Group field theories generating polyhedral complexes Johannes Th - - PowerPoint PPT Presentation

Group field theories generating polyhedral complexes

Johannes Th¨ urigen

w.i.p. with D. Oriti, J. Ryan

Max-Planck Institute for Gravitational Physics, Potsdam (Albert-Einstein Institute)

July 17, 2014

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 1 / 13

Dynamics of LQG: Group field theory (GFT)

Standard GFT: QFT of a field on a group φ : G ×D − → R S[φ] =

• [dg] φ(g1) K(g1, g2) φ(g2) +
• i∈I

λi

• [dg] Vi
• {ge}Ei

e∈Ei

φ(ge) , Perturbative expansion of state sum = sum over spin foams ZGFT =

• Dφ e−S[φ] =
• Γ

1 C(Γ)

i∈I

(−λi)Vi A(Γ)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 2 / 13

GFT for arbitrary graphs?

Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13

GFT for arbitrary graphs?

Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT Goal here: Define the general class of combinatorial complexes relevant for LQG and Spin Foams Discuss equivalent diagrammatic representations for them Present two types of GFTs generating those in the perturbative sum

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13

GFT for arbitrary graphs?

Apparent difference between canonical and covariant approaches to LQG: States in canonical LQG labeled by graphs of arbitrary valence Covariant theories developed in simplicial setting Generalization of spin foams to arbitrary valence exists [KKL ’10] Slightly more challenging for GFT Goal here: Define the general class of combinatorial complexes relevant for LQG and Spin Foams Discuss equivalent diagrammatic representations for them Present two types of GFTs generating those in the perturbative sum Caveat: Focus here on (dual) polyhedral 2-complexes (just polygons)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 3 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02] Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn.

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces)

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

1 2 3 4 5

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

more general than piecewise linear (e.g. 11-cell)

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

1 2 3 4 5

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

1 2 3 4 5

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

235 125 415 345 1234

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract n-polytope is a poset P

[McMullen,Schulte ’02]

(P1) P contains a least and greatest face f−1, fn. (P2) Each maximal chain has length n + 1 (P3) P is strongly connected (sequence of faces) (P4) Homogeneity degrees ki = 2, i = 1, .., n − 1

(for f < g of dimension i-1 and i+1 there are exactly ki i-faces h in P such that f < h < g)

more general than piecewise linear (e.g. 11-cell) dual polytope: invert partial order boundary is a closed manifold → generalize!

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

235 125 415 345 1234

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” An abstract polyhedral n-complex is a poset P (P1) P contains least and greatest face f−1, fn+1. (P2) Each maximal chain has length n + 2 (P3) (P is strongly connected) (P4’) Homogeneity degrees ki = 2, i = 1, .., n − 1 more general than piecewise linear (e.g. 11-cell) dual polytope branching possible!

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

235 125 415 345 1234

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Mathematical background: Abstract polyhedral complexes

”Combinatorial complexes = complexes of abstract polytopes” A generalized abstract polyhedral n-complex is a poset (P1) P contains least and greatest face f−1, fn+1. (P2) Each maximal chain has length n + 2 (P3) (P is strongly connected) (P4”) Face degree kf

i ∈ {1, 2}, i = 1, .., n − 1

more general than piecewise linear (e.g. 11-cell) dual polytope branching possible!

∅ 1 2 3 4 12 23 34 41 1234 5 15 25 35 45 125 235 345 415 12345

235 125 415 345 1234

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 4 / 13

Representations of Polyhedral 2-complexes

LQG/SF: 2-complexes, boundary 1-complexes finite abstract 2-polytopes are polygons alternative representation: stranded diagrams

(faces represented by strands)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 5 / 13

Representations of Polyhedral 2-complexes

LQG/SF: 2-complexes, boundary 1-complexes finite abstract 2-polytopes are polygons alternative representation: stranded diagrams

(faces represented by strands)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 5 / 13

Representations of Polyhedral 2-complexes

LQG/SF: 2-complexes, boundary 1-complexes finite abstract 2-polytopes are polygons alternative representation: stranded diagrams

