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

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

Triangulation Boundary of a 4-simplex: 5 tetrahedron and 10 triangles 2 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

Boundary Graph: Triangulation 1 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)

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, …

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…

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.

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/2 labels of SL(2, ℂ) irreps 𝑗= ℤ/2 labels of SU(2)/SU(1,1) discrete irreps − 1 2 +𝑖 𝑠∈ℝ labels of SU(1,1) continous irreps

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’]

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

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!

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

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

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

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

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

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,….

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)

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

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′

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 )

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 =

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 =

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

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

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 ….

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