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Spinor Gravity A.Hebecker,C.Wetterich

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

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

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

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action contains 2d powers of spinors !

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symmetries General coordinate transformations (diffeomorphisms) General coordinate transformations (diffeomorphisms) Spinor ψ(x) : transforms as scalar Spinor ψ(x) : transforms as scalar Vielbein : transforms as vector Vielbein : transforms as vector Action S : invariant 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)

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

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

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

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

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Vielbein and metric Generating functional

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If regularized functional measure can be defined (consistent with diffeomorphisms) Non- perturbative definition of quantum gravity

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Effective action W=ln Z Gravitational field equation

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Gravitational field equation and energy momentum tensor Special case : effective action depends only on metric

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Symmetries dictate general form of gravitational field equation diffeomorphisms !

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

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Approximative computation of field equation

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

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

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

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Functional measure needs regularization !

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

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

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

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New gravitational degree of freedom for local Lorentz-symmetry: H is gauge degree of freedom matrix notation : standard vielbein :

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new invariants ( only global Lorentz symmetry ) : derivative terms for H mn

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Gravity with global Lorentz symmetry has additional field

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

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linear approximation ( weak gravity ) for β = 0 : only new massless field c μν

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

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

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Newtonian gravity

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

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

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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?

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geometry One can define new curvature free connection Torsion

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

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

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vielbein Complex formulation

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

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

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

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Rather realistic model known d=18 : first step : brane compactifcation d=18 : first step : brane compactifcation d=6, SO(12) theory : d=6, SO(12) theory : second step : monopole compactification second step : monopole compactification d=4 with three generations, d=4 with three generations, including generation symmetries including generation symmetries SSB of generation sym: realistic mass and mixing hierarchies for quarks and leptons 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)

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

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

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

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