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Functional renormalization group approach to the continuum limit of Group Field Theories Daniele Oriti Albert Einstein Institute 8th International Conference on the Exact Renormalization Group ERG2016 ICTP , Trieste, Italy, EU 23/09/2016 Plan of
Daniele Oriti Albert Einstein Institute 8th International Conference on the Exact Renormalization Group ERG2016 ICTP , Trieste, Italy, EU 23/09/2016
(Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)
QFT of spacetime, not defined on spacetime
(Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)
QFT of spacetime, not defined on spacetime a QFT for the building blocks of (quantum) space
(Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)
QFT of spacetime, not defined on spacetime a QFT for the building blocks of (quantum) space
ϕ : G×d → C
Quantum field theories over group manifold G (or corresponding Lie algebra) relevant classical phase space for “GFT quanta”:
(T ∗G)×d ' (g ⇥ G)×d
can reduce to subspaces in specific models depending on conditions on the field
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
example: d=4 d is dimension of “spacetime-to-be”; for gravity models, G = local gauge group of gravity (e.g. Lorentz group)
b b b b
1 2 3 4
N
arguments of GFT field:
bi ∈ su(2)
| b | ~ J = irrep of SU(2) ~ “area of triangles”
very general framework; interest rests on specific models/use (most interesting QG models are for Lorentz group in 4d)
(Boulatov, Ooguri, De Pietri, Freidel, Krasnov, Rovelli, Perez, DO, Livine, Baratin, ……)
QFT of spacetime, not defined on spacetime a QFT for the building blocks of (quantum) space
ϕ : G×d → C
Quantum field theories over group manifold G (or corresponding Lie algebra) relevant classical phase space for “GFT quanta”:
(T ∗G)×d ' (g ⇥ G)×d
can reduce to subspaces in specific models depending on conditions on the field
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
example: d=4 d is dimension of “spacetime-to-be”; for gravity models, G = local gauge group of gravity (e.g. Lorentz group)
b b b b
1 2 3 4
N
arguments of GFT field:
bi ∈ su(2)
| b | ~ J = irrep of SU(2) ~ “area of triangles”
a QFT for the building blocks of (quantum) space (d=4)
Fock vacuum: “no-space” (“emptiest”) state | 0 >
a QFT for the building blocks of (quantum) space (d=4)
Fock vacuum: “no-space” (“emptiest”) state | 0 > single field “quantum”: spin network vertex or tetrahedron (“building block of space”)
g g g g
1 2 3 4
g g g g
1 2 3 4
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
a QFT for the building blocks of (quantum) space (d=4)
Fock vacuum: “no-space” (“emptiest”) state | 0 > generic quantum state: arbitrary collection of spin network vertices (including glued ones) or tetrahedra (including glued ones)
j1 j2 j3 j4 j5 j6 j7 j8 j9 j10 j11 j12 j13 j14 j15 j16 j17 j18 j19 j20 j21 j22 j23single field “quantum”: spin network vertex or tetrahedron (“building block of space”)
g g g g
1 2 3 4
g g g g
1 2 3 4
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
a QFT for the building blocks of (quantum) space (d=4)
classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
a QFT for the building blocks of (quantum) space
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
a QFT for the building blocks of (quantum) space
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
simplest example (case d=4): simplicial setting specific combinatorics depends on model a QFT for the building blocks of (quantum) space
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
combinatorics of field arguments in interaction: gluing of 5 tetrahedra across common triangles, to form 4-simplex (“building block of spacetime”) simplest example (case d=4): simplicial setting specific combinatorics depends on model a QFT for the building blocks of (quantum) space
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
! ! ! ! ! !
!
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simplest example (case d=4): simplicial setting specific combinatorics depends on model a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum
Γ
a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices)
Γ
a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices)
Γ
Feynman amplitudes (model-dependent): equivalently:
spin networks ~ covariant LQG)
(with group+Lie algebra variables)
Reisenberger,Rovelli, ’00
a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices)
Γ
Feynman amplitudes (model-dependent): equivalently:
spin networks ~ covariant LQG)
(with group+Lie algebra variables)
Reisenberger,Rovelli, ’00
a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices)
Γ
Feynman amplitudes (model-dependent): equivalently:
spin networks ~ covariant LQG)
(with group+Lie algebra variables)
Reisenberger,Rovelli, ’00
a QFT for the building blocks of (quantum) space
Feynman perturbative expansion around trivial vacuum Feynman diagrams (obtained by convoluting propagators with interaction kernels) = = stranded diagrams dual to cellular complexes of arbitrary topology (simplicial case: simplicial complexes obtained by gluing d-simplices)
Γ
Feynman amplitudes (model-dependent): equivalently:
spin networks ~ covariant LQG)
(with group+Lie algebra variables)
Reisenberger,Rovelli, ’00
GFT as lattice quantum gravity: dynamical triangulations + quantum Regge calculus a QFT for the building blocks of (quantum) space
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
G12 1 2 3 4 G23 G34 G14 G13 G24
DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
1 g1
3
g1
1
g1
2
g4
2
g4
3
g4
1
4
2
g3 g3
3
g3
1
3 g
2 2
g3
2
g2
1
2 G12 1 2 3 4 G23 G34 G14 G13 G24
DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
1 g1
3
g1
1
g1
2
g4
2
g4
3
g4
1
4
2
g3 g3
3
g3
1
3 g
2 2
g3
2
g2
1
2 G12 1 2 3 4 G23 G34 G14 G13 G24
GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
1 g1
3
g1
1
g1
2
g4
2
g4
3
g4
1
4
2
g3 g3
3
g3
1
3 g
2 2
g3
2
g2
1
2
GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
GFT Hilbert space = Fock space of open spin network vertices - contains any LQG state (all spin networks) any LQG observable has a 2nd quantised, GFT counterpart choice of LQG dynamics (Hamiltonian constraint operator) translates into choice of GFT action
DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc]) DO, J. Ryan, J. Thurigen, ‘14
(LQG spin network states ~ many-particles states, “particle” ~ spin network vertex) second quantized version of Loop Quantum Gravity but dynamics not derived from canonical quantization of GR
(DO, 1310.7786 [gr-qc])
QFT methods (i.e. GFT reformulation of LQG and spin foam models) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
DO, J. Ryan, J. Thurigen, ‘14
Tensor models
Non-commutative geometry
Spin foam models Simplicial gravity path integrals (e.g. quantum Regge calculus) (causal) Dynamical Triangulations
