Asymptotic Analysis of Spin Foam Amplitude with Timelike Triangles:

Towards emerging gravity from spin foam models

Hongguang Liu

Miami 2018

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

Dual Graph

1

2

3

4

5

5 valent vertex and boundary

Boundary graph of a vertex:5 node and 10 links

Dual graph

Boundary Graph: Triangulation 1

2

3

4

5

12

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+Simplicityconstraint

• Lorentzian theory:

Gauge group !: SL(2, ℂ)

• Boundary gauge fixing: fix the normal perpendicular to thetetrahedron

• Time gauge ) = 1,0,0,0 �EPRL models)

• Space gauge ) = 0,0,0,1 (Conrady-Hybrida Extension)

Barrett, Rovelli, Dittrich, Engle, Livine, Freidel, Han, Dupris, Conrady, …

-.

/0

/1

/2/3

/4

-̃.

.

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

Known predictions: Gravitational wavers, Kasner solution, etc…

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

Timelike

Summary

• Asymptotic analysis of spin foam model with timelike or spaceliketriangles.

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

• Critical configuration are simplicial geometry (possibelydegenerate).

• There will be no degenerate sector in the critical configuration with Sorkin like configuration.

• Asymptotic formula! ~#$%&'( + #*%*&'(

+( = ! Regge action on on the simplicial complex.

Conrady-Hnybida Extension:EPRL/FK with timelike triangles

! ∈ ℝ, % ∈ ℤ/2 labels of SL(2, ℂ) irreps

* = ,ℤ/2 labels of SU(2)/SU(1,1) discrete irreps

−12 + 0 1 ∈ ℝ labels of SU(1,1) con8nous irreps

“time” gauge and “space” gauge• Timelike tetrahedron with normals 2 = 0,0,0,1 , Stabilize group SU(1,1)• Spacelike tetrahedron with normals 2 = 1,0,0,0 , Stabilize group SU(2)

Y map ℋ5 →ℋ(8,9) : physical Hilbert space

Area spectrum:

SFM Amplitude !"(1,1) generators: $⃑ = ('(, *+, *,)

. /+ /,

Coherent states: on each triangle is defined as (with eigenstate of *+)

Ψ12 ∈ ℋ 5,6 = 78 9:(;)| ⟩>, −@, + , .BCDEBFD8 9:±(;)| ⟩>, ±> , @HIJDEBFD

, ; ∈ KL(1,1)/KL(2)

*+ eigenstates: *+| ⟩>, P, ± = P | ⟩>, P, ± , P ∈ ℝ

The amplitude now given by

Amplitude appears in integration form

• Actions K for timelike triangles are pure imaginary!• ½ order singularity appears in denominator h

Av(K) =

Z

SL(2,C)

Y

e

dgveY

f2t

Z

CP1

⌦zvf

hvefhve0f

�eSvf+ + eSvf+ + eSvf+ + eSvf+

�Y

f2s

Z

CP1

⌦zvf eSvf�sp

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Spacelike face action[Kaminski 17’]

'( eigenstates: '(| ⟩>, C = C| ⟩>,C , m ∈ ℤ/2

Timelike face action

Basic variables and gauge transformations

Gauge transformations

Actions

!"# ∈ ℂ& ', (") ∈ *+(2, ℂ), Z")# = (")2 ̅!"# ∈ ℂ45)#± ∈ ℂ4 s.t. 57, 58 = 1, 57, 57 = 58, 58 = 0.

Variables

Null basis in ℂ4

Asymptotics of the amplitudestationary phase approximation

• Semiclassical limit of spinfoam models:• Area spectrum: !" = $%&'"/2• Keep area fixed & Take $%& → 0• Results in scaling '"~-" → ∞ uniformly. Large - / asymptotics

• Recall integration form of the amplitude

0 linear in /" and pure imaginary ½ order singularity appears in denominator h

Stationary phase approximation with branch point!

/ = − 2& + 4 -, n~6-

SU(1,1) continuous series

Av(K) =

Z

SL(2,C)

Y

e

dgveY

f2t

Z

CP1

⌦zvf

hvefhve0f

�eSvf+ + eSvf+ + eSvf+ + eSvf+

�Y

f2s

Z

CP1

⌦zvf eSvf�sp

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Stationary phase analysiswith branch points

Stationary phase approximation with branch point:• Critical point locates at the branch point

Critical point !": solutions of #$% ! = 0

Our case

• Critical point and branch point are separated ✘

!(: ½ order singularity

Asymptotics of integration ): Λ → ∞

Multivariable case: iterate evaluation

Critical point equations

Variation respect to !"# : gluing condition on edges $ → $′

Decomposition of '"(# using )(#± : change of variables

'"(# = ,"(# )(#± + ."(#)(#∓'"(# ∈ ℂ2,"(# ∈ ℂ."(# ∈ ℂ

3"(

3"(45

6(

6(47

Z"(# = 3"(9 ̅!"# Impose:

Variation respect to !"# : gluing condition on vertices $ → $′

Variation respect to '(" : closure on faces )