(faces represented by strands)

subdivision → edges/faces represented by vertices

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 5 / 13

Representations of Polyhedral 2-complexes

LQG/SF: 2-complexes, boundary 1-complexes finite abstract 2-polytopes are polygons alternative representation: stranded diagrams

(faces represented by strands)

subdivision → edges/faces represented by vertices subdivided cells adjacent to bulk vertex define spin foam atoms

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 5 / 13

Atomic decomposition

A spin foam atom consists of a bulk vertex v boundary vertices V = {¯ vi, ...}, isomorphic to bulk edges (v¯ vi) bisection vertices V = {ˆ vij, ...}, isomorphic to face wedges (v¯ vi ˆ vij¯ vj) boundary (half)edges (¯ vi ˆ vij), (¯ vjˆ vij)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 6 / 13

Atomic decomposition

A spin foam atom consists of a bulk vertex v boundary vertices V = {¯ vi, ...}, isomorphic to bulk edges (v¯ vi) bisection vertices V = {ˆ vij, ...}, isomorphic to face wedges (v¯ vi ˆ vij¯ vj) boundary (half)edges (¯ vi ˆ vij), (¯ vjˆ vij) interpretation as dual to (local) D-polytope possible, but not unique

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 6 / 13

Atomic decomposition

A spin foam atom consists of a bulk vertex v boundary vertices V = {¯ vi, ...}, isomorphic to bulk edges (v¯ vi) bisection vertices V = {ˆ vij, ...}, isomorphic to face wedges (v¯ vi ˆ vij¯ vj) boundary (half)edges (¯ vi ˆ vij), (¯ vjˆ vij) interpretation as dual to (local) D-polytope possible, but not unique Spin foam atoms are isomorphic to bisected closed graphs (cf. [KKL ’10, KLP ’12])

(which are isomorphic to closed graphs)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 6 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches GFT: Wick contractions of fields φ(g¯

vˆ v1, .., g¯ vˆ vn)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches GFT: Wick contractions of fields φ(g¯

vˆ v1, .., g¯ vˆ vn)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches GFT: Wick contractions of fields φ(g¯

vˆ v1, .., g¯ vˆ vn)

Boundary given by patches which are not bonded/ by deletion of

internal (closed) strands

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches GFT: Wick contractions of fields φ(g¯

vˆ v1, .., g¯ vˆ vn)

Boundary given by patches which are not bonded/ by deletion of

internal (closed) strands

Any generalized polyhedral (2-)complex has a decomposition into atoms, and equivalently is a bonding of atoms

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Foams as molecules from atoms

Spin foams are bondings of atoms along patches GFT: Wick contractions of fields φ(g¯

vˆ v1, .., g¯ vˆ vn)

Boundary given by patches which are not bonded/ by deletion of

internal (closed) strands

Any generalized polyhedral (2-)complex has a decomposition into atoms, and equivalently is a bonding of atoms

All of this works for generalized polyhedral complexes of arbitrary dimension

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 7 / 13

Generalizing the combinatorics of simplicial GFT

No obstacle for GFT of polyhedral interactions with simplicial boundary

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 8 / 13

Generalizing the combinatorics of simplicial GFT

No obstacle for GFT of polyhedral interactions with simplicial boundary Important: criteria, analytical/numerical control

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 8 / 13

Generalizing the combinatorics of simplicial GFT

No obstacle for GFT of polyhedral interactions with simplicial boundary Important: criteria, analytical/numerical control Polytopes effectively generated anyway: Ex.: Uncoloring of colored GFT (bipartite simplex gluings) [Bonzom, Gurau, Rivasseau ’12]

2 1 2 1 2 1 3 2 1 2 1 2 1 3 3 3 2 2 1 3 1 2 1 3 3 2 1 2 1 2 1 3 3 3 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 8 / 13

Generalizing the combinatorics of simplicial GFT

No obstacle for GFT of polyhedral interactions with simplicial boundary Important: criteria, analytical/numerical control Polytopes effectively generated anyway: Ex.: Uncoloring of colored GFT (bipartite simplex gluings) [Bonzom, Gurau, Rivasseau ’12] Decomposition generalizes to GFT without colors: Any regular boundary atoms generated by simplicial interaction