—-> renormalizability of GFT for given discrete gravity path integral/spin foam amplitudes
—-> in particular, those with geometric interpretation (e.g. diffeomorphisms)
Ben Geloun, ’11; Girelli, Livine, ’11; Baratin, Girelli, DO, ‘11 Kegeles, DO, ‘15
—-> renormalizability of GFT for given discrete gravity path integral/spin foam amplitudes
—-> in particular, those with geometric interpretation (e.g. diffeomorphisms)
Ben Geloun, ’11; Girelli, Livine, ’11; Baratin, Girelli, DO, ‘11 Kegeles, DO, ‘15
controlling quantum dynamics of more and more interacting degrees of freedom
(as in QFT for condensed matter systems….) new analytic tools from QFT embedding
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s
few QG d.o.f.s in classical approx.! (e.g. discrete/lattice gravity) General Relativity! (continuum spacetime) full Quantum Gravity
N
h few QG d.o.f.s! (e.g. simple LQG spinnets)
continuum approximation very different (conceptually, technically) from classical approximation
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s
few QG d.o.f.s in classical approx.! (e.g. discrete/lattice gravity) General Relativity! (continuum spacetime) full Quantum Gravity
N
h few QG d.o.f.s! (e.g. simple LQG spinnets)
continuum approximation very different (conceptually, technically) from classical approximation N-direction (collective behaviour of many interacting degrees of freedom): continuum approximation h-direction: classical approximation
new (non-geometric, non-spatio-temporal) physical degrees of freedom (“building blocks”) for space-time new direction to explore: number of fundamental degrees of freedom (quantum) continuum, geometric space-time should be recovered in the regime of large number N of non-spatio-temporal d.o.f.s
few QG d.o.f.s in classical approx.! (e.g. discrete/lattice gravity) General Relativity! (continuum spacetime) full Quantum Gravity
N
h few QG d.o.f.s! (e.g. simple LQG spinnets)
continuum approximation very different (conceptually, technically) from classical approximation N-direction (collective behaviour of many interacting degrees of freedom): continuum approximation h-direction: classical approximation “well-understood” in spin foam models and discrete gravity
Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s
Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s
collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions
which of the macroscopic phases is described by a smooth geometry with matter fields?
Renormalization Group is crucial tool for taking into account the physics of more and more d.o.f.s in specific GFT case:
RG flow of effective actions & phase structure & phase transitions
collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions
which of the macroscopic phases is described by a smooth geometry with matter fields?
the issue:
controlling quantum dynamics of more and more interacting degrees of freedom
the issue:
up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations controlling quantum dynamics of more and more interacting degrees of freedom
the issue:
up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations controlling quantum dynamics of more and more interacting degrees of freedom need control over parameter space
(full theory space) expect different phases and phase transitions as result of quantum dynamics (what are the phases of LQG?)
Koslowski, ’07; DO, ‘07
the issue:
up to infinite refinement (infinite number of degrees of freedom), at least in simple approximations controlling quantum dynamics of more and more interacting degrees of freedom need control over parameter space
(full theory space) expect different phases and phase transitions as result of quantum dynamics (what are the phases of LQG?)
Koslowski, ’07; DO, ‘07
AL vacuum KS vacuum DG vacuum (or BF vacuum)
GFT condensate phase transitions ?
Ashtekar, Lewandowski, ’94 Koslowski, Sahlmann, ’10 Dittrich, Geiller, ’14, ‘15 Gielen, DO, Sindoni, ’13 Kegeles, DO, Tomlin, to appear
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
controlling the combinatorics of GFT Feynman diagrams to control the structure of divergences need to adapt/redefine many QFT notions: connectedness, subgraph contraction, Wick ordering, ….. key difficulties:
general strategy: treat GFTs as ordinary QFTs defined on Lie group manifold use group structures (Killing form, topology, etc) to define notion of scale and to set up mode integration subtleties of quantum gravity context at the level of interpretation scales: defined by propagator: e.g. spectrum of Laplacian on G = indexed by group representations
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
S(ϕ, ϕ) = 1 2 Z [dgi]ϕ(gi)K(gi)ϕ(gi) + λ D! Z [dgia]ϕ(gi1)....ϕ(¯ giD)V(gia, ¯ giD) + c.c.
controlling the combinatorics of GFT Feynman diagrams to control the structure of divergences need to adapt/redefine many QFT notions: connectedness, subgraph contraction, Wick ordering, ….. key difficulties: most results for Tensorial GFTs
locality principle and soft breaking of locality:
locality principle and soft breaking of locality: tensor invariant interactions
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
locality principle and soft breaking of locality: tensor invariant interactions
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
kinetic term = e.g. Laplacian on G
d
⇤
⇥=1
∆⇥ ⇥1
propagator
locality principle and soft breaking of locality: tensor invariant interactions
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
kinetic term = e.g. Laplacian on G
d
⇤
⇥=1
∆⇥ ⇥1
propagator “coloring” allows control over topology of Feynman diagrams
locality principle and soft breaking of locality: tensor invariant interactions
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
indexed by bipartite d-colored graphs (“bubbles”) ~ dual to d-cells with triangulated boundary
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
kinetic term = e.g. Laplacian on G
d
⇤
⇥=1
∆⇥ ⇥1
propagator require generalization of notions of “connectedness”, “contraction of high subgraphs”, “locality”, Wick ordering, …. taking into account internal structure of Feynman graphs, full combinatorics of dual cellular complex, results from crystallization theory (dipole moves) “coloring” allows control over topology of Feynman diagrams
example of Feynman diagram
“contraction of internal line” ~ dipole contraction
consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
b b b b
1 2 3 4
N
arguments of GFT field:
bi ∈ su(2)
gravity case: d=4 | b | ~ J = irrep of SU(2) ~ “area of triangles” from LQG from Regge calculus consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
“geometric” interpretation of the RG flow?
b b b b
1 2 3 4
N
arguments of GFT field:
bi ∈ su(2)
gravity case: d=4 | b | ~ J = irrep of SU(2) ~ “area of triangles” from LQG from Regge calculus consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
“geometric” interpretation of the RG flow?
