Group field theory: a quantum field theory for the atoms of space - - PowerPoint PPT Presentation
Group field theory: a quantum field theory for the atoms of space Daniele Oriti Albert Einstein Institute Conference Frontiers in Fundamental Physics XIV Workshop on Quantum Gravity Marseille 17/07/2014
we already know: GFT <---> Spin Foams - actually: GFT = Spin Foams
(DO, 1310.7786 [gr-qc])
DO, ’06, ’11
the reformulation provides powerful new tools to address open issues in (L)QG (Part II-III) GFT renormalization Effective (quantum) cosmology from GFT condensates
(DO, 1310.7786 [gr-qc])
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
(states effectively defined on “infinitely refined graph”)
G12 1 2 3 4 G23 G34 G14 G13 G24
plus gauge invariance at vertices
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
modified class of quantum states:
can generalise
DO, J. Ryan, J. Thuerigen, ‘14
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
plus gauge invariance at d-valent vertices modified class of quantum states:
can generalise
DO, J. Ryan, J. Thuerigen, ‘14
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
plus gauge invariance at d-valent vertices turned into Hilbert space by:
γV
d ⊂ HV
1, ..., g1 d; ....; gV 1 , ..., gV d
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
modified class of quantum states:
can generalise
DO, J. Ryan, J. Thuerigen, ‘14
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
plus gauge invariance at d-valent vertices turned into Hilbert space by:
γV
d ⊂ HV
1, ..., g1 d; ....; gV 1 , ..., gV d
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
modified class of quantum states:
can generalise
DO, J. Ryan, J. Thuerigen, ‘14
embedding
γV
d ⇢ HV ' L2
G12 1 2 3 4 G23 G34 G14 G13 G24
embedding
γV
d ⇢ HV ' L2
embedding
γV
d ⇢ HV ' L2
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
embedding
γV
d ⇢ HV ' L2
embedding
γV
d ⇢ HV ' L2
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
embedding
γV
d ⇢ HV ' L2
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
same in other basis: fluxes, spins
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
“gluing” of spin network vertices: imposition of symmetry, specific linear combination embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
same in other basis: fluxes, spins
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
“gluing” of spin network vertices: imposition of symmetry, specific linear combination every cylindrical function is contained in new Hilbert space embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
same in other basis: fluxes, spins
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
“gluing” of spin network vertices: imposition of symmetry, specific linear combination every cylindrical function is contained in new Hilbert space embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
same in other basis: fluxes, spins
1 2 3 4 g1
1
g1
2
g1
3
g2
1
g
2 2
g3
2
g3
1
g3
3 2
g3 g4
1
g4
2
g4
3
“gluing” of spin network vertices: imposition of symmetry, specific linear combination every cylindrical function is contained in new Hilbert space embedding
γV
d ⇢ HV ' L2
wave function for many open spin net vertices wave function for closed graph
ij }) =
e∈E(Γ)
G
ij Γ({ga i ↵ab ij ; gb j↵ab ij }) = ΨΓ({ga i (gb j)−1})
⇢ { if [(i a) (j b)] 2 E(Γ),
same in other basis: fluxes, spins LQG kinematical scalar product for given graph (with V vertices) is restriction of scalar product for V open-vertices states
γV
d ⊂ HV
embedding with standard Haar measure
γV
d ⊂ HV
embedding with standard Haar measure however, generic cylindrical functions for different graphs are embedded differently in new Hilbert space:
γV
d ⊂ HV
embedding with standard Haar measure however, generic cylindrical functions for different graphs are embedded differently in new Hilbert space:
γV
d ⊂ HV
embedding with standard Haar measure however, generic cylindrical functions for different graphs are embedded differently in new Hilbert space:
Γ =
i
1...⇥ V ˜ Γ
Γ⇥ =
i
1...⇥ V ˜ Γ
i∈V
(⇧
d
a=1
mana(ga)Cj1...jd;I n1..nd
(⇧
d
a=1
a
related by unitary transformation (NC Fourier transform, Peter-Weyl decomposition)
standard procedure for writing same states in 2nd quantized form
standard procedure for writing same states in 2nd quantized form
d
V
V
d
d
standard procedure for writing same states in 2nd quantized form
Γ (⇥
Γ (⇥
symmetry under vertex relabeling bosonic statistics
d
V
V
d
d
standard procedure for writing same states in 2nd quantized form
Γ (⇥
Γ (⇥
symmetry under vertex relabeling bosonic statistics (assumption!)
