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    Multisymplectic Geometry and Loop QuantumGravity

    Toward a Covariant Canonical Quantum Gravity

    Dimitri Vey

    LUTH, University of Paris 7

    June 14, 2010

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

    2/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    Contents

    1 Conceptuals problems in quantum gravity

    2 Loop Quantum Gravity

    3 Covariant fields Theories

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

    3/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    Quantizations problemsWhere we stand today?Universal Hamiltonian formalism

    Contents

    1 Conceptuals problems in quantum gravityQuantizations problemsWhere we stand today?Universal Hamiltonian formalism

    2 Loop Quantum GravityThe road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    3

    Covariant fields TheoriesHamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    Quantizations problemsWhere we stand today?Universal Hamiltonian formalism

    Quantizations problems

    Quantum gravity (QG) attempts to unify quantum mechanics(QM) with general relativity (GR) namely a theory which

    reduces to ordinary QM (G 0) and to Einstein GR ( 0)QM is defined on a fixed background (non-dynamic) structure

    QFT (relativistic quantum field theory) Minkowski spacetimeis the fixed background of the theory

    GR great insight : there is no fixed spacetime background ,spacetime is dynamic (diffeomorphism invariance)

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    Quantizations problemsWhere we stand today?Universal Hamiltonian formalism

    Where we stand today?

    Mathematical fundations of QFT :

    (i) Covariant Quantization Feynman integral involving Lagrangian functional, manifestely relativistic(ii) Canonical quantization method, based on Hamiltonian

    formulation of the dynamics of classical fields

    Space and time are treated asymetrically

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    Quantizations problemsWhere we stand today?Universal Hamiltonian formalism

    Therefore, Heart of Quantum Gravity is looking for a covariantcanonical quantization and in such a context, we are intersested in

    the so called multisymplectic formalism.

    A motivation to study the Universal Hamiltonian formalism is

    to apply it in the context of integrable systems

    to analyse the canonical structure of the physical theories, for

    instance general relativity and string theory with the aim toquantify those theories.

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

    7/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    Contents

    1 Conceptuals problems in quantum gravityQuantizations problemsWhere we stand today?Universal Hamiltonian formalism

    2 Loop Quantum GravityThe road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    3

    Covariant fields TheoriesHamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    The road ahead : LQG

    Non pertubative canonical quantization : LQG in a nutshell

    Hamiltonian quantisation of pure 4D Gravity : M = RNon-perturbative quantisation : question of renormalisationavoided

    Background independent quantisation : no background metricneeded in the spirit of Einstein theory of General Relativity

    Dimensional reduction of Yang-Mills theory : symplecticstructure and phase space in LQG (SL(2,C) Ashtekar

    connection)

    General Relativity in the prism of Gauge theory, first classconstraints generate symmetries.

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    http://goforward/http://find/http://goback/
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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    Hamiltonian Formulation of General Relativity : ADM

    Einstein-Hilbert action functional of the metric g :

    SEH[g] =

    Md4xL =

    Md4x

    g R =M

    d4xg gR[g]

    Euler-Lagrange (Einstein) equations obtained varying action w.r.t themetric variables g .

    GR is a constrained hamiltonian system

    GR : Generally covariant theory (diffeomorphism invariance)

    Vanishing Hamiltonian. Hamiltonian : linear combinaison of first class constraints(generate Gauges Transformations)

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    C l bl i i Th d h d LQG

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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    First order formalism and Palatini action

    Palatini first-order formulation of GR : independantvariables (, e)

    connection 1-form = IJ

    JIJdx

    then curvature 2-form F() tetrad field (local flat frame)eI = eIdx

    Palatini action with tetrad: functional of (e, )

    S[, e] = 12 M IJKLeI

    eJ

    FKL()

    Canonical analysis of Palatini action reproduces ADM results

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    C t l bl i t it Th d h d LQG

    http://goforward/http://find/http://goback/
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    11/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    From Palatini action to Ashtekar Gravity

    Palatini Gravity as SO(1,3) gauge theory

    Geometrical Interpretation (IJ , eI): We are dealing with a

    vector Lorentz bundle over the spacetime manifold SO(1, 3) M

    Palatini Gravity GR as gauge theory with gauge group givenby the Lorentz group, SO(1, 3)

    Ashtekar self-dual spin connection

    Ashtekar reexpressed GR in terms of variables S[e,A] withAIJ [] =

    IJ 1/2i

    MNIJ

    MN so called Ashtekar-Sen connection.

    SL(2,C) double cover SO(1, 3) : sl(2,C) complexification of the Liealgebra so(1, 3)

    Key Point Ashtekar variables : simplify constraints of GR

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity The road ahead : LQG

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

    12/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    Ashtekar Variables, Loop formulation of GR

    New Configuration variable : Ashtekar Variables (Aia, Eai )

    SU(2)-connection Aia = ia + 0ia (on a 3-manifold ) Densitized triad Eai = 1/2ijkabceejbekcCanonical variables : {Aia(x), Ebj (y)} = (8G)baij3(x, y)Constraints generate symetries : Gauss constraint G (new degrees of freedom in Ashtekar Variables) Spatial diffeomorphisms constraint H Hamiltonian constraint H (time reparameterizations)

    G = H = H = 0

    Dynamics ? Spacetime splitting breaks manifest covariance(complexity of

    H)

    Theory is not really dynamical.

