Spin network functions as “many-particles” states G dV /G V � d ⇢ H V ' L 2 � g 1 , ..., ~ g V � � 3 ' ~ H ˜ embedding γ V ⇢ { Z Y j ) − 1 } ) Ψ Γ ( { G ab d ↵ ab ij � Γ ( { g a i ↵ ab ij ; g b j ↵ ab ij } ) = Ψ Γ ( { g a i ( g b ij } ) = if [( i a ) ( j b )] 2 E ( Γ ), G e ∈ E ( Γ ) on E ( Γ ) ⇢ ( { 1 , . . . , V } ⇥ { 1 , . . . , d } ) 1 g 1 g 1 3 wave function 1 2 g 1 for closed graph wave function for many open spin net vertices 3 g 4 1 g 2 same in other basis: fluxes, spins 2 g 4 2 g 4 2 2 1 g 4 g 3 2 2 g 3 1 g 3 g 3 3 3
Spin network functions as “many-particles” states G dV /G V � d ⇢ H V ' L 2 � g 1 , ..., ~ g V � � 3 ' ~ H ˜ embedding γ V ⇢ { Z Y j ) − 1 } ) Ψ Γ ( { G ab d ↵ ab ij � Γ ( { g a i ↵ ab ij ; g b j ↵ ab ij } ) = Ψ Γ ( { g a i ( g b ij } ) = if [( i a ) ( j b )] 2 E ( Γ ), G e ∈ E ( Γ ) on E ( Γ ) ⇢ ( { 1 , . . . , V } ⇥ { 1 , . . . , d } ) 1 g 1 g 1 3 wave function 1 2 g 1 for closed graph wave function for many open spin net vertices 3 g 4 1 g 2 same in other basis: fluxes, spins 2 g 4 2 g 4 2 2 “gluing” of spin network vertices: imposition of 1 g 4 g 3 2 symmetry, specific linear combination 2 g 3 1 g 3 g 3 3 3
Spin network functions as “many-particles” states G dV /G V � d ⇢ H V ' L 2 � g 1 , ..., ~ g V � � 3 ' ~ H ˜ embedding γ V ⇢ { Z Y j ) − 1 } ) Ψ Γ ( { G ab d ↵ ab ij � Γ ( { g a i ↵ ab ij ; g b j ↵ ab ij } ) = Ψ Γ ( { g a i ( g b ij } ) = if [( i a ) ( j b )] 2 E ( Γ ), G e ∈ E ( Γ ) on E ( Γ ) ⇢ ( { 1 , . . . , V } ⇥ { 1 , . . . , d } ) 1 g 1 g 1 3 wave function 1 2 g 1 for closed graph wave function for many open spin net vertices 3 g 4 1 g 2 same in other basis: fluxes, spins 2 g 4 2 g 4 2 2 “gluing” of spin network vertices: imposition of 1 g 4 g 3 2 symmetry, specific linear combination 2 g 3 1 g 3 g 3 3 every cylindrical function is contained in new Hilbert space 3
Spin network functions as “many-particles” states G dV /G V � d ⇢ H V ' L 2 � g 1 , ..., ~ g V � � 3 ' ~ H ˜ embedding γ V ⇢ { Z Y j ) − 1 } ) Ψ Γ ( { G ab d ↵ ab ij � Γ ( { g a i ↵ ab ij ; g b j ↵ ab ij } ) = Ψ Γ ( { g a i ( g b ij } ) = if [( i a ) ( j b )] 2 E ( Γ ), G e ∈ E ( Γ ) on E ( Γ ) ⇢ ( { 1 , . . . , V } ⇥ { 1 , . . . , d } ) 1 g 1 g 1 3 wave function 1 2 g 1 for closed graph wave function for many open spin net vertices 3 g 4 1 g 2 same in other basis: fluxes, spins 2 g 4 2 g 4 2 2 “gluing” of spin network vertices: imposition of 1 g 4 g 3 2 symmetry, specific linear combination 2 g 3 1 g 3 g 3 3 every cylindrical function is contained in new Hilbert space 3 Any LQG state can be written in terms of “many-vertices” states
Spin network functions as “many-particles” states G dV /G V � d ⇢ H V ' L 2 � g 1 , ..., ~ g V � � 3 ' ~ H ˜ embedding γ V ⇢ { Z Y j ) − 1 } ) Ψ Γ ( { G ab d ↵ ab ij � Γ ( { g a i ↵ ab ij ; g b j ↵ ab ij } ) = Ψ Γ ( { g a i ( g b ij } ) = if [( i a ) ( j b )] 2 E ( Γ ), G e ∈ E ( Γ ) on E ( Γ ) ⇢ ( { 1 , . . . , V } ⇥ { 1 , . . . , d } ) 1 g 1 g 1 3 wave function 1 2 g 1 for closed graph wave function for many open spin net vertices 3 g 4 1 g 2 same in other basis: fluxes, spins 2 g 4 2 g 4 2 2 “gluing” of spin network vertices: imposition of 1 g 4 g 3 2 symmetry, specific linear combination 2 g 3 1 g 3 g 3 3 every cylindrical function is contained in new Hilbert space 3 Any LQG state can be written in terms of “many-vertices” states LQG kinematical scalar product for given graph (with V vertices) is restriction of scalar product for V open-vertices states
Spin network functions as “many-particles” states d ⊂ H V embedding H ˜ with standard Haar measure γ V
Spin network functions as “many-particles” states d ⊂ H V embedding H ˜ with standard Haar measure γ V M H ext H V however, generic cylindrical functions for di ff erent graphs = d are embedded di ff erently in new Hilbert space: V
Spin network functions as “many-particles” states d ⊂ H V embedding H ˜ with standard Haar measure γ V M H ext H V however, generic cylindrical functions for di ff erent graphs = d are embedded di ff erently in new Hilbert space: V • states associated to di ff erent graphs with same number of vertices are NOT orthogonal • states associated to graphs with di ff erent number of vertices ARE orthogonal • no cylindrical consistency, no projective limit