d
V
V
d
d
standard procedure for writing same states in 2nd quantized form
Γ (⇥
Γ (⇥
symmetry under vertex relabeling bosonic statistics (assumption!)
d
V
V
d
d
⇥ (⇤
⇥ ∗ ⇥ (⇤
standard procedure for writing same states in 2nd quantized form
Γ (⇥
Γ (⇥
symmetry under vertex relabeling bosonic statistics (assumption!)
d
V
V
d
d
⇥ (⇤
⇥ ∗ ⇥ (⇤
which create/annihilate spin network vertices
Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, diffeo-invariant vacuum state)
Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, diffeo-invariant vacuum state) single field “quantum”: spin network vertex or tetrahedron (“building block of space”)
g g g g
1 2 3 4
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅
⌅ ⌅ ⌅ ⌅ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
generic quantum state: arbitrary collection of spin network vertices (including glued ones) or tetrahedra (including glued ones) Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, diffeo-invariant vacuum state) single field “quantum”: spin network vertex or tetrahedron (“building block of space”)
g g g g
1 2 3 4
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅
⌅ ⌅ ⌅ ⌅ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
generic quantum state: arbitrary collection of spin network vertices (including glued ones) or tetrahedra (including glued ones) Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, diffeo-invariant vacuum state)
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
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅
⌅ ⌅ ⌅ ⌅ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
any LQG operator can be written in 2nd quantized form
any LQG operator can be written in 2nd quantized form “2-body” operator (acts on single-vertex, does not create new vertices)
⇥ ,⇥
⇥ ˆ
=
any LQG operator can be written in 2nd quantized form “(n+m)-body” operator (acts on spin network with n vertices, gives spin network with m vertices)
1...d⇤
n ⌅
1, ...,⇤
n) ⌅
1)...⌅
n)
“2-body” operator (acts on single-vertex, does not create new vertices)
⇥ ,⇥
⇥ ˆ
=
any LQG operator can be written in 2nd quantized form “(n+m)-body” operator (acts on spin network with n vertices, gives spin network with m vertices)
1...d⇤
n ⌅
1, ...,⇤
n) ⌅
1)...⌅
n)
“2-body” operator (acts on single-vertex, does not create new vertices)
⇥ ,⇥
⇥ ˆ
=
basic field operators and the set of observables as functions of them define the quantum kinematics of the corresponding GFT
can use general correspondence for operators to rewrite also any dynamical quantum equation for LQG states in 2nd quantized form
can use general correspondence for operators to rewrite also any dynamical quantum equation for LQG states in 2nd quantized form assume quantum dynamics is encoded in “physical projector equation”:
d
can use general correspondence for operators to rewrite also any dynamical quantum equation for LQG states in 2nd quantized form projector operator will in general decompose into 2-body, 3-body, ...., n-body operators (weighted by (coupling) constants), i.e. will have non-zero “matrix elements” involving 2, 3, ..., n spin network vertices
1, ..., ⇤
n⇤ = Pn+m (⇤
1, ..., ⇤
n)
assume quantum dynamics is encoded in “physical projector equation”:
d
can use general correspondence for operators to rewrite also any dynamical quantum equation for LQG states in 2nd quantized form projector operator will in general decompose into 2-body, 3-body, ...., n-body operators (weighted by (coupling) constants), i.e. will have non-zero “matrix elements” involving 2, 3, ..., n spin network vertices
1, ..., ⇤
n⇤ = Pn+m (⇤
1, ..., ⇤
n)
assume quantum dynamics is encoded in “physical projector equation”:
d
the same quantum dynamics can be expressed in 2nd quantized form (using general map for operators):
⇥
⌥
n,m/n+m=2
n+m ⇤ ⇧ ⌥
{⇥ ,⇥ }
ˆ c†
⇥ 1...ˆ
c†
⇥ m Pn+m (⌅
⇥1, ..., ⌅ ⇥m, ⌅ ⇥
1, ..., ⌅
⇥
n) ˆ
c⇥
c⇥
⌅ ⌃ |Ψ = ⌥
⇥
c†
⇥ ˆ
c⇥
|Ψ ⇥
⌥
n,m/n+m=2
n+m
g1...d⌅ gm d⌅ g
1...d⌅
g
n
⇤†(⌅ g1)... ⇤†(⌅ gm)Pn+m (⌅ g1, ...,⌅ gm,⌅ g
1, ...,⌅
g
n)
⇤(⌅ g
1)...