for < J > small, < n > large, A finite (not too small), and small curvature
b b b b
1 2 3 4
N
arguments of GFT field:
bi ∈ su(2)
gravity case: d=4 | b | ~ J = irrep of SU(2) ~ “area of triangles” from LQG from Regge calculus consistent with cosmological interpretation of classical GFT fields and with results of GFT condensate cosmology (see later)
]
g1 g2 g3 g
1
g
2
g
3
h1 h2 h3
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
step by step, towards renormalizable 4d gravity models:
—> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models ———> generic (and robust?) asymptotic freedom
Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 Ben Geloun, Rivasseau, ’11 Carrozza, DO, Rivasseau, ’12. ‘13 Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 Ben Geloun, ’12; Carrozza, ‘14
many important lessons (e.g. learnt to deal with combinatorics and topology of spin foam complex)
]
g1 g2 g3 g
1
g
2
g
3
h1 h2 h3
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
step by step, towards renormalizable 4d gravity models:
—> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models ———> generic (and robust?) asymptotic freedom
Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 Ben Geloun, Rivasseau, ’11 Carrozza, DO, Rivasseau, ’12. ‘13 Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 Ben Geloun, ’12; Carrozza, ‘14
main open issues:
understand in full the geometric interpretation of UV/IR and of RG flow
many important lessons (e.g. learnt to deal with combinatorics and topology of spin foam complex)
]
g1 g2 g3 g
1
g
2
g
3
h1 h2 h3
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
step by step, towards renormalizable 4d gravity models:
—> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models ———> generic (and robust?) asymptotic freedom
Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 Ben Geloun, Rivasseau, ’11 Carrozza, DO, Rivasseau, ’12. ‘13 Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 Ben Geloun, ’12; Carrozza, ‘14
recent results:
main open issues:
understand in full the geometric interpretation of UV/IR and of RG flow
many important lessons (e.g. learnt to deal with combinatorics and topology of spin foam complex)
]
g1 g2 g3 g
1
g
2
g
3
h1 h2 h3
S(ϕ, ϕ) =
tbIb(ϕ, ϕ) .
step by step, towards renormalizable 4d gravity models:
—> non-abelian ( SU(2) ) ——> SO(4) or SO(3,1) with simplicity constraints: first results on BC-like 4d models ———> generic (and robust?) asymptotic freedom
Ben Geloun, Bonzom, ’11; Ben Geloun, ‘13 Ben Geloun, Rivasseau, ’11 Carrozza, DO, Rivasseau, ’12. ‘13 Lahoche, DO, ’15; Carrozza, Lahoche, DO, ‘16 Ben Geloun, ’12; Carrozza, ‘14
the GFT proposal:
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
controlling the continuum limit ~ evaluating GFT path integral (in some non-perturbative approximation)
Benedetti, Ben Geloun, DO, Martini, Lahoche, Carrozza, Douarte, …. Freidel, Louapre, Noui, Magnen, Smerlak, Gurau, Rivasseau, Tanasa, Dartois, Delpouve, …..
(computing full SF sum)
the GFT proposal:
Z = Z DϕDϕ ei Sλ(ϕ,ϕ) = X
Γ
λNΓ sym(Γ) AΓ
controlling the continuum limit ~ evaluating GFT path integral (in some non-perturbative approximation) two directions:
non-perturbative resummation of perturbative (SF) series variety of techniques:
Benedetti, Ben Geloun, DO, Martini, Lahoche, Carrozza, Douarte, …. Freidel, Louapre, Noui, Magnen, Smerlak, Gurau, Rivasseau, Tanasa, Dartois, Delpouve, …..
(computing full SF sum)
recent results:
FRG for (tensorial) GFT models
recent results:
FRG for (tensorial) GFT models (similar to matrix model but distinctively field-theoretic)
Eichhorn, Koslowski, ‘14
recent results:
FRG for (tensorial) GFT models (similar to matrix model but distinctively field-theoretic)
Eichhorn, Koslowski, ‘14
0.00 0.01 0.02 0.03 0.04 1.0 0.8 0.6 0.4 0.2 0.0 0.2
ΛN mN
generically (so far): two FPs (Gaussian-UV, Wilson-Fisher-IR) asymptotic freedom
Benedetti, Ben Geloun, DO, ’14 ; Ben Geloun, Martini, DO, ’15, ’16, Benedetti, Lahoche, ’15; Douarte, DO, ‘16 Krajewski, Toriumi, ‘14
regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
ΓN[ϕ, ϕ] = sup
J,J
⇢ Tr(J · ϕ) + Tr(J · ϕ) WN[J, J] ∆SN[ϕ, ϕ]
k k
k
Wetterich equation:
k
regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
ΓN[ϕ, ϕ] = sup
J,J
⇢ Tr(J · ϕ) + Tr(J · ϕ) WN[J, J] ∆SN[ϕ, ϕ]
k k
k
boundary conditions:
Γk=0[ϕ, ϕ] = Γ[ϕ, ϕ] , Γk=Λ[ϕ, ϕ] = S[ϕ, ϕ] re ϕ = hφi.
Wetterich equation:
k
regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
ΓN[ϕ, ϕ] = sup
J,J
⇢ Tr(J · ϕ) + Tr(J · ϕ) WN[J, J] ∆SN[ϕ, ϕ]
k k
k
boundary conditions:
Γk=0[ϕ, ϕ] = Γ[ϕ, ϕ] , Γk=Λ[ϕ, ϕ] = S[ϕ, ϕ] re ϕ = hφi.
Wetterich equation:
k
regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
ΓN[ϕ, ϕ] = sup
J,J
⇢ Tr(J · ϕ) + Tr(J · ϕ) WN[J, J] ∆SN[ϕ, ϕ]
k k
k
computing the effective action solving the Wetterich equation amounts to solving the GFT path integral
boundary conditions:
Γk=0[ϕ, ϕ] = Γ[ϕ, ϕ] , Γk=Λ[ϕ, ϕ] = S[ϕ, ϕ] re ϕ = hφi.
