Spinor GravitySpinor Gravity

A.Hebecker,C.Wetterich

Unified TheoryUnified Theoryof fermions and bosonsof fermions and bosons

Fermions fundamentalFermions fundamental Bosons compositeBosons composite

Alternative to supersymmetryAlternative to supersymmetry Bosons look fundamental at large Bosons look fundamental at large

distances, distances, e.g. hydrogen atom, helium nucleuse.g. hydrogen atom, helium nucleus Graviton, photon, gluons, W-,Z-bosons , Graviton, photon, gluons, W-,Z-bosons ,

Higgs scalar : all compositeHiggs scalar : all composite

Geometrical degrees of Geometrical degrees of freedomfreedom

ΨΨ(x) : spinor field ( Grassmann (x) : spinor field ( Grassmann variable)variable)

vielbein : fermion bilinearvielbein : fermion bilinear

Gauge bosons, scalars …

from vielbein components in higher dimensions(Kaluza,Klein)

concentrate first on gravity

actionaction

contains 2d powers of spinors !

symmetriessymmetries General coordinate transformations General coordinate transformations

(diffeomorphisms)(diffeomorphisms) Spinor Spinor ψψ(x) : transforms (x) : transforms

as scalaras scalar Vielbein : transforms Vielbein : transforms

as vectoras vector Action S : invariantAction S : invariantK.Akama,Y.Chikashige,T.Matsuki,H.Terazawa (1978)

K.Akama (1978)A.Amati,G.Veneziano (1981)G.Denardo,E.Spallucci (1987)

Lorentz- transformationsLorentz- transformationsGlobal Lorentz transformations:Global Lorentz transformations: spinor spinor ψψ vielbein transforms as vector vielbein transforms as vector action invariantaction invariant

Local Lorentz transformations:Local Lorentz transformations: vielbein does vielbein does notnot transform as vector transform as vector inhomogeneous piece, missing covariant inhomogeneous piece, missing covariant

derivativederivative

Gravity with Gravity with globalglobal and and not not locallocal Lorentz Lorentz

symmetry ?symmetry ?

Compatible with Compatible with observation !observation !

How to get gravitational How to get gravitational field equations ?field equations ?

How to determine How to determine vielbein and metric ?vielbein and metric ?

Functional integral Functional integral formulation formulation

of gravityof gravity

CalculabilityCalculability ( at least in principle)( at least in principle) Quantum gravityQuantum gravity Non-perturbative formulationNon-perturbative formulation

Vielbein and metricVielbein and metric

Generating functional

IfIf regularized functional regularized functional measuremeasure

can be definedcan be defined(consistent with (consistent with

diffeomorphisms)diffeomorphisms)

Non- perturbative Non- perturbative definition of definition of quantum quantum

gravitygravity

Effective actionEffective action

W=ln Z

Gravitational field equation

Gravitational field equationGravitational field equationand energy momentum and energy momentum

tensortensor

Special case : effective action depends only on metric

Symmetries dictate Symmetries dictate general form of general form of

gravitational field gravitational field equationequation

diffeomorphisms !diffeomorphisms !

Unified theory in higher Unified theory in higher dimensionsdimensions

and energy momentum and energy momentum tensortensor

No additional fields – no genuine sourceNo additional fields – no genuine source JJμμ

m m : expectation values different from : expectation values different from vielbein, e.g. incoherent fluctuationsvielbein, e.g. incoherent fluctuations

Can account for matter or radiation in Can account for matter or radiation in effective four dimensional theory effective four dimensional theory ( including gauge fields as higher ( including gauge fields as higher dimensional vielbein-components)dimensional vielbein-components)

Approximative computation Approximative computation

of field equationof field equation

Loop- and Loop- and Schwinger-Dyson- Schwinger-Dyson-

equationsequations

Terms with two derivatives

covariant derivative

expected

new !

has no spin connection !

Fermion determinant in Fermion determinant in background fieldbackground field

Comparison with Einstein gravity : totally antisymmetric part of spin connection is missing !

Ultraviolet divergenceUltraviolet divergencenew piece from missing totally antisymmetricspin connection :

naïve momentum cutoff Λ :

Ω → K

Functional measure needs Functional measure needs regularization !regularization !

Assume diffeomorphism symmetry Assume diffeomorphism symmetry preserved :preserved :

relative coefficients become relative coefficients become calculablecalculable

B. De Witt

D=4 :

τ=3

New piece from violation of local Lorentz – symmetry !

Gravity with Gravity with globalglobal and and not not locallocal Lorentz Lorentz

symmetry ?symmetry ?

Compatible with Compatible with observation !observation !