⇤(⌅ g
n)
⇥ |Ψ = =
g ⇤(⌅ g) ⇤(⌅ g) |Ψ
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language first candidate (“microcanonical ensemble”): Zm =
|s> is arbitrary basis
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language first candidate (“microcanonical ensemble”): Zm =
|s> is arbitrary basis
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language first candidate (“microcanonical ensemble”): Zm =
|s> is arbitrary basis
more general context (abstract structures, no continuum, topology change, …) suggest more general ansatz (“canonical ensemble”):
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language first candidate (“microcanonical ensemble”): Zm =
|s> is arbitrary basis
more general context (abstract structures, no continuum, topology change, …) suggest more general ansatz (“canonical ensemble”):
with different numbers of vertices (“grandcanonical ensemble”):
F − µ b N)|s⟩
partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language first candidate (“microcanonical ensemble”): Zm =
|s> is arbitrary basis
more general context (abstract structures, no continuum, topology change, …) suggest more general ansatz (“canonical ensemble”):
with different numbers of vertices (“grandcanonical ensemble”):
F − µ b N)|s⟩ this is the expression leading most directly to GFTs and Spin Foam models
introducing a basis of 2nd quantized coherent states for the field operator, gives:
s
F − µ b N)|si ⌘
where the quantum corrected action is:
introducing a basis of 2nd quantized coherent states for the field operator, gives:
s
F − µ b N)|si ⌘
where the quantum corrected action is: this is the GFT partition function with classical GFT action: S
= ⌃ d⇤ g ⇥†(⇤ g) ⇥(⇤ g) − −
⇥
⇧
n,m/n+m=2
n+m ⇤⌃ d⇤ g1...d⇤ gm d⇤ g
1...d⇤
g
n ⇥†(⇤
g1)...⇥†(⇤ gm) Vn+m (⇤ g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n) ⇥(⇤
g
1)...⇥(⇤
g
n)
⌅ Vn+m (⇤ g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n) = Pn+m (⇤
g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n)
. Sardelli, ’08
the GFT interaction term is the Spin Foam vertex amplitude
introducing a basis of 2nd quantized coherent states for the field operator, gives:
s
F − µ b N)|si ⌘
where the quantum corrected action is: this is the GFT partition function with classical GFT action: S
= ⌃ d⇤ g ⇥†(⇤ g) ⇥(⇤ g) − −
⇥
⇧
n,m/n+m=2
n+m ⇤⌃ d⇤ g1...d⇤ gm d⇤ g
1...d⇤
g
n ⇥†(⇤
g1)...⇥†(⇤ gm) Vn+m (⇤ g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n) ⇥(⇤
g
1)...⇥(⇤
g
n)
⌅ Vn+m (⇤ g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n) = Pn+m (⇤
g1, ...,⇤ gm,⇤ g
1, ...,⇤
g
n)
. Sardelli, ’08
the GFT interaction term is the Spin Foam vertex amplitude quantum corrections give new interaction terms or renormalisation of existing ones
test construction in “known” example: 3d quantum gravity (euclidean)
test construction in “known” example: 3d quantum gravity (euclidean) Hamiltonian and diffeo constraints impose flatness of gravity holonomy general matrix elements of projector operator:
Γ⇥ = ΨΓ|
ΓΓ,Γ
Γ⇥ (independent) closed loops
test construction in “known” example: 3d quantum gravity (euclidean) such action decomposes into an action on 2, 4, 6,... spin network vertices (glued to form closed graphs, because of gauge invariance of P - graphs formed by an odd number of spin net vertices do not arise) Hamiltonian and diffeo constraints impose flatness of gravity holonomy general matrix elements of projector operator:
Γ⇥ = ΨΓ|
ΓΓ,Γ
Γ⇥ (independent) closed loops
test construction in “known” example: 3d quantum gravity (euclidean) such action decomposes into an action on 2, 4, 6,... spin network vertices (glued to form closed graphs, because of gauge invariance of P - graphs formed by an odd number of spin net vertices do not arise) Hamiltonian and diffeo constraints impose flatness of gravity holonomy general matrix elements of projector operator:
Γ⇥ = ΨΓ|
ΓΓ,Γ
Γ⇥ (independent) closed loops
1, ...,
n) = Pn+m (
1, ...,
n)
this in turn should give possible GFT interaction terms
. Sardelli, ’08
test construction in “known” example: 3d quantum gravity (euclidean) such action decomposes into an action on 2, 4, 6,... spin network vertices (glued to form closed graphs, because of gauge invariance of P - graphs formed by an odd number of spin net vertices do not arise) we expect these to give rise to the known Boulatov GFT model for 3d QG Hamiltonian and diffeo constraints impose flatness of gravity holonomy general matrix elements of projector operator:
Γ⇥ = ΨΓ|
ΓΓ,Γ
Γ⇥ (independent) closed loops
1, ...,
n) = Pn+m (
1, ...,
n)
this in turn should give possible GFT interaction terms
. Sardelli, ’08
indeed....