Wetterich equation:
k
Wetterich equation expanded in field powers, with all possible contractions; truncation matching classical action system of flow equations is generically non-homogeneous, because of combinatorial patterns of contractions for compact groups, it is also non-autonomous, due to hidden scale (size of group) regularised path integral: ZN[J, J] = eWN [J,J] =
Z dφdφ eS[φ,φ]∆SN [φ,φ]+Tr(J·φ)+Tr(J·φ)
k
k k
Rk(p, p0) = θ(k2 − Σsp2
s )Zk(k2 − Σsp2 s )δ(p − p0)
( regulator cutting off IR modes (UV well-defined with appropriate choice of IR regulator)
∆SN[φ, φ] = Tr(φ · RN · φ) = X
P,P0
φP RN(P; P0) φP0
k k k
ΓN[ϕ, ϕ] = sup
J,J
⇢ Tr(J · ϕ) + Tr(J · ϕ) WN[J, J] ∆SN[ϕ, ϕ]
k k
k
computing the effective action solving the Wetterich equation amounts to solving the GFT path integral
φ φ φ φ φ φ φ φ φ φ φ φ
the model:
S[φ, φ] = (2π)d Z
R×d[dxi]d i=1 φ(x1, . . . , xd)
✓
X
s=1
4s + µ ◆ φ(x1, . . . , xd) +λ 2 (2π)2d Z
R×2d[dxi]d i=1[dx0 j]d j=1
φ(x1, x2, . . . , xd)φ(x0
1, x2, . . . , xd)φ(x0 1, x0 2, . . . , x0 d)φ(x1, x0 2, . . . , x0 d)
n
G = R
φ φ φ φ φ φ φ φ φ φ φ φ
the model:
S[φ, φ] = (2π)d Z
R×d[dxi]d i=1 φ(x1, . . . , xd)
✓
X
s=1
4s + µ ◆ φ(x1, . . . , xd) +λ 2 (2π)2d Z
R×2d[dxi]d i=1[dx0 j]d j=1
φ(x1, x2, . . . , xd)φ(x0
1, x2, . . . , xd)φ(x0 1, x0 2, . . . , x0 d)φ(x1, x0 2, . . . , x0 d)
n
G = R Γk[ϕ, ϕ] = Z
R⇥d[dpi]d i=1 ϕ12...d(Zk
X
s
p2
s + µk)ϕ12...d
+ λk 2 Z
R⇥2d[dpi]d i=1[dp0 j]d j=1
ϕ12...dϕ102...dϕ1020...d0ϕ120...d0 + sym n 1, 2, . . . , d
φ φ φ φ φ φ φ φ φ φ φ φ
the model:
S[φ, φ] = (2π)d Z
R×d[dxi]d i=1 φ(x1, . . . , xd)
✓
X
s=1
4s + µ ◆ φ(x1, . . . , xd) +λ 2 (2π)2d Z
R×2d[dxi]d i=1[dx0 j]d j=1
φ(x1, x2, . . . , xd)φ(x0
1, x2, . . . , xd)φ(x0 1, x0 2, . . . , x0 d)φ(x1, x0 2, . . . , x0 d)
n
G = R
step 1: compactly configuration space to U(1)^d, with step 2: determine (non-standard) scaling of coupling constants step 3: take non-compact limit so to regularise the most divergent contributions to the RG flow la V = ⇣
2π l
⌘d
Γk[ϕ, ϕ] = Z
R⇥d[dpi]d i=1 ϕ12...d(Zk
X
s
p2
s + µk)ϕ12...d
+ λk 2 Z
R⇥2d[dpi]d i=1[dp0 j]d j=1
ϕ12...dϕ102...dϕ1020...d0ϕ120...d0 + sym n 1, 2, . . . , d
scaling of couplings:
Zk = Z klχkχ, µk = µkZ klχk2χ, λk = λkZ
2 klξk4ξ
scaling of couplings:
Zk = Z klχkχ, µk = µkZ klχk2χ, λk = λkZ
2 klξk4ξ
(regularized) flow equations: non-autonomous, non-homogeneous; matches TGFT on U(1)^d
8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : ηk = λklξkσ l2χk2(2−χ)(1 + µk)2 n (ηk − χ) h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd−1 ld−1 + 2(d − 1)k l i + 2 h (d − 1)k l + π
d−1 2
ΓE ⇣
d−1 2
⌘ kd−1 ld−1 io + χ β(µk) = − d λklξkσ l2χk6−2χ(1 + µk)2 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4 3 k3 l i + 2 h 2k3 l + π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 io − ηkµk − (2 − χ)µk β(λk) = 2λ
2 klξkσ
l2χk6−2χ(1 + µk)3 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4(2d − 1) 3 k3 l + 2δd,3k2i + 2 h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 + 2(2d − 1)k3 l + 2δd,3k2io − 2ηkλk − σλk (49)
scaling of couplings:
Zk = Z klχkχ, µk = µkZ klχk2χ, λk = λkZ
2 klξk4ξ
most divergent contributions finite for:
ξ = 2χ + (d − 1)
and redefined anomalous dimension:g η0
k = ηk − χ:
ηk = 1 Zk β(Zk) = 1 Zk β(Zk) + χ
(regularized) flow equations: non-autonomous, non-homogeneous; matches TGFT on U(1)^d
8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : ηk = λklξkσ l2χk2(2−χ)(1 + µk)2 n (ηk − χ) h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd−1 ld−1 + 2(d − 1)k l i + 2 h (d − 1)k l + π
d−1 2
ΓE ⇣
d−1 2
⌘ kd−1 ld−1 io + χ β(µk) = − d λklξkσ l2χk6−2χ(1 + µk)2 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4 3 k3 l i + 2 h 2k3 l + π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 io − ηkµk − (2 − χ)µk β(λk) = 2λ
2 klξkσ
l2χk6−2χ(1 + µk)3 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4(2d − 1) 3 k3 l + 2δd,3k2i + 2 h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 + 2(2d − 1)k3 l + 2δd,3k2io − 2ηkλk − σλk (49)
scaling of couplings:
Zk = Z klχkχ, µk = µkZ klχk2χ, λk = λkZ
2 klξk4ξ
most divergent contributions finite for:
ξ = 2χ + (d − 1)
and redefined anomalous dimension:g η0
k = ηk − χ:
ηk = 1 Zk β(Zk) = 1 Zk β(Zk) + χ
(regularized) flow equations: non-autonomous, non-homogeneous; matches TGFT on U(1)^d
8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : ηk = λklξkσ l2χk2(2−χ)(1 + µk)2 n (ηk − χ) h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd−1 ld−1 + 2(d − 1)k l i + 2 h (d − 1)k l + π
