Martin Bojowald- Quantum geometry and quantum dynamics at the Planck scale

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    Quantum geometry and quantum dynamics

    at the Planck scale

    Martin Bojowald

    The Pennsylvania State UniversityInstitute for Gravitation and the Cosmos

    University Park, PA

    Loop Quantum Cosmology p.

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    Space-time in canonical quantum gravity

    Canonical formulations provide insights in underlyingsymmetries, corresponding to general covariance for gravity.

    Quantization implies correction terms which can change theunderlying symmetries or even provide new quantum degrees offreedom.

    In loop quantum gravity, corrections arise from quantumgeometry (spatial structure) as well as quantum dynamics.

    Main recent developments (model systems or perturbations):

    Consistent deformations of classical gravity. Effective description to derive interacting quantum states

    and quantum corrections in equations of motion.

    Loop Quantum Cosmology p.

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    Canonical gravity

    Infinite dimensional phase space of fields qab (spatial metric) and

    pab (momenta, related to extrinsic curvature).Other components N and Na of the space-time metric

    ds2 = gdxdx = N2dt2 + qab(dxa + Nadt)(dxb + Nbdt)

    are not dynamical since N and Na do not occur in the action.

    Thus, momenta pN = S/N and pNa vanish identically, and somust pN and pNa . Implies constraints

    C = det q16G (3)R 16Gdet q (pabpab12(paa)2) = 0 , Ca = 2Dbpba = 0

    Diffeomorphism constraint D[Na] = d3xNaCa generates

    spatial diffeomorphisms, Hamiltonian constraint C completesspace-time transformations.

    Loop Quantum Cosmology p.

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    Constrained dynamics

    Dynamics of general relativity in canonical formulation

    determined by constraints: H[N, Na] =

    d3x(N C+ NaCa) = 0

    for all multiplier functions N, Na.

    Generates equations of motion f = {f, H[N, Na]} for any phasespace function f(q, p), dot refers to time gauge as given by lapseN and shift Na.

    Consistency requirement:

    H[M, Ma] = {H[M, Ma], H[N, Na]} = 0 must vanish for all Nand Na for fields satisfying the constraints

    first class constraint algebra, to be realized also afterincluding quantum corrections. Quite restrictive.

    Loop Quantum Cosmology p.

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    Constraint algebra

    Specific constraint algebra for gravity: hypersurfacedeformations

    {H[N1], H[N2]} = D qab(N1bN2 N2bN1)

    N

    N

    NN

    N

    2

    2

    1

    1

    a

    Basic information about space-time manifold. Not very sensitive

    to dynamics, e.g. higher curvature actions.

    Loop Quantum Cosmology p.

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    Quantum corrections

    After quantization, constraints will change and show quantumspace-time structure without direct reference to manifold orcoordinates.

    Three types of corrections, in general equally important:

    Entire states evolve which spread and deform. Quantumfluctuations, correlations and higher moments are independent

    variables back-reactingon expectation values.

    In loop quantum gravity, holonomieshe(A) = Pexp(e A

    iaidt) as non-local, non-linear functions

    imply higher order corrections.

    In loop quantum gravity, fluxes quantizing metric havediscrete spectra containing zero. Inverse metric components

    receive corrections for discrete (lattice-like) states with smallelementary areas.

    Loop Quantum Cosmology p.

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    Quantum Friedmann equation

    [PRL 100 (2008) 221301]a

    a

    2=

    8G

    3

    1 Q

    crit

    1

    2

    1 Q

    crit ( P) +(

    P)2

    ( + P)2 2

    where P is pressure and parameterizes quantum correlations,

    Q := + 0crit + ( P)k=0

    k+1( P)k/( + P)k

    with fluctuation parameters k; crit = 3/8G2 with scale .Simple behavior if = P (free, massless scalar): bounSingh.[P. Singh, PRD 73 (2006) 063508] Also if = 0 when Q = crit.

    But: so far no consistent inhomogeneous formulation withholonomy corrections.

    Loop Quantum Cosmology p.