Spin network functions as “many-particles” states d ⊂ H V embedding H ˜ with standard Haar measure γ V M H ext H V however, generic cylindrical functions for di ff erent graphs = d are embedded di ff erently in new Hilbert space: V • states associated to di ff erent graphs with same number of vertices are NOT orthogonal • states associated to graphs with di ff erent number of vertices ARE orthogonal • no cylindrical consistency, no projective limit each state can be decomposed in products of “single-particle” (vertices) basis states: ⇥ ⇥ � 1 ... ⇥ � V ⇥ ⇥ � 1 ... ⇥ � V � � ⇥ | ⇥ ˜ Γ � = | ⇤ � 1 � ..... | ⇤ � V � � g | ⇥ ˜ Γ ⇥ = � ⇤ g i | ⇤ � i ⇥ ˜ ˜ Γ Γ � i ⇥ � i ⇥ i ∈ V d � ⇥ ⌃ ⇧ D j a m a n a ( g a ) C j 1 ...j d ; I ⇥ i = � � ( ⇧ g ) = ⇥ ⇧ g i | ⇧ ⇥ i ⇤ = ⇧ j i , ⇧ m i , I ⇤ ⇤ n 1 ..n d a =1 related by unitary transformation (NC Fourier transform, d ⇤⇧ ⌅ � ⇥ ⇧ ⌃ Peter-Weyl decomposition) ⇥ i = � � ( ⇧ g ) = ⇥ ⇧ g i | ⇧ ⇥ i ⇤ = E g a ( X a ) ⌅ � ⇥ ⇧ X i ⇤ ⇤ X a a a =1
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form ⇣ ⌘ H (1) ⌦ · · · H ( V ) H V ' F ( H d ) = M M result: H ext = d d d V V
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form ⇣ ⌘ H (1) ⌦ · · · H ( V ) H V ' F ( H d ) = M M result: H ext = d d d V V symmetry under vertex relabeling bosonic statistics Γ ( ⇥ g V ) = � ˜ Γ ( ⇥ g V ) � ˜ g 1 , ..., ⇥ g i , ..., ⇥ g j , ..., ⇥ g 1 , ..., ⇥ g j , ..., ⇥ g i , ..., ⇥
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form ⇣ ⌘ H (1) ⌦ · · · H ( V ) H V ' F ( H d ) = M M result: H ext = (assumption!) d d d V V symmetry under vertex relabeling bosonic statistics Γ ( ⇥ g V ) = � ˜ Γ ( ⇥ g V ) � ˜ g 1 , ..., ⇥ g i , ..., ⇥ g j , ..., ⇥ g 1 , ..., ⇥ g j , ..., ⇥ g i , ..., ⇥
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form ⇣ ⌘ H (1) ⌦ · · · H ( V ) H V ' F ( H d ) = M M result: H ext = (assumption!) d d d V V symmetry under vertex relabeling bosonic statistics Γ ( ⇥ g V ) = � ˜ Γ ( ⇥ g V ) � ˜ g 1 , ..., ⇥ g i , ..., ⇥ g j , ..., ⇥ g 1 , ..., ⇥ g j , ..., ⇥ g i , ..., ⇥ can define conjugate bosonic field operators: c † � � ⇥ † ( g 1 , .., g d ) ≡ ˆ ⇥ † ( ⇤ ⇥ ( g 1 , .., g d ) ≡ ˆ ˆ ⇥ ( ⇤ g ) = ˆ � ( ⇤ g ) ˆ g ) = ˆ � � ∗ � ( ⇤ g ) � � ⇥ c ⇥ ⇥ ⇥ ⇥ � ⇥ � � ⇥ � ⇥ ⇥ ⇥ � = ⇥ j, ⇥ m, I � = ⇥ X or
2nd quantized reformulation: kinematics standard procedure for writing same states in 2nd quantized form ⇣ ⌘ H (1) ⌦ · · · H ( V ) H V ' F ( H d ) = M M result: H ext = (assumption!) d d d V V symmetry under vertex relabeling bosonic statistics Γ ( ⇥ g V ) = � ˜ Γ ( ⇥ g V ) � ˜ g 1 , ..., ⇥ g i , ..., ⇥ g j , ..., ⇥ g 1 , ..., ⇥ g j , ..., ⇥ g i , ..., ⇥ can define conjugate bosonic field operators: c † � � ⇥ † ( g 1 , .., g d ) ≡ ˆ ⇥ † ( ⇤ ⇥ ( g 1 , .., g d ) ≡ ˆ ˆ ⇥ ( ⇤ g ) = ˆ � ( ⇤ g ) ˆ g ) = ˆ � � ∗ � ( ⇤ g ) � � ⇥ c ⇥ ⇥ ⇥ � ⇥ ⇥ � � ⇥ � ⇥ ⇥ ⇥ � = ⇥ j, ⇥ m, I � = ⇥ X or which create/annihilate spin network vertices
Spin networks in 2nd quantization
Spin networks in 2nd quantization Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, di ff eo-invariant vacuum state)
Spin networks in 2nd quantization Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, di ff eo-invariant vacuum state) single field “quantum”: spin network vertex or tetrahedron (“building block of space”) g 4 � g 2 � ⇥ ⌅ � � ⇥ ⌅ g � 3 g � ⇥ ⌅ g 1 � 1 � ⇥ ⌅ ϕ † ( g 1 , g 2 , g 3 , g 4 ) | ⇤⌅ = | ⌅ ˆ • � � ⇥ ⌅ ⇥ ⌅ g 3 � ⇥ ⌅ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⇥ ⌅ g 4 ⌅ ⇥ ⇥ ⌅ g ⌅ ⇥ 2 ⇥ ⌅ ⌅ ⇥ ⇥ ⇥ ⌅ ⇥ ⇥
Spin networks in 2nd quantization Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, di ff eo-invariant vacuum state) single field “quantum”: spin network vertex or tetrahedron (“building block of space”) g 4 � g 2 � ⇥ ⌅ � � ⇥ ⌅ g � 3 g � ⇥ ⌅ g 1 � 1 � ⇥ ⌅ ϕ † ( g 1 , g 2 , g 3 , g 4 ) | ⇤⌅ = | ⌅ ˆ • � � ⇥ ⌅ ⇥ ⌅ g 3 � ⇥ ⌅ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⇥ ⌅ g 4 ⌅ ⇥ ⇥ ⌅ g ⌅ ⇥ 2 ⇥ ⌅ ⌅ ⇥ ⇥ ⇥ ⌅ ⇥ ⇥ generic quantum state: arbitrary collection of spin network vertices (including glued ones) or tetrahedra (including glued ones)