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
gives the identity kernel
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
g1 g4' g1' g4 g6' g
2'
g5' g2 g3
6
g g5 g3'
⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
2
⇥ δ
1
⇥ δ
6 G−1 5
⇥ = ... = = ⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
gives the identity kernel
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
g1 g4' g1' g4 g6' g
2'
g5' g2 g3
6
g g5 g3'
⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
2
⇥ δ
1
⇥ δ
6 G−1 5
⇥ = ... = = ⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
exactly usual tetrahedral interaction term of Boulatov GFT gives the identity kernel
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
g1 g4' g1' g4 g6' g
2'
g5' g2 g3
6
g g5 g3'
⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
2
⇥ δ
1
⇥ δ
6 G−1 5
⇥ = ... = = ⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
exactly usual tetrahedral interaction term of Boulatov GFT gives the identity kernel
g1 g1' g4' g4 g3' g3 g5 g5' g2 g
2' 6
g g6'
⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ (G2G1) δ (G6G5) δ (G3G5G4G2) = ... = = ⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
g1 g4' g1' g4 g6' g
2'
g5' g2 g3
6
g g5 g3'
⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
2
⇥ δ
1
⇥ δ
6 G−1 5
⇥ = ... = = ⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
exactly usual tetrahedral interaction term of Boulatov GFT gives the identity kernel
g1 g1' g4' g4 g3' g3 g5 g5' g2 g
2' 6
g g6'
⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ (G2G1) δ (G6G5) δ (G3G5G4G2) = ... = = ⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
so-called “pillow” interaction term, also considered in Boulatov GFT
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
g1 g4' g1' g4 g6' g
2'
g5' g2 g3
6
g g5 g3'
⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
2
⇥ δ
1
⇥ δ
6 G−1 5
⇥ = ... = = ⇤ [dgidgi] ϕ123 ϕ345 ϕ526 ϕ641 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
exactly usual tetrahedral interaction term of Boulatov GFT gives the identity kernel
g1 g1' g4' g4 g3' g3 g5 g5' g2 g
2' 6
g g6'
⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ (G2G1) δ (G6G5) δ (G3G5G4G2) = ... = = ⇤ [dgidgi] ϕ123 ϕ356 ϕ546 ϕ621 δ
1
⇥ δ
2
⇥ δ
3
⇥ δ
4
⇥ δ
5
⇥ δ
6
⇥ Gi = gig−1
i
ϕijk = ϕ (gi, gj, gk)
can then compute diagrams of order 6,8,.... - GFT action will in general contain infinite number of interactions so-called “pillow” interaction term, also considered in Boulatov GFT
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
quantum dynamics:
“combinatorial non-locality” in pairing of field arguments classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields)
quantum dynamics: Feynman amplitudes (model-dependent):
networks)
(with group+Lie algebra variables)
Reisenberger,Rovelli, ’00
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence
insights:
equations (scalar product)
. Sardelli, ’08
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence
insights:
equations (scalar product)
. Sardelli, ’08
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
insights:
equations (scalar product)
. Sardelli, ’08
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence 1. making sense of quantum dynamics
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
insights:
equations (scalar product)
. Sardelli, ’08
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence 1. making sense of quantum dynamics
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
insights:
equations (scalar product)
. Sardelli, ’08
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence 1. making sense of quantum dynamics
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
alternative: spin foam (lattice) refinement/coarse graining
(B. Bahr, B. Dittrich, ’09, ’10; B. Bahr, B. Dittrich, F . Hellmann, W. Kaminski, ‘12)
Renormalization Group is crucial tool (mathematical, conceptual, physical)
but taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
Renormalization Group is crucial tool (mathematical, conceptual, physical)
but taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”) “number of d.o.f.s” N vs “scale”
because of background geometry
Renormalization Group is crucial tool (mathematical, conceptual, physical)
but taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”) “number of d.o.f.s” N vs “scale”
because of background geometry
in specific GFT case: fundamental formulation of QG d.o.f.s given by a QFT, defined perturbatively around the “no-space” vacuum - need to prove consistency of the theory:
Renormalization Group is crucial tool (mathematical, conceptual, physical)
but taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”) “number of d.o.f.s” N vs “scale”
because of background geometry
in specific GFT case: fundamental formulation of QG d.o.f.s given by a QFT, defined perturbatively around the “no-space” vacuum - need to prove consistency of the theory:
if achieved (and GR emerges in continuum limit): a renormalizable quantum field theory of gravity (full background independence: a QFT for the non-spatio-temporal “atoms of space”)
Renormalization Group is crucial tool (mathematical, conceptual, physical)
taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
Renormalization Group is crucial tool (mathematical, conceptual, physical)
taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
separated by phase transitions, depends on value of coupling constants
what are the macroscopic phases? which one is effectively described by a smooth geometry with matter fields? which one do we live in?
Renormalization Group is crucial tool (mathematical, conceptual, physical)
taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
separated by phase transitions, depends on value of coupling constants
what are the macroscopic phases? which one is effectively described by a smooth geometry with matter fields? which one do we live in? idea of “geometrogenesis” in LQG/GFT : continuum geometric physics in new (condensate?) phase in canonical LQG context:
in covariant SF/GFT context:
DO, 0710.3276 [gr-qc]
also in tensor models
Renormalization Group is crucial tool (mathematical, conceptual, physical)
taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
separated by phase transitions, depends on value of coupling constants
what are the macroscopic phases? which one is effectively described by a smooth geometry with matter fields? which one do we live in? idea of “geometrogenesis” in LQG/GFT : continuum geometric physics in new (condensate?) phase in canonical LQG context:
in covariant SF/GFT context:
DO, 0710.3276 [gr-qc]
also in tensor models
in GFT context: need to prove dynamically the phase transition to non-degenerate (e.g. condensate) phase
Renormalization Group is crucial tool (mathematical, conceptual, physical)
taking into account the physics of more and more d.o.f.s (“flow” of the system across different “scales”)
separated by phase transitions, depends on value of coupling constants
what are the macroscopic phases? which one is effectively described by a smooth geometry with matter fields? which one do we live in? idea of “geometrogenesis” in LQG/GFT : continuum geometric physics in new (condensate?) phase in canonical LQG context:
in covariant SF/GFT context:
DO, 0710.3276 [gr-qc]
also in tensor models
in GFT context: need to prove dynamically the phase transition to non-degenerate (e.g. condensate) phase experience and results in tensor models and GFTs
preliminary understanding:
(hard cut-off of fields, or heat-kernel regularisation of propagator, in representation space)
]
g1 g2 g3 g
1
g
2
g
3
h1 h2 h3
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools)
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models
indexed by d-colored “bubbles”
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions
indexed by d-colored “bubbles”
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions general enough class of models
indexed by d-colored “bubbles”
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions general enough class of models
indexed by d-colored “bubbles”
(g8, g9, g10, g11)(g12, g9, g10, g11)(g12, g7, g6, g4)
Laplacian kinetic term (or its power “a”)
d
⇥=1
propagator =
example of Feynman diagram (d=4) 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)
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) scales: defined by propagator: spectrum of Laplacian = indexed by group representations or Lie algebra elements “p”
i=0 Ci
M−2i
s=1 p2a s +µ) , ∀i ≥ 0 , M > 1
s=1 pa s +µ2)
i=0 Ci;
this allows systematic power counting
locality: melonic diagrams with melonic boundaries are dominant
systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models scales: defined by propagator: spectrum of Laplacian = indexed by group representations or Lie algebra elements “p”
i=0 Ci
M−2i
s=1 p2a s +µ) , ∀i ≥ 0 , M > 1
s=1 pa s +µ2)
i=0 Ci;
this allows systematic power counting
locality: melonic diagrams with melonic boundaries are dominant
many results:
(requires more subtle analysis of combinatorics of diagrams, crucial role of rank of incidence matrix between edges and faces of Feynman diagrams)
pre-geometric content seems to improve renormalizability; asymptotic freedom seems generic
what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase? AL vacuum
