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The Group Field Theory Approach to Quantum Gravity Or: Quantum Gravity As a Quantum field Theory of Simplicial Geometry?A

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The Group Field Theory Approach to Quantum Gravity Or: Quantum Gravity As a Quantum field Theory of Simplicial Geometry?A (TALKING TO HIS BRAIN) “Ok, brain, let’s face it: I don’t like you and you don’t like me... but let’s get through this thing... ...and then I can continue killing you with beer...” Homer Simpson The Group Field Theory approachto Quantum Gravity – p. 1/3 The Group Field Theory approach to Quantum Gravity or: Quantum Gravity as a quantum field theory of simplicial geometry?a Daniele Oriti Department of Applied Mathematics and Theoretical Physics University of Cambridge General reviews: J. Baez, Lect. Notes Phys. 543, 25-94 (2000), gr-qc/9905087; D. Oriti, Rept. Prog. Phys. 64, 1489 (2001), gr-qc/0106091; A. Perez, Class. Quant. Grav. 20, R43 (2003), gr-qc/0301113; D. Oriti, gr-qc/0311066; C. Rovelli, Quantum Gravity, CUP (2005), L. Freidel, hep-th/0505016 The Group Field Theory approachto Quantum Gravity – p. 2/3 Plan of the talk Past dreams: Sum-over-histories formulation of Quantum Gravity and 3rd Quantization Recent struggles: Matrix Models, Dynamical Triangulations, Spin Foam Models Present hopes: the Group Field Theory formalism, what it is and how to interpret it Future struggles, or how to make the dream come true and the hopes fulfilled The Group Field Theory approachto Quantum Gravity – p. 3/3 Past dreams: ———- sum-over-histories formulation of Quantum Gravity and the idea of “third quantization” The Group Field Theory approachto Quantum Gravity – p. 4/3 QG from sum-over-geometries histories: 4-geometries gµν Partition function: iSgr(g,M) ZQG = Dge {g(M)} States/boundary data: 3-geometries hab Transition amplitudes for boundary data: ( M) Z(h(S),h(S )) = DgeiSgr g, {g|h,h} Cosmological setting......physical meaning? Can be made local? The Group Field Theory approachto Quantum Gravity – p. 5/3 Generalization Why not to consider also topology as a dynamical variable? + + +........ The Group Field Theory approachto Quantum Gravity – p. 6/3 Generalization Why not to consider also topology as a dynamical variable? + + +........ simply: ( M) Z(h(S),h(S )) = w(M) DgeiSgr g, ? M {g|h,h } The Group Field Theory approachto Quantum Gravity – p. 6/3 3rd quantization? push for more: a “3rd quantized”formulation, with the topology changing processes described dynamically as an “interaction of quantum universes”(S. Coleman, S. Giddings, A. Strominger, M. McGuigan, ...) The Group Field Theory approachto Quantum Gravity – p. 7/3 3rd quantization? push for more: a “3rd quantized”formulation, with the topology changing processes described dynamically as an “interaction of quantum universes”(S. Coleman, S. Giddings, A. Strominger, M. McGuigan, ...) define a scalar field theory in superspace (space of all 3-geometries) for φ(hij), The Group Field Theory approachto Quantum Gravity – p. 7/3 3rd quantization? push for more: a “3rd quantized”formulation, with the topology changing processes described dynamically as an “interaction of quantum universes”(S. Coleman, S. Giddings, A. Strominger, M. McGuigan, ...) define a scalar field theory in superspace (space of all 3-geometries) for φ(hij), by the action: 1 ( )=− D ( )∇ ( )+ D n S φ 2 hij φ hij φ hij λ hij φ , where ∇ is the Wheeler-DeWitt operator, with a (non-local) interaction term in superspace The Group Field Theory approachto Quantum Gravity – p. 7/3 3rd quantization? and study the quantum theory defined by: Z = DφeiS(φ) The Group Field Theory approachto Quantum Gravity – p. 8/3 3rd quantization? and study the quantum theory defined by: Z = DφeiS(φ) through its perturbative expansion in Feynman diagrams The Group Field Theory approachto Quantum Gravity – p. 8/3 3rd quantization? and study the quantum theory defined by: Z = DφeiS(φ) through its perturbative expansion in Feynman diagrams the (perturbative) 3rd quantized vacuum of the theory will be the “no-spacetime state” The Group Field Theory approachto Quantum Gravity – p. 8/3 3rd quantization? and study the quantum theory defined by: Z = DφeiS(φ) through its perturbative expansion in Feynman diagrams the (perturbative) 3rd quantized vacuum of the theory will be the “no-spacetime state” the amplitude for the free propagation will be given by the usual quantum gravity path integral for trivial topology, and the other Feynman graphs of the theory will describe topology-changing processes The Group Field Theory approachto Quantum Gravity – p. 8/3 Recent struggles: ——– Matrix Models, Dynamical Triangulations, Spin