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields Theories

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

    13/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    Contents

    1 Conceptuals problems in quantum gravityQuantizations problemsWhere we stand today?Universal Hamiltonian formalism

    2 Loop Quantum GravityThe road ahead : LQGHamiltonian Formulation of General Relativity : ADMFirst order formalism and Ashtekar Variables

    3 Covariant fields TheoriesHamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields TheoriesQ f fi

    http://goforward/http://find/http://goback/
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    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    In a finite dimensional classical system, the motivation for

    choosing the cotangent bundle as a mathematical model forphase space lay in the possibility of identifying elements ofTM with initial data for the dynamical evolution.

    Analogously, in a field theory we would expect the state spaceto consist of all Cauchy data for the system

    In the canonical approach to a standard field theory, the canonicalvariables are defined on space like hypersurfaces

    Space and time are treated asymetrically, and thus we have a noncovariance scheme

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields TheoriesTh i i i Q i i f fi ld

    http://goforward/http://find/http://goback/
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    15/19

    Conceptuals problems in quantum gravityLoop Quantum Gravity

    Covariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    The main motivation : Quantization of fields

    The most useful objects for that are the Poisson brackets.

    {pi, qj} = ji leads to [pi, q

    j] = ijidF

    dt= {H,F} leads to

    dF

    dt= i[H, F]

    The main idea is to construct a Hamiltonian description of classical fieldstheory compatible with Principles of special and general relativity and string theories more generally any effort towards understanding gravitation

    Since space-time should merge out from the dynamics Need a description without any space-time/field splitting a priori. There is no space-time structure given a priori.

    Space-time coordinates should instead merge out from the analysis of what arethe observable quantities and from the dynamics.

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields TheoriesTh i ti ti Q ti ti f fi lds

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    p p q g yLoop Quantum Gravity

    Covariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    The Hamilton equations : system of first order equations

    Let the Legendre map

    : TY TY(y, v)

    y, L

    v(y, v)

    Assume it is inversible and denote its inverse.

    1 : TY TY(q, p) (q,V(q, p)))

    Hamiltonian function H(q, p) = p,V(q, p) L(q,V(q, p)).

    The Hamilton equations are

    dqi

    dt(t) =

    H

    pi(q(t), p(t)),

    dpidt

    (t) = H

    qi(q(t), p(t)).

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fields

    http://goforward/http://find/http://goback/
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    p p q g yLoop Quantum Gravity

    Covariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    General variational problems

    Question: Is there a similar of the Hamiltonian theory for fields in

    E := {u : X Y}

    which are critical points of

    L(u) :=

    L(u, du),

    In this Lagrangian functional of maps u we have

    X = a m dimensional manifold (space-time) = dx0 dx1 ... dxm1 = a volume form on XY = a n dimensional manifold (fields) ?

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravity Hamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fields

    http://goforward/http://find/http://goback/
  • 8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity

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    Loop Quantum GravityCovariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    General variational problems

    to be able to treat more general variational problems: the study ofm-dimensional submanifolds chosen in

    E := {m Nm+n}

    which are critical points of

    L(m) :=

    m

    L(q,Tqm),

    ( is a m-form on Nm+n).

    Particular examples are when m is the graph in Nm+n = X Y of some some

    map u : X Y or a section of a bundle. It may be interesting to assume less structure. We want to state a completedemocracy between time, space and internal variables : (revendications ofKaluza-Klein theories, supergravity in 11 D and superstrings in 10 D.)

    Then the distinction between these variables should be a consequence of

    dynamical equations.Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

    Conceptuals problems in quantum gravityL Q G i

    Hamiltonian Formalism and fields TheoriesThe main motivation : Quantization of fields

    http://goforward/http://find/http://goback/
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    Loop Quantum GravityCovariant fields Theories

    The main motivation : Quantization of fieldsThe Hamilton equations : system of first order equationsGeneral variational problems

    General variational problems

    The analogue of tangent bundle in mechanics is the Grassmann bundle GrnNof oriented n-dimensional subspaces of tangent spaces to N.

    The analogue of the cotangent bundle in mechanics is nTN.Note that dimGrnN = n + k + nk so dimnTN = n + k + (n+k)!

    n!k!>

    dimGrnN + 1 unless n = 1 (classical mechanics) or k = 1 (submanifolds arehypersurfaces).

    Generalized Hamilton system of equations :

    u

    x =H

    p (x, u, p)

    p

    x=

    H

    u(x, u, p).

    Dimitri Vey Multisymplectic Geometry and Loop Quantum Gravity

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