what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase? AL vacuum
ALh0|ES|0iAL = 0
totally degenerate geometry (emptiest state) connection highly fluctuating diffeo invariant
what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase? AL vacuum KS vacuum
KSh0|ES|0iKS = ES
non-degenerate geometry (triad condensate) connection highly fluctuating diffeo covariant
ALh0|ES|0iAL = 0
totally degenerate geometry (emptiest state) connection highly fluctuating diffeo invariant
what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase? AL vacuum KS vacuum
KSh0|ES|0iKS = ES
non-degenerate geometry (triad condensate) connection highly fluctuating diffeo covariant
ALh0|ES|0iAL = 0
totally degenerate geometry (emptiest state) connection highly fluctuating diffeo invariant
DG vacuum (or BF vacuum)
DGh0|F(A)|0iDG = 0
non-degenerate flat connection metric highly fluctuating diffeo covariant simplicial context
AL vacuum KS vacuum DG vacuum (or BF vacuum) what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase?
AL vacuum KS vacuum DG vacuum (or BF vacuum)
what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase?
AL vacuum KS vacuum DG vacuum (or BF vacuum)
GFT condensate what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase?
AL vacuum KS vacuum DG vacuum (or BF vacuum)
GFT condensate phase transitions what is the LQG/GFT continuum phase structure? what is the physical, geometric LQG/GFT phase?
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system
simple candidates for physical (geometric) vacuum: GFT condensates
DO, L. Sindoni, 1010.5149 [gr-qc]; S. Gielen, DO, L. Sindoni, 1303.3576 [gr-qc], 1311.1238 [gr-qc];
what is their definition? do they have a continuum geometric interpretation? what is their effective quantum dynamics? does it relate to GR?
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
described by single collective wave function (depending on homogeneous anisotropic geometric data)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
described by single collective wave function (depending on homogeneous anisotropic geometric data)
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data)
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data)
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
following procedures of standard BEC described by single collective wave function (depending on homogeneous anisotropic geometric data)
similar equations obtained in non-linear extension of LQC (Bojowald et al. ’12) similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
e3 e1 e2
describes geometric tetrahedron
requires: tetrahedra flat enough
e3 e1 e2
describes geometric tetrahedron
i
jkeA j eB k
requires: tetrahedra flat enough
N
m=1
e3 e1 e2
describes geometric tetrahedron
i
jkeA j eB k
requires: tetrahedra flat enough
N
m=1
e3 e1 e2
describes geometric tetrahedron
choosing embedding point and 3 reference vectors: m ⌃⇤
i
jkeA j eB k
requires: tetrahedra flat enough
N
m=1
e3 e1 e2
describes geometric tetrahedron
choosing embedding point and 3 reference vectors: m ⌃⇤
i
jkeA j eB k
requires: tetrahedra flat enough
N
m=1
e3 e1 e2
describes geometric tetrahedron
choosing embedding point and 3 reference vectors: m ⌃⇤
i
jkeA j eB k
many results in LQG, simplicial geometry
requires: tetrahedra flat enough
N
m=1
e3 e1 e2
describes geometric tetrahedron
choosing embedding point and 3 reference vectors: m ⌃⇤
i
jkeA j eB k
many results in LQG, simplicial geometry
requires: tetrahedra flat enough
kl⇥j mn ˜
i
that is, they are the metric coefficients for the metric “sampled” at N points
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
N
m=1
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
N
m=1
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
N
m=1
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
N
m=1
that is, they are the metric coefficients for the metric “sampled” at N points
i.e. all GFT quanta are labelled by the same (gauge invariant) data
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
N
m=1
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
that is, they are the metric coefficients for the metric “sampled” at N points
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
a simple choice of quantum GFT condensate (homogeneous continuum quantum space) single-particle condensate (Gross-Pitaevskii approximation)
⇥
dipole used in SF cosmology (Bianchi, Rovelli, Vidotto, ‘10)
follow closely procedure used in real BECs
1303.3576 [gr-qc], 1311.1238 [gr-qc]
follow closely procedure used in real BECs
⇥
1303.3576 [gr-qc], 1311.1238 [gr-qc]
follow closely procedure used in real BECs
⇥
