ef fectiv e f ield theory fr om spin f oam models :3d e...

Effective field theory from spin Foam models :3d example

Laurent Freidel

Pennstate2007

Background independentLoop Quantum Gravity in a nutshell

•Hamiltonian quantisation: gravity is a gauge theory SU(2) Yang-mills phase space (A,E) + constraints

•Eigenstates of Geometrical operators, Area, Voldiscrete spectra quantized space geometry

•Dynamics: encoded in spin foam models allowing the computation of transition amplitudes between Spin networks states: quantum spacetime geometry

•Kinematical Hilbert space is spanned by spin network: graph colored by su(2) rep

+ ...

s

s’

i2

j5

j6j4 i4

j1

j3

j2

i3

i1

2

(!, je, iv) (15)

"(!,je,iv)(A) (16)

= trj(h!(A)) (17)

!Area"!,je,iv = 8!"l2P!

e"R#!

"je(je + 1) "!,je,iv (18)

!Area"(!,je,iv) =

#8!"l2P

!

e"R#!

"je(je + 1)$ %& '

("(!,je,iv) (19)

)Vol"!,je,iv =

#*8!"l2P

3 !

v"R#!

vje,iv

("!,je,iv (20)

F (A) ! E (21)

!Area(H) = A (22)

H = tr([E, E]F (A)) (23)

!

i

Hj(i) (24)

+Dg e

ilP

S(g)(25)

KF (j) ! kF (lP dj) (26)

[Xi!, Xj] = ilP #ijkX

k (27)

m" sin($m)/$ (28)

S3 # SU(2) (29)

X # %P (30)

Wave function

Background independence: what the quantum geometry is at Planck scale cannot be postulated its needs to be determined dynamically

Spin Foam models: transition amplitude

The dynamics results in a succession of evolution moves on a can be encoded by a colored singular surface S (2d complex) interpolating between initial and final spin networks

Transition amplitudes between spin network states are defined by

!s, s!"phys =!

F :s"s!

A[F ], (11)

where the notation anticipates the interpretation of such amplitudes as defining the physicalscalar product. The domain of the previous sum is left unspecified at this stage. We shalldiscuss this question further in Section 6. This last equation is the spin foam counterpartof equation (9). This definition remains formal until we specify what the set of allowedspin foams in the sum are and define the corresponding amplitudes.

l

l

j

k

j

l

k

q

q

o

p

p

os

m

n

j

k

# j

j

j

k

k

k

l

l

l

p

oq

q

p

o m

n s

j

k

l

m

ns

Figure 3: A typical path in a path integral version of loop quantum gravity is givenby a series of transitions through di!erent spin-network states representing a state of 3-geometries. Nodes and links in the spin network evolve into 1-dimensional edges and faces.New links are created and spins are reassigned at vertexes (emphasized on the right). The‘topological’ structure is provided by the underlying 2-complex while the geometric degreesof freedom are encoded in the labeling of its elements with irreducible representations andintertwiners.

In standard quantum mechanics the path integral is used to compute the matrix ele-ments of the evolution operator U(t). It provides in this way the solution for dynamicssince for any kinematical state " the state U(t)" is a solution to Schrodinger’s equation.Analogously, in a generally covariant theory the path integral provides a device for con-structing solutions to the quantum constraints. Transition amplitudes represent the matrixelements of the so-called generalized ‘projection’ operator P (Sections 3.1 and 6.3) suchthat P" is a physical state for any kinematical state ". As in the case of the vectorconstraint the solutions of the scalar constraint correspond to distributional states (zerois in the continuum part of its spectrum). Therefore, Hphys is not a proper subspace of Hand the operator P is not a projector (P 2 is ill defined)8. In Section 4 we give an explicitexample of this construction.

The background-independent character of spin foams is manifest. The 2-complex can bethought of as representing ‘space-time’ while the boundary graphs as representing ‘space’.

8In the notation of the previous section states in Hphys are elements of Cyl!.

12

faces of S are colored by spins (quanta of area)

edges of S are colored by invariant maps (quanta of 3d volume)

2 complex,triangulation coloring, int

geometryface,edge vert, local

amplitudes

Quantum transition amplitude; physical scalar product

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

AS(#0, #1) =!

jf ,ie

"

f

Af (jf )"

e

Ae(jf , ie)"

v

Av(jf , ie) (0.40)

3

initial, finalgeometry

2

(!, je, iv) (15)

"(!,je,iv)(A) (16)

= trj(h!(A)) (17)

!Area"!,je,iv = 8!"l2P!

e"R#!

"je(je + 1) "!,je,iv (18)

!Area"(!,je,iv) =

#8!"l2P

!

e"R#!

"je(je + 1)$ %& '

("(!,je,iv) (19)

)Vol"!,je,iv =

#*8!"l2P

3 !

v"R#!

vje,iv

("!,je,iv (20)

F (A) ! E (21)

!Area(H) = A (22)

H = tr([E, E]F (A)) (23)

!

i

Hj(i) (24)

A(!0, !1) =!

