1. Asymptotic Analysis of Spin Foam Amplitude with Timelike Triangles: Towards emerging gravity from spin foam models
Hongguang Liu
Centre de Physique Theorique, Marseille, France
Miami 2018
In collaboration with Muxin Han

2. Outline Introduction Amplitude and asymptotic analysis
Geometric interpretation
Amplitude at critical configurations
Conclusion

3. Triangulation Boundary of a 4-simplex: 5 tetrahedron and 10 triangles2 3 4 5
Boundary of a 4-simplex:
5 tetrahedron and 10 triangles
The building block: 4 simplex
Dual graph
Dual Graph
Boundary graph of a vertex:
5 node and 10 links 1 2 3 4 5
5 valent vertex and boundary

4. Boundary Graph: Triangulation1 2 3 4 5
Each link dual to a triangle :
Colored by spin! 1 2 3 4 5
Gluing triangles
Spin network of boundary graph
Each node dual to a tetrahedron.
Gauge invariance: an intertwiner (rank-4 invariant tensor)

5. Spin foam models
A state sum model on inspired from BF action+Simplicity constraint
Lorentzian theory:
Gauge group 𝐺:SL(2,ℂ)
Boundary gauge fixing: fix the normal perpendicular to the tetrahedron
Time gauge 𝑢= 1,0,0,0 （EPRL models)
Space gauge 𝑢= 0,0,0,1 (Conrady-Hybrida Extension)
𝑒 𝑓
𝑔 1 𝑓 𝑒𝑓
𝑔 2
𝑔 5
𝑔 3
𝑔 4
Barrett, Rovelli, Dittrich, Engle, Livine, Freidel, Han, Dupris, Conrady, …

6. Motivation: (3+1)-D Regge calculus Model
A discrete formulation of General relativity.
Discrete geometry: Sorkin triangulation
Initial slice triangulation (spacelike tetrahedra).
dragging vertices forward to construct the spacetime triangulation.
Result : spacetime triangulation with
Every 4-simplex contains both timelike and spacelike tetrahedra (3 timelike, 2 spacelike)
Every timelike tetrahedron contains both timelike and spacelike triangles
Timelike
(1 + 1)-dimensional analogue of the Sorkin scheme
Known predictions: Gravitational wavers, Kasner solution, etc…

7. Summary
Asymptotic analysis of spin foam model with timelike or spacelike triangles.
The asymptotics of the amplitude is dominated by critical configurations.
Critical configuration are simplicial geometry (possibely degenerate).
There will be no degenerate sector in the critical configuration with Sorkin like configuration.
Asymptotic formula
𝐴 ~ 𝑁 + 𝑒 𝑖 𝑆 𝒦 + 𝑁 − 𝑒 −𝑖 𝑆 𝒦
𝑆 𝒦 =𝐴 Regge action on on the simplicial complex.

8. Conrady-Hnybida Extension: EPRL/FK with timelike triangles
“time” gauge and “space” gauge
Timelike tetrahedron with normals 𝑢= 0,0,0,1 ,Stabilize group SU(1,1)
Spacelike tetrahedron with normals 𝑢= 1,0,0,0 ,Stabilize group SU(2)
Y map ℋ 𝑗 → ℋ (𝜌,𝑛) : physical Hilbert space
Area spectrum:
𝜌 ∈ℝ,𝑛∈ℤ/ labels of SL(2, ℂ) irreps
𝑗= ℤ/ labels of SU(2)/SU(1,1) discrete irreps − 1 2 +𝑖 𝑠∈ℝ labels of SU(1,1) continous irreps

9. SFM Amplitude 𝔰𝔲(1,1) generators: 𝐹 =( 𝐽 3 , 𝐾 1 , 𝐾 2 )
𝐾 1 eigenstates: 𝐾 1 | 𝑗,𝜆,± = 𝜆 | 𝑗,𝜆,± , 𝜆∈ℝ 𝑡
𝑥 1
𝑥 2
𝐽 3 eigenstates: 𝐽 3 | 𝑗,𝑚 = 𝑚| 𝑗,𝑚 , m∈ℤ/2
Coherent states: on each triangle is defined as (with eigenstate of 𝐾 1 )
Ψ 𝑒𝑓 ∈ ℋ 𝜌,𝑛 = 𝑌 𝐷 𝑗 (𝑣)| 𝑗,−𝑠,+ , 𝑡𝑖𝑚𝑒𝑙𝑖𝑘𝑒 𝑌 𝐷 𝑗± (𝑣)| 𝑗,±𝑗 , 𝑠𝑝𝑎𝑐𝑒𝑙𝑖𝑘𝑒 , 𝑣∈𝑆𝑈(1,1)/𝑆𝑈(2)
The amplitude now given by
Amplitude appears in integration form
Timelike face action
Actions 𝑆 for timelike triangles are pure imaginary!
½ order singularity appears in denominator h
Spacelike face action
[Kaminski 17’]