Spin networks in 2nd quantization Fock vacuum: “no-space” (“emptiest”) state | 0 > ~ AL LQG vacuum (this is the natural background independent, di ff eo-invariant vacuum state) single field “quantum”: spin network vertex or tetrahedron (“building block of space”) g 4 � g 2 � ⇥ ⌅ � � ⇥ ⌅ g � 3 g � ⇥ ⌅ g 1 � 1 � ⇥ ⌅ ϕ † ( g 1 , g 2 , g 3 , g 4 ) | ⇤⌅ = | ⌅ ˆ • � � ⇥ ⌅ ⇥ ⌅ g 3 � ⇥ ⌅ ⇤⇤⇤⇤⇤⇤⇤⇤⇤ ⇥ ⌅ g 4 ⌅ ⇥ ⇥ ⌅ g ⌅ ⇥ 2 ⇥ ⌅ ⌅ ⇥ ⇥ ⇥ ⌅ ⇥ ⇥ generic quantum state: arbitrary collection of spin network vertices (including glued ones) or tetrahedra (including glued ones) j 15 j 14 j 19 j 18 j 22 j 17 j 20 j 16 j 8 j 21 j 12 j 13 j 11 j 23 j 10 j 2 j 3 j 6 j 7 j 9 j 1 j 5 j 4
2nd quantized reformulation: kinematics - observables \ ˆ any LQG operator can be written in 2nd quantized form O = O ( E, A )
2nd quantized reformulation: kinematics - observables \ ˆ any LQG operator can be written in 2nd quantized form O = O ( E, A ) “2-body” operator (acts on single-vertex, does not create new vertices) ⌃ � | ⌃ � � ⇤ = O 2 ( ⇤ � � ) � O 2 � ⇥ ⇤ O 2 | ⇤ � , ⇤ ⌅ ⇤ � ⇥ † ⇥ g � ⇧ c † ⌃ ⇥ † ( ⇤ � � ) ˆ � � = g � ) ⇧ g � ) � O 2 = O 2 ( ⇤ � ˆ g ) O 2 ( ⇤ ⇥ ( ⇤ ⇥ , ⇧ ⇧ � , ⇤ d ⇤ g d ⇤ g, ⇤ c ⇥ ⇥ � , ⇥ ⇥ � �
2nd quantized reformulation: kinematics - observables \ ˆ any LQG operator can be written in 2nd quantized form O = O ( E, A ) “2-body” operator (acts on single-vertex, does not create new vertices) ⌃ � | ⌃ � � ⇤ = O 2 ( ⇤ � � ) � O 2 � ⇥ ⇤ O 2 | ⇤ � , ⇤ ⌅ ⇤ � ⇥ † ⇥ g � ⇧ c † ⌃ ⇥ † ( ⇤ � � ) ˆ � � = g � ) ⇧ g � ) � O 2 = O 2 ( ⇤ � ˆ g ) O 2 ( ⇤ ⇥ ( ⇤ ⇥ , ⇧ ⇧ � , ⇤ d ⇤ g d ⇤ g, ⇤ c ⇥ ⇥ � , ⇥ ⇥ � � “(n+m)-body” operator (acts on spin network with n vertices, gives spin network with m vertices) � � m | � � � � � � � � � O n + m � ⇥ ⇤ � 1 , ...., ⇤ O n + m | ⇤ 1 , ..., ⇤ n ⇤ = O n + m ( ⇤ � 1 , ..., ⇤ � m , ⇤ 1 , ..., ⇤ n ) � ⇤ � ⇥ † ⇥ � ⇥ † ( ⇤ ⇥ † ( ⇤ g � g � g � g � g � g � � O n + m ⇥ , ˆ ˆ = g 1 ) ... ⌅ g m ) O n + m ( ⇤ n ) ⌅ ⇥ ( ⇤ 1 ) ... ⌅ ⇥ ( ⇤ n ) d ⇤ g 1 ...d ⇤ g m d ⇤ 1 ...d ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ n ⌅
2nd quantized reformulation: kinematics - observables \ ˆ any LQG operator can be written in 2nd quantized form O = O ( E, A ) “2-body” operator (acts on single-vertex, does not create new vertices) ⌃ � | ⌃ � � ⇤ = O 2 ( ⇤ � � ) � O 2 � ⇥ ⇤ O 2 | ⇤ � , ⇤ ⌅ ⇤ � ⇥ † ⇥ g � ⇧ c † ⌃ ⇥ † ( ⇤ � � ) ˆ � � = g � ) ⇧ g � ) � O 2 = O 2 ( ⇤ � ˆ g ) O 2 ( ⇤ ⇥ ( ⇤ ⇥ , ⇧ ⇧ � , ⇤ d ⇤ g d ⇤ g, ⇤ c ⇥ ⇥ � , ⇥ ⇥ � � “(n+m)-body” operator (acts on spin network with n vertices, gives spin network with m vertices) � � m | � � � � � � � � � O n + m � ⇥ ⇤ � 1 , ...., ⇤ O n + m | ⇤ 1 , ..., ⇤ n ⇤ = O n + m ( ⇤ � 1 , ..., ⇤ � m , ⇤ 1 , ..., ⇤ n ) � ⇤ � ⇥ † ⇥ � ⇥ † ( ⇤ ⇥ † ( ⇤ g � g � g � g � g � g � � O n + m ⇥ , ˆ ˆ = g 1 ) ... ⌅ g m ) O n + m ( ⇤ n ) ⌅ ⇥ ( ⇤ 1 ) ... ⌅ ⇥ ( ⇤ n ) d ⇤ g 1 ...d ⇤ g m d ⇤ 1 ...d ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ n ⌅ basic field operators and the set of observables as functions of them define the quantum kinematics of the corresponding GFT
2nd quantized reformulation: dynamics can use general correspondence for operators to rewrite also any dynamical quantum equation for LQG states in 2nd quantized form
2nd quantized reformulation: dynamics 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”: ⇣ ⌘ � ˆ P � ˆ ˆ � | Ψ ⇥ � H ext F | Ψ i = | Ψ i = 0 P | Ψ ⇥ = | Ψ ⇥ or: I d
2nd quantized reformulation: dynamics 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”: ⇣ ⌘ � ˆ P � ˆ ˆ � | Ψ ⇥ � H ext F | Ψ i = | Ψ i = 0 P | Ψ ⇥ = | Ψ ⇥ or: I d 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 ⇥ ⇤ � � 2 � P 2 + � 3 � P Ψ ⇤ = | Ψ ⇤ � P 3 + .... | Ψ ⇤ = | Ψ ⇤ ⇥ m | � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ ⇤ P n + m | ⇤ n ⇤ = P n + m ( ⇤ n ) ⇥ 1 , ...., ⇤ 1 , ..., ⇤ ⇥ 1 , ..., ⇤ ⇥ m , ⇤ 1 , ..., ⇤
2nd quantized reformulation: dynamics 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”: ⇣ ⌘ � ˆ P � ˆ ˆ � | Ψ ⇥ � H ext F | Ψ i = | Ψ i = 0 P | Ψ ⇥ = | Ψ ⇥ or: I d 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 ⇥ ⇤ � � 2 � P 2 + � 3 � P Ψ ⇤ = | Ψ ⇤ � P 3 + .... | Ψ ⇤ = | Ψ ⇤ ⇥ m | � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ ⇤ P n + m | ⇤ n ⇤ = P n + m ( ⇤ n ) ⇥ 1 , ...., ⇤ 1 , ..., ⇤ ⇥ 1 , ..., ⇤ ⇥ m , ⇤ 1 , ..., ⇤ the same quantum dynamics can be expressed in 2nd quantized form (using general map for operators): ⇤ ⌅ ⌥ ⇥ ⇧ ⌥ ⌥ c † c † c † ⌃ | Ψ � = ⇥ � ⇥ � ˆ � 1 ... ˆ � m P n + m ( ⌅ n ) ˆ 1 ... ˆ ˆ � ˆ � | Ψ � � n + m ⇥ 1 , ..., ⌅ ⇥ m , ⌅ 1 , ..., ⌅ c ⇥ c ⇥ c ⇥ � � � � ⇥ ⇥ ⇥ n n,m/n + m =2 { ⇥ � , ⇥ � � } � ⇥ �� ⇥ ⌥ ⇥ ⇤ † ( ⌅ ⇤ † ( ⌅ g � g � g � g � g � g � g 1 ) ... g m ) P n + m ( ⌅ n ) ⇤ ( ⌅ 1 ) ... ⇤ ( ⌅ n ) | Ψ � = � n + m d ⌅ g 1 ...d ⌅ g m d ⌅ 1 ...d ⌅ g 1 , ..., ⌅ g m , ⌅ 1 , ..., ⌅ n n,m/n + m =2 � = ⇤ ( ⌅ g ) ⇤ ( ⌅ g ) | Ψ � d ⌅ g