S

AS(!0, !1) (25)

+Dg e

ilP

S(g)(26)

KF (j) ! kF (lP dj) (27)


k (28)

m" sin($m)/$ (29)

S3 # SU(2) (30)

non perturbative and combinatorial definition of quantum gravity amplitudes

Spin Foam models: constraint BF

The choice of local amplitudes determines the dynamics either by exponentiation of the hamiltonian constraint

Or more efficiently via a spacetime (feynman integral) approach

One uses the fact that gravity can be written as a constraint topological BF theory.

Plebanski, Ooguri, Reisenberger, L.F, Krasnov, De Pietri, Perez,...

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

AS(#0, #1) =!

jf ,ie

"

f

Af (jf )"

e

Ae(jf , ie)"

v

Av(jf , ie) (0.40)

SPlebanski =

#Bi # F i(A), + constr. Bi #Bj = $!ij (0.41)

Bi #Bj = $!ij (0.42)

3

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

AS(#0, #1) =!

jf ,ie

"

f

Af (jf )"

e

Ae(jf , ie)"

v

Av(jf , ie) (0.40)

SPlebanski =


Bi #Bj = $!ij (0.42)

3

with constraints

We can construct an exact finite path integral representation for the topological theory in terms of a state sum model.

Impose on the amplitudes the constraints that lead to GR•In 3D exact quantisation: Ponzano-regge model•In 4D this leads to the Barrett Crane model G=SO(4), representations (jL,jR) constraint is jL=jR

exciting new developpements on his front Engle, Rovelli, Livine, Speziale, L.F, Krasnov,...

Spin Foam model: Coupling to matter

•We need to construct a sufficient large set of physical observables allowing the interpretation of these amplitudes?

At the quantum level we want to know how quantum gravity affects and modify the rules of quantum field theory, what are the QG corrections to

usual field theory and usual Feynman rules?

•Low energy limits: How does a classical spacetime emerge from quantum geometry and a corresponding set of transition amplitude?

operationally one needs matters in order to define what is geometry, At the classical level we study the motion of test particles.

•Technically very hard to construct these observables in pure gravity

Any theory of QG can be view from the point of view of matter as a dimensionfull deformation of Field Theory.

Spin Foam model: Coupling to matter

Integrate out quantum gravity fluctuation: since spin foam models are combinatorial objects the direct coupling to fields is hard. The strategy is to first construct the coupling of quantum

gravity to quantum particles i-e Feynman diagrams and then reconstruct the effective field theory.

Z =Z

DgD!eiS[!,g]+ ilp

S[g]

Z =Z

D!eiSe f f [!]

= !"

C"!I"(lp)observables of

quantum geometry

Expand in Feynman diagrams

ZD!

= !"

C"

ZDgI"(g)e

ilp

S[g]

• Explicitely realized in 2+1 dimensions

Unification: Matter as a charged topological defect!

computed using spin foam

eiµdxµ !i

µdxµ

e.o.m: pure gravity F =0

In the first order formalism gravity is described in term of:

a frame field a spin connection

2+1 Euclidean Gravity

particles F(!)i = 4"G pi #(x)

create a conical singularity with deficit angle

! = "m, " = 4#G

S[e,!] =1

16"G

Zei!F(!)i

Deser, Jackiw, t hooft

Ponzano-Regge Model

• First quantum gravity model ever written in 1968

• Background independent and non perturbative finite definition of

Euclidean quantum gravity amplitude in 2+1d

• Its kinematics is the one of usual loop quantum gravity

• Can be shown to be the continuum limit of a discretization of gravity

preserving diffeo symmetry

• Coupling to matter uniquely fixed L.F, Louapre

• Equivalent with (Witten) Chern-Simons quantization when it exists

3

m ! sin(!m)/! (31)

S3 " SU(2) (32)

X " "P (33)

P 2 # 1/!2 (34)

GN ! 0 (35)

g = # + h (36)

$hab(x)hcd(y)% " P abcd

|x& y|2 (37)

s = (!, je, iv) (38)

g = (q, k) (39)

(a/a)2 = 8$G/3%(1& %/%crit) (40)

(a/a)2 = 8$G/3(%& %1(v))(%2(v)& %)/%crit (41)

" > 0 (42)

• Chose a triangulation of M dual to a spin foam with boundary spin networks

• Color edges of by SU(2) representations

Sum over internal geometries

Ponzano-Regge model

SU(2) 6j symbol

je

Z! is independent of the internal triangulation finite after proper gauge fixing of diffeo symmetry

spin foam formulation of quantum gravity amplitude

Z!( jin, jout) = "{ je}

#e

d je #t

!jet1

jet2jet3

jet4jet5

jet6

"

j =llP

physical scalar product between spin network states

!

