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Spinor Gravity

Spinor Gravity. A.Hebecker,C.Wetterich. Unified Theory of fermions and bosons. Fermions fundamental Bosons composite Alternative to supersymmetry Bosons look fundamental at large distances, e.g. hydrogen atom, helium nucleus

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Spinor Gravity

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  1. Spinor Gravity A.Hebecker,C.Wetterich

  2. Unified Theoryof fermions and bosons Fermions fundamental Bosons composite • Alternative to supersymmetry • Bosons look fundamental at large distances, e.g. hydrogen atom, helium nucleus • Graviton, photon, gluons, W-,Z-bosons , Higgs scalar : all composite

  3. Geometrical degrees of freedom • Ψ(x) : spinor field ( Grassmann variable) • vielbein : fermion bilinear

  4. Gauge bosons, scalars … from vielbein components in higher dimensions (Kaluza,Klein) concentrate first on gravity

  5. action contains 2d powers of spinors !

  6. symmetries • General coordinate transformations (diffeomorphisms) • Spinor ψ(x) : transforms as scalar • Vielbein : transforms as vector • Action S : invariant K.Akama,Y.Chikashige,T.Matsuki,H.Terazawa (1978) K.Akama (1978) A.Amati,G.Veneziano (1981) G.Denardo,E.Spallucci (1987)

  7. Lorentz- transformations Global Lorentz transformations: • spinor ψ • vielbein transforms as vector • action invariant Local Lorentz transformations: • vielbein does not transform as vector • inhomogeneous piece, missing covariant derivative

  8. Gravity with global and not local Lorentz symmetry ?Compatible with observation !

  9. How to get gravitational field equations ?How to determine vielbein and metric ?

  10. Functional integral formulation of gravity • Calculability ( at least in principle) • Quantum gravity • Non-perturbative formulation

  11. Vielbein and metric Generating functional

  12. If regularized functional measurecan be defined(consistent with diffeomorphisms)Non- perturbative definition of quantum gravity

  13. Effective action W=ln Z Gravitational field equation

  14. Gravitational field equationand energy momentum tensor Special case : effective action depends only on metric

  15. Symmetries dictate general form of gravitational field equationdiffeomorphisms !

  16. Unified theory in higher dimensionsand energy momentum tensor • No additional fields – no genuine source • Jμm : expectation values different from vielbein, e.g. incoherent fluctuations • Can account for matter or radiation in effective four dimensional theory ( including gauge fields as higher dimensional vielbein-components)

  17. Approximative computation of field equation

  18. Loop- and Schwinger-Dyson- equations Terms with two derivatives expected new ! covariant derivative has no spin connection !

  19. Fermion determinant in background field Comparison with Einstein gravity : totally antisymmetric part of spin connection is missing !

  20. Ultraviolet divergence new piece from missing totally antisymmetric spin connection : Ω→ K naïve momentum cutoff Λ :

  21. Functional measure needs regularization !

  22. Assume diffeomorphism symmetry preserved :relative coefficients become calculable B. De Witt D=4 : τ=3 New piece from violation of local Lorentz – symmetry !

  23. Gravity with global and not local Lorentz symmetry ?Compatible with observation !

  24. Phenomenology, d=4 Most general form of effective action which is consistent with diffeomorphism and global Lorentz symmetry Derivative expansion new not in one loop

  25. New gravitational degree of freedom for localLorentz-symmetry: H is gauge degree of freedom matrix notation : standard vielbein :

  26. new invariants ( only global Lorentz symmetry ) :derivative terms for Hmn

  27. Gravity with global Lorentz symmetry has additional field

  28. Local Lorentz symmetry not tested! loop and SD- approximation : β =0 new invariant ~ τ is compatible with all present tests !

  29. linear approximation ( weak gravity ) for β = 0 : only new massless field cμν

  30. Post-Newtonian gravity No change in lowest nontrivial order in Post-Newtonian-Gravity ! beyond linear gravity

  31. most general bilinear term dilatation mode σ is affected ! For β≠ 0 : linear and Post-Newtonian gravity modified !

  32. Newtonian gravity

  33. Schwarzschild solution no modification for β = 0 ! strong experimental bound on β !

  34. cosmology general isotropic and homogeneous vielbein : only the effective Planck mass differs between cosmology and Newtonian gravity if β = 0 Otherwise : same cosmological equations !

  35. Modifications only for β ≠ 0 !Valid theory with global instead of local Lorentz invariance for β = 0 ! General form in one loop / SDE : β = 0 Can hidden symmetry be responsible?

  36. geometry One can define new curvature free connection Torsion

  37. Time space asymmetry • Difference in signature from spontaneous symmetry breaking • With spinors : signature depends on signature of Lorentz group • Unified setting with complex orthogonal group: • Both euclidean orthogonal group and minkowskian Lorentz group are subgroups • Realized signature depends on ground state !

  38. Complex orthogonal group d=16 , ψ : 256 – component spinor , real Grassmann algebra

  39. vielbein Complex formulation

  40. Invariant action (complex orthogonal group, diffeomorphisms ) For τ = 0 : local Lorentz-symmetry !!

  41. Unification in d=16 or d=18 ? • Start with irreducible spinor • Dimensional reduction of gravity on suitable internal space • Gauge bosons from Kaluza-Klein-mechanism • 12 internal dimensions : SO(10) x SO(3) gauge symmetry – unification + generation group • 14 internal dimensions : more U(1) gener. sym. (d=18 : anomaly of local Lorentz symmetry ) L.Alvarez-Gaume,E.Witten

  42. Chiral fermion generations • Chiral fermion generations according to chirality index (C.W. , Nucl.Phys. B223,109 (1983) ; E. Witten , Shelter Island conference,1983 ) • Nonvanishing index for brane (noncompact internal space ) (C.W. , Nucl.Phys. B242,473 (1984) ) • Wharping : d=4 mod 8 possible (C.W. , Nucl.Phys. B253,366 (1985) )

  43. Rather realistic model known • d=18 : first step : brane compactifcation • d=6, SO(12) theory : • second step : monopole compactification • d=4 with three generations, including generation symmetries • SSB of generation sym: realistic mass and mixing hierarchies for quarks and leptons C.W. , Nucl.Phys. B244,359( 1984) ; 260,402 (1985) ; 261,461 (1985) ; 279,711 (1987)

  44. Unification of bosons and fermions Unification of all interactions ( d >4 ) Non-perturbative ( functional integral ) formulation Manifest invariance under diffeomophisms SStrings Sp.Grav. ok ok ok ok - ok - ok Comparison with string theory

  45. Finiteness/regularization Uniqueness of ground state/ predictivity No dimensionless parameter SStrings Sp.Grav. ok - - ? ok ? Comparison with string theory

  46. conclusions • Unified theory based only on fermions seems possible • Quantum gravity – if functional measure can be regulated • Does realistic higher dimensional model exist?

  47. end

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