Spinor Gravity

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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 - PowerPoint PPT Presentation

Transcript of Spinor Gravity

  • Spinor GravityA.Hebecker,C.Wetterich

  • Unified Theoryof fermions and bosons Fermions fundamental Bosons composite

    Alternative to supersymmetryBosons look fundamental at large distances, e.g. hydrogen atom, helium nucleusGraviton, photon, gluons, W-,Z-bosons , Higgs scalar : all composite

  • Geometrical degrees of freedom

    (x) : spinor field ( Grassmann variable)vielbein : fermion bilinear

  • Gauge bosons, scalars

    from vielbein components in higher dimensions(Kaluza,Klein)

    concentrate first on gravity

  • actioncontains 2d powers of spinors !

  • symmetriesGeneral coordinate transformations (diffeomorphisms)Spinor (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)

  • Lorentz- transformationsGlobal Lorentz transformations: spinor vielbein transforms as vector action invariant

    Local Lorentz transformations:vielbein does not transform as vectorinhomogeneous piece, missing covariant derivative

  • Gravity with global and not local Lorentz symmetry ?

    Compatible with observation !

  • How to get gravitational field equations ?

    How to determine vielbein and metric ?

  • Functional integral formulation of gravityCalculability ( at least in principle)Quantum gravityNon-perturbative formulation

  • Vielbein and metricGenerating functional

  • If regularized functional measurecan be defined(consistent with diffeomorphisms)

    Non- perturbative definition of quantum gravity

  • Effective actionW=ln ZGravitational field equation

  • Gravitational field equationand energy momentum tensorSpecial case : effective action depends only on metric

  • Symmetries dictate general form of gravitational field equation

    diffeomorphisms !

  • Unified theory in higher dimensionsand energy momentum tensorNo additional fields no genuine sourceJm : 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)

  • Approximative computation of field equation

  • Loop- and Schwinger-Dyson- equationsTerms with two derivativescovariant derivative expectednew !has no spin connection !

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

  • Ultraviolet divergencenew piece from missing totally antisymmetricspin connection :nave momentum cutoff : K

  • Functional measure needs regularization !

  • Assume diffeomorphism symmetry preserved :

    relative coefficients become calculableB. De WittD=4 :

    =3New piece from violation of local Lorentz symmetry !

  • Gravity with global and not local Lorentz symmetry ?

    Compatible with observation !

  • Phenomenology, d=4Most general form of effective action which is consistent with diffeomorphism and global Lorentz symmetry

    Derivative expansionnewnot in one loop

  • New gravitational degree of freedomfor local Lorentz-symmetry:H is gauge degree of freedommatrix notation :standard vielbein :

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

  • Gravity with global Lorentz symmetry has additional field

  • Local Lorentz symmetry not tested!

    loop and SD- approximation : =0

    new invariant ~ is compatible with all present tests !

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

  • Post-Newtonian gravityNo change in lowest nontrivial order in Post-Newtonian-Gravity !beyond linear gravity

  • most general bilinear termdilatation mode is affected ! For 0 : linear and Post-Newtonian gravity modified !

  • Newtonian gravity

  • Schwarzschild solutionno modification for = 0 !strong experimental bound on !

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

  • Modifications only for 0 !

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

  • geometryOne can define new curvature free connectionTorsion

  • Time space asymmetryDifference in signature from spontaneous symmetry breakingWith spinors : signature depends on signature of Lorentz group

    Unified setting with complex orthogonal group:Both euclidean orthogonal group and minkowskian Lorentz group are subgroupsRealized signature depends on ground state !

  • Complex orthogonal groupd=16 , : 256 component spinor , real Grassmann algebra

  • vielbeinComplex formulation

  • Invariant actionFor = 0 : local Lorentz-symmetry !!(complex orthogonal group, diffeomorphisms )

  • Unification in d=16 or d=18 ?Start with irreducible spinorDimensional 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

  • Chiral fermion generationsChiral 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) )

  • Rather realistic model knownd=18 : first step : brane compactifcation

    d=6, SO(12) theory :second step : monopole compactification

    d=4 with three generations, including generation symmetriesSSB 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)

  • Comparison with string theory

    Unification of bosons and fermionsUnification of all interactions ( d >4 )Non-perturbative ( functional integral ) formulationManifest invariance under diffeomophisms SStrings Sp.Grav. ok ok

    ok ok

    - ok

    - ok

  • Comparison with string theory

    Finiteness/regularization Uniqueness of ground state/ predictivity

    No dimensionless parameter

    SStrings Sp.Grav. ok -

    - ?

    ok ?

  • conclusionsUnified theory based only on fermions seems possibleQuantum gravity if functional measure can be regulatedDoes realistic higher dimensional model exist?

  • end