Gluon propagator fromYang-Mills spin foamsFlorian ConradyIGPG Penn StateQuantum Gravity in the Ameri as IIIIGPG Penn StateAugust 25, 2006

Plan of talk1. Brief review: gravitons from spin foams2. Why onsider Yang-Mills spin foams?3. SU(2) YM theory and its representations4. Gluons from Yang-Mills spin foams5. Comparison to Rovelli's approa h6. Con lusion

1

1. Brief review:Gravitons from spin foamsWork by: Rovelli, Modesto, Bian hi, Speziale,Willis, Livine . . .

2

De�nition of the propagator

PSfrag repla ements 4d bulk bounding3-sphereboundaryspin network s

x

y

Gabcdq (x,y) =

∑

s W [s] hab(x) hcd(y) Ψq[s]∑

s W [s] Ψq[s],

3

Ingredients

Sum over spin foam amplitude for a given boundary spin network s:W [s] =

1

Z

∫

Dφ fs(φ) e− R φ2− λ5!

Rφ5

Field insertions:

hab(~x) = (det g)gab(~x) − δab = Eai(~x)Ebi(~x) − δab

4

Ingredients

The boundary state is a Gaussian that is peaked at a ba kground geometry:Ψq[s] = CΓ exp

−1

2

∑

ll′

αll′jl − jl

(0)

(

jl(0))

k2

jl′ − jl′(0)

(

jl′(0))

k2

+ i∑

l

Φ(0)l jl

The phase term determines the average momentum of the state.

Asymptoti formula for 10j symbol:B(jij) =

∑

σ

P (σ)1

2

[

ei SRegge(σ)+i k π/4 + e−i SRegge(σ)−i k π/4]

+ D(jik) 5

Re ipe• perturbative expansion of group �eld theory→ sum over omplexes + sum over spin foams on ea h omplex• lowest order: omplex = 4-simplex• asymptoti formula for 10j symbols• expansion of Regge a tion• boundary state �pi ks out� one of the three terms→ graviton propagator

6

2. Why onsider Yang-Millsspin foams?• di�eren e to gravity: �xed ba kground• similarity: have to extra t physi s from spin foam amplitudes• advantage: we already know what the result should be

; test if our methods make sense• also inherent interest: on�nement problem

7

3. SU(2) YM theory and itsrepresentations

8

SU(2) latti e gauge theory in d = 3

Partition fun tion:

Z =

∫(

∏

e⊂κdUe

)

exp(

− β S(U))

Coupling:

β =4

ag2+

1

3

The a tion is a sum of fa e/plaquette a tions:S =

∑

f⊂κ

Sf

9

Spin foam representationThe spin foam representation results from the latti e pathintegral by 1. expanding the plaquette a tions in loops, and2. integrating out the onne tion variables Ue. There areseveral di�erent ways to do this.In d = 3 we an do it in su h a way that the amplitudesbe ome 6j-symbols. [Anishetty, Cheluvaraja, Sharat handra℄The ubes of the dual latti e κ∗ are split into tetrahedra,giving a triangulation T :

PSfrag repla ements4d bulkbounding3-sphereboundaryspin network

j1

j2

j3

j4j5

j6 triangulation T

10

Spin foam representationZ =

∑

~

(

∏

e⊂T

(2je + 1)

)(

∏

t⊂T

At

)(

∏

e⊂κ∗

e−2β je(je+1)

)

At = (−1)

P

i ji

{

j1 j2 j3j4 j5 j6

}

= ±

PSfrag repla ements4d bulkbounding3-sphereboundaryspin network

j1

j2

j3

j4j5

j6

triangulation

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

j1

j2

j3

j4j5

j6

11

3d SU(2) YM theory as a deformation of 3d gravity

3d SU(2) Yang-Mills theory an be seen as a deformation of 3d gravity:

ZYM =∑

~

(

∏

e⊂T

(2je + 1)

)(

∏

t⊂T

At

)(

∏

e⊂κ∗

e−2β je(je+1)

)

ZBF =∑

~

(

∏

e⊂T

(2je + 1)

)(

∏

t⊂T

At

)

12

4. Gluons from Yang-Mills spin foams13

Diakonov & PetrovDiakonov & Petrov suggested an argument for extra tinggluons from spin foams: [hep-th/9912268℄• repla e sums over spins by integrals• use asymptoti formula for 6j symbols• stationary phase approximation→ �atness onstraint on spin foams• solve onstraint→ gluons

14

Present workWe extend this argument by in luding stati quarks assour es and e�e ts due to dis reteness of spin.• Poisson summation formula→ trade dis reteness of spins for topologi al ex itations• use asymptoti formula for 6j symbols• stationary phase approximation→ �atness onstraint on spin foams+ defe t reated by sour e• solve onstraint→ gluons + topologi al ex itations→ stati potential

15

Topologi al ex itations

In this talk we ignore dis reteness e�e ts and repla e the sums over spins simplyby integrals.The proper treatment of dis reteness leads to additional degrees of freedom thatare similar to monopoles in U(1), and may be of interest for the on�nementproblem.

