ASYMPTOTIC ANALYSIS OF SPIN FOAM AMPLITUDE WITH TIMELIKE TRIANGLES:

TOWARDS EMERGING GRAVITY FROM SPIN FOAM MODELS

Hongguang Liu

ILQGS 19

Based on arXiv:1810.09042

In collaboration with Muxin Han

Centre de Physique Theorique, Marseille, France

Outline ■ Introduction

■ Amplitude and asymptotic analysis

■ Geometric interpretation

■ Amplitude at critical configurations

■ Conclusion

Triangulation

The building block: 4 simplex

Boundary of a 4-simplex:

5 tetrahedron and 10

triangles

1

2

34

5

1

2

3

4

5

5 valent vertex

and boundary

Boundary graph of a vertex:

5 node and 10 links

Dual graph

Picture from the book covariant loop quantum gravity (C. Rovelli F. Vidotto)

Boundary Graph: Triangulation

1

2

3

4

5

Gluing triangles

Each link dual to a triangle :

Colored by spin!

Each node dual to a tetrahedron.

Gauge invariance: an intertwiner

(rank-4 invariant tensor)

Spin network of boundary graph

Spin foam models

■ A state sum model on inspired from BF action

■ Lorentzian theory:

Gauge group

■ 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)

1

2

3

4

5

[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 and final slice (Cauchy surface)

triangulation (spacelike tetrahedra).

– dragging vertices forward to construct

the spacetime triangulation.

Result : spacetime triangulation with

• Every 4-simplex contains both

timelike and spacelike tetrahedra

(e.g. 3 timelike, 2 spacelike)

• Every timelike tetrahedron

contains both timelike and

spacelike triangles

Known predictions: Gravitational wavers, Kasner solution, FLRW, etc…

(1 + 1)-dimensional analogue of the Sorkin scheme

Timelike

[Gentle, Miller, Sorkin, Williams, Barrett, Collins, …

Asymptotics of spin foam models: What we have so far?

■ Spacelike tetrahedron (Euclidean model and Lorentzian model with Time gauge)

– Barrett-Crane model [Barrett and Steele, 03’]

– FK model [Conrady and L. Freidel, 08’]

– EPRL model [Barrett et al. 09’]

– EPRL model with many 4 -simplices [Han and Zhang 11’]

■ Timelike tetrahedron with all faces spacelike [Kaminski et al, 17’]

■ Results:

– Asymptotics of the amplitude is dominated by critical configurations.

– Critical configurations are simplicial geometry (possibly degenerate)

– Asymptotic limit related to Regge action (discrete GR)

Asymptotics of spin foam models: What we have so far?

■ Spacelike tetrahedron (Euclidean model and Lorentzian model with Time gauge)

– Barrett-Crane model [Barrett and Steele, 03’]

– FK model [Conrady and L. Freidel, 08’]

– EPRL model [Barrett et al. 09’]

– EPRL model with many 4 -simplices [Han and Zhang 11’]

■ Timelike tetrahedron with all faces spacelike [Kaminski et al, 17’]

■ Results:

– Asymptotics of the amplitude is dominated by critical configurations.

– Critical configurations are simplicial geometry (possibly degenerate)

– Asymptotic limit related to Regge action (discrete GR)

Non of them contains timelike triangles!!!In all examples the Regge geometries contain timelike triangles.

Summary

■ Asymptotic analysis of spin foam model with timelike triangles.

■ The asymptotics of the amplitude is dominated by critical configurations.

■ Critical configuration are again simplicial geometry.

■ There will be no degenerate sector in the critical configuration.

■ Asymptotic formula

𝐴 ~𝑁+𝑒𝑖𝑆𝒦 + 𝑁−𝑒

−𝑖𝑆𝒦

𝑆𝒦 Regge action on on the simplicial complex.

