Lattice Spinor Gravity
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Lattice Spinor Lattice Spinor GravityGravity
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Quantum gravityQuantum gravity
Quantum field theoryQuantum field theory Functional integral formulationFunctional integral formulation
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Symmetries are crucialSymmetries are crucial Diffeomorphism symmetryDiffeomorphism symmetry
( invariance under general coordinate ( invariance under general coordinate transformations )transformations )
Gravity with fermions : local Lorentz Gravity with fermions : local Lorentz symmetrysymmetry
Degrees of freedom less important :Degrees of freedom less important :
metric, vierbein , spinors , random triangles ,metric, vierbein , spinors , random triangles ,
conformal fields…conformal fields…
Graviton , metric : collective degrees of freedom Graviton , metric : collective degrees of freedom
in theory with diffeomorphism symmetryin theory with diffeomorphism symmetry
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Regularized quantum Regularized quantum gravitygravity
①① For finite number of lattice points : For finite number of lattice points : functional integral should be well definedfunctional integral should be well defined
②② Lattice action invariant under local Lattice action invariant under local Lorentz-transformationsLorentz-transformations
③③ Continuum limit exists where Continuum limit exists where gravitational interactions remain presentgravitational interactions remain present
④④ Diffeomorphism invariance of continuum Diffeomorphism invariance of continuum limit , and geometrical lattice origin for limit , and geometrical lattice origin for thisthis
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Spinor gravitySpinor gravity
is formulated in terms of is formulated in terms of fermionsfermions
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Unified TheoryUnified Theoryof fermions and bosonsof fermions and bosons
Fermions fundamentalFermions fundamental Bosons collective degrees of freedomBosons collective degrees of freedom
Alternative to supersymmetryAlternative to supersymmetry Graviton, photon, gluons, W-,Z-bosons , Higgs Graviton, photon, gluons, W-,Z-bosons , Higgs
scalar : all are collective degrees of freedom scalar : all are collective degrees of freedom ( composite )( composite )
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
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Massless collective fields Massless collective fields or bound states –or bound states –
familiar if dictated by familiar if dictated by symmetriessymmetries
for chiral QCD :for chiral QCD :
Pions are massless bound Pions are massless bound states of states of
massless quarks !massless quarks ! for strongly interacting electrons :for strongly interacting electrons :
antiferromagnetic spin wavesantiferromagnetic spin waves
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Gauge bosons, scalars …
from vielbein components in higher dimensions(Kaluza, Klein)
concentrate first on gravity
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Geometrical degrees of Geometrical degrees of freedomfreedom
ΨΨ(x) : spinor field ( Grassmann (x) : spinor field ( Grassmann variable)variable)
vielbein : fermion bilinearvielbein : fermion bilinear
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Possible ActionPossible Action
contains 2d powers of spinors d derivatives contracted with ε - tensor
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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)
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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
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1) Gravity with 1) Gravity with globalglobal and not local Lorentz and not local Lorentz
symmetry ?symmetry ?Compatible with Compatible with
observation !observation ! 2)2) Action with Action with locallocal Lorentz Lorentz symmetry ? symmetry ? Can be Can be constructed !constructed !
Two alternatives :
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Spinor gravity with Spinor gravity with local Lorentz symmetrylocal Lorentz symmetry
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Spinor degrees of Spinor degrees of freedomfreedom
Grassmann variablesGrassmann variables Spinor indexSpinor index Two flavorsTwo flavors Variables at every space-time pointVariables at every space-time point
Complex Grassmann variablesComplex Grassmann variables
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Action with local Lorentz Action with local Lorentz symmetrysymmetry
A : product of all eight spinors , maximal number , totally antisymmetric
D : antisymmetric product of four derivatives ,L is totally symmetricLorentz invariant tensor
Double index
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Symmetric four-index Symmetric four-index invariantinvariant
Symmetric invariant bilinears
Lorentz invariant tensors
Symmetric four-index invariant
Two flavors needed in four dimensions for this construction
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Weyl spinorsWeyl spinors
= diag ( 1 , 1 , -1 , -1 )
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Action in terms of Weyl - Action in terms of Weyl - spinorsspinors
Relation to previous formulation
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SO(4,C) - symmetrySO(4,C) - symmetry
Action invariant for arbitrary complex transformation parameters ε !
Real ε : SO (4) - transformations
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Signature of timeSignature of time
Difference in signature between space and time :
only from spontaneous symmetry breaking , e.g. byexpectation value of vierbein – bilinear !
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Minkowski - actionMinkowski - action
Action describes simultaneously euclidean and Minkowski theory !
SO (1,3) transformations :
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Emergence of geometryEmergence of geometry
Euclidean vierbein bilinearMinkowski -vierbein bilinear
GlobalLorentz - transformation
vierbeinmetric
/ Δ
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Can action can be Can action can be reformulated in terms of reformulated in terms of
vierbein bilinear ?vierbein bilinear ?