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language � ⟨ s | δ ( � first candidate (“microcanonical ensemble”): Z m = F ) | s ⟩ |s> is arbitrary basis s
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language � ⟨ s | δ ( � first candidate (“microcanonical ensemble”): Z m = F ) | s ⟩ |s> is arbitrary basis s only states solving dynamical constraint contribute (natural from continuum canonical theory)
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language � ⟨ s | δ ( � first candidate (“microcanonical ensemble”): Z m = F ) | s ⟩ |s> is arbitrary basis s only states solving dynamical constraint contribute (natural from continuum canonical theory) � more general context (abstract structures, no continuum, topology change, …) F | s ⟩ ⟨ s | e − b suggest more general ansatz (“canonical ensemble”): Z c = s
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language � ⟨ s | δ ( � first candidate (“microcanonical ensemble”): Z m = F ) | s ⟩ |s> is arbitrary basis s only states solving dynamical constraint contribute (natural from continuum canonical theory) � more general context (abstract structures, no continuum, topology change, …) F | s ⟩ ⟨ s | e − b suggest more general ansatz (“canonical ensemble”): Z c = s or, introducing a new parameter weighting di ff erently quantum states � ⟨ s | e − ( b F − µ b N ) | s ⟩ with di ff erent numbers of vertices (“grandcanonical ensemble”): Z g = s
2nd quantized reformulation: dynamics partition function (and correlations) of GFT is then obtained from partition function (and correlations) for spin networks, recast in 2nd quantised language � ⟨ s | δ ( � first candidate (“microcanonical ensemble”): Z m = F ) | s ⟩ |s> is arbitrary basis s only states solving dynamical constraint contribute (natural from continuum canonical theory) � more general context (abstract structures, no continuum, topology change, …) F | s ⟩ ⟨ s | e − b suggest more general ansatz (“canonical ensemble”): Z c = s or, introducing a new parameter weighting di ff erently quantum states � ⟨ s | e − ( b F − µ b N ) | s ⟩ with di ff erent numbers of vertices (“grandcanonical ensemble”): Z g = s this is the expression leading most directly to GFTs and Spin Foam models
2nd quantized reformulation: dynamics
2nd quantized reformulation: dynamics introducing a basis of 2nd quantized coherent states for the field operator, Z h s | e − ( b F − µ b N ) | s i ⌘ gives: X D ϕ D ϕ e − S eff ( ϕ , ϕ ) Z g = s S eff ( ϕ , ϕ ) = S ( ϕ , ϕ ) + O ( ~ ) = h ϕ | b F | ϕ i + O ( ~ ) where the quantum corrected action is: h ϕ | ϕ i
2nd quantized reformulation: dynamics introducing a basis of 2nd quantized coherent states for the field operator, Z h s | e − ( b F − µ b N ) | s i ⌘ gives: X D ϕ D ϕ e − S eff ( ϕ , ϕ ) Z g = s S eff ( ϕ , ϕ ) = S ( ϕ , ϕ ) + O ( ~ ) = h ϕ | b F | ϕ i + O ( ~ ) where the quantum corrected action is: h ϕ | ϕ i this is the GFT partition function with classical GFT action: ⌃ ⇥ , ⇥ † ⇥ g ⇥ † ( ⇤ � S = d ⇤ g ) ⇥ ( ⇤ g ) − ⇥ ⇤⌃ ⌅ ⇧ n ⇥ † ( ⇤ g 1 ) ... ⇥ † ( ⇤ g � g � g � g � g � g � � n + m d ⇤ g 1 ...d ⇤ g m d ⇤ 1 ...d ⇤ g m ) V n + m ( ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ n ) ⇥ ( ⇤ 1 ) ... ⇥ ( ⇤ n ) − n,m/n + m =2 g � g � g � g � V n + m ( ⇤ n ) = P n + m ( ⇤ n ) g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ the GFT interaction term is the Spin Foam vertex amplitude E. Alesci, K. Noui, F . Sardelli, ’08