Conrady-Hnybida Extension:EPRL/FK with timelike triangles

𝜌 ∈ ℝ, 𝑛 ∈ ℤ/2 labels of SL(2, ℂ) irreps

𝑗 = ቐℤ/2

−1

2+ 𝑖 𝑠 ∈ ℝ

labels of SU(1,1) irreps

From “time” gauge to “space” gauge• Timelike tetrahedron with normals 𝑢 = 0,0,0,1• Stabilize group SU(1,1)

Y map ℋ𝑗 →ℋ(𝜌,𝑛) : physical Hilbert space

Area spectrum:

Conrady-Hnybida Extension:Boundary tetrahedron

A timelike tetrahedron can contain both timelike and spacelike triangles!

All triangles spacelikeAll triangles timelike

timelike

spacelike

Mixed

Spacelike action[Kaminski et al]

Sorkin Scheme in Lorentzian Regge Calculus

timelike action

t

y

x

SFM Amplitude

𝔰𝔲(1,1) generators: റ𝐹 = (𝐽3, 𝐾1, 𝐾2)

𝑡 𝑥1 𝑥2

Coherent states: on each triangle is defined as (with eigenstate of 𝐾1 and 𝐽3 )

Ψ𝑒𝑓 ∈ ℋ 𝜌,𝑛 = ൝𝑌 𝐷𝑗(𝑣)| ۧ𝑗, −𝑠, + , 𝑡𝑖𝑚𝑒𝑙𝑖𝑘𝑒

𝑌 𝐷𝑗±(𝑣)| ۧ𝑗, ±𝑗 , 𝑠𝑝𝑎𝑐𝑒𝑙𝑖𝑘𝑒, 𝑣 ∈ 𝑆𝑈(1,1)

𝐾1 eigenstates: 𝐾1| ۧ𝑗, 𝜆, ± = 𝜆 | ۧ𝑗, 𝜆, ± , 𝜆 ∈ ℝ

The amplitude now given by

Amplitude appears in integration form (in a large j approximation)

• Actions for timelike triangles are pure imaginary!

• ½ order singularity appears in denominator hSpacelike face action[Kaminski 17’]

𝐽3 eigenstates: 𝐽3| ۧ𝑗, 𝑚 = 𝑚| ۧ𝑗,𝑚 , 𝑚 ∈ ℤ/2

Timelike face action

timelike

spacelike

Basic variables and gauge transformations

Actions

𝑧𝑣𝑓 ∈ ℂ𝑃 1, 𝑔𝑣𝑒 ∈ 𝑆𝐿(2, ℂ), Z𝑣𝑒𝑓 = 𝑔𝑣𝑒† ҧ𝑧𝑣𝑓 ∈ ℂ2, 𝑣𝑒𝑓 ∈ 𝑆𝑈(1,1),

𝑙𝑒𝑓± , 𝜉𝑒𝑓

± ∈ ℂ2, 𝑙𝑒𝑓± = 𝑣𝑒𝑓

1±1

, 𝜉𝑒𝑓+ = 𝑣𝑒𝑓

10

, 𝜉𝑒𝑓− = 𝑣𝑒𝑓

0−1

s.t. 𝑙+, 𝑙− = 𝜉+, 𝜉+ = 𝜉−, 𝜉− = 1, 𝜉+, 𝜉− = 𝑙+, 𝑙+ = 𝑙−, 𝑙− = 0.

Variables

Null basis in ℂ2

• Spacelike action for spacelike triangles, complex [Kaminski et al.]

Pure imaginary!!

• Timelike action for timelike triangles

SU(1,1) inner product

Basic variables and gauge transformations

Actions

• Spacelike action for spacelike triangles, complex [Kaminski et al.]

Pure imaginary!!

• Timelike action for timelike triangles

Gauge transformations

Asymptotics of the amplitudestationary phase approximation

■ Semiclassical limit of spinfoam models:

– Area spectrum: 𝐴𝑓 = 𝑙𝑝2𝑛𝑓/2

– Keep area fixed & Take 𝑙𝑝2 → 0

– Results in scaling 𝑛𝑓~𝑠𝑓 → ∞ uniformly. Large - 𝑗 asymptotics

■ Recall integration form of the amplitude (timelike face part only)

𝑆 linear in 𝑗𝑓 and pure imaginary ½ order singularity appears in denominator h

Stationary phase approximation with branch

point!

