2 Unified Theory of fermions and bosons Fermions fundamentalBosons compositeAlternative to supersymmetryBosons look fundamental at large distances,e.g. hydrogen atom, helium nucleusGraviton, 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 componentsin higher dimensions(Kaluza,Klein)concentrate first on gravity
6 symmetries General coordinate transformations (diffeomorphisms) Spinor ψ(x) : transforms as scalarVielbein : transforms as vectorAction S : invariantK.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 ψvielbein transforms as vectoraction invariantLocal Lorentz transformations:vielbein does not transform as vectorinhomogeneous piece, missing covariant derivative
8 Gravity with global and not local Lorentz symmetry Gravity with global and not local Lorentz symmetry ? Compatible with observation !
9 How to get gravitational field equations How to get gravitational field equations ? How to determine vielbein and metric ?
10 Functional integral formulation of gravity Calculability( at least in principle)Quantum gravityNon-perturbative formulation
12 If regularized functional measure can be defined (consistent with diffeomorphisms) Non- perturbative definition of quantum gravity
13 Effective actionW=ln ZGravitational field equation
14 Gravitational field equation and energy momentum tensor Special case : effective action depends only on metric
15 Symmetries dictate general form of gravitational field equation diffeomorphisms !
16 Unified theory in higher dimensions and energy momentum tensor No additional fields – no genuine sourceJμm : expectation values different from vielbein, e.g. incoherent fluctuationsCan account for matter or radiation in effective four dimensional theory ( including gauge fields as higher dimensional vielbein-components)
33 Schwarzschild solution no modification for β = 0 !strong experimental bound on β !
34 cosmology Otherwise : same cosmological equations ! general isotropic andhomogeneous vielbein :only the effective Planck mass differsbetween cosmology and Newtonian gravityif β = 0Otherwise : same cosmological equations !
35 Modifications only for β ≠ 0 Modifications only for β ≠ 0 ! Valid theory with global instead of local Lorentz invariance for β = 0 !General form in one loop / SDE : β = 0Can hidden symmetry be responsible?
36 geometryOne can define new curvature free connectionTorsion
37 Time space asymmetryDifference in signature from spontaneous symmetry breakingWith spinors : signature depends on signature of Lorentz groupUnified setting with complex orthogonal group:Both euclidean orthogonal group and minkowskian Lorentz group are subgroupsRealized signature depends on ground state !
40 Invariant action For τ = 0 : local Lorentz-symmetry !! (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 spaceGauge bosons from Kaluza-Klein-mechanism12 internal dimensions : SO(10) x SO(3) gauge symmetry – unification + generation group14 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 tochirality 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 compactifcationd=6, SO(12) theory :second step : monopole compactificationd=4 with three generations,including generation symmetriesSSB of generation sym: realistic mass and mixing hierarchies for quarks and leptonsC.W. , Nucl.Phys. B244,359( 1984) ; 260,402 (1985) ; 261,461 (1985) ; 279,711 (1987)
44 Comparison with string theory Unification of bosons and fermionsUnification of all interactions ( d >4 )Non-perturbative( functional integral ) formulationManifest invariance under diffeomophismsSStrings Sp.Grav.ok okokok
45 Comparison with string theory SStrings Sp.Grav.ok?ok ?Finiteness/regularizationUniqueness of ground state/predictivityNo dimensionless parameter
46 conclusions Unified theory based only on fermions seems possible Quantum gravity –if functional measure can be regulatedDoes realistic higher dimensional model exist?