2nd quantized reformulation: dynamics introducing a basis of 2nd quantized coherent states for the field operator, Z h s | e − ( b F − µ b N ) | s i ⌘ gives: X D ϕ D ϕ e − S eff ( ϕ , ϕ ) Z g = s S eff ( ϕ , ϕ ) = S ( ϕ , ϕ ) + O ( ~ ) = h ϕ | b F | ϕ i + O ( ~ ) where the quantum corrected action is: h ϕ | ϕ i this is the GFT partition function with classical GFT action: ⌃ ⇥ , ⇥ † ⇥ g ⇥ † ( ⇤ � S = d ⇤ g ) ⇥ ( ⇤ g ) − ⇥ ⇤⌃ ⌅ ⇧ n ⇥ † ( ⇤ g 1 ) ... ⇥ † ( ⇤ g � g � g � g � g � g � � n + m d ⇤ g 1 ...d ⇤ g m d ⇤ 1 ...d ⇤ g m ) V n + m ( ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ n ) ⇥ ( ⇤ 1 ) ... ⇥ ( ⇤ n ) − n,m/n + m =2 g � g � g � g � V n + m ( ⇤ n ) = P n + m ( ⇤ n ) g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ g 1 , ..., ⇤ g m , ⇤ 1 , ..., ⇤ the GFT interaction term is the Spin Foam vertex amplitude E. Alesci, K. Noui, F . Sardelli, ’08 quantum corrections give new interaction terms or renormalisation of existing ones
2nd quantized reformulation: dynamics - 3d example test construction in “known” example: 3d quantum gravity (euclidean)
2nd quantized reformulation: dynamics - 3d example test construction in “known” example: 3d quantum gravity (euclidean) Hamiltonian and di ff eo constraints impose flatness of gravity holonomy general matrix elements of projector operator: K. Noui, A. Perez, ’04 � � Ψ Γ | ⇥ P | Ψ ⇥ δ ( H f ) | Ψ ⇥ Γ � ⇥ = � Ψ Γ | Γ � ⇥ f ⇤ ˜ Γ � Γ , Γ � (independent) closed loops
2nd quantized reformulation: dynamics - 3d example test construction in “known” example: 3d quantum gravity (euclidean) Hamiltonian and di ff eo constraints impose flatness of gravity holonomy general matrix elements of projector operator: K. Noui, A. Perez, ’04 � � Ψ Γ | ⇥ P | Ψ ⇥ δ ( H f ) | Ψ ⇥ Γ � ⇥ = � Ψ Γ | Γ � ⇥ f ⇤ ˜ Γ � Γ , Γ � (independent) closed loops 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)
2nd quantized reformulation: dynamics - 3d example test construction in “known” example: 3d quantum gravity (euclidean) Hamiltonian and di ff eo constraints impose flatness of gravity holonomy general matrix elements of projector operator: K. Noui, A. Perez, ’04 � � Ψ Γ | ⇥ P | Ψ ⇥ δ ( H f ) | Ψ ⇥ Γ � ⇥ = � Ψ Γ | Γ � ⇥ f ⇤ ˜ Γ � Γ , Γ � (independent) closed loops 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) this in turn should give possible GFT interaction terms g � g � g � g � V n + m ( � n ) = P n + m ( � n ) E. Alesci, K. Noui, F . Sardelli, ’08 g 1 , ..., � g m , � 1 , ..., � g 1 , ..., � g m , � 1 , ..., �
2nd quantized reformulation: dynamics - 3d example test construction in “known” example: 3d quantum gravity (euclidean) Hamiltonian and di ff eo constraints impose flatness of gravity holonomy general matrix elements of projector operator: K. Noui, A. Perez, ’04 � � Ψ Γ | ⇥ P | Ψ ⇥ δ ( H f ) | Ψ ⇥ Γ � ⇥ = � Ψ Γ | Γ � ⇥ f ⇤ ˜ Γ � Γ , Γ � (independent) closed loops 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) this in turn should give possible GFT interaction terms g � g � g � g � V n + m ( � n ) = P n + m ( � n ) E. Alesci, K. Noui, F . Sardelli, ’08 g 1 , ..., � g m , � 1 , ..., � g 1 , ..., � g m , � 1 , ..., � we expect these to give rise to the known Boulatov GFT model for 3d QG
2nd quantized reformulation: dynamics - 3d example
2nd quantized reformulation: dynamics - 3d example indeed....
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices)
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel g 1' g 1 ⇤ G 3 G 5 G − 1 G 2 G 6 G − 1 G 4 G − 1 6 G − 1 � ⇥ � ⇥ � ⇥ [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ = ... = δ δ 2 1 5 g 6' g 4' g 2 g 3 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 4 g 3' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) i � g g 2' 6 g 5 g 5'
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel g 1' g 1 ⇤ G 3 G 5 G − 1 G 2 G 6 G − 1 G 4 G − 1 6 G − 1 � ⇥ � ⇥ � ⇥ [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ = ... = δ δ 2 1 5 g 6' g 4' g 2 g 3 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 4 g 3' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) i � g g 2' 6 g 5 exactly usual tetrahedral interaction term of Boulatov GFT g 5'
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel g 1' g 1 ⇤ G 3 G 5 G − 1 G 2 G 6 G − 1 G 4 G − 1 6 G − 1 � ⇥ � ⇥ � ⇥ [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ = ... = δ δ 2 1 5 g 6' g 4' g 2 g 3 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 4 g 3' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) i � g g 2' 6 g 5 exactly usual tetrahedral interaction term of Boulatov GFT g 5' g 3' g 3 ⇤ g [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ ( G 2 G 1 ) δ ( G 6 G 5 ) δ ( G 3 G 5 G 4 G 2 ) = ... = g 5 6 g 2 g 1 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 6' g 5' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) g i � g 1' 2' g 4 g 4'
