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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