Lattice Spinor Lattice Spinor GravityGravity

Quantum gravityQuantum gravity

Quantum field theoryQuantum field theory Functional integral formulationFunctional integral formulation

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

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

Spinor gravitySpinor gravity

is formulated in terms of is formulated in terms of fermionsfermions

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

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

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

Possible ActionPossible Action

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 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 :

Spinor gravity with Spinor gravity with local Lorentz symmetrylocal Lorentz symmetry

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

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

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

Weyl spinorsWeyl spinors

= diag ( 1 , 1 , -1 , -1 )

Action in terms of Weyl - Action in terms of Weyl - spinorsspinors

Relation to previous formulation

SO(4,C) - symmetrySO(4,C) - symmetry

Action invariant for arbitrary complex transformation parameters ε !

Real ε : SO (4) - transformations

Signature of timeSignature of time

Difference in signature between space and time :

only from spontaneous symmetry breaking , e.g. byexpectation value of vierbein – bilinear !

Minkowski - actionMinkowski - action

Action describes simultaneously euclidean and Minkowski theory !

SO (1,3) transformations :

Emergence of geometryEmergence of geometry

Euclidean vierbein bilinearMinkowski -vierbein bilinear

GlobalLorentz - transformation

vierbeinmetric

/ Δ

Can action can be Can action can be reformulated in terms of reformulated in terms of

vierbein bilinear ?vierbein bilinear ?

No suitable W exists

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 ?

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

Vierbein and metricVierbein 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 for vierbein

similar for metric

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

Lattice spinor gravityLattice spinor gravity

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

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 )

cellscells

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

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

Lattice derivativesLattice derivatives

and cell averagesand cell averages

express spinors in derivatives and express spinors in derivatives and averagesaverages

Bilinears and lattice Bilinears and lattice derivativesderivatives

Action in terms ofAction in terms of lattice derivatives lattice derivatives

Continuum limitContinuum limit

Lattice distance Δ drops out in continuum limit !

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

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

Lattice action and Lattice action and functional measure functional measure of spinor gravity are of spinor gravity are

lattice diffeomorphism lattice diffeomorphism invariant !invariant !

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

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 )

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 ?

end

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)

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 d=4 mod 4 possible for ‘‘extended extended

Lorentz symmetryLorentz symmetry’’ ( otherwise only d = ( 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.

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- - 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.

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