Spinor Gravity A.Hebecker,C.Wetterich. Unified Theory of fermions and bosons Fermions fundamental...
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Transcript of Spinor Gravity A.Hebecker,C.Wetterich. Unified Theory of fermions and bosons Fermions fundamental...
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- transformations
Global 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 Can account for matter or radiation in effective four dimensional theory ( including effective four dimensional theory ( including gauge fields as higher dimensional vielbein-gauge 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 divergence
new 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 from Idea : difference in signature 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 if functional measure can be regulatedregulated
Does realistic higher dimensional Does realistic higher dimensional model exist ?model exist ?
Local Lorentz symmetry not verified Local Lorentz symmetry not verified by observationby observation
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=4
Most 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
new
not 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 gravity
No 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
GeometryGeometry
One can define new curvature free connection
Torsion