



           Choice for F automatically gives fermions (spinors)
           ‘inner structure’ (i.e. color etc).

               Can naturally coincide with the SM particle content.

           Spectral action gives:
               General Relativity + Standard Model + more (              )

            A geometrical understanding of the Standard Model!
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   8
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12    9
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12    9
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to

                                                        Value  of  
                                                     (Theory  ‘lives’  here)




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                             9
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to

                                                         Value  of  
                                                      (Theory  ‘lives’  here)

                                                     Values  of  coupling  
                                                     constants




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                              9
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to

                                                         Value  of  
                                                      (Theory  ‘lives’  here)

                                                     Values  of  coupling  
                                                     constants




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                              9
Wednesday, May 30, 2012
Standard Model | Predictions
         Bonus: spectral action leads to

                                                         Value  of  
                                                      (Theory  ‘lives’  here)

                                                     Values  of  coupling  
                                                     constants




                          Higgs  mass  predic#on:  158  -­‐  173  GeV

Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                              9
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                     10
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...

              ... such that action is invariant under supersymmetry
              transformations.




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                     10
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...

              ... such that action is invariant under supersymmetry
              transformations.

              MSSM: N = 1 supersymmetric version of SM:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                     10
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...

              ... such that action is invariant under supersymmetry
              transformations.

              MSSM: N = 1 supersymmetric version of SM:
                 Chiral fermions (quarks, leptons)

                 Gauge bosons (gluons, ...)




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                     10
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...

              ... such that action is invariant under supersymmetry
              transformations.

              MSSM: N = 1 supersymmetric version of SM:
                 Chiral fermions (quarks, leptons)   Sfermions (squarks, sleptons)

                 Gauge bosons (gluons, ...)          Gauginos (gluinos, ...)




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   10
Wednesday, May 30, 2012
Supersymmetry & MSSM
              N = 1 supersymmetry: each particle in the theory has a
              superpartner, differing in spin by 1/2...

              ... such that action is invariant under supersymmetry
              transformations.

              MSSM: N = 1 supersymmetric version of SM:
                 Chiral fermions (quarks, leptons)   Sfermions (squarks, sleptons)

                 Gauge bosons (gluons, ...)          Gauginos (gluinos, ...)

              Possible solution for various (theoretical) problems
              (e.g. provides dark matter candidate).
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                   10
Wednesday, May 30, 2012
NCG & Supersymmetry
  Complication:
  Different origins of
  particles




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12   11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:
                    SUSY:           Which actions are supersymmetric?




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:
                    SUSY:           Which actions are supersymmetric?
                     NCG:           For which noncommutative geometries is the
                                    spectral action supersymmetric?




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:
                    SUSY:           Which actions are supersymmetric?
                     NCG:           For which noncommutative geometries is the
                                    spectral action supersymmetric?
  Status:            ✓ SYM  
                     ✓ super-QCD
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                      SUSY   NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:
                    SUSY:           Which actions are supersymmetric?
                     NCG:           For which noncommutative geometries is the
                                    spectral action supersymmetric?
  Status:            ✓ SYM  
                     ✓ super-QCD
                          General: soon to be finished.
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                 11
Wednesday, May 30, 2012
NCG & Supersymmetry
                                                                         SUSY       NCG
  Complication:                                  Chiral fermions
  Different origins of                           Sfermions (scalar)
  particles                                      Gauge bosons
                                                 Gauginos
  Moreover: different questions:
                    SUSY:           Which actions are supersymmetric?
                     NCG:           For which noncommutative geometries is the
                                    spectral action supersymmetric?
  Status:            ✓ SYM                                         Up next:
                     ✓ super-QCD                                   ?   The Higgs mass.
                          General: soon to be finished.
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12                                        11
Wednesday, May 30, 2012
I Thank you.




Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12       12
Wednesday, May 30, 2012

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Noncommutative geometry and supersymmetry

  • 1. Non[com,mut]ative geometry & Supersymmetry yrtemmysrepuS Is | Thijs van den Broek From | Radboud University Nijmegen / Nikhef With | Walter van Suijlekom & Wim Beenakker Wednesday, May 30, 2012
  • 2. Geometry & physics Classical physics: flat space, time Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 3. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 4. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 5. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Metric Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 6. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Metric Einstein equations: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 7. Geometry & physics Classical physics: flat space, time General relativity (Einstein, 1916): Curved spacetime Metric Einstein equations: Or: Einstein - Hilbert action Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 Wednesday, May 30, 2012
  • 8. ? Does this set-up allow for generalisations to noncommutative spaces Wednesday, May 30, 2012 3
  • 9. Noncommutative geometry | Basics Analogy: QM: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 10. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 11. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 12. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 13. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 14. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Noncommutative space (...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 15. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Noncommutative space (...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 16. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Determines metric Noncommutative space (...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 17. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 18. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 19. Noncommutative geometry | Basics Analogy: QM: acting on wave functions. Extend geometry to noncommutative geometry: Fermions / spinors Determines metric Noncommutative space (...) GRT is a special example. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 4 Wednesday, May 30, 2012
  • 20. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 21. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 22. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 23. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 24. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 25. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 26. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 27. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 28. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 29. NCG | Details To be a bit more precise, we’re working with in the case of GRT: 5 Wednesday, May 30, 2012
  • 30. NCG | Action Wish: An action Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 31. NCG | Action Wish: An action Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 32. NCG | Action Wish: An action That is fixed by the noncommutative geometry Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 33. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 34. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 35. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 36. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 37. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 38. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 39. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 40. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 41. NCG | Action Wish: An action That is fixed by the noncommutative geometry NCG Ac#on  /   Feynman   Cross Lagrangian rules sec#ons   ‘Spectral action’: Analogy: GRT: gives Einstein - Hilbert action + more ( ) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 6 Wednesday, May 30, 2012
  • 42. ? How different are the internal and external degrees of freedom of a particle? Wednesday, May 30, 2012
  • 43. Standard Model | Set up The geometry: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 44. Standard Model | Set up The geometry: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 45. Standard Model | Set up The geometry: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 46. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 47. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 48. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 49. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content. Spectral action gives: General Relativity + Standard Model + more ( ) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 50. Standard Model | Set up “Gauge group” The geometry: (internal degrees of freedom) Curved spacetime (external degrees of freedom) Choice for F automatically gives fermions (spinors) ‘inner structure’ (i.e. color etc). Can naturally coincide with the SM particle content. Spectral action gives: General Relativity + Standard Model + more ( ) A geometrical understanding of the Standard Model! Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 8 Wednesday, May 30, 2012
  • 51. Standard Model | Predictions Bonus: spectral action leads to Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 52. Standard Model | Predictions Bonus: spectral action leads to Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 53. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 54. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constants Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 55. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constants Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 56. Standard Model | Predictions Bonus: spectral action leads to Value  of   (Theory  ‘lives’  here) Values  of  coupling   constants Higgs  mass  predic#on:  158  -­‐  173  GeV Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 9 Wednesday, May 30, 2012
  • 57. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 58. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 59. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 60. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Gauge bosons (gluons, ...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 61. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Sfermions (squarks, sleptons) Gauge bosons (gluons, ...) Gauginos (gluinos, ...) Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 62. Supersymmetry & MSSM N = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2... ... such that action is invariant under supersymmetry transformations. MSSM: N = 1 supersymmetric version of SM: Chiral fermions (quarks, leptons) Sfermions (squarks, sleptons) Gauge bosons (gluons, ...) Gauginos (gluinos, ...) Possible solution for various (theoretical) problems (e.g. provides dark matter candidate). Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 10 Wednesday, May 30, 2012
  • 63. NCG & Supersymmetry Complication: Different origins of particles Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 64. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 65. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 66. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 67. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 68. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   ✓ super-QCD Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 69. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   ✓ super-QCD General: soon to be finished. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 70. NCG & Supersymmetry SUSY NCG Complication: Chiral fermions Different origins of Sfermions (scalar) particles Gauge bosons Gauginos Moreover: different questions: SUSY: Which actions are supersymmetric? NCG: For which noncommutative geometries is the spectral action supersymmetric? Status: ✓ SYM   Up next: ✓ super-QCD ? The Higgs mass. General: soon to be finished. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 11 Wednesday, May 30, 2012
  • 71. I Thank you. Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12 12 Wednesday, May 30, 2012