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    Inverse metric corrections

    [with G. Hossain, M. Kagan, S. Shankaranarayanan: PRD 78 (2008) 063547]Corrected Hamiltonian, perturbative around FRW:

    HQgrav :=1

    16Gd3xNH

    (0) + (2)

    H(0) +

    HQ(2)

    + NHQ(1)

    with H(0) = 6H2a and

    HQ(1) =

    4(1 + f)

    HacjK

    jc

    (1 + g)

    H2

    a

    jcEcj +

    2

    a

    cjEcj

    HQ(2) = aKjc Kkd ckdj a(Kjc cj)2 2H

    aEcjK

    jc

    H2

    2a3 E

    c

    jE

    d

    k

    k

    c

    j

    d + H2

    4a3 (E

    c

    j

    j

    c)

    2

    (1 + h) jk

    2a3(cE

    cj )(dE

    dk)

    0

    0.5

    1

    1.5

    2

    2.5

    0 0.5 1 1.5 2

    (r)

    r=1/2r=3/4

    r=1r=3/2

    r=2

    Loop Quantum Cosmology p.

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    Anomaly cancellation

    First class algebra to second order if 2f + g = 0 and

    h

    f +

    a

    a

    = 0

    f g 2a fa

    a

    a= 0

    f + g

    a

    g

    a+

    a

    a= 0

    1

    6

    a

    Ecja3

    +(2)

    (Eai )(aj

    ci cjai ) = 0

    as well as a condition for matter correction functions in terms of2 (if matter is present).

    All coefficients fixed in terms of , which can be derived in

    models.

    Loop Quantum Cosmology p.

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    Quantum constraint algebra

    Anomaly-free constraints including quantum corrections exist:consistent deformation. Underlying discreteness (inverse metriccorrections) does not destroy general covariance.

    Constraint algebra of hypersurface deformations quantumcorrected(contains inverse metric):

    {HQ

    [N1], HQ

    [N2]} = D 2N a1/2a(N2 N1)

    N

    N

    NN

    N

    2

    2

    1

    1

    a

    Loop Quantum Cosmology p.1

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    Quantum constraint algebra

    Anomaly-free constraints including quantum corrections exist:consistent deformation. Underlying discreteness (inverse metriccorrections) does not destroy general covariance.

    Constraint algebra of hypersurface deformations quantumcorrected(contains inverse metric):

    {HQ

    [N1], HQ

    [N2]} = D 2N a1/2a(N2 N1)

    Provides consistent cosmological perturbation equations:

    Hamiltonian constraint, diffeomorphism constraint and evolutionequations.

    Quantum corrections to constraints also change gauge invariant

    variables. Consistent perturbation equations can be formulatedin terms of only gauge invariant quantities.

    Loop Quantum Cosmology p.1

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    Corrected perturbation equations

    c

    + H(1 + f)

    = G

    c

    GI

    (2

    ) 3H(1 + f) + H(1 + f)= 4G

    (1 + f3)

    GI 2(1 + f1) + a2V,()GI

    + H

    2

    1 a

    2

    d

    da

    + (1 + f)

    +2 H + H21 + a

    2

    df

    da a

    2

    d

    da(1 + f)

    = 4G

    GI a2V,()GI

    Now being used for cosmological applications (power spectrum),e.g. non-conservation of power on large scales.

    Loop Quantum Cosmology p.1

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    Number of atoms

    Corrections depend on discrete building blocks. Spatialgeometry subdivided dynamically. (Avoids macroscopic lattice.)

    Elementary size 0 (coordinate length), total volume V0. Number

    of atoms of geometry: N= V0/30, densityN/V0 = 30 .Basic holonomies exp(i0a/N) and fluxes =

    20a

    2/2P ifgeometry nearly isotropic.

    V0

    0l0

    0.5

    1

    1.5

    2

    2.5

    0 0.5 1 1.5 2

    (r)

    r=1/2r=3/4

    r=1r=3/2

    r=2

    Loop Quantum Cosmology p.1

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    Number of atoms

    Corrections depend on discrete building blocks. Spatialgeometry subdivided dynamically. (Avoids macroscopic lattice.)

    Elementary size 0 (coordinate length), total volume V0. Number

    of atoms of geometry: N= V0/30, densityN/V0 = 30 .Basic holonomies exp(i0a/N) and fluxes =

    20a

    2/2P ifgeometry nearly isotropic.

    Holonomy corrections when curvature a/N k = (N/V0)1/3.Inverse metric corrections large when a a = (N/V0)1/3P.

    (For a a, physical vertex densityN/a3

    V0 = (a/a)3

    /3

    P nearone per Planck volume.)

    Classical range: a/N k, a a, determined by vertexdensity

    N/V0 of quantum geometry state.

    Loop Quantum Cosmology p.1

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