10. Basic variables and gauge transformations
Actions
Variables
𝑧 𝑣𝑓 ∈ ℂ𝑃 1 , 𝑔 𝑣𝑒 ∈𝑆𝐿(2,ℂ), Z 𝑣𝑒𝑓 = 𝑔 𝑣𝑒 † 𝑧 𝑣𝑓 ∈ ℂ 2
𝑙 𝑒𝑓 ± ∈ ℂ 2 s.t. 𝑙 + , 𝑙 − =1, 𝑙 + , 𝑙 + = 𝑙 − , 𝑙 − =0.
Null basis in ℂ 2
Gauge transformations

11. Asymptotics of the amplitude stationary phase approximation
Semiclassical limit of spinfoam models:
Area spectrum: 𝐴 𝑓 = 𝑙 𝑝 2 𝑛 𝑓 /2
Keep area fixed & Take 𝑙 𝑝 2 →0
Results in scaling 𝑛 𝑓 ~𝑠 𝑓 →∞ uniformly. Large - 𝑗 asymptotics
Recall integration form of the amplitude
SU(1,1) continuous series
𝑗=− 1 2 +𝑖 𝑠, n~𝛾𝑠
𝑆 linear in 𝑗 𝑓 and pure imaginary
½ order singularity appears in denominator h
Stationary phase approximation with branch point!

12. Stationary phase analysis with branch points
Asymptotics of integration 𝐼: Λ→∞
Critical point 𝑥 𝑐 : solutions of 𝛿 𝑥 𝑆 𝑥 =0
𝑥 0 : ½ order singularity
Stationary phase approximation with branch point:
Critical point locates at the branch point
Our case ✔
Critical point and branch point are separated ✘
Multivariable case: iterate evaluation

13. Critical point equations
Decomposition of 𝑍 𝑣𝑒𝑓 using 𝑙 𝑒𝑓 ± : change of variables
𝑍 𝑣𝑒𝑓 = 𝜁 𝑣𝑒𝑓 𝑙 𝑒𝑓 ± + 𝛼 𝑣𝑒𝑓 𝑙 𝑒𝑓 ∓
𝜁 𝑣𝑒𝑓 ∈ℂ
𝑍 𝑣𝑒𝑓 ∈ ℂ 2
𝛼 𝑣𝑒𝑓 ∈ℂ
Z 𝑣𝑒𝑓 = 𝑔 𝑣𝑒 † 𝑧 𝑣𝑓 Impose:
Variation respect to 𝑧 𝑣𝑓 : gluing condition on edges 𝑒 →𝑒′
𝑔 𝑣𝑒
𝑔 𝑣𝑒′ 𝑣
𝑡 𝑒′ 𝑓
𝑡 𝑒

14. Critical point equations for a general timelike tetrahedron
Variation respect to 𝑙 𝑒𝑓 : gluing condition on vertices 𝑣→𝑣′
𝑡 𝑒 1 𝑣′ 𝑣 𝑒
Variation respect to 𝑔 𝑣𝑒 : closure on faces 𝑓
null vectors 1 2 3 4 5
Timelike vectors
[Kaminski 17’]
spacelike vectors
Check the paper for cases when all triangles are timelike

15. Geometric interpretation of critical configuration
Critical point solutions
Geometrical interpretation
Simplicial geometry
ℂ 2 ,SL(2,ℂ)
ℝ (1,3) , SO(1,3)
Define maps ℂ 2 → ℝ (1,3) , 𝜋:SL(2,ℂ)→SO(1,3)
With gauge transformation 𝑔 →−𝑔, we can always gauge fix 𝐺= 𝜋(𝑔)∈ SO + (1,3)
Geometric vector and bivectors
𝑢= 0,0,0,1
spin 1
normal of triangles in a tetrahedron

16. Geometric solution for a tetrahedron with both timelike and spacelike triangles
A non-degenerate tetrahedron geometry exists only when timelike triangles is with action 𝑆 𝑣𝑓+
We define a bivector
with vectors and areas
Critical point equations
Parallel transport
Simplicity condition
Closure