𝑗 = −1

2+ 𝑖 𝑠, 𝑛~𝛾𝑠, 𝜌 ~ − 𝑠

SU(1,1) continuous series

Stationary phase analysiswith branch points

Stationary phase approximation with branch point:

• Critical point locates at the branch point

Critical point 𝑥𝑐: solutions of 𝛿𝑥𝑆 𝑥 = 0

Our case

• Critical point and branch point are separated

✔

✘

𝑥0: ½ order singularity

Asymptotics of integration 𝐼: Λ → ∞

Multivariable case: iterate evaluation.

Measure factor is more invovled.

Critical point equationsfor a general timelike tetrahedron

Variation respect to 𝑧𝑣𝑓 : gluing condition on edges 𝑒 → 𝑒′

Decomposition of 𝑍𝑣𝑒𝑓 using 𝑙𝑒𝑓±

: change of variables

𝑍𝑣𝑒𝑓 = 휁𝑣𝑒𝑓 𝑙𝑒𝑓± + 𝛼𝑣𝑒𝑓𝑙𝑒𝑓

∓

𝑍𝑣𝑒𝑓 ∈ ℂ2휁𝑣𝑒𝑓 ∈ ℂ

𝛼𝑣𝑒𝑓 ∈ ℂ

𝑔𝑣𝑒

𝑔𝑣𝑒′𝑣

𝑡𝑒

𝑡𝑒′

𝑓

Z𝑣𝑒𝑓 = 𝑔𝑣𝑒† ҧ𝑧𝑣𝑓 impose:

𝑍𝑣𝑒𝑓 = 휁𝑣𝑒𝑓 𝜉𝑒𝑓± + 𝛼𝑣𝑒𝑓𝜉𝑒𝑓

±

Real condition of spacelike action : 𝛼𝑣𝑒𝑓 = 0

Spacelike:

Timelike:

Variation respect to 𝑙𝑒𝑓 : gluing condition on vertices 𝑣 → 𝑣′

Variation respect to 𝑔𝑣𝑒 : closure on triangules 𝑓

timelike vectorsspacelike vectors

null vectors

𝑒

𝑡𝑒

12

3

4

5

𝑣

𝑣′

Check the paper for cases when all triangles are timelike

Timelike action:

Spacelike action: Real Condition

Critical point equationsfor a general timelike tetrahedron

Geometric interpretationof critical configuration

■ Define maps ℂ2 → 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

spin 1

Critical point solutions Geometrical interpretationSimplicial geometry

ℂ2, SL(2, ℂ) ℝ(1,3), SO(1,3)

𝑢 = 0,0,0,1

normal of triangles in a tetrahedron

Geometric solutionfor a tetrahedron with both timelike and spacelike triangles

A non-degenerate tetrahedron geometry exists only when timelike triangles is with action 𝑆𝑣𝑓+

We define a bivector

with vectors and areas

Critical point equationsParallel transport

ClosureSimplicity condition

𝐵𝑒𝑓 𝑣 = 𝐵𝑒′𝑓 𝑣 = 𝐵𝑓 𝑣

𝑁𝑒 𝑣 ⋅ 𝐵𝑒𝑓 𝑣 = 0 𝑓∈𝑡𝑒

𝜖𝑒𝑓 𝑣 𝐵𝑒𝑓 𝑣 = 0

Define:

Geometric Reconstructionsingle non-degenerate 4 simplex

A geometrical 4-simplex

1

2

34

5

Geometric solutions

Outpointing normals:𝑁𝑖Δ Normals 𝑁𝑖

𝑁𝑖 = ±𝑁𝑖Δ

Bivectors 𝐵𝑖𝑗Δ Bivectors 𝐵𝑖𝑗

𝐵𝑖𝑗 = ±𝐵𝑖𝑗Δ

Geometric rotations:

𝐺Δ ∈ 𝑂(1,3)Geometrical solution

𝐺 ∈ 𝑆𝑂(1,3)

𝐺𝑖 = 𝐺𝑖Δ𝐼𝑠𝑖(𝐼𝑅𝑢)

𝑠

Length matching condition

Orientation matching condition

Determine tetrahedron

edge lengths uniquely

[Barrett et al, Han et al, Kaminski et al,….

Cut

(non-degenerate)

every 4 out of 5 linearly independent

Geometric Reconstructionsingle 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 𝐺𝑣𝑒 , such that the

bivectors 𝐵𝑓 𝑣 =∗ 𝐺𝑣𝑒𝑣𝑒𝑓⋀𝐺𝑣𝑒𝑢 correspond to bivectors of a reconstructed 4

simplex as

And normals

The two gauge equivalence classes of geometric solutions are related by

𝐵𝑓(𝑣) = 𝑟(𝑣)𝐵𝑓Δ(𝑣), 𝑟 𝑣 = 𝐼𝑠(𝑣) = ±1

𝑁𝑒(𝑣) = 𝐼𝑠𝑒 𝑣 𝑁𝑒Δ(𝑣)=±𝑁𝑒

Δ(𝑣)

Geometric ReconstructionSimplicial complex with many 4 simplicies

𝑁𝑒(𝑣) = 𝐺𝑣 𝑣′𝑁𝑒(𝑣′)

Reconstruction theorem

Gluing condition 𝑒 = (𝑣, 𝑣′)

Consistent Orientation:

𝑣: [𝑝0, 𝑝1,𝑝2, 𝑝3, 𝑝4]

𝑣′: −[𝑝0′, 𝑝1,𝑝2, 𝑝3, 𝑝4]𝑠𝑔𝑛 𝑉 𝑣 𝑟(𝑣) = 𝑐𝑜𝑛𝑠𝑡

𝐵𝑓(𝑣) = 𝑟(𝑣)𝐵𝑓Δ(𝑣)

Reconstruction at 𝑣, 𝑣′

𝑁𝑒(𝑣) = 𝐼𝑠𝑒(𝑣)𝑁𝑒Δ(𝑣)

The simplicial complex 𝒦 can be subdivided into sub-complexes

𝒦1, … ,𝒦2, such that (1) each 𝒦𝑖 is a simplicial complex with boundary, (2)

within each sub-complex 𝒦𝑖 , 𝑠𝑔𝑛 𝑉 𝑣 is a constant. Then there exist

𝐺𝑓∆ ∈ 𝑆𝑂 1,3 = 𝐼σ𝑣∈𝑓 1𝐼σ𝑣,𝑒∈𝑓 𝑠𝑒 𝑣 𝐺𝑓 = ±𝐺𝑓.

such that 𝐺𝑓∆ are the discrete spin connection compatible with the co-frame.

For two non gauge inequivalent geometric solutions with different 𝑟:෪𝐺𝑓 = 𝑅𝑢𝐺𝑓𝑅𝑢

𝐵𝑓(𝑣) = 𝐺𝑣 𝑣′𝐵𝑓(𝑣′)𝐺𝑣 𝑣′

−1

𝑣′

𝑣

Degenerate Solutions?Degenerate Condition: All normals are parallel to each other: 𝐺𝑣𝑒 ∈ 𝑆𝑂(1,2)

if the 4-simplex contains both timelike and spacelike tetrahedra

Can not be degenerate!!