2 1 2 1 2 1 3 2 1 2 1 2 1 3 3 3 2 2 1 3 1 2 1 3 3 2 1 2 1 2 1 3 3 3 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 8 / 13

Generalizing the combinatorics of simplicial GFT

No obstacle for GFT of polyhedral interactions with simplicial boundary Important: criteria, analytical/numerical control Polytopes effectively generated anyway: Ex.: Uncoloring of colored GFT (bipartite simplex gluings) [Bonzom, Gurau, Rivasseau ’12] Decomposition generalizes to GFT without colors: Any regular boundary atoms generated by simplicial interaction Prop.: Any atom with regular boundary graph can be decomposed into simplicial atoms

2 1 2 1 2 1 3 2 1 2 1 2 1 3 3 3 2 2 1 3 1 2 1 3 3 2 1 2 1 2 1 3 3 3 Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 8 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

S[{φ(l)}] =

• l
• [dg]φ(l)(g 1)K(l)(g 1, g 2)φ(l)(g 2) +
• i∈I

λi

• [dg]Vi
• {g ¯

v}Vi ¯ v∈Vi

φ(l)(g ¯

v)

. . .

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

S[{φ(l)}] =

• l
• [dg]φ(l)(g 1)K(l)(g 1, g 2)φ(l)(g 2) +
• i∈I

λi

• [dg]Vi
• {g ¯

v}Vi ¯ v∈Vi

φ(l)(g ¯

v)

. . . extension to D-polytopes not specified

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

S[{φ(l)}] =

• l
• [dg]φ(l)(g 1)K(l)(g 1, g 2)φ(l)(g 2) +
• i∈I

λi

• [dg]Vi
• {g ¯

v}Vi ¯ v∈Vi

φ(l)(g ¯

v)

. . . extension to D-polytopes not specified infinite field species l ∈ N formally possible

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

S[{φ(l)}] =

• l
• [dg]φ(l)(g 1)K(l)(g 1, g 2)φ(l)(g 2) +
• i∈I

λi

• [dg]Vi
• {g ¯

v}Vi ¯ v∈Vi

φ(l)(g ¯

v)

. . . extension to D-polytopes not specified infinite field species l ∈ N formally possible GFT generating known Spin Foams [KKL ’10]

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Multi-species GFT

Straightforward generalization to irregular boundary graphs:

[Reisenberger, Rovelli ’01]

Extend field content φ(l) : G ×l − → R

S[{φ(l)}] =

• l
• [dg]φ(l)(g 1)K(l)(g 1, g 2)φ(l)(g 2) +
• i∈I

λi

• [dg]Vi
• {g ¯

v}Vi ¯ v∈Vi

φ(l)(g ¯

v)

. . . extension to D-polytopes not specified infinite field species l ∈ N formally possible GFT generating known Spin Foams [KKL ’10] practical usefullness?

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 9 / 13

Virtual edges

Correspondence to regular graphs k odd: Any graph can be obtained from a k-regular graph by contraction/deletion of edges labeled as virtual k even: Any graph with even valencies Advantage: Only patches with fixed valency needed!

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 10 / 13

Dually weighted GFT

Fields with indices m¯

vˆ v = 0, 1, .., M: φm¯

v (g¯

v) = φm¯

vˆ v1 ,...,m¯ vˆ vD (g¯

vˆ v1, ..., g¯ vˆ vD )

Meaning: edge physical m = 0, virtual m > 1

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 11 / 13

Dually weighted GFT

Fields with indices m¯

vˆ v = 0, 1, .., M: φm¯

v (g¯

v) = φm¯

vˆ v1 ,...,m¯ vˆ vD (g¯

vˆ v1, ..., g¯ vˆ vD )

Meaning: edge physical m = 0, virtual m > 1 S[φ] =

• m1,m2
• [dg]φm1(g 1)K(g 1, g 2)K(M)

m1,m2φm2(g 2)

+

• i∈I

λi

• {m¯

v }V

• [dg]Vi
• {g ¯

v}Vi

• V(M)