Phenomenology, d=4Phenomenology, d=4Most general form of effective action Most general form of effective action

which is consistent with which is consistent with diffeomorphism and diffeomorphism and

global global Lorentz symmetryLorentz symmetry

Derivative expansionDerivative expansion

newnot in one loop

New gravitational degree of New gravitational degree of freedomfreedom

for local Lorentz-symmetry:H is gauge degree of freedom

matrix notation :

standard vielbein :

new invariants ( only new invariants ( only globalglobal Lorentz Lorentz symmetry ) :symmetry ) :

derivative terms for Hderivative terms for Hmnmn

Gravity with Gravity with global Lorentz symmetry global Lorentz symmetry

has additional fieldhas additional field

Local Lorentz symmetry not Local Lorentz symmetry not tested!tested!

loop and SD- approximation : loop and SD- approximation : ββ =0 =0

new invariant ~ new invariant ~ ττ is compatible with all is compatible with all present tests !present tests !

linear approximation ( weak linear approximation ( weak gravity )gravity )

for β = 0 : only new massless field cμν

Post-Newtonian gravityPost-Newtonian gravity

No change in lowest nontrivial order in Post-Newtonian-Gravity !

beyond linear gravity

most general bilinear most general bilinear termterm

dilatation mode σ is affected !

For β ≠ 0 : linear and Post-Newtonian gravity modified !

Newtonian gravityNewtonian gravity

Schwarzschild solutionSchwarzschild solution

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

cosmologycosmology

only the effective Planck mass differs between cosmology and Newtonian gravity if β = 0

general isotropic and homogeneous vielbein :

Otherwise : same cosmological equations !

Modifications only for Modifications only for ββ ≠ ≠ 0 !0 !

Valid theory with global Valid theory with global instead of local Lorentz instead of local Lorentz invariance for invariance for ββ = 0 ! = 0 !

General form in one loop / SDE : β = 0 Can hidden symmetry be responsible?

geometrygeometryOne can define new curvature free connection

Torsion

Time space asymmetryTime space asymmetry Difference in signature from spontaneous Difference in signature from spontaneous

symmetry breakingsymmetry breaking With spinors : signature depends on With spinors : signature depends on

signature of Lorentz groupsignature of Lorentz group

Unified setting with Unified setting with complex orthogonal complex orthogonal groupgroup::

Both Both euclideaneuclidean orthogonal group and orthogonal group and minkowskianminkowskian Lorentz group are subgroups Lorentz group are subgroups

Realized signature depends on ground Realized signature depends on ground state !state !

Complex orthogonal Complex orthogonal groupgroup

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

vielbeinvielbein

Complex formulation

Invariant actionInvariant action

For τ = 0 : local Lorentz-symmetry !!

(complex orthogonal group, diffeomorphisms )

Unification in d=16 or Unification in d=16 or d=18 ?d=18 ?

Start with irreducible spinorStart with irreducible spinor Dimensional reduction of gravity on suitable Dimensional reduction of gravity on suitable

internal spaceinternal space Gauge bosons from Kaluza-Klein-mechanismGauge bosons from Kaluza-Klein-mechanism 12 internal dimensions : SO(10) x SO(3) 12 internal dimensions : SO(10) x SO(3)

gauge symmetry – unification + generation gauge symmetry – unification + generation groupgroup

14 internal dimensions : more U(1) gener. 14 internal dimensions : more U(1) gener. sym.sym.

(d=18 : anomaly of local Lorentz symmetry )(d=18 : anomaly of local Lorentz symmetry )L.Alvarez-Gaume,E.Witten

Chiral fermion Chiral fermion generationsgenerations

Chiral fermion generations according to Chiral fermion generations according to chirality indexchirality 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 possibleWharping : d=4 mod 8 possible (C.W. , Nucl.Phys. B253,366 (1985) )(C.W. , Nucl.Phys. B253,366 (1985) )

Rather realistic model Rather realistic model knownknown

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 compactificationsecond step : monopole compactification

d=4 with three generations, d=4 with three generations, including generation symmetriesincluding generation symmetries SSB of generation sym: realistic mass and SSB of generation sym: realistic mass and

mixing hierarchies for quarks and leptonsmixing hierarchies for quarks and leptons

C.W. , Nucl.Phys. B244,359( 1984) ; 260,402 (1985) ; 261,461 (1985) ; 279,711 (1987)

Comparison with string Comparison with string theorytheory

Unification of bosons Unification of bosons and fermionsand fermions

Unification of all Unification of all interactions ( d >4 )interactions ( d >4 )

Non-perturbative Non-perturbative ( functional integral ) ( functional integral )

formulationformulation Manifest invariance Manifest invariance

under diffeomophismsunder diffeomophisms

SStrings SStrings Sp.Grav.Sp.Grav.

ok okok ok

ok okok ok

- - ok ok

- - ok ok

Comparison with string Comparison with string theorytheory

Finiteness/Finiteness/regularizationregularization

Uniqueness of ground Uniqueness of ground

state/ state/ predictivitypredictivity

No dimensionless No dimensionless parameterparameter

SStrings SStrings Sp.Grav.Sp.Grav.

ok ok - -

-- ? ?

ok ?ok ?

conclusionsconclusions Unified theory based only on Unified theory based only on

fermions seems possiblefermions seems possible Quantum gravity – Quantum gravity – if functional measure can be if functional measure can be

regulatedregulated Does realistic higher dimensional Does realistic higher dimensional

model exist?model exist?

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