No suitable W exists
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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,
vierbein and metric ?vierbein and metric ?
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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
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Vierbein and metricVierbein and metric
Generating functional
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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
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Effective actionEffective action
W=ln Z
Gravitational field equation for vierbein
similar for metric
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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 for metric : curvature scalar R + additional terms
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Lattice spinor gravityLattice spinor gravity
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Lattice regularizationLattice regularization
Hypercubic latticeHypercubic lattice Even sublattice Even sublattice Odd sublatticeOdd sublattice
Spinor degrees of freedom on points Spinor degrees of freedom on points of odd sublatticeof odd sublattice
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Lattice actionLattice action
Associate cell to each point y of even Associate cell to each point y of even sublattice sublattice
Action: sum over cellsAction: sum over cells
For each cell : twelve spinors For each cell : twelve spinors located at nearest neighbors of y located at nearest neighbors of y ( on odd sublattice )( on odd sublattice )
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cellscells
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Local SO (4,C ) symmetryLocal SO (4,C ) symmetry
Basic SO(4,C) invariant building blocksBasic SO(4,C) invariant building blocks
Lattice actionLattice action
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Lattice symmetriesLattice symmetries
Rotations by Rotations by ππ/2 in all lattice planes/2 in all lattice planes
Reflections of all lattice coordinatesReflections of all lattice coordinates
Diagonal reflections e.g zDiagonal reflections e.g z11↔z↔z22
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Lattice derivativesLattice derivatives
and cell averagesand cell averages
express spinors in derivatives and express spinors in derivatives and averagesaverages
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Bilinears and lattice Bilinears and lattice derivativesderivatives
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Action in terms ofAction in terms of lattice derivatives lattice derivatives
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Continuum limitContinuum limit
Lattice distance Δ drops out in continuum limit !
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Regularized quantum Regularized quantum gravitygravity
For finite number of lattice points : For finite number of lattice points : functional integral should be well definedfunctional integral should be well defined
Lattice action invariant under local Lattice action invariant under local Lorentz-transformationsLorentz-transformations
Continuum limit exists where Continuum limit exists where gravitational interactions remain presentgravitational interactions remain present
Diffeomorphism invariance of continuum Diffeomorphism invariance of continuum limit , and geometrical lattice origin for limit , and geometrical lattice origin for thisthis
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Lattice diffeomorphism Lattice diffeomorphism invarianceinvariance
Lattice equivalent of diffeomorphism symmetry Lattice equivalent of diffeomorphism symmetry in continuumin continuum
Action does not depend on positioning of Action does not depend on positioning of lattice points in manifold , once formulated in lattice points in manifold , once formulated in terms of lattice derivatives and average fields terms of lattice derivatives and average fields in cellsin cells
Arbitrary instead of regular latticesArbitrary instead of regular lattices Continuum limit of lattice diffeomorphism Continuum limit of lattice diffeomorphism
invariant action is invariant under general invariant action is invariant under general coordinate transformationscoordinate transformations
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Lattice action and Lattice action and functional measure functional measure of spinor gravity are of spinor gravity are
lattice diffeomorphism lattice diffeomorphism invariant !invariant !
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Gauge symmetriesGauge symmetries
Proposed action for lattice gravity has also Proposed action for lattice gravity has also
chiral SU(2) x SU(2) local gauge symmetry chiral SU(2) x SU(2) local gauge symmetry
in continuum limit , in continuum limit ,
acting on flavor indices.acting on flavor indices.
Lattice action : Lattice action :
only global gauge symmetry realizedonly global gauge symmetry realized
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Next tasksNext tasks
Compute effective action for Compute effective action for composite metriccomposite metric
Verify presence of Einstein-Hilbert Verify presence of Einstein-Hilbert term ( curvature scalar )term ( curvature scalar )
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ConclusionsConclusions
Unified theory based only on fermions Unified theory based only on fermions seems possibleseems possible
Quantum gravity – Quantum gravity –
functional measure can be regulatedfunctional measure can be regulated Does realistic higher dimensional Does realistic higher dimensional
unifiedunified
model exist ?model exist ?
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end
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Gravitational field equationGravitational field equationand energy momentum and energy momentum
tensortensor
Special case : effective action depends only on metric
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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)
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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
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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
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vielbeinvielbein
Eμm = δμ
m : SO(1,15) - symmetryhowever : Minkowski signature not singled out in action !
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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 !
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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)
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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*
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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
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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
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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
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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 d=4 mod 4 possible for ‘‘extended extended
Lorentz symmetryLorentz symmetry’’ ( otherwise only d = ( otherwise only d = 2 mod 8 )2 mod 8 )
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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)
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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
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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 okok
-- ? ?
ok ?ok ?