applied to (coherent) GFT condensate state, gives equation for “wave function”:
i] ˜
i)σ(g0 i) + λ
from truncation of SD equations for GFT model
1303.3576 [gr-qc], 1311.1238 [gr-qc]
follow closely procedure used in real BECs non-linear and non-local extension of quantum cosmology-like equation for “collective wave function
⇥
applied to (coherent) GFT condensate state, gives equation for “wave function”:
i] ˜
i)σ(g0 i) + λ
from truncation of SD equations for GFT model
1303.3576 [gr-qc], 1311.1238 [gr-qc]
follow closely procedure used in real BECs non-linear and non-local extension of quantum cosmology-like equation for “collective wave function
⇥
applied to (coherent) GFT condensate state, gives equation for “wave function”:
i] ˜
i)σ(g0 i) + λ
from truncation of SD equations for GFT model
1303.3576 [gr-qc], 1311.1238 [gr-qc]
similar equations obtained in non-linear extension of LQC (Bojowald et al. ’12)
microscopic (single vertex, 1st quantized) variables:
k ˆ
microscopic (single vertex, 1st quantized) variables:
k ˆ
macroscopic (2nd quantized) variables:
a = iκ
a t
microscopic (single vertex, 1st quantized) variables:
k ˆ
macroscopic (2nd quantized) variables:
a = iκ
a t
satisfying:
a , [
microscopic (single vertex, 1st quantized) variables:
k ˆ
macroscopic (2nd quantized) variables:
a = iκ
a t
entering effective (semiclassical) cosmological equations via expectation values:
a i
satisfying:
a , [
microscopic (single vertex, 1st quantized) variables:
k ˆ
macroscopic (2nd quantized) variables:
a = iκ
a t
entering effective (semiclassical) cosmological equations via expectation values:
a i
satisfying:
a , [
macroscopic geometric conjugate variables are instead:
a = h b
a i
(as we are considering GFT hydrodynamics)
the cosmological holonomies have automatically a dependence on N = average number of microscopic cells/ building blocks forming the condensate generically, effective cosmological equations will carry a dependence on <N> when expressed in terms
the cosmological holonomies have automatically a dependence on N = average number of microscopic cells/ building blocks forming the condensate generically, effective cosmological equations will carry a dependence on <N> when expressed in terms
the cosmological holonomies have automatically a dependence on N = average number of microscopic cells/ building blocks forming the condensate this is the GFT condensate counterpart of the “lattice refinement” in LQC generically, effective cosmological equations will carry a dependence on <N> when expressed in terms
the cosmological holonomies have automatically a dependence on N = average number of microscopic cells/ building blocks forming the condensate this is the GFT condensate counterpart of the “lattice refinement” in LQC generically, effective cosmological equations will carry a dependence on <N> when expressed in terms
is introduced to interpret the connection as being integrated along a finite path in and expressed in a given coordinate system;
fundamental GFT quanta with same average volume) :
the cosmological holonomies have automatically a dependence on N = average number of microscopic cells/ building blocks forming the condensate this is the GFT condensate counterpart of the “lattice refinement” in LQC generically, effective cosmological equations will carry a dependence on <N> when expressed in terms
exact relation between <N> and cosmological variables depends on quantum state
is introduced to interpret the connection as being integrated along a finite path in and expressed in a given coordinate system;
fundamental GFT quanta with same average volume) :
derivation of (quantum) cosmological equations from GFT quantum dynamics very general it rests on:
derivation of (quantum) cosmological equations from GFT quantum dynamics very general it rests on:
general features:
derivation of (quantum) cosmological equations from GFT quantum dynamics very general it rests on:
general features:
derivation of (quantum) cosmological equations from GFT quantum dynamics very general it rests on:
general features:
exact form of equations depends on specific model considered if GFT dynamics involves Laplacian kinetic term, then FRW equation is contained in effective cosmological dynamics for GFT condensate, with QG corrections
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit)
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit) 1. making sense of quantum dynamics
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit) 1. making sense of quantum dynamics
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s, i.e. many and refined graphs (continuum limit) 1. making sense of quantum dynamics