16

Potential between test quarks

Stati �test� quarks an be represented by a re tangular Wilson loop

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

C

T

R

q

q

1/2

The potential between the quarks is extra ted from the expe tation value of theWilson loop:

V (R) = limT→∞

− 1

Tln 〈WC〉

17

Wilson loop in the spin foam representation

Qualitatively:

〈WjC〉 = sum over all spin foams that are bounded by the Wilson loop

More pre isely:〈WjC〉 =

1

Z

∑

~

(

∏

e⊂T

(2je + 1)

)

∏

t⊂T ′

At

(

∏

v⊂C

A9jv

)(

∏

e⊂κ∗

e−2β je(je+1)

)

18

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

Wilson loop Clatti e κ

j

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

Wilson looplatti e

j1

j2

j3

j4j5

j6

j′2

j′5

modi�edtriangulation T ′

A9jv = ±

j1 j2 j3j′5 j j5j6 j′2 j4

= ±PSfrag repla ements4d bulkbounding3-sphereboundaryspin network

triangulationWilson looplatti e

modi�edtriangulationj1

j2

j3

j4

j5

j6

j′2j′5

j

19

Starting point〈WjC〉 =

1

Z

∑

~

(

∏

e⊂T

(2je + 1)

)

∏

t⊂T ′

At

(

∏

v⊂C

A9jv

)(

∏

e⊂κ∗

e−2β je(je+1)

)

a small, β large ; typi al spins large. We therefore approximate

2je + 1 ≈ 2je , je (je + 1) ≈ j2e − 1/4We repla e the sums over spins by integrals:

∑

je

−→∫ ∞

0

dje

20

Asymptoti formula for 6j symbolsAmplitude of a tetrahedron:

At = (−1)P

i ji

{

j1 j2 j3j4 j5 j6

}

Interpret the spin assignments ji as edge lengths ji + 1/2of the tetrahedron.Then, for large ji and a non-degenerate tetrahedronAt ≈

1√6πV

(

eiIR + e−iIR)

.

Here, V is the volume of the tetrahedron, and IR is the ontribution of the tetrahedron to the Regge a tion�adis retized version of ∫ d3x√

g R on triangulations.21

Stationary phase approximation

Given the asymptoti formula, we expe t an os illatory behaviour of the gravity+ sour e amplitude, ompared to a slowly hanging fa tore−2

β j2e .

; repla e path integral by a path integral over stationary points of the gravity +sour e part, i.e.

∫

Dj Agrav+source(j)Arest(j) ≈∫

Djcl Agrav+source(jcl) Arest(jcl)where the spin foams jcl are stationary points of the gravity + sour e part. 22

Outside the loop

How are these stationary points hara terized?For tetrahedra away from the loop C, the amplitude orresponds to pure 3dgravity.

; stationary points orrespond to the �at geometry.; Regge a tion IR = 0 in asymptoti formula.We ignore the volume fa tors, and set the amplitude At = 1.We assume the same for degenerate tetrahedra.

23

Along the loop

We have no asymptoti formula for the 9j-symbolPSfrag repla ements4d bulkbounding3-sphereboundaryspin network

triangulationWilson looplatti e

modi�edtriangulationj1

j2

j3

j4

j5

j6

j′2j′5

j

.

We know, however, that the ji are typi ally mu h larger than the j of the sour e,and that the 9j-symbol is zero unless j′i, ji and j satisfy the triangle inequality.The simplest possibility: 9j-symbol has just the e�e t of ensuring the triangleinequality and an be otherwise treated like the 6j-symbols, i.e.

A9jv ≈

1 , |ji − j| ≤ j′i ≤ ji + j ,

0 , otherwise .

24

Flatness onstraint

The stationary points are given by spin foams that are �at outside of C andsatisfy the additional triangle inequality|ji − j| ≤ j′i ≤ ji + j ,along C.What do we mean by a ��at� spin foam?The part of the spin foam outside C an be embedded in �at Eu lidean R3 su hthat ea h edge has length ji + 1/2.

25

Embedding in R3 olor spa e

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

Wilson looplatti emodi�edtriangulation

T ′ R3

be

je|be|=je+1/2

Ea h edge e of the triangulation T ′ is mapped into a di�eren e ve tor be in R3of length je + 1/2.Along losed urves the ve tors be lose, i.e. db = 0.Let us all the target spa e R3 olor spa e. 26

Defe t along C

Along C we do not have �atness. We only have theadditional triangle inequalities.