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel g 1' g 1 ⇤ G 3 G 5 G − 1 G 2 G 6 G − 1 G 4 G − 1 6 G − 1 � ⇥ � ⇥ � ⇥ [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ = ... = δ δ 2 1 5 g 6' g 4' g 2 g 3 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 4 g 3' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) i � g g 2' 6 g 5 exactly usual tetrahedral interaction term of Boulatov GFT g 5' g 3' g 3 ⇤ g [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ ( G 2 G 1 ) δ ( G 6 G 5 ) δ ( G 3 G 5 G 4 G 2 ) = ... = g 5 6 g 2 g 1 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 6' g 5' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) g i � g 1' 2' g 4 so-called “pillow” interaction term, also considered in Boulatov GFT g 4' L. Freidel, D. Louapre, ’02
2nd quantized reformulation: dynamics - 3d example indeed.... using gauge invariance of GFT fields (i.e. of spin net vertices) gives the identity kernel g 1' g 1 ⇤ G 3 G 5 G − 1 G 2 G 6 G − 1 G 4 G − 1 6 G − 1 � ⇥ � ⇥ � ⇥ [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ = ... = δ δ 2 1 5 g 6' g 4' g 2 g 3 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 45 ϕ 5 � 2 � 6 ϕ 6 � 4 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 4 g 3' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) i � g g 2' 6 g 5 exactly usual tetrahedral interaction term of Boulatov GFT g 5' g 3' g 3 ⇤ g [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ ( G 2 G 1 ) δ ( G 6 G 5 ) δ ( G 3 G 5 G 4 G 2 ) = ... = g 5 6 g 2 g 1 ⇤ g 1 g − 1 g 2 g − 1 g 3 g − 1 g 4 g − 1 g 5 g − 1 g 6 g − 1 � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ � ⇥ = [ dg i dg i � ] ϕ 123 ϕ 3 � 56 ϕ 5 � 4 � 6 � ϕ 62 � 1 � δ δ δ δ δ δ 1 � 2 � 3 � 4 � 5 � 6 � g 6' g 5' G i = g i g − 1 ϕ ijk = ϕ ( g i , g j , g k ) g i � g 1' 2' g 4 so-called “pillow” interaction term, also considered in Boulatov GFT g 4' L. Freidel, D. Louapre, ’02 can then compute diagrams of order 6,8,.... - GFT action will in general contain infinite number of interactions
Group field theories classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields) S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! “combinatorial non-locality” in pairing of field arguments quantum dynamics: λ N Γ Z D ϕ D ϕ e i S λ ( ϕ , ϕ ) X Z = = sym ( Γ ) A Γ Γ
Group field theories classical action: kinetic (quadratic) term + (higher order) interaction (convolution of GFT fields) S ( ϕ , ϕ ) = 1 Z [ dg i ] ϕ ( g i ) K ( g i ) ϕ ( g i ) + λ Z [ dg ia ] ϕ ( g i 1 ) .... ϕ (¯ g iD ) V ( g ia , ¯ g iD ) + c.c. 2 D ! “combinatorial non-locality” in pairing of field arguments quantum dynamics: λ N Γ Z D ϕ D ϕ e i S λ ( ϕ , ϕ ) X Z = = sym ( Γ ) A Γ Γ Feynman amplitudes (model-dependent): � • spin foam models (sum-over-histories of spin networks) Reisenberger,Rovelli, ’00 � • lattice path integrals (with group+Lie algebra variables) A. Baratin, DO, ‘11
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence insights: � • direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT) � • SF vertex is elementary matrix element of projector operator E. Alesci, K. Noui, F . Sardelli, ’08 � • GFT/SF partition function (transition amplitude) contains more than canonical projector equations (scalar product) L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence insights: � • direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT) � • SF vertex is elementary matrix element of projector operator E. Alesci, K. Noui, F . Sardelli, ’08 � • GFT/SF partition function (transition amplitude) contains more than canonical projector equations (scalar product) L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13 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)
