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Introduction Renormalization Group Flows for Quantum Gravity Vincent Rivasseau Laboratoire de Physique théorique Université Paris-Sud and Perimeter Institute Rencontre de Physique des Particules Institut Henri Poincaré, 16 Janvier 2015 Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Quantum Gravity as Field Theory? + Feynman’s quantization: functional integral =? Quantizing Gravity ' Randomizing Geometry Z R Z ' Dg e S AEH (g ) But what is the measure Dg ? On which underlying space-time S? Should one also sum over topologies? How to put sources for observables? How to connect to our classical space-time and to general relativity? Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Main Problem Quantizing gravity by simply adding quantum fluctuations of the space-time metric around a background (eg flat R 4 space-time) is not perturbatively renormalizable. Renormalization is not just about removing infinities. In the modern (Wilsonian) point of view, the renormalization group (RG) is a flow of the action in some theory space, and represents the way physics changes with observation scale. Usually it can be probed quite efficiently via numerical methods; functional renormalization group equation (FRGE), lattice Monte Carlo... Many other issues: background independence, Lorentz invariance, topology change, locality/non-locality and/or extended objects (strings)... Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Main Problem Quantizing gravity by simply adding quantum fluctuations of the space-time metric around a background (eg flat R 4 space-time) is not perturbatively renormalizable. Renormalization is not just about removing infinities. In the modern (Wilsonian) point of view, the renormalization group (RG) is a flow of the action in some theory space, and represents the way physics changes with observation scale. Usually it can be probed quite efficiently via numerical methods; functional renormalization group equation (FRGE), lattice Monte Carlo... Many other issues: background independence, Lorentz invariance, topology change, locality/non-locality and/or extended objects (strings)... Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Main Problem Quantizing gravity by simply adding quantum fluctuations of the space-time metric around a background (eg flat R 4 space-time) is not perturbatively renormalizable. Renormalization is not just about removing infinities. In the modern (Wilsonian) point of view, the renormalization group (RG) is a flow of the action in some theory space, and represents the way physics changes with observation scale. Usually it can be probed quite efficiently via numerical methods; functional renormalization group equation (FRGE), lattice Monte Carlo... Many other issues: background independence, Lorentz invariance, topology change, locality/non-locality and/or extended objects (strings)... Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Main Problem Quantizing gravity by simply adding quantum fluctuations of the space-time metric around a background (eg flat R 4 space-time) is not perturbatively renormalizable. Renormalization is not just about removing infinities. In the modern (Wilsonian) point of view, the renormalization group (RG) is a flow of the action in some theory space, and represents the way physics changes with observation scale. Usually it can be probed quite efficiently via numerical methods; functional renormalization group equation (FRGE), lattice Monte Carlo... Many other issues: background independence, Lorentz invariance, topology change, locality/non-locality and/or extended objects (strings)... Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Mainstream Approaches String Theory (Green, Schwarz, Witten, many others...) Unification of all interactions. Particles are vibrating modes of strings, and closed strings include naturally the gravitational sector. Requires supersymmetry. Background independence and sum over topologies might occur in M theory. Matrix Models (David, Kazakov, Kontsevich...) Well controlled discretized quantization of 2d quantum gravity. Sum over topologies well-understood. Closely related to renormalizable non-commutative quantum field theory (NCQFT) (Grosse-Wulkenhaar). Continuum version, CFT coupled to Liouville 2d gravity (Belavin, Knizhnik, Polyakov, Zamlodchikov...) Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Mainstream Approaches String Theory (Green, Schwarz, Witten, many others...) Unification of all interactions. Particles are vibrating modes of strings, and closed strings include naturally the gravitational sector. Requires supersymmetry. Background independence and sum over topologies might occur in M theory. Matrix Models (David, Kazakov, Kontsevich...) Well controlled discretized quantization of 2d quantum gravity. Sum over topologies well-understood. Closely related to renormalizable non-commutative quantum field theory (NCQFT) (Grosse-Wulkenhaar). Continuum version, CFT coupled to Liouville 2d gravity (Belavin, Knizhnik, Polyakov, Zamlodchikov...) Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Mainstream Approaches String Theory (Green, Schwarz, Witten, many others...) Unification of all interactions. Particles are vibrating modes of strings, and closed strings include naturally the gravitational sector. Requires supersymmetry. Background independence and sum over topologies might occur in M theory. Matrix Models (David, Kazakov, Kontsevich...) Well controlled discretized quantization of 2d quantum gravity. Sum over topologies well-understood. Closely related to renormalizable non-commutative quantum field theory (NCQFT) (Grosse-Wulkenhaar). Continuum version, CFT coupled to Liouville 2d gravity (Belavin, Knizhnik, Polyakov, Zamlodchikov...) Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Many Alternative Approaches Loop quantum gravity (LQG, Rovelli, Smolin, Thiemann...): background independent Hamiltonian, first-quantized formalism (spin networks, spin foams). Group field theory (GFT, Boulatov, Freidel, Oriti): second quantized version of LQG, introduces dynamics and sums over topologies. Causal dynamical triangulations (CDT, Ambjorn, Loll...): Gluing rules for space-time cells. These gluing rules and the causality condition may not be represented by a quantum field theory action. Uses mostly numerical tools. No sum over topologies. Asymptotic Safety (Weinberg, Reuter, Percacci, Saueressig...): searches for a non-trivial ultraviolet fixed point in the theory space of all diffeo-invariant functions of the metric. Uses mostly numerical tools. No background independence (no sum over topologies). Tensor Models (Gurau, Ben Geloun, Bonzom, R...): a recently revived proposal. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Many Alternative Approaches Loop quantum gravity (LQG, Rovelli, Smolin, Thiemann...): background independent Hamiltonian, first-quantized formalism (spin networks, spin foams). Group field theory (GFT, Boulatov, Freidel, Oriti): second quantized version of LQG, introduces dynamics and sums over topologies. Causal dynamical triangulations (CDT, Ambjorn, Loll...): Gluing rules for space-time cells. These gluing rules and the causality condition may not be represented by a quantum field theory action. Uses mostly numerical tools. No sum over topologies. Asymptotic Safety (Weinberg, Reuter, Percacci, Saueressig...): searches for a non-trivial ultraviolet fixed point in the theory space of all diffeo-invariant functions of the metric. Uses mostly numerical tools. No background independence (no sum over topologies). Tensor Models (Gurau, Ben Geloun, Bonzom, R...): a recently revived proposal. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Many Alternative Approaches Loop quantum gravity (LQG, Rovelli, Smolin, Thiemann...): background independent Hamiltonian, first-quantized formalism (spin networks, spin foams). Group field theory (GFT, Boulatov, Freidel, Oriti): second quantized version of LQG, introduces dynamics and sums over topologies. Causal dynamical triangulations (CDT, Ambjorn, Loll...): Gluing rules for space-time cells. These gluing rules and the causality condition may not be represented by a quantum field theory action. Uses mostly numerical tools. No sum over topologies. Asymptotic Safety (Weinberg, Reuter, Percacci, Saueressig...): searches for a non-trivial ultraviolet fixed point in the theory space of all diffeo-invariant functions of the metric. Uses mostly numerical tools. No background independence (no sum over topologies). Tensor Models (Gurau, Ben Geloun, Bonzom, R...): a recently revived proposal. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Many Alternative Approaches Loop quantum gravity (LQG, Rovelli, Smolin, Thiemann...): background independent Hamiltonian, first-quantized formalism (spin networks, spin foams). Group field theory (GFT, Boulatov, Freidel, Oriti): second quantized version of LQG, introduces dynamics and sums over topologies. Causal dynamical triangulations (CDT, Ambjorn, Loll...): Gluing rules for space-time cells. These gluing rules and the causality condition may not be represented by a quantum field theory action. Uses mostly numerical tools. No sum over topologies. Asymptotic Safety (Weinberg, Reuter, Percacci, Saueressig...): searches for a non-trivial ultraviolet fixed point in the theory space of all diffeo-invariant functions of the metric. Uses mostly numerical tools. No background independence (no sum over topologies). Tensor Models (Gurau, Ben Geloun, Bonzom, R...): a recently revived proposal. