Quantum Gravity and emergent metric Quantum Gravity and emergent metric

Quantum Quantum Gravity Gravity

and emergent and emergent metric metric

Why do we need Why do we need quantum gravity for quantum gravity for

cosmology ?cosmology ? gravitational equations provide gravitational equations provide

fundamental framework for cosmologyfundamental framework for cosmology gravity is coupled to quantum matter gravity is coupled to quantum matter

and radiationand radiation energy momentum tensor is a quantum energy momentum tensor is a quantum

objectobject can one have an equation with classical can one have an equation with classical

metric field on one side and a quantum metric field on one side and a quantum object on the other side ?object on the other side ?

Can one have an equation with Can one have an equation with classical metric field on one side classical metric field on one side

and a quantum object on the other and a quantum object on the other side ?side ?

yes : equation for expectation values !

metric is expectation metric is expectation value of quantum fieldvalue of quantum field

One only needs to assume that One only needs to assume that somesome quantum quantum theory exists for which an observable with theory exists for which an observable with properties of metric exists and has a nonzero properties of metric exists and has a nonzero expectation valueexpectation value

formalism : quantum effective action – exact field formalism : quantum effective action – exact field equations follow from variation of action equations follow from variation of action functionalfunctional

IfIf effective action takes form of Einstein-Hilbert effective action takes form of Einstein-Hilbert action ( with cosmological constant ) the Einstein action ( with cosmological constant ) the Einstein field equations followfield equations follow

This would be sufficient for cosmology !This would be sufficient for cosmology !

Einstein gravityEinstein gravity

Is Einstein Hilbert action sufficient ?Is Einstein Hilbert action sufficient ? It cannot be the exact effective action It cannot be the exact effective action

for a theory of quantum gravity !for a theory of quantum gravity ! Can it be a sufficiently accurate Can it be a sufficiently accurate

approximation for the quantum approximation for the quantum effective action ?effective action ?

Answer to this question needs Answer to this question needs consistent theory of consistent theory of quantum quantum gravity gravity !!

Einstein gravity as effective Einstein gravity as effective theory for large distance theory for large distance scales or small momentascales or small momenta

diffeomorphism symmetrydiffeomorphism symmetry derivative expansionderivative expansion

zero derivatives : cosmological constantzero derivatives : cosmological constant

two derivatives : curvature scalar Rtwo derivatives : curvature scalar R

four derivatives : Rfour derivatives : R22, two more tensor , two more tensor structuresstructures

higher derivatives are expected to be higher derivatives are expected to be induced by quantum fluctuations induced by quantum fluctuations

short distance short distance modificationsmodifications

coefficient Rcoefficient R22 order one order one

( typical quantum contribution 1/16( typical quantum contribution 1/16ππ22 ) : ) :

higher order derivative terms play a role only higher order derivative terms play a role only onceonce

curvature scalar is of the order of squared curvature scalar is of the order of squared PlanckPlanck

massmass

singularity of black holes , inflationary singularity of black holes , inflationary cosmologycosmology

no analytic behavior expected : Rno analytic behavior expected : R22 ln(R) etc. ln(R) etc.

long distance long distance modifications ??modifications ??

non- local termsnon- local terms f( R ) with huge coefficients of Taylor f( R ) with huge coefficients of Taylor

expansionexpansion

this could modify late time behavior of this could modify late time behavior of cosmology and be related to dark energycosmology and be related to dark energy

possible explanation why cosmological possible explanation why cosmological constant is zero or small ?constant is zero or small ?

need for quantum gravityneed for quantum gravity

before judgment one needs at least before judgment one needs at least one consistent model of quantum one consistent model of quantum gravitygravity

will it be unique ? probably not !will it be unique ? probably not !

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

scalar gravityscalar gravity

quantum field theory for scalarsquantum field theory for scalars d=2 , two complex fields i=1,2d=2 , two complex fields i=1,2 non-linear sigma-modelnon-linear sigma-model

diffeomorphism symmetry of actiondiffeomorphism symmetry of action

with D.Sexty

lattice regularizationlattice regularization

collective metric collective metric observableobservable

metric correlation metric correlation functionfunction

response of metric to response of metric to sourcesource

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

Geometrical degrees of Geometrical degrees of freedomfreedom

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

vielbein : fermion bilinearvielbein : fermion bilinear

Emergence of geometryEmergence of geometry

vierbeinmetric

/ Δ

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)A.Hebecker, C.Wetterich

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

Gravitational field equationGravitational field equationand energy momentum and energy momentum

tensortensor

Special case : effective action depends only on 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 ( invariant )( invariant )

Reflections of all lattice coordinates Reflections of all lattice coordinates ( odd )( odd )

Diagonal reflections e.g zDiagonal reflections e.g z11↔z↔z2 2 ( odd )( odd )

Lattice derivativesLattice derivatives

and cell averagesand cell averages

express spinors in terms of derivatives and express spinors in terms of 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

Positioning of lattice Positioning of lattice pointspoints

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

lattice diffeomorphism lattice diffeomorphism invariant !invariant !

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 term

( curvature scalar )( 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

Lattice derivativesLattice derivatives

Cell average :

Lattice diffeomorphism Lattice diffeomorphism invarianceinvariance

ContinuumLimit :

Lattice diffeomorphism Lattice diffeomorphism transformationtransformation

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)

Gauge symmetriesGauge symmetries

Proposed action for lattice spinor gravity Proposed action for lattice spinor gravity has also 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

Gauge bosons, scalars …

from vielbein components in higher dimensions(Kaluza, Klein)

concentrate first on gravity