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 Composite bosons look fundamental at large Composite bosons look fundamental at large

distances, distances, e.g. hydrogen atom, helium nucleus, pionse.g. hydrogen atom, helium nucleus, pions Characteristic scale for compositeness : Planck Characteristic scale for compositeness : Planck

massmass Graviton, photon, gluons, W-,Z-bosons , Higgs Graviton, photon, gluons, W-,Z-bosons , Higgs

scalar : all compositescalar : all composite

massless bound states –massless bound states –familiar if dictated by familiar if dictated by

symmetriessymmetries

In chiral QCD :In chiral QCD : Pions are massless bound Pions are massless bound

states of states of massless quarks !massless quarks !

Gauge bosons, scalars …

from vielbein components in higher dimensions(Kaluza, Klein)

concentrate first on gravity

Geometrical degrees of Geometrical degrees of freedomfreedom

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

vielbein : fermion bilinearvielbein : fermion bilinear

ActionAction

contains 2d powers of spinors d derivatives contracted with ε - tensor

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)D.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

1) Gravity with 1) Gravity with globalglobal and not and not locallocal Lorentz Lorentz

symmetry ?symmetry ?Compatible with Compatible with

observation !observation ! 2) Action with 2) Action with locallocal Lorentz Lorentz symmetry ? symmetry ? Can be Can be constructed !constructed !

Two alternatives :

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

How to determine How to determine geometry of space-time, geometry of space-time,

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

Symmetries dictate general Symmetries dictate general form of effective action and form of effective action and gravitational field equationgravitational field equation

diffeomorphisms !diffeomorphisms !

Effective action : curvature scalar R + additional terms

Gravitational field equationGravitational field equationand energy momentum and energy momentum

tensortensor

Special case : effective action depends only on metric

Unified theory in higher Unified theory in higher dimensionsdimensions

and energy momentum and energy momentum tensortensor

Only spinors , no additional fields – no genuine Only spinors , no additional fields – no genuine sourcesource

JJμμm m : expectation values different from vielbein : expectation values different from vielbein

and and incoherentincoherent fluctuations fluctuations

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

Approximative computation Approximative computation

of field equationof field equation

for action

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 reflects violation of local Lorentz – symmetry !

Gravity with Gravity with globalglobal and not and not local local Lorentz symmetry :Lorentz symmetry :

Compatible with observation Compatible with observation !!

No observation constrains No observation constrains additional term in effective additional term in effective

action that violates local action that violates local Lorentz symmetry ( ~ Lorentz symmetry ( ~ ττ ) )

Action with Action with locallocal Lorentz Lorentz symmetry can be symmetry can be

constructed !constructed !

Time space asymmetryTime space asymmetry unified treatment of time and space –unified treatment of time and space –

but important difference between but important difference between time and space due to time and space due to signaturesignature

Origin ?Origin ?

Time space asymmetry fromTime space asymmetry fromspontaneous symmetry spontaneous symmetry

breakingbreaking Idea : difference in signature Idea : difference in signature

from from spontaneous symmetry breakingspontaneous symmetry 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 groupcomplex orthogonal group:: Both Both euclideaneuclidean orthogonal group and orthogonal group and

minkowskian minkowskian Lorentz group are subgroupsLorentz group are subgroups Realized signature depends on ground state !Realized signature depends on ground state !

C.W. , PRL , 2004

Complex orthogonal Complex orthogonal groupgroup

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

ρ ,τ : antisymmetric 128 x 128 matrices

SO(16,C)

Compact part : ρNon-compact part : τ

vielbeinvielbein

Eμm = δμ

m : SO(1,15) - symmetryhowever : Minkowski signature not singled out in action !

Formulation of action Formulation of action invariantinvariant

under SO(16,C)under SO(16,C)

Even invariant under larger Even invariant under larger symmetry groupsymmetry group

SO(128,C)SO(128,C)

Local symmetry !Local symmetry !

complex formulationcomplex formulation so far real Grassmann so far real Grassmann

algebraalgebra introduce complex structure introduce complex structure

byby

σ is antisymmetric 128 x 128 matrix , generates SO(128,C)

Invariant actionInvariant action

For τ = 0 : local Lorentz-symmetry !!

(complex orthogonal group, diffeomorphisms )

invariants with respect to SO(128,C) and therefore also with respect to subgroup SO (16,C)

contractions with δ and ε – tensors

no mixed terms φ φ*

Generalized Lorentz Generalized Lorentz symmetrysymmetry

Example d=16 : SO(128,C) instead of Example d=16 : SO(128,C) instead of SO(1,15)SO(1,15)

Important for existence of chiral spinors Important for existence of chiral spinors in in

effective four dimensional theory aftereffective four dimensional theory after dimensional reduction of higher dimensional reduction of higher

dimensionaldimensional gravitygravity

S.Weinberg

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

Ground state with Ground state with appropriate isometries:appropriate isometries:

guarantees massless guarantees massless gauge bosons and gauge bosons and

graviton in spectrumgraviton in spectrum

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

geometriesgeometries (noncompact internal space )(noncompact internal space ) C.W. , Nucl.Phys. B242,473 (1984) C.W. , Nucl.Phys. B242,473 (1984) and wharpingand wharping C.W. , Nucl.Phys. B253,366 (1985) C.W. , Nucl.Phys. B253,366 (1985) d=4 mod 4 possible for ‘extended d=4 mod 4 possible for ‘extended

Lorentz symmetry’ ( otherwise only d = Lorentz symmetry’ ( otherwise only d = 2 mod 8 )2 mod 8 )

Rather realistic model Rather realistic model knownknown

d=18 : first step : brane compactifcation d=18 : first step : brane compactifcation

d=6, SO(12) theory : ( anomaly free )d=6, SO(12) theory : ( anomaly free ) 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 symmetry: realistic SSB of generation symmetry: realistic

mass and mixing hierarchies for quarks mass and mixing hierarchies for quarks and leptons and leptons

(except large Cabibbo angle)(except large Cabibbo angle)C.W. , Nucl.Phys. B244,359( 1984) ; B260,402 (1985) ; B261,461 (1985) ; B279,711 (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 fermions Unified theory based only on fermions

seems possibleseems possible Quantum gravity – Quantum gravity – if functional measure can be regulatedif functional measure can be regulated Does realistic higher dimensional Does realistic higher dimensional

model exist ?model exist ? Local Lorentz symmetry not verified by Local Lorentz symmetry not verified by

observationobservation

Local Lorentz symmetry Local Lorentz symmetry not verified by not verified by observation !observation !

Gravity with Gravity with globalglobal and not and not local local Lorentz symmetry :Lorentz symmetry :

Compatible with observation Compatible with observation !!

No observation constrains No observation constrains additional term in effective additional term in effective

action that violates local action that violates local Lorentz symmetry ( ~ Lorentz symmetry ( ~ ττ ) )

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 SG

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 massless has additional massless

field !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 Linear approximation ( weak gravity )( weak gravity )

for β = 0 : only new massless field cμν

cμν couples only to spin ( antisymmetric part of energy momentum tensor )test would need source with macroscopic spin and test particle with macroscopic spin

Post-Newtonian gravityPost-Newtonian gravityNo change in lowest nontrivial order in Post-Newtonian-Gravity !

beyond linear gravity !

Schwarzschild and cosmological solutions : not modified !

Second possible invariant Second possible invariant ( ~( ~ββ ) )

strongly constrained by strongly constrained by observation !observation !

dilatation mode σ is affected !

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

most general bilinear term :

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?

end

GeometryGeometryOne can define new curvature free connection

Torsion