17. Geometric Reconstruction single non-degenerate 4 simplex
Cut
A geometrical 4-simplex
Geometric solutions
(non-degenerate)
𝑁 𝑖 =± 𝑁 𝑖 Δ
Outpointing normals: 𝑁 𝑖 Δ
Normals 𝑁 𝑖
every 4 out of 5 linearly independent
Bivectors 𝐵 𝑖𝑗 Δ
Bivectors 𝐵 𝑖𝑗
𝐵 𝑖𝑗 =± 𝐵 𝑖𝑗 Δ
Geometric rotations: 𝐺 Δ ∈𝑂(1,3)
Geometrical solution
𝐺∈𝑆𝑂(1,3)
𝐺 𝑖 = 𝐺 𝑖 Δ 𝐼 𝑠 𝑖 ( 𝐼 𝑅 𝑢 ) 𝑠 1 2 3 4 5
Determine tetrahedron edge lengths uniquely
Length matching condition
𝜂 𝐼𝐽 𝑁 𝑖 𝐼 𝐵 𝑖𝑗 𝐽𝐾 =0
Orientation matching condition
[Barrett et al, Han et al, Kaminski et al,….

18. Geometric Reconstruction single 4 simplex
Reconstruction theorem
The non-degenerate geometrical solution exists if and only if the lengths and orientations matching conditions are satisfied.
There will be two gauge in-equivalent geometric solutions 𝐺 𝑣𝑒 , such that the bivectors 𝐵 𝑓 𝑣 =∗ 𝐺 𝑣𝑒 𝑣 𝑒𝑓 ⋀ 𝐺 𝑣𝑒 𝑢 correspond to bivectors of a reconstructed 4 simplex as
And normals
The two gauge equivalence classes of geometric solutions are related by
𝐵 𝑓 (𝑣)=𝑟 (𝑣)𝐵 𝑓 Δ (𝑣), 𝑟 𝑣 = 𝐼 𝑠(𝑣) =±1
𝑁 𝑒 (𝑣)= 𝐼 𝑠 𝑒 𝑣 𝑁 𝑒 Δ (𝑣)=± 𝑁 𝑒 Δ (𝑣)

19. Geometric Reconstruction Simplicial complex with many 4 simplicies
Gluing condition 𝑒=(𝑣, 𝑣 ′ )
Reconstruction at 𝑣, 𝑣 ′
𝑁 𝑒 (𝑣)= 𝐺 𝑣 𝑣 ′ 𝑁 𝑒 ( 𝑣 ′ )
𝑁 𝑒 (𝑣)= 𝐼 𝑠 𝑒 (𝑣) 𝑁 𝑒 Δ (𝑣)
𝐵 𝑓 (𝑣)= 𝐺 𝑣 𝑣 ′ 𝐵 𝑓 ( 𝑣 ′ ) 𝐺 𝑣 𝑣′ −1
𝐵 𝑓 (𝑣)=𝑟(𝑣) 𝐵 𝑓 Δ (𝑣)
Consistent Orientation:
𝑣:[ 𝑝 0 , 𝑝 1, 𝑝 2 , 𝑝 3 , 𝑝 4 ]
𝑣 ′ :−[ 𝑝 0 ′, 𝑝 1, 𝑝 2 , 𝑝 3 , 𝑝 4 ]
𝑠𝑔𝑛 𝑉 𝑣 𝑟(𝑣)=𝑐𝑜𝑛𝑠𝑡
Reconstruction theorem
The simplicial complex 𝒦 can be subdivided into sub-complexes 𝒦 1 ,…, 𝒦 2 ,  such that (1) each 𝒦 𝑖 is a simplicial complex with boundary, (2) within each sub-complex 𝒦 𝑖 , 𝑠𝑔𝑛 𝑉 𝑣 is a constant. Then there exist
𝐺 𝑓 ∆ ∈𝑆𝑂 1,3 = 𝐼 𝑣∈𝑓 1 𝐼 𝑣,𝑒∈𝑓 𝑠 𝑒 𝑣 𝐺 𝑓 =± 𝐺 𝑓 .
such that 𝐺 𝑓 ∆ are the discrete spin connection compatible with the co-frame.
For two non gauge inequivalent geometric solutions with different 𝑟:
𝐺 𝑓 = 𝑅 𝑢 𝐺 𝑓 𝑅 𝑢