d−1 2
ΓE ⇣
d−1 2
⌘ kd−1 ld−1 io + χ β(µk) = − d λklξkσ l2χk6−2χ(1 + µk)2 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4 3 k3 l i + 2 h 2k3 l + π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 io − ηkµk − (2 − χ)µk β(λk) = 2λ
2 klξkσ
l2χk6−2χ(1 + µk)3 n (η − χ) h π
d−1 2
ΓE ⇣
d+3 2
⌘ kd+1 ld−1 + 4(2d − 1) 3 k3 l + 2δd,3k2i + 2 h π
d−1 2
ΓE ⇣
d+1 2
⌘ kd+1 ld−1 + 2(2d − 1)k3 l + 2δd,3k2io − 2ηkλk − σλk (49)
now can take thermodynamic limit
8 > > > > > > > > > > > > < > > > > > > > > > > > > : ηk = 2π
d−1 2
ΓE ⇣
d1 2
⌘ λk (1 + µk)2 h ηk d − 1 + 1 i β(µk) = −2d π
d−1 2
ΓE ⇣
d+1 2
⌘ λk (1 + µk)2 h ηk d + 1 + 1 i − ηkµk − 2µk β(λk) = 4π
d−1 2
ΓE ⇣
d+1 2
⌘ λ
2 k
(1 + µk)3 h ηk d + 1 + 1 i − 2ηkλk − (5 − d)λk
flow equations for couplings: autonomous, still non-homogeneous
8 > > > > > > > > > > > > < > > > > > > > > > > > > : ηk = 2π
d−1 2
ΓE ⇣
d1 2
⌘ λk (1 + µk)2 h ηk d − 1 + 1 i β(µk) = −2d π
d−1 2
ΓE ⇣
d+1 2
⌘ λk (1 + µk)2 h ηk d + 1 + 1 i − ηkµk − 2µk β(λk) = 4π
d−1 2
ΓE ⇣
d+1 2
⌘ λ
2 k
(1 + µk)3 h ηk d + 1 + 1 i − 2ηkλk − (5 − d)λk
flow equations for couplings: autonomous, still non-homogeneous
0.000 0.005 0.010 0.015 0.020
0.0
λN μN
0.000 0.005 0.010 0.015 0.020λN μN
d=3 d=4
8 > > > > > > > > > > > > < > > > > > > > > > > > > : ηk = 2π
d−1 2
ΓE ⇣
d1 2
⌘ λk (1 + µk)2 h ηk d − 1 + 1 i β(µk) = −2d π
d−1 2
ΓE ⇣
d+1 2
⌘ λk (1 + µk)2 h ηk d + 1 + 1 i − ηkµk − 2µk β(λk) = 4π
d−1 2
ΓE ⇣
d+1 2
⌘ λ
2 k
(1 + µk)3 h ηk d + 1 + 1 i − 2ηkλk − (5 − d)λk
flow equations for couplings: autonomous, still non-homogeneous
general features independent of rank-d: Gaussian-UV FP, Wilson-Fisher-IR FP asymptotic freedom
2nd non-G IR FP at negative coupling
0.000 0.005 0.010 0.015 0.020
0.0
λN μN
0.000 0.005 0.010 0.015 0.020λN μN
d=3 d=4
similar model with gauge invariance (imposed in both kinetic and interaction terms):
Γk[ϕ, ϕ] = Z dp ϕ(p) h ZkΣsp2
s + µk
i ϕ(p)δ(Σp) + λk 2 Z dpdp0 ϕ12...dϕ102...dϕ1020...d0ϕ120...d0δ(Σp)δ(Σp0)δ(p0
1 + p2 + · · · + pd)δ(p1 + p0 2 + · ·
similar model with gauge invariance (imposed in both kinetic and interaction terms):
Γk[ϕ, ϕ] = Z dp ϕ(p) h ZkΣsp2
s + µk
i ϕ(p)δ(Σp) + λk 2 Z dpdp0 ϕ12...dϕ102...dϕ1020...d0ϕ120...d0δ(Σp)δ(Σp0)δ(p0
1 + p2 + · · · + pd)δ(p1 + p0 2 + · ·
8 > > > > > > > > > > > > < > > > > > > > > > > > > : η0
k =
dλk (1 + µk)2 π
d−2 2
(d − 1)
3 2
n η0
k
1 ΓE ⇣
d 2
⌘ + 2 ΓE ⇣
d2 2
⌘
dλk (1 + µk)2 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
k + 2)µk
βd6=4(λk) = 2λ
2 k
(1 + µk)3 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
kλk + (d − 6)λk
similar RG flow equations, different scaling dimensions of couplings:
similar model with gauge invariance (imposed in both kinetic and interaction terms):
Γk[ϕ, ϕ] = Z dp ϕ(p) h ZkΣsp2
s + µk
i ϕ(p)δ(Σp) + λk 2 Z dpdp0 ϕ12...dϕ102...dϕ1020...d0ϕ120...d0δ(Σp)δ(Σp0)δ(p0
1 + p2 + · · · + pd)δ(p1 + p0 2 + · ·
8 > > > > > > > > > > > > < > > > > > > > > > > > > : η0
k =
dλk (1 + µk)2 π
d−2 2
(d − 1)
3 2
n η0
k
1 ΓE ⇣
d 2
⌘ + 2 ΓE ⇣
d2 2
⌘
dλk (1 + µk)2 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
k + 2)µk
βd6=4(λk) = 2λ
2 k
(1 + µk)3 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
kλk + (d − 6)λk
similar RG flow equations, different scaling dimensions of couplings:
μ
d=4
similar model with gauge invariance (imposed in both kinetic and interaction terms):
Γk[ϕ, ϕ] = Z dp ϕ(p) h ZkΣsp2
s + µk
i ϕ(p)δ(Σp) + λk 2 Z dpdp0 ϕ12...dϕ102...dϕ1020...d0ϕ120...d0δ(Σp)δ(Σp0)δ(p0
1 + p2 + · · · + pd)δ(p1 + p0 2 + · ·
8 > > > > > > > > > > > > < > > > > > > > > > > > > : η0
k =
dλk (1 + µk)2 π
d−2 2
(d − 1)
3 2
n η0
k
1 ΓE ⇣
d 2
⌘ + 2 ΓE ⇣
d2 2
⌘
dλk (1 + µk)2 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
k + 2)µk
βd6=4(λk) = 2λ
2 k
(1 + µk)3 π
d−2 2
√ d − 1 n η0
k
1 ΓE ⇣
d+2 2
⌘ + 2 ΓE ⇣
d 2
⌘
kλk + (d − 6)λk
similar RG flow equations, different scaling dimensions of couplings:
μ
d=4
again, general features independent of rank-d: Gaussian-UV FP (asymptotic freedom), Wilson-Fisher-IR FP symmetric phase + broken or condensate phase 2nd non-G IR FP at negative coupling
the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)?
Quantum Gravity problem: identify microscopic d.o.f. of quantum spacetime and their fundamental dynamics various models: loop quantum cosmology, .... derive effective (QG-inspired) models for fundamental (quantum) cosmology: explain features of early Universe, obtain testable QG predictions the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)?