PSfrag repla ements4d bulkbounding3-sphereboundaryspin network

triangulation

Wilson looplatti e

modi�edtriangulation

R

T ′ R3

f

e be

e′b′e

JfAround fa es f ⊂ T ′ dual to C the ve tors be do not lose, in general. We only have the triangle inequality∣

∣

∣|be| − j

∣

∣

∣. |b′e| ≤ |be| + j .

We represent this by a de� it ve torJf =

j nf , f dual to C ,

0 , otherwise . 27

Defe t along C

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

Wilson looplatti emodi�edtriangulation

RR

T ′ R3

f

e be

e′b′e

Jf

Thus, we an express �atness + triangle inequalities along C by the single ondition

db = J .

28

From integrals over spins to integral overembeddings

PSfrag repla ements4d bulkbounding3-sphereboundaryspin network

x

triangulationWilson looplatti e

modi�edtriangulationR

RR

T ′ R3

bxµjxµ

|bxµ|=jxµ+1/2µWe repla e the integrals over �at spin foams by an integral over embeddings of

T ′ in the olor spa e R3.〈WjC〉 =

1

Z

∫R3

(

∏

xµd3bxµ

)

∫

S2dn δ (db − J) exp

[

− 1

2β

∑

x

b2xµ

]

29

Solving the onstraint

General solution to onstraint:

bxµ = ∇µϕx + bxµ ,where ϕ is an R3-valued s alar on T ′, and b is a parti ular solution to theinhomogeneous equation.The parti ular solution an be hosen asbxρ = ǫρµνuµ (u · ∇)−1Jxνwhere u is a unit ve tor in one of the latti e dire tions (say u = 1).

30

Solving the onstraintϕ is determined (up to a onstant) by b and b ; an repla e onstrained integralover b by an un onstrained integral over ϕ:〈WjC〉 =

1

Z

∫(

∏

xdϕx

)∫

S2dn exp

[

− 1

4β

∑

x

(

∇ρ ϕx + ǫρµνuµ (u · ∇)−1Jxν

)2

]

ϕ plays the role of the gluons: it has 3 degrees of freedom orresponding to 1physi al degree of freedom per gluon in d = 3.

31

PotentialGaussian integration over ϕ yields

〈WjC〉 =1

Z

∫

S2dn exp

−2

β

∑

xy

JaxµGx,yJ

ayν

,

where Gx,y is the latti e propagator.Re alling that

J = j nC and β =4

ag2,we obtain

〈WjC〉 = exp

−j2 a g2

2

∑

xy

CxµGx,yCyµ

.

32

Potential

Up to latti e orre tions this gives

V (R) = j2g2

π/a∫

−π/a

d2k

(2π)2

(

1

k2− eik1R

k2

)

,

whi h is roughly in agreement with the tree-level perturbative al ulation:

V (R) = j(j + 1) g2

∫

d2k

(2π)2

(

1

k2− eik1R

k2

)

33

Physi al pi ture

In the absen e of a sour e, the gluon �eld has the a tionS =

1

4β

∑

x

(∇µ ϕx)2

.

If we Wi k-rotate ba k to + −−−, the lassi al solutions are waves.Re all that the ve tors bxµ = ∇µϕx des ribe the di�eren e ve tors of theembedded triangulation.; ϕ de�nes the embedding T ′ → R3!

34

Gluons as waves of spin

PSfrag repla ements4d bulkbounding3-sphereboundaryspin networktriangulation

Wilson looplatti emodi�edtriangulation

RR

R

R3R3

69

50

42

68

56

39

A wave in ϕ orresponds to a wave of spins on the triangulation!35

Comparison to Rovelli's approa h

Rovelli et al. 3d SU(2) YMsour e �eld insertions Wilson loopapproximation asymptoti s of 10jexpansion around ba kground asymptoti s of 6jexpansion around ba kgroundtriangulation indep. GFT expansion re�nement limit oupling �xed G tuning of g36

6. Con lusion

We have presented a heuristi derivation of gluons from spin foams in 3d SU(2)Yang-Mills theory, extending earlier work by Diakonov & Petrov.Questions:

• how ould one develop a systemati perturbation theory?• generalization to d > 3 and SU(N)?The omparison gravity/YM suggests an alternative to the group �eld theoryapproa h: a re�nement limit with a tuned Plan k length.

37

Referen es

C.Rovelli, Graviton propagator from ba kground-independent quantum gravitygr-q /0508124E.Bian hi, L.Modesto, C.Rovelli, S.Speziale, Graviton propagator in loopquantum gravity gr-q /0604044D.Diakonov, V.Petrov, Yang-Mills theory in three dimensions as quantumgravity theory hep-th/9912268F.Conrady, Gluons and topologi al ex itations from spin foams in preparation38