Foam Models The Group Field Theory approachto Quantum Gravity – p. 9/3 Matrix Models for 2d quantum gravity define for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N The Group Field Theory approachto Quantum Gravity – p. 10/3 Matrix Models for 2d quantum gravity define for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N and the quantum theory via Z = dM e− S(M), defined as a Feynman series in λ The Group Field Theory approachto Quantum Gravity – p. 10/3 Matrix Models for 2d quantum gravity define for an N × N hermitian matrix Mij the action 1 λ S(M)= tr(M 2) − √ tr(M 3) 2 3 N and the quantum theory via Z = dM e− S(M), defined as a Feynman series in λ Feynman expansion produces trivalent (fat) graphs of ALL topologies M ij i M M ij j ji i j k M jk M ki The Group Field Theory approachto Quantum Gravity – p. 10/3 Matrix Models for 2d quantum gravity these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i M M ij j ji i j k M jk M ki The Group Field Theory approachto Quantum Gravity – p. 11/3 Matrix Models for 2d quantum gravity these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i M M ij j ji i j k M jk M ki definition of 2d quantum gravity as a sum over ALL 2d triangulations of ALL topologies The Group Field Theory approachto Quantum Gravity – p. 11/3 Matrix Models for 2d quantum gravity these graphs are dual to TRIANGULATIONS of orientable closed surfaces M ij i M M ij j ji i j k M jk M ki definition of 2d quantum gravity as a sum over ALL 2d triangulations of ALL topologies that means: 1 Z = dM e− S(M) = λn2(T )N χ(T ) sym(T ) T The Group Field Theory approachto Quantum Gravity – p. 11/3 Dynamical triangulations idea: triangulate a given continuum manifold → spacetime obtained by gluing D-simplices The Group Field Theory approachto Quantum Gravity – p. 12/3 Dynamical triangulations idea: triangulate a given continuum manifold → spacetime obtained by gluing D-simplices assume all edge lengths are equal to a; geometric degrees of freedom are encoded in the combinatorics of the triangulation The Group Field Theory approachto Quantum Gravity – p. 12/3 Dynamical triangulations idea: triangulate a given continuum manifold → spacetime obtained by gluing D-simplices assume all edge lengths are equal to a; geometric degrees of freedom are encoded in the combinatorics of the triangulation theory (both Riemannian and Lorentzian) defined by: 1 ( Λ) Z(Λ,a)= eiSRegge T,a, sym(T ) T where T are triangulations (fixed topology), and SRegge is Regge action for simplicial gravity The Group Field Theory approachto Quantum Gravity – p. 12/3 Dynamical triangulations in Lorentzian case one distinguishes between spacelike edges (length square a2) and timelike ones (length square −a2), and imposes some additional restrictions... The Group Field Theory approachto Quantum Gravity – p. 13/3 Dynamical triangulations in Lorentzian case one distinguishes between spacelike edges (length square a2) and timelike ones (length square −a2), and imposes some additional restrictions... then one looks for continuum limit (smooth phase, with correct dimensionality) as a → 0 and the triangulation is refined (and Λ gets renormalised) The Group Field Theory approachto Quantum Gravity – p. 13/3 Dynamical triangulations in Lorentzian case one distinguishes between spacelike edges (length square a2) and timelike ones (length square −a2), and imposes some additional restrictions... then one looks for continuum limit (smooth phase, with correct dimensionality) as a → 0 and the triangulation is refined (and Λ gets renormalised) recent results: it seems that in Lorentzian setting, for trivial topology, there are indications of good continuum limit The Group Field Theory approachto Quantum Gravity – p. 13/3 Spin Foam models: the general idea spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces states/boundary data → spin networks Ψ=graphs with links labeled by irreps J The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces states/boundary data → spin networks Ψ=graphs with links labeled by irreps J combinatorial and representation data encode geometric degrees of freedom The Group Field Theory approachto Quantum Gravity – p. 14/3 Spin Foam models: the general idea spacetime manifold → combinatorial 2-complex σ (vertices, links, faces) histories → 2-complexes with representations J of Lorentz group assigned to faces states/boundary data → spin networks Ψ=graphs with links labeled by irreps J combinatorial and representation data encode geometric degrees of freedom Spin Foam model (M. Reisenberger, J. Baez): Z = w(σ) Af (Jf ) Ae(Jf|e) Av(Jf|v) σ|Ψ,Ψ {J} f e v The Group Field Theory approachto Quantum Gravity – p.
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