Quantum Gravity problem: identify microscopic d.o.f. of quantum spacetime and their fundamental dynamics various models: loop quantum cosmology, .... derive effective (QG-inspired) models for fundamental (quantum) cosmology: explain features of early Universe, obtain testable QG predictions
also work by:
the issue: identify relevant phase for effective continuum geometry extract effective continuum dynamics and relate it to GR is GR a good effective description of LQG/SF/GFT in some approximation (in one continuum phase)?
re-thinking the “Cosmological Principle”: “every point is equivalent to any other” ~ homogeneity of space
re-thinking the “Cosmological Principle”: “every point is equivalent to any other” ~ homogeneity of space really means: a certain approximation is assumed valid: universe is in state where inhomogeneities can be neglected, in relation to dynamics of homogeneous modes ~ universe is in state where effects on largest wavelengths of shorter wavelengths is negligible ~ can neglect wavelengths (much) shorter than scale factor
re-thinking the “Cosmological Principle”: “every point is equivalent to any other” ~ homogeneity of space really means: a certain approximation is assumed valid: universe is in state where inhomogeneities can be neglected, in relation to dynamics of homogeneous modes ~ universe is in state where effects on largest wavelengths of shorter wavelengths is negligible ~ can neglect wavelengths (much) shorter than scale factor very similar in spirit to hydrodynamic approximation: dynamics of microscopic degrees of freedom can be neglected + effects of small wavelengths can be neglected degrees of freedom of local region can describe whole of system (in a coarse grained, statistical sense) i.e. whole universe (dynamics) well-approximated by local patch (dynamics)
re-thinking the “Cosmological Principle”: “every point is equivalent to any other” ~ homogeneity of space really means: a certain approximation is assumed valid: universe is in state where inhomogeneities can be neglected, in relation to dynamics of homogeneous modes ~ universe is in state where effects on largest wavelengths of shorter wavelengths is negligible ~ can neglect wavelengths (much) shorter than scale factor very similar in spirit to hydrodynamic approximation: dynamics of microscopic degrees of freedom can be neglected + effects of small wavelengths can be neglected degrees of freedom of local region can describe whole of system (in a coarse grained, statistical sense) i.e. whole universe (dynamics) well-approximated by local patch (dynamics) cosmology is (non-linear) dynamics for such density and for geometric (global) observables computed from it end result of (any) proper construction: basic variable is “single-patch density” with arguments the geometric data of minisuperspace
key strategy: coarse graining of QG configurations coarse graining of QG (quantum) dynamics
very difficult in general (see comparatively simpler problem of coarse graining classical GR) (see also analogous problem in condensed matter theory) key strategy: coarse graining of QG configurations coarse graining of QG (quantum) dynamics
very difficult in general (see comparatively simpler problem of coarse graining classical GR) (see also analogous problem in condensed matter theory) key strategy: coarse graining of QG configurations coarse graining of QG (quantum) dynamics
quantum condensates (BEC) effective hydrodynamics directly read out of microscopic quantum dynamics (in simplest approximation)
DO, L. Sindoni, E. Wilson-Ewing, ’16; M. De Cesare, M. Sakellariadou, ‘16
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation
DO, L. Sindoni, E. Wilson-Ewing, ’16; M. De Cesare, M. Sakellariadou, ‘16
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation Quantum GFT condensates are continuum homogeneous (quantum) spaces
DO, L. Sindoni, E. Wilson-Ewing, ’16; M. De Cesare, M. Sakellariadou, ‘16
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation Quantum GFT condensates are continuum homogeneous (quantum) spaces
DO, L. Sindoni, E. Wilson-Ewing, ’16; M. De Cesare, M. Sakellariadou, ‘16
|σ⌦ := exp (ˆ σ) |0⌦
⌅ e σ(gIk) = σ(gI)
⇥
| ⇧ | ⇧
ˆ σ := ⌅ d4g σ(gI) ˆ ϕ†(gI)
superposition of infinitely many SN dofs e.g. (simplest): GFT field coherent state described by single collective wave function (depending on homogeneous anisotropic geometric data)
σ (D) D ' {geometries of tetrahedron} ' ' {continuum spatial geometries at a point} ' ' minisuperspace of homogeneous geometries
e3 e1 e2
describes geometric tetrahedron (closure + simplicity constraints)
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
BAB
i
= ⇥i
jkeA j eB k
e3 e1 e2
describes geometric tetrahedron (closure + simplicity constraints)
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
BAB
i
= ⇥i
jkeA j eB k
many results in LQG, simplicial geometry
|BI(m) :=
N
⌃
m=1
ˆ ˜ ⇤†(B1(m), . . . , B4(m))|0
e3 e1 e2
describes geometric tetrahedron (closure + simplicity constraints)
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
BAB
i
= ⇥i
jkeA j eB k
many results in LQG, simplicial geometry
|BI(m) :=
N
⌃
m=1
ˆ ˜ ⇤†(B1(m), . . . , B4(m))|0
e3 e1 e2
describes geometric tetrahedron (closure + simplicity constraints)
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
BAB
i
= ⇥i
jkeA j eB k
many results in LQG, simplicial geometry
construct:
gij = 1 8 tr(B1B2B3)⇥i
kl⇥j mn ˜
Bkm ˜ Bln ,
s ˜ Bij := BAB
i
Bj AB in terms of the biv
interpretation: spatial metric coefficients (and conjugate variables) “sampled” at N points
BI(m) ↔ gij(xm) ↔ ai(xm) gI(m) ↔ Kij(xm) ↔ pai(xm)
|BI(m) :=
N
⌃
m=1
ˆ ˜ ⇤†(B1(m), . . . , B4(m))|0
e3 e1 e2
describes geometric tetrahedron (closure + simplicity constraints)
ϕ(g1, g2, g3, g4) ↔ ϕ(B1, B2, B3, B4) → C
BAB
i
= ⇥i
jkeA j eB k
many results in LQG, simplicial geometry
gij(m) = gij(k) ⌅k, m = 1, . . . , N.
i.e. all GFT quanta are labelled by the same (gauge invariant) data
construct:
gij = 1 8 tr(B1B2B3)⇥i
kl⇥j mn ˜
Bkm ˜ Bln ,
s ˜ Bij := BAB
i
Bj AB in terms of the biv
interpretation: spatial metric coefficients (and conjugate variables) “sampled” at N points
BI(m) ↔ gij(xm) ↔ ai(xm) gI(m) ↔ Kij(xm) ↔ pai(xm)
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Ψ
1 N!
N
Y
m=1
Φ(Bi(m))
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Ψ
1 N!
N
Y
m=1
Φ(Bi(m))
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Ψ
1 N!
N
Y
m=1
Φ(Bi(m))
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Ψ
1 N!
N
Y
m=1
Φ(Bi(m))
quantum GFT condensates are continuum homogeneous (quantum) spaces
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Ψ
1 N!
N
Y
m=1
Φ(Bi(m))
quantum GFT condensates are continuum homogeneous (quantum) spaces
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation Quantum GFT condensates are continuum homogeneous (quantum) spaces described by single collective wave function (depending on homogeneous anisotropic geometric data)
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation problem 2: extract from fundamental theory an effective macroscopic dynamics for such states Quantum GFT condensates are continuum homogeneous (quantum) spaces described by single collective wave function (depending on homogeneous anisotropic geometric data)
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation problem 2: extract from fundamental theory an effective macroscopic dynamics for such states Quantum GFT condensates are continuum homogeneous (quantum) spaces following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data)
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation problem 2: extract from fundamental theory an effective macroscopic dynamics for such states Quantum GFT condensates are continuum homogeneous (quantum) spaces following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data) QG (GFT) analogue of Gross-Pitaevskii hydrodynamic equation in BECs is non-linear and non-local extension of (loop) quantum cosmology equation for collective wave function
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation problem 2: extract from fundamental theory an effective macroscopic dynamics for such states Quantum GFT condensates are continuum homogeneous (quantum) spaces following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data) QG (GFT) analogue of Gross-Pitaevskii hydrodynamic equation in BECs is non-linear and non-local extension of (loop) quantum cosmology equation for collective wave function
cosmology as QG hydrodynamics!!!
