Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum...
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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
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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
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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
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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
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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
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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
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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
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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
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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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
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
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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
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
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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
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
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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
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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
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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 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
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8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity
17/19
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
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8/3/2019 Dimitri Vey- Multisymplectic Geometry and Loop Quantum Gravity: Toward a Covariant Canonical Quantum Gravity
18/19
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
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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
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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