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence insights: � • direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT) � • SF vertex is elementary matrix element of projector operator E. Alesci, K. Noui, F . Sardelli, ’08 � • GFT/SF partition function (transition amplitude) contains more than canonical projector equations (scalar product) L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13 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 2. understanding continuum phase structure 3. extracting e ff ective continuum dynamics
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence insights: � • direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT) � • SF vertex is elementary matrix element of projector operator E. Alesci, K. Noui, F . Sardelli, ’08 � • GFT/SF partition function (transition amplitude) contains more than canonical projector equations (scalar product) L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13 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 GFT renormalization 2. understanding continuum phase structure 3. extracting e ff ective continuum dynamics
Summary: GFT as QFT reformulation of LQG LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence insights: � • direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT) � • SF vertex is elementary matrix element of projector operator E. Alesci, K. Noui, F . Sardelli, ’08 � • GFT/SF partition function (transition amplitude) contains more than canonical projector equations (scalar product) L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13 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 GFT renormalization alternative: spin foam (lattice) 2. understanding continuum phase structure refinement/coarse graining (B. Bahr, B. Dittrich, ’09, ’10; B. Bahr, B. 3. extracting e ff ective continuum dynamics Dittrich, F . Hellmann, W. Kaminski, ‘12)
II. � Group field theory renormalisation
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences” (so, relevant even with some “natural QG cut-off”), but taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”)
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences” (so, relevant even with some “natural QG cut-off”), 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” � � in continuum spacetime physics, “number of d.o.f.s” translates to energy/distance scale, � because of background geometry � � in QG, only first notion makes sense � � still, Renormalization Group is right tool, but needs to be adapted to background independent context
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences” (so, relevant even with some “natural QG cut-off”), 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” � � in continuum spacetime physics, “number of d.o.f.s” translates to energy/distance scale, � because of background geometry � � in QG, only first notion makes sense � � still, Renormalization Group is right tool, but needs to be adapted to background independent context 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: � � perturbative GFT renormalizability
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences” (so, relevant even with some “natural QG cut-off”), 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” � � in continuum spacetime physics, “number of d.o.f.s” translates to energy/distance scale, � because of background geometry � � in QG, only first notion makes sense � � still, Renormalization Group is right tool, but needs to be adapted to background independent context 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: � � perturbative GFT renormalizability 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 of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences”, but � taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”)
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences”, but � taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”) • for our QG models (LQG/spin foams, tensor models), do not expect to have a unique continuum limit � � collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions, depends on value of coupling constants � � � • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), � what are the macroscopic phases? � which one is effectively described by a smooth geometry with matter fields? which one do we live in?