{m¯

v }V

• ¯

v∈Vi

φ(g ¯

v)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 11 / 13

Dually weighted GFT

Fields with indices m¯

vˆ v = 0, 1, .., M: φm¯

v (g¯

v) = φm¯

vˆ v1 ,...,m¯ vˆ vD (g¯

vˆ v1, ..., g¯ vˆ vD )

Meaning: edge physical m = 0, virtual m > 1 S[φ] =

• m1,m2
• [dg]φm1(g 1)K(g 1, g 2)K(M)

m1,m2φm2(g 2)

+

• i∈I

λi

• {m¯

v }V

• [dg]Vi
• {g ¯

v}Vi

• V(M)

{m¯

v }V

• ¯

v∈Vi

φ(g ¯

v)

using a matrix A, limM→∞ 1

M TrAq = δq,2:

K(M)

m1m2 = D i=1 K(M) m1,i m2,i =

1 A

• m1,i m2,i

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 11 / 13

Dually weighted GFT

Fields with indices m¯

vˆ v = 0, 1, .., M: φm¯

v (g¯

v) = φm¯

vˆ v1 ,...,m¯ vˆ vD (g¯

vˆ v1, ..., g¯ vˆ vD )

Meaning: edge physical m = 0, virtual m > 1 S[φ] =

• m1,m2
• [dg]φm1(g 1)K(g 1, g 2)K(M)

m1,m2φm2(g 2)

+

• i∈I

λi

• {m¯

v }V

• [dg]Vi
• {g ¯

v}Vi

• V(M)

{m¯

v }V

• ¯

v∈Vi

φ(g ¯

v)

using a matrix A, limM→∞ 1

M TrAq = δq,2:

K(M)

m1m2 = D i=1 K(M) m1,i m2,i =

1 A

• m1,i m2,i

All physical quantities in the limit limM→∞ M → ∞

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 11 / 13

Dually weighted GFT

Fields with indices m¯

vˆ v = 0, 1, .., M: φm¯

v (g¯

v) = φm¯

vˆ v1 ,...,m¯ vˆ vD (g¯

vˆ v1, ..., g¯ vˆ vD )

Meaning: edge physical m = 0, virtual m > 1 S[φ] =

• m1,m2
• [dg]φm1(g 1)K(g 1, g 2)K(M)

m1,m2φm2(g 2)

+

• i∈I

λi

• {m¯

v }V

• [dg]Vi
• {g ¯

v}Vi

• V(M)

{m¯

v }V

• ¯

v∈Vi

φ(g ¯

v)

using a matrix A, limM→∞ 1

M TrAq = δq,2:

K(M)

m1m2 = D i=1 K(M) m1,i m2,i =

1 A

• m1,i m2,i

All physical quantities in the limit limM→∞ M → ∞ Result: Virtual edges are not dynamical, propagation as composite objects

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 11 / 13

Gravitational models, dually weighted

No obstacle for including the established amplitudes Ex.: eprl edge operator on edge ¯ v: K(g v1¯

v, g v2¯ v) =

• G ×2 dhv1¯

vdhv2¯ v

• (¯

vˆ v)

• Jˆ

v∈J

trJˆ

v

• g v1¯

vˆ vh−1 v1¯ v SJˆ

v,N¯ v hv2¯

vg −1 v2¯ vˆ v

• (SJ,N simplicity operator, J set of γ–simple reps of g = so(4) ∼

= su(2)+ × su(2)−) imposes simplicity on ”triangulated” atom alternative: trivial factor δ(g v1¯

vˆ vh−1 v1¯ v)δ(hv2¯ vg −1 v2¯ vˆ v) on virtual links

modification of geometricity for higher valency?

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 12 / 13

Conclusions

combinatorial compatibility of any LQG and GFT shown higher-valent GFT models: multi-species, dual weighting precise understanding of generated complexes

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 13 / 13

Conclusions

combinatorial compatibility of any LQG and GFT shown higher-valent GFT models: multi-species, dual weighting precise understanding of generated complexes Stage is set to analyze these models effective interactions, relevant operators ... (though from a QFT perspective the generalization seems rather unnecessary and complicated)

Johannes Th¨ urigen (AEI Potsdam) Polyhedral GFT July 17, 2014 13 / 13