follow closely procedure used in real BECs single-particle GFT condensate:
|σ⌦ := exp (ˆ σ) |0⌦
⌅ e σ(gIk) = σ(gI)
⇥
| ⇧ | ⇧
ˆ σ := ⌅ d4g σ(gI) ˆ ϕ†(gI)
PRL, arXiv:1303.3576 [gr-qc]; JHEP, arXiv:1311.1238 [gr-qc]
superposition of infinitely many SN dofs
follow closely procedure used in real BECs single-particle GFT condensate:
|σ⌦ := exp (ˆ σ) |0⌦
⌅ e σ(gIk) = σ(gI)
⇥
| ⇧ | ⇧
ˆ σ := ⌅ d4g σ(gI) ˆ ϕ†(gI)
PRL, arXiv:1303.3576 [gr-qc]; JHEP, arXiv:1311.1238 [gr-qc]
superposition of infinitely many SN dofs
applied to (coherent) GFT condensate state, gives equation for “wave function”:
Z [dg0
i] ˜
K(gi, g0
i)σ(g0 i) + λ
δ ˜ V δϕ(gi)|ϕ⌘σ = 0
from truncation of SD equations for GFT model basically (up to some approximations), the “classical GFT eqns”
similar equations to M. Bojowald et al., arXiv:1210.8138 [gr-qc]
follow closely procedure used in real BECs single-particle GFT condensate:
|σ⌦ := exp (ˆ σ) |0⌦
⌅ e σ(gIk) = σ(gI)
⇥
| ⇧ | ⇧
ˆ σ := ⌅ d4g σ(gI) ˆ ϕ†(gI)
PRL, arXiv:1303.3576 [gr-qc]; JHEP, arXiv:1311.1238 [gr-qc]
superposition of infinitely many SN dofs no perturbative (spin foam) expansion - infinite superposition of SF amplitudes
applied to (coherent) GFT condensate state, gives equation for “wave function”:
Z [dg0
i] ˜
K(gi, g0
i)σ(g0 i) + λ
δ ˜ V δϕ(gi)|ϕ⌘σ = 0
from truncation of SD equations for GFT model basically (up to some approximations), the “classical GFT eqns”
similar equations to M. Bojowald et al., arXiv:1210.8138 [gr-qc]
follow closely procedure used in real BECs non-linear and non-local extension of quantum cosmology-like equation for “collective wave function” QG (GFT) analogue of Gross-Pitaevskii hydrodynamic equation in BECs single-particle GFT condensate:
|σ⌦ := exp (ˆ σ) |0⌦
⌅ e σ(gIk) = σ(gI)
⇥
| ⇧ | ⇧
ˆ σ := ⌅ d4g σ(gI) ˆ ϕ†(gI)
PRL, arXiv:1303.3576 [gr-qc]; JHEP, arXiv:1311.1238 [gr-qc]
superposition of infinitely many SN dofs no perturbative (spin foam) expansion - infinite superposition of SF amplitudes
applied to (coherent) GFT condensate state, gives equation for “wave function”:
Z [dg0
i] ˜
K(gi, g0
i)σ(g0 i) + λ
δ ˜ V δϕ(gi)|ϕ⌘σ = 0
from truncation of SD equations for GFT model basically (up to some approximations), the “classical GFT eqns”
similar equations to M. Bojowald et al., arXiv:1210.8138 [gr-qc]
G=SU(2), dynamics encodes embedding into SL(2,C) ~ simplicity constraints)
Engle,Pereira, Rovelli, Livine, ’07; Freidel, Krasnov, ‘07 DO, Sindoni, Wilson-Ewing, ‘16
ˆ ϕ(gv) → ˆ ϕ(gv, φ).
− V5(gva, φa) = V5(gva) Y δ(φa − φ1). → → − K2(gv1, gv2, φ1, φ2) = K2(gv1, gv2, (φ1 − φ2)2), Y
G=SU(2), dynamics encodes embedding into SL(2,C) ~ simplicity constraints)
Engle,Pereira, Rovelli, Livine, ’07; Freidel, Krasnov, ‘07 DO, Sindoni, Wilson-Ewing, ‘16
ˆ ϕ(gv) → ˆ ϕ(gv, φ).
− V5(gva, φa) = V5(gva) Y δ(φa − φ1). → → − K2(gv1, gv2, φ1, φ2) = K2(gv1, gv2, (φ1 − φ2)2), Y
|σi ⇠ exp ✓Z dgvdφ σ(gv, φ)ˆ φ†(gv, φ) ◆ |0i, des, σ(gv, φ) ! σj(φ) s network node exists.
G=SU(2), dynamics encodes embedding into SL(2,C) ~ simplicity constraints)
Engle,Pereira, Rovelli, Livine, ’07; Freidel, Krasnov, ‘07 DO, Sindoni, Wilson-Ewing, ‘16
ˆ ϕ(gv) → ˆ ϕ(gv, φ).
− V5(gva, φa) = V5(gva) Y δ(φa − φ1). → → − K2(gv1, gv2, φ1, φ2) = K2(gv1, gv2, (φ1 − φ2)2), Y
|σi ⇠ exp ✓Z dgvdφ σ(gv, φ)ˆ φ†(gv, φ) ◆ |0i, des, σ(gv, φ) ! σj(φ) s network node exists.
G=SU(2), dynamics encodes embedding into SL(2,C) ~ simplicity constraints)
Engle,Pereira, Rovelli, Livine, ’07; Freidel, Krasnov, ‘07 DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
functions A, B, w define the details of the EPRL model
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
σj(φ) = ρj(φ)eiθj(φ)
Ej ≈ (ρ0
j)2 + ρ2 j(θ0 j)2 − m2 jρ2.
