Download - Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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Page 1: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice Spinor Lattice Spinor GravityGravity

Page 2: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Quantum gravityQuantum gravity

Quantum field theoryQuantum field theory Functional integral formulationFunctional integral formulation

Page 3: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 4: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 5: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Spinor gravitySpinor gravity

is formulated in terms of is formulated in terms of fermionsfermions

Page 6: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 7: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 8: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Gauge bosons, scalars …

from vielbein components in higher dimensions(Kaluza, Klein)

concentrate first on gravity

Page 9: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Geometrical degrees of Geometrical degrees of freedomfreedom

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

vielbein : fermion bilinearvielbein : fermion bilinear

Page 10: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Possible ActionPossible Action

contains 2d powers of spinors d derivatives contracted with ε - tensor

Page 11: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 12: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 13: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 :

Page 14: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Spinor gravity with Spinor gravity with local Lorentz symmetrylocal Lorentz symmetry

Page 15: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 16: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 17: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 18: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Weyl spinorsWeyl spinors

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

Page 19: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Relation to previous formulation

Page 20: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Action invariant for arbitrary complex transformation parameters ε !

Real ε : SO (4) - transformations

Page 21: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Signature of timeSignature of time

Difference in signature between space and time :

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

Page 22: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Minkowski - actionMinkowski - action

Action describes simultaneously euclidean and Minkowski theory !

SO (1,3) transformations :

Page 23: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Emergence of geometryEmergence of geometry

Euclidean vierbein bilinearMinkowski -vierbein bilinear

GlobalLorentz - transformation

vierbeinmetric

/ Δ

Page 24: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

vierbein bilinear ?vierbein bilinear ?

No suitable W exists

Page 25: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 ?

Page 26: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 27: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Vierbein and metricVierbein and metric

Generating functional

Page 28: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 29: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Effective actionEffective action

W=ln Z

Gravitational field equation for vierbein

similar for metric

Page 30: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 31: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice spinor gravityLattice spinor gravity

Page 32: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 33: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 )

Page 34: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

cellscells

Page 35: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 36: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 )

Page 37: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice derivativesLattice derivatives

and cell averagesand cell averages

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

Page 38: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Bilinears and lattice Bilinears and lattice derivativesderivatives

Page 39: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Action in terms ofAction in terms of lattice derivatives lattice derivatives

Page 40: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Continuum limitContinuum limit

Lattice distance Δ drops out in continuum limit !

Page 41: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 42: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 43: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

lattice diffeomorphism lattice diffeomorphism invariant !invariant !

Page 44: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice action for bosons Lattice action for bosons in d=2in d=2

Page 45: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Positioning of lattice Positioning of lattice pointspoints

Page 46: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice derivativesLattice derivatives

Cell average :

Page 47: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice diffeomorphism Lattice diffeomorphism invarianceinvariance

ContinuumLimit :

Page 48: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Lattice diffeomorphism Lattice diffeomorphism transformationtransformation

Page 49: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Effective action is Effective action is diffeomorphism diffeomorphism

symmetricsymmetric

Same for effective action for graviton !

Page 50: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

lattice diffeomorphism lattice diffeomorphism invariant !invariant !

Page 51: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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

Page 52: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 )

Page 53: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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 ?

Page 54: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

end

Page 55: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

Gravitational field equationGravitational field equationand energy momentum and energy momentum

tensortensor

Special case : effective action depends only on metric

Page 56: Lattice Spinor Gravity Lattice Spinor Gravity. Quantum gravity Quantum field theory Quantum field theory Functional integral formulation Functional integral.

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)