20. 𝐺 𝑣𝑒 𝑣 𝑒𝑓 = 𝐺 𝑣𝑒′ 𝑣 𝑒′𝑓 , 𝑓 𝜀 𝑒𝑓 𝑣 𝑣 𝑒𝑓 =0
Degenerate Solutions?
Degenerate Condition: All normals are parallel to each other: 𝐺 𝑣𝑒 ∈𝑆𝑂(1,2)
if the 4-simplex contains both timelike and spacelike tetrahedra
Can not be degenerate!!
Vector geometries
𝐺 𝑣𝑒 𝑣 𝑒𝑓 = 𝐺 𝑣𝑒′ 𝑣 𝑒′𝑓 , 𝑓 𝜀 𝑒𝑓 𝑣 𝑣 𝑒𝑓 =0
Flipped signature solution
1-1 correspondence to solutions in split signature space 𝑀′ with (-,+,+,-)
pair of two non-gauge equivalent vector geometries 𝐺 𝑣𝑒 ± ,
Geometric 𝑆𝑂(𝑀′) non-degenerate solution 𝐺′ 𝑣𝑒 .
The two vector geometries are obtained from 𝑆𝑂(𝑀′) solutions with map Φ ± :
Φ ± 𝐺 ′ 𝑣𝑒 = 𝐺 𝑣𝑒 ±
Geometric 𝑆𝑂(𝑀′) solution 𝐺′ 𝑣𝑒 degenerate: vector geometries 𝐺 𝑣𝑒 ± gauge equivalent
Reconstruction
shape matching
Reconstruction 4-simplex in 𝑀′

21. Action at critical configuration for timelike triangles
Recall stationary phase formula
Action after decomposition: A phase
Asymptotics of amplitude: determine 𝜃 𝑒 ′ 𝑣𝑒𝑓 and 𝜙 𝑒 ′ 𝑣𝑒𝑓 at critical points
U(1) ambiguity from boundary coherent states Ψ=𝐷 𝑣(𝑁) 𝑒 𝑖 𝑢 𝐾 1 | 𝑗,−𝑠
𝑒 𝑆 ~ Ψ 𝐷( 𝑔 −1 𝑔′) Ψ′
Phase difference at two critical points ∆𝑆=𝑆 𝐺 −𝑆( 𝐺 )

22. Phase difference on one 4 simplex with boundary triangles
Boundary tetrahedron normals
𝑁 𝑒′
𝑁 𝑒
𝑔 𝑣𝑒
𝑔 𝑣𝑒′ 𝑣
Reconstruction theorem
Two solutions for given boundary data.
Reflection
𝐺 𝑓 𝑣 = 𝐺 𝑒′𝑣 𝐺 𝑣𝑒
Rotation in a spacelike plane
From critical point equations
Reconstruction =

23. Phase difference on simplicial complex with many 4 simplicies
Face holonomy: 𝐺 𝑓 𝑣 = 𝑣 𝐺 𝑒′𝑣 𝐺 𝑣𝑒
Reconstruction theorem: for two gauge inequivalent solutions
(with different uniform orientation 𝑟)
𝐺 𝑓 = 𝑅 𝑢 𝐺 𝑓 𝑅 𝑢
Now the normals become
Rotation in a spacelike plane
parallel transported vector along the face
From critical point equations
Reconstruction =

24. Phase difference 𝐴 𝑓 =𝛾 𝑠 𝑓 = 𝑛 𝑓 /2∈𝑍/2 Now phase difference
𝑖𝜋 ambiguity relates to the lift ambiguity!
Some of them may be absorbed to gauge transformations 𝑔 𝑣𝑒 → −𝑔 𝑣𝑒
Fixed at each vertex
No. of simplicies
in the bulk
𝜃 𝑓 here a rotation angle
Related to dihedral angles Θ 𝑓 𝑣 = 𝜋− 𝜃 𝑓 (𝑣)
deficit angles 𝜖 𝑓 𝑣 =2𝜋− 𝑣∈𝑓 Θ 𝑓 𝑣 =(1− 𝑚 𝑓 )𝜋− 𝜃 𝑓 (𝑣)
The total phase difference
When 𝑚 𝑓 even, Regge action up to a sign
In all known examples of Regge calculus, 𝑚 𝑓 is even

25. Degenerate solutions
From critical point equations: two non-gauge equivalent solution 𝑔 𝑣𝑒 ±
Degenerate: 𝑔 𝑣𝑒 ± ∈𝑆𝑈 1,1
1-1 Correspondence to flipped signature non-degenerate solutions:
Lift ambiguity
bivectors in flipped signature space =
Regge action

26. Conclusion & Outlook
The asymptotic analysis of spin foam model is now complete (with non degenerate boundaries)
The asymptotics of the amplitude is dominated by critical configurations which are simplicial geometry
The model excludes degenerate geometry sector.
The asymptotic limit of the amplitude is related to Regge action
Outlooks
Continuum limit
Results in Lorentzian Regge calculus: emerging gravity from spin foams (Kasner cosmology model with Han.)
Black holes in semi-classical limit of spin foam models
Calculation of the Propagators ….

27. Thanks for your attention