Timelike vectors[Kaminski 17’] spacelike vectors

null vectors

*+"

12

3

4

5

1

$$′

Check the paper for cases when all triangles are timelike

Critical point equationsfor a general timelike tetrahedron

Geometric interpretationof critical configuration

• Define maps ℂ" → ℝ(&,(), *: SL(2, ℂ) → SO(1,3)• With gauge transformation 2 → −2, we can always gauge fix 4 =*(2) ∈ SO7(1,3)

• Geometric vector and bivectorsspin 1

Critical point solutions Geometrical interpretationSimplicial geometryℂ", SL(2, ℂ) ℝ(&,(), SO(1,3)

8 = 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 !"#$We define a bivector

with vectors and areas

Critical point equations

Parallel transport

ClosureSimplicity condition

Geometric Reconstructionsingle non-degenerate 4 simplex

A geometrical 4-simplex

1

2

34

5

Geometric solutions

Outpointing normals:&'( Normals &'&' = ±&'(

Bivectors +',( Bivectors +',+', = ±+',(

Geometric rotations:-( ∈ /(1,3) Geometrical solution

- ∈ 3/(1,3)-' = -'(456(478)5

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 theoremThe 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 !"# , such that the bivectors $% & =∗ !"#&#%⋀!"#* correspond to bivectors of a reconstructed 4 simplex as

And normals

The two gauge equivalence classes of geometric solutions are related by

$%(&) = -(&)$%.(&), - & = /0(") = ±13#(&) = /04 " 3#.(&)=±3#.(&)

Geometric ReconstructionSimplicial complex with many 4 simplicies

!"($) = '( ()!"($*)

Reconstruction theorem

Gluing condition + = ($, $*)

Consistent Orientation: $: [/0, /1,/2, /3, /4]

$*: −[/0′, /1,/2, /3, /4]

89: ; $ <($) = =>:8?

@A($) = '( ()@A($*)'( (*

B1@A($) = <($)@A

C($)

Reconstruction at $, $*

!"($) = DEF(()!"C($)

The simplicial complex G can be subdivided into sub-complexes G1,… ,G2, such that (1) each GI is a simplicial complex with boundary, (2) within each sub-complex GI , 89: ; $ is a constant. Then there exist

'A∆∈ LM 1,3 = D

∑Q∈R 1D∑Q,F∈R EF ( 'A = ±'A.

such that 'A∆ are the discrete spin connection compatible with the co-frame.For two non gauge inequivalent geometric solutions with different <:

T'A = UV'AUV

Degenerate Solutions?Degenerate Condition: All normals are parallel to each other: !"# ∈ %&(1,2)if the 4-simplex contains both timelike and spacelike tetrahedra

Can not be degenerate!!Vector geometries

!"# ,#- = !"#/,#/-, ∑- 1#- , ,#- = 0

1-1 correspondence to solutions in split signature space 3′ with (-,+,+,-)• pair of two non-gauge equivalent vector geometries !"#± ,• Geometric %&(3′) non-degenerate solution !′"#.The two vector geometries are obtained from %&(3′) solutions with map Φ±:

Φ± !/"# = !"#±

Geometric %&(3′) solution !′"# degenerate: vector geometries !"#± gauge equivalent

Flipped signature solution

Reconstruction Reconstruction 4-simplex in 3′shape matching

Action at critical configurationfor timelike triangles

Action after decomposition: A phase

Asymptotics of amplitude: determine !"#$"% and &"#$"% at critical points

U(1) ambiguity from boundary coherent states Ψ = ) *(,)./ 0 12 | ⟩5, −8

Phase difference at two critical points ∆: = : ; − :( <;)

Recall stationary phase formula

.= ~ Ψ )(?@A?′) Ψ′

Phase differenceon one 4 simplex with boundary triangles

!"#

!"$%"

$%"#&

From critical point equations

Reconstruction theoremTwo solutions for given boundary data.

'( & = '"#%'%" Reflection

=

Reconstruction

Boundary tetrahedron normals

Rotation in a spacelike plane

Phase differenceon simplicial complex with many 4 simplicies

Face holonomy: !" # = ∏& !'(&!&'

)!" = *+!"*+

Reconstruction theorem: for two gauge inequivalent solutions(with different uniform orientation ,)

From critical point equations

=

Reconstruction

Now the normals become

parallel transported vector along the face

Rotation in a spacelikeplane

Phase difference

Now phase difference

!" ambiguity relates to the lift ambiguity! Some of them may be absorbed to gauge transformations #$% → −#$%

() = +,) = -)/2 ∈ 1/2

2) here a rotation angle

Related to dihedral angles Θ) 4 = " − 2)(4)deficit angles 7) 4 = 2" − ∑$∈) Θ) 4 = (1 − :))" − 2)(4)

Fixed at each vertex

The total phase difference

No. of simpliciesin the bulk

When :) even, Regge action up to a sign In all known examples of Regge calculus, :) is even

Degenerate solutionsFrom critical point equations: two non-gauge equivalent solution !"#±

Degenerate: !"#± ∈ &' 1,11-1 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

• 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

• ….

Thanks for your attention