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Many Alternative Approaches Loop quantum gravity (LQG, Rovelli, Smolin, Thiemann...): background independent Hamiltonian, first-quantized formalism (spin networks, spin foams). Group field theory (GFT, Boulatov, Freidel, Oriti): second quantized version of LQG, introduces dynamics and sums over topologies. Causal dynamical triangulations (CDT, Ambjorn, Loll...): Gluing rules for space-time cells. These gluing rules and the causality condition may not be represented by a quantum field theory action. Uses mostly numerical tools. No sum over topologies. Asymptotic Safety (Weinberg, Reuter, Percacci, Saueressig...): searches for a non-trivial ultraviolet fixed point in the theory space of all diffeo-invariant functions of the metric. Uses mostly numerical tools. No background independence (no sum over topologies). Tensor Models (Gurau, Ben Geloun, Bonzom, R...): a recently revived proposal. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Renormalization Group RG flows from ultra-violet to infra-red in some (large) theory space: - local products of field operators at the same point, with possibly low order derivative couplings... - Einsteinian theory space: diffeo-invariant functions of the metric Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Renormalization Group RG flows from ultra-violet to infra-red in some (large) theory space: - local products of field operators at the same point, with possibly low order derivative couplings... - Einsteinian theory space: diffeo-invariant functions of the metric Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Renormalization Group RG flows from ultra-violet to infra-red in some (large) theory space: - local products of field operators at the same point, with possibly low order derivative couplings... - Einsteinian theory space: diffeo-invariant functions of the metric Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Renormalization Group RG flows from ultra-violet to infra-red in some (large) theory space: - local products of field operators at the same point, with possibly low order derivative couplings... - Einsteinian theory space: diffeo-invariant functions of the metric Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Renormalization Group RG flows from ultra-violet to infra-red in some (large) theory space: - local products of field operators at the same point, with possibly low order derivative couplings... - Einsteinian theory space: diffeo-invariant functions of the metric Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Asymptotic Safety Typical simulations with FRGE in the truncated Einsteinian theory space show a resilient ultraviolet fixed point (Reuter, Percacci, Saueressig, Benedetti...) Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Asymptotic Safety Typical simulations with FRGE in the truncated Einsteinian theory space show a resilient ultraviolet fixed point (Reuter, Percacci, Saueressig, Benedetti...) Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Models The tensor track builds upon the success of matrix models. Loop String Quantum Gravity Group Random Field Theory Matrices Theory Tensor Track Dynamical Asymptotic Triangulations Safety It is an improvement of group field theory which allows for its renormalization. It proposes a theory space different from that of asymptotic safety. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Models The tensor track builds upon the success of matrix models. Loop String Quantum Gravity Group Random Field Theory Matrices Theory Tensor Track Dynamical Asymptotic Triangulations Safety It is an improvement of group field theory which allows for its renormalization. It proposes a theory space different from that of asymptotic safety. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Models The tensor track builds upon the success of matrix models. Loop String Quantum Gravity Group Random Field Theory Matrices Theory Tensor Track Dynamical Asymptotic Triangulations Safety It is an improvement of group field theory which allows for its renormalization. It proposes a theory space different from that of asymptotic safety. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Tensor Track It is a quantum field theory of space time, not on space-time, which can incorporate background independence sum over topologies renormalizability asymptotic freedom It relies on a new theory space, based on U(N)⊗D invariant interactions. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Symmetry breaking ∃! Hilbert space `2 (Z). U(N) invariance can be broken. vector models => matrix models => tensor models Smaller symmetry means there are more invariants available for interactions Vector Models have exactly one connected invariant interaction, of degree 2 namely the scalar product φ̄ · φ. Matrix Models (N = N1 N2 , U(N1 N2 ) broken to U(N1 ) ⊗ U(N2 )) have infinitely many connected invariant interactions, one at every (even) degree, namely Tr (MM † )p . Tensor Models (N = N1 N2 N3 · · · , U(N1 N2 N3 · · · ) broken to U(N1 ) ⊗ U(N2 ) ⊗ U(N3 ) · · · ) have even much more invariants => richer theory space. Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Counting Tensor Invariants Tensor interactions = regular D-edge-colored connected bipartite graphs can be counted for D = 1, 2, 3, 4 · · · Z1c (n) = 1, 0, 0, 0, 0, ... Z2c (n) = 1, 1, 1, 1, 1, 1, 1... Z3c (n) = 1, 3, 7, 26,197, 624, 4163... Tr(MM † )n T 2 = 2 3 T Z4c (n) Φ̄ · Φ 1 2 2 3 3 3 1, 7, 141, 604, 13753... 1 1 1 2 2 1 1 T 2 2 2 3 T 1 1 2 3 33 1 1 33 3 11 22 2 3 3 2 1 1 3 1 1 2 2 2 2 1 1 2 Vincent Rivasseau 3 Renormalization Group Flows for Quantum Gravity Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Tensors and Quantum Gravity The Feynman graphs of tensor models are dual to colored triangulations, pondered according to the Einstein-Hilbert action with cosmological constant, hence can be considered a simplification of Regge calculus (1962): SRegge = Λ X vol(σD ) − σD 1 X vol(σD−2 ) δ(σD−2 ) 16πG σ D−2 Discretized Einstein Hilbert action on an equilateral triangulation with QD equilateral D-simplices and QD−2 (D − 2)-simplices is simply AG (N) = e κ1 QD−2 −κ2 QD On the Feynman dual graph: QD → n, number of vertices; QD−2 → F , number of faces, hence Regge action for equilateral simplices becomes AG (N) = λn N F the natural amplitudes of tensor models! Vincent Rivasseau Renormalization Group Flows for Quantum Gravity (1) Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Results on tensor models Their 1/N expansion (Gurau) is not of topological nature (as it was in D = 2). It leads to a new program for studying random geometry in d ≥ 3 through multiple scalings. Single scaling at any D and double scaling at d ≤ 6 have been solved and lead to branched polymers (Gurau et al...). Simple tensor field theories with tensor interactions and propagators of the inverse Laplacian type can be renormalized in many cases (up to rank/dimension 6) (Ben Geloun, Riv...) They can incorporate loop quantum gravity data (Carrozza, Oriti, Riv...) Generically they are asymptotically free, as shown by one and two loop computations (Ben Geloun, Samary, Carrozza...). Recently asymptotic freedom has been very clearly shown using an FRGE (Benedetti, BenGeloun and Oriti, arXiv.1411.3180). Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction Numerical Study of Tensor Track Flows Quartic Models with single coupling and Mass term, Small N 0.2 0.0 mN -0.2 -0.4 -0.6 -0.8 -1.0 0.00 0.01 Vincent Rivasseau 0.02 0.03 0.04 ΛRenormalization N Group Flows for Quantum Gravity Introduction Numerical Study of Tensor Track Flows Quartic Melonic Models with single coupling and Mass term, Large N 0.0 -0.2 mN -0.4 -0.6 -0.8 -1.0 0.000 0.002 0.004 0.006 0.008 ΛN Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction The conjectured physical picture The big bang could be emergence of (classical) space-time through a sequence of phase transitions (geometrogenesis). geometrogenesis The tensor flow could describe such transitions and ultimately connect in the infrared to the Einsteinian ultraviolet fixed points studied by Asymptotic Safety. Thursday, July 12, 2012 Vincent Rivasseau Renormalization Group Flows for Quantum Gravity Introduction The conjectured physical picture The big bang could be emergence of (classical) space-time through a sequence of phase transitions (geometrogenesis). geometrogenesis The tensor flow could describe such transitions and ultimately connect in the infrared to the Einsteinian ultraviolet fixed points studied by Asymptotic Safety. Thursday, July 12, 2012 Vincent Rivasseau Renormalization Group Flows for Quantum Gravity