Qj ≈ ρ2
j θ0 j.
h m2
j = Bj/Aj.
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
σj(φ) = ρj(φ)eiθj(φ)
Ej ≈ (ρ0
j)2 + ρ2 j(θ0 j)2 − m2 jρ2.
Qj ≈ ρ2
j θ0 j.
h m2
j = Bj/Aj.
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
V (φ) = X
j
Vj¯ σj(φ)σj(φ) = X
j
Vjρj(φ)2,
universe volume (at fixed “time”)
e Vj ∼ j3/2`3
Pl
vity acting on an
σj(φ) = ρj(φ)eiθj(φ)
Ej ≈ (ρ0
j)2 + ρ2 j(θ0 j)2 − m2 jρ2.
Qj ≈ ρ2
j θ0 j.
h m2
j = Bj/Aj.
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
therefore πφ = hσ|ˆ πφ(φ)|σi definition of t i = ~ P
j Qj
re the Gross-P
momentum of scalar field (at fixed “time”) constant of motion ~ continuity equation
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
V (φ) = X
j
Vj¯ σj(φ)σj(φ) = X
j
Vjρj(φ)2,
universe volume (at fixed “time”)
e Vj ∼ j3/2`3
Pl
vity acting on an
σj(φ) = ρj(φ)eiθj(φ)
Ej ≈ (ρ0
j)2 + ρ2 j(θ0 j)2 − m2 jρ2.
Qj ≈ ρ2
j θ0 j.
h m2
j = Bj/Aj.
DO, Sindoni, Wilson-Ewing, ‘16
Aj∂2
φσj(φ) Bjσj(φ) + wj¯
σj(φ)4 = 0.
interaction terms sub-dominant (dilute-gas approx., consistent with simple approximation of vacuum state)
Ej = Aj|∂φσj(φ)|2 Bj|σj(φ)|2 + 2 5Re
Qj = i 2 h ¯ σj(φ)∂φσj(φ) σj(φ)∂φ¯ σj(φ) i .
therefore πφ = hσ|ˆ πφ(φ)|σi definition of t i = ~ P
j Qj
re the Gross-P
momentum of scalar field (at fixed “time”) constant of motion ~ continuity equation
ρ00
j − Q2 j
ρ3
j
− m2
jρj ≈ 0,
V (φ) = X
j
Vj¯ σj(φ)σj(φ) = X
j
Vjρj(φ)2,
universe volume (at fixed “time”)
e Vj ∼ j3/2`3
Pl
vity acting on an
σj(φ) = ρj(φ)eiθj(φ)
Ej ≈ (ρ0
j)2 + ρ2 j(θ0 j)2 − m2 jρ2.
Qj ≈ ρ2
j θ0 j.
h m2
j = Bj/Aj.
energy density of scalar field (at fixed “time”)
ρ = π2
φ
2V 2 = ~2(P
j Qj)2
2(P
j Vjρ2 j)2,
DO, Sindoni, Wilson-Ewing, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
DO, Sindoni, Wilson-Ewing, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
9j / ρj(φ) 6= 0 8φ
j Vjρ2 j,
remains positive at all times generic quantum bounce!
DO, Sindoni, Wilson-Ewing, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
9j / ρj(φ) 6= 0 8φ
j Vjρ2 j,
remains positive at all times generic quantum bounce!
DO, Sindoni, Wilson-Ewing, ‘16 + primordial accelleration
De Cesare, Sakellariadou, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
9j / ρj(φ) 6= 0 8φ
j Vjρ2 j,
remains positive at all times generic quantum bounce!
n ρ2
j |Ej|/m2 j and ρ4 j Q2 j/m2 j (
rge volume, but of small space-tim ✓ V 0 3V ◆2 = 2 P
j Vj mj ρ2 j
3 P
j Vjρ2 j
!2
P V 00 V = 4 P
j Vjm2 jρ2 j
P
j Vjρ2 j
.
eqns if m2
j ≈ 3GN
DO, Sindoni, Wilson-Ewing, ‘16 + primordial accelleration
De Cesare, Sakellariadou, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
9j / ρj(φ) 6= 0 8φ
j Vjρ2 j,
remains positive at all times generic quantum bounce!
n ρ2
j |Ej|/m2 j and ρ4 j Q2 j/m2 j (
rge volume, but of small space-tim ✓ V 0 3V ◆2 = 2 P
j Vj mj ρ2 j
3 P
j Vjρ2 j
!2
P V 00 V = 4 P
j Vjm2 jρ2 j
P
j Vjρ2 j
.
eqns if m2
j ≈ 3GN
σj(φ) = 0, for all j 6= jo.
✓ V 0 3V ◆2 = 4πG 3 ✓ 1 ρ ρc ◆ + VjoEjo 9V ,
✓ ◆ h ρc = 6πG~2/V 2
jo ⇠ (6π/j3
that the second term is a quan
LQC-like modified dynamics!
DO, Sindoni, Wilson-Ewing, ‘16 + primordial accelleration
De Cesare, Sakellariadou, ‘16
effective dynamics for volume - generalised Friedmann equations: ✓ V 0 3V ◆2 = B B @ 2 P
j Vj ⇢j
r Ej −
Q2
j
ρ2
j + m2
j⇢2 j
3 P
j Vj⇢2 j
1 C C A
2
@ V 00 V = 2 P
j Vj
h Ej + 2m2
j⇢2 j
i P
j Vj⇢2 j
9j / ρj(φ) 6= 0 8φ
j Vjρ2 j,
remains positive at all times generic quantum bounce!
n ρ2
j |Ej|/m2 j and ρ4 j Q2 j/m2 j (
rge volume, but of small space-tim ✓ V 0 3V ◆2 = 2 P
j Vj mj ρ2 j
3 P
j Vjρ2 j
!2
P V 00 V = 4 P
j Vjm2 jρ2 j
P
j Vjρ2 j
.
eqns if m2
j ≈ 3GN
σj(φ) = 0, for all j 6= jo.
✓ V 0 3V ◆2 = 4πG 3 ✓ 1 ρ ρc ◆ + VjoEjo 9V ,
✓ ◆ h ρc = 6πG~2/V 2
jo ⇠ (6π/j3
that the second term is a quan
LQC-like modified dynamics!
DO, Sindoni, Wilson-Ewing, ‘16 + primordial accelleration
De Cesare, Sakellariadou, ‘16 Gielen, ‘16
can show that 1) generic solutions approximate such simple condensates at late times 2) GFT interactions can make primordial acceleration last enough e-folds to avoid need for inflation
De Cesare, Pithis, Sakellariadou, ‘16