Vector geometries

𝐺𝑣𝑒 𝑣𝑒𝑓 = 𝐺𝑣𝑒′𝑣𝑒′𝑓, σ𝑓 휀𝑒𝑓 𝑣 𝑣𝑒𝑓= 0

1-1 correspondence to solutions in split signature space 𝑀′ with (-,+,+,-)

• pair of two non-gauge equivalent vector geometries 𝐺𝑣𝑒±

,

• Geometric 𝑆𝑂(𝑀′) non-degenerate solution 𝐺′𝑣𝑒.

The two vector geometries are obtained from 𝑆𝑂(𝑀′) solutions with map Φ±:

Φ± 𝐺′𝑣𝑒 = 𝐺𝑣𝑒

±

Geometric 𝑆𝑂(𝑀′) solution 𝐺′𝑣𝑒 degenerate: vector geometries 𝐺𝑣𝑒±

gauge

equivalent

Flipped signature solution

Reconstruction Reconstruction 4-simplex in 𝑀′shape matching

Action at critical configurationtimelike triangles

Action after decomposition: A phase

Asymptotics of amplitude: determine 𝜃𝑒′𝑣𝑒𝑓 and 𝜙𝑒′𝑣𝑒𝑓 at critical points

U(1) ambiguity from boundary coherent states Ψ = 𝐷 𝑣(𝑁)𝑒𝑖 𝑢 𝐾1 | ۧ𝑗, −𝑠

Phase difference at two critical points ∆𝑆 = 𝑆 𝐺 − 𝑆( ෨𝐺)

Recall stationary phase formula

𝑒𝑆 ~ Ψ 𝐷(𝑔−1𝑔′) Ψ′

Phase differenceon one 4 simplex with boundary triangles

𝑁𝑒′

𝑁𝑒

𝑔𝑣𝑒

𝑔𝑣𝑒′𝑣

From critical point equations

Reconstruction theorem

Two solutions for given boundary data.

𝐺𝑓 𝑣 = 𝐺𝑒′𝑣𝐺𝑣𝑒Reflection

=

Reconstruction

Boundary tetrahedron normals

Rotation in a

spacelike plane

Phase differenceon simplicial complex with many 4 simplices

Face holonomy: 𝐺𝑓 𝑣 = ς𝑣𝐺𝑒′𝑣𝐺𝑣𝑒

෪𝐺𝑓 = 𝑅𝑢𝐺𝑓𝑅𝑢

Reconstruction theorem: for two gauge inequivalent solutions

(with different uniform orientation 𝑟)

From critical point equations

=

Reconstruction

Now the normals become

parallel transported vector along the face

Rotation

in a

spacelike

plane

Phase difference

Now phase difference

𝑖𝜋 ambiguity relates to the lift ambiguity!

Some of them may be absorbed to gauge transformations 𝑔𝑣𝑒 → −𝑔𝑣𝑒

𝐴𝑓 = 𝛾𝑠𝑓 = 𝑛𝑓/2 ∈ 𝑍/2

𝜃𝑓 here a rotation angle

Related to dihedral angles Θ𝑓 𝑣 = 𝜋 − 𝜃𝑓(𝑣)

deficit angles 𝜖𝑓 𝑣 = 2𝜋 − σ𝑣∈𝑓Θ𝑓 𝑣 = (1 − 𝑚𝑓)𝜋 − 𝜃𝑓(𝑣)

Fixed at each vertex

The total phase difference

No. of simplicies

in the bulk

When 𝑚𝑓 even, Regge action up to a sign

In all known examples of Regge calculus, 𝑚𝑓 is even

Degenerate solutions

From critical point equations: two non-gauge equivalent solution 𝑔𝑣𝑒±

Degenerate: 𝑔𝑣𝑒± ∈ 𝑆𝑈 1,1

1-1 Correspondence to flipped signature non-degenerate solutions:

=

Lift ambiguity

Regge action

bivectors in flipped signature space

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 and Results in Lorentzian Regge calculus: emerging gravity

from spin foams (Kasner and FLRW cosmology model with Han.)

• Black holes in semi-classical limit of spin foam models

• Graviton propagator

• Causal structure?

• ….

Thanks for your attention