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences”, but � taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”) • for our QG models (LQG/spin foams, tensor models), do not expect to have a unique continuum limit � � collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions, depends on value of coupling constants � � � • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), � 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: also in tensor models � T. Koslowski, 0709.3465 [gr-qc] DO, 0710.3276 [gr-qc] V. Rivasseau, ‘13
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences”, but � taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”) • for our QG models (LQG/spin foams, tensor models), do not expect to have a unique continuum limit � � collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions, depends on value of coupling constants � � � • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), � 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: also in tensor models � T. Koslowski, 0709.3465 [gr-qc] DO, 0710.3276 [gr-qc] V. Rivasseau, ‘13 in GFT context: need to prove dynamically the phase transition to non-degenerate (e.g. condensate) phase
Renormalization of (tensorial) GFTs: motivations Renormalization Group is crucial tool (mathematical, conceptual, physical) � � renormalization is not about “curing or hiding divergences”, but � taking into account the physics of more and more d.o.f.s � (“flow” of the system across different “scales”) • for our QG models (LQG/spin foams, tensor models), do not expect to have a unique continuum limit � � collective behaviour of (interacting) fundamental d.o.f.s should lead to different macroscopic phases, separated by phase transitions, depends on value of coupling constants � � � • for a non-spatio-temporal QG system (e.g. LQG in GFT formulation), � 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: also in tensor models � T. Koslowski, 0709.3465 [gr-qc] DO, 0710.3276 [gr-qc] V. Rivasseau, ‘13 in GFT context: need to prove dynamically the phase transition to non-degenerate (e.g. condensate) phase V. Bonzom, R. Gurau, A. Riello, V. Rivasseau, ’11; experience and results in tensor models and GFTs A. Baratin, S. Carrozza, DO, J. Ryan, M. Smerlak, ‘13
Renormalization of (tensorial) GFTs: a brief review preliminary understanding: � � power counting and radiative corrections in GFT models � (hard cut-off of fields, or heat-kernel regularisation of propagator, in representation space) � � � • 3d (non-abelian) (colored) Boulatov model (BF theory): � � • partial power counting and scaling theorems � � L. Freidel, R. Gurau, DO, ’09; J. Magnen, K. Noui, V. Rivasseau, M. Smerlak, ’09; J. Ben Geloun, J. Magnen, V. Rovasseau, ‘10 ; S. Carrozza, DO, ’11,’12 � • radiative corrections of 2-point function: need for Laplacian kinetic term � � ] J. Ben Geloun, V. Bonzom, ‘11 � h 1 g 1 � g � 1 g � g 2 2 h 2 � g � g 3 3 h 3 � • super-renormalizability in abelian case (with Laplacian) � � J. Ben Geloun, ‘13 � • 4d gravity models � � • radiative correction of 2-point function in EPRL-FK model J. Ben Geloun, R. Gurau, V. Rivasseau, ‘10; T. Krajewski, J. Magnen, V. Rivasseau, A. Tanasa, P. Vitale, ’10; A. Riello, ‘13
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools)
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . b ∈ B indexed by d-colored “bubbles” � [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 )
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . b ∈ B indexed by d-colored “bubbles” � [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 )
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . b ∈ B indexed by d-colored “bubbles” � [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 ) general enough class of models
Renormalization of (tensorial) GFTs: a brief review systematic renormalizability analysis of tensorial GFT models (crucial use of tensor models tools) renormalization ingredients and class of models tracial locality - tensor invariant interactions � S ( ϕ , ϕ ) = t b I b ( ϕ , ϕ ) . b ∈ B indexed by d-colored “bubbles” � [ d g i ] 12 � ( g 1 , g 2 , g 3 , g 4 ) � ( g 1 , g 2 , g 3 , g 5 ) � ( g 8 , g 7 , g 6 , g 5 ) � ( g 8 , g 9 , g 10 , g 11 ) � ( g 12 , g 9 , g 10 , g 11 ) � ( g 12 , g 7 , g 6 , g 4 ) general enough class of models ⇥ � 1 � d m 2 − Laplacian kinetic term � ⇤ ∆ ⇥ propagator = (or its power “a”) ⇥ =1
GFT renormalization 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)
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