!

• is a Feynman graph supported on edges of

• Amplitude for a Scalar particle coupled to gravity

!

Particle insertion

!

particle insertion

particle= e.v of an observable in topological state sum

L.F, D. Louapre, Barrett

I!(")

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"

dPed"e eitr(XeGe)eiSP (Xe,Pe,!e) (20)

I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)"

t

(21)

2

6 Laurent Freidel

to vertices of the triangulation, such that the variation

!Xe = !e! (1.9)

leaves the action (1.3) invariant. The discrete covariant derivative reduceto the usual derivative !e! " !se #!te when the gauge field is abelianand the symmetry is due to the discrete Bianchi identity.

Since this symmetry is non compact we need to gauge fixed it in orderto define the partition function and expectation values of observables.A natural gauge fixing consists in choosing a collection of edges T whichform a tree (no loops) and which is maximal (connected and which goesthrough all vertices). We then arbitarily fix the value of Xe for all edgese $ T . In the continuum this gauge fixing amounts to choose a vectorfield v (the tree) and fix the value of ei

µvµ, that is to chose an ‘axial’gauge.

Taking this gauge fixing and the Faddev-Popov determinant into ac-count in the derivation (1.5, 1.10) we obtain the gauge fixed Ponzano-Regge model

Z!,T,j0 =!

{je}

"

e

dje

"

e!T

!je,j0e

(dj0e)2

"

t

#je1 je2 je3

je3 je5 je6

$. (1.10)

As a consistency test it can be shown that Z!,T,j0 = ZGF! is independent

of the choice of maximal tree T and gauge fixing parameter j0.

1.4 Coupling matter to quantum gravity

In order to couple particle to matter fields we first construct the couplingof gravity to Feynman integrals since as we are going to see there is anatural and unambiguous way to couple the Ponzano-regge model toFeynman integrals.

We use the fact that Feynman integrals can be written as a worldlineintegral [11], that is if " is a (closed for simplicity) Feynman graph itsFeynman integral is given by

I"(e) =%D""Dx"Dp" eiS! . (1.11)

where

S"(x", p","") =12

!

e!"

%

ed# tr

&peet #

"e

2(p2

e # µ2e)

'. (1.12)

" is a Lagrange multiplier field which is the worldline frame field and is

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)

$ %& '

"

t

(21)

2

is the usual Feynman propagatorevaluated on a Planckian lattice

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)

$ %& '

"

t

(21)

O" ="

e!"

KF (je)dje

(22)

KF (G) =!

d"ei!

“P 2(G)"( sin !m

! )2”

=i

P 2(G)"(

sin "m"

)2 " i$(23)

Pi(G) # tr(G%i)2i&

&2P 2(G) $ 1 (24)

K(G) =#

j!NdjKF (j)'j(G) djKF (j) =

!dG K(G)'j(G) (25)

KF (j) =2i&2

cos &m

eidj(m+i#)

dj=

&3

4(kF (dj) (26)

2

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

3

computation from a discretization of SG+SPart

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)

I!(g) =

! "

v

dxv

"

e

GF (xse , xte ; g) =

! "

e

DxeD!e eiP

e SP (pe,xe,!e) (0.18)

! > 0 !

I! =

!Dg eiS(g)I!(g) =

!DgDpeD!e eiS(g)ei

Pe SP (pe,xe,!e) (0.19)

" = 4#G

m "

$! = !

"

I"(") =

! "

e

dXe

"

f

dgf

"

e!!

dPed!e etr(XeGe)eiSP (Xe,Pe,!e) (0.20)

I"(") =

! "

f

dgf

"

e!!

KF (Ge)"

e/!!

%(Ge) =#

je

"

e

dje

"

e!!

KF (je)

$ %& '

"

t

(0.21)

O! ="

e!!

KF (je)

dje

(0.22)

KF (G) =

!d!e

i!“P 2(G)"( sin !m

! )2

”

=i

P 2(G)"(

sin "m"

)2 " i&(0.23)

Pi(G) # tr(G'i)

2i""2P 2(G) $ 1 (0.24)

K(G) =#

j!NdjKF (j)(j(G) djKF (j) =

!dG K(G)(j(G) (0.25)

KF (j) =2i"2

cos "m

ei"dj(m+i#)

dj=

"3

4# cos "mkF ("dj) (0.26)

gf Ge (0.27)""%"

$e=v

G±1e = 1 (0.28)

KF (g1) =i

P 2(g1)"(

sin "m"

)2 " i&(0.29)

#

j

(0.30)

P ie # tr(Ge'

i) (0.31)

2

3

m ! sin(!m)/! (31)

S3 " SU(2) (32)

X " "P (33)

P 2 # 1/!2 (34)

GN ! 0 (35)

g = # + h (36)


|x& y|2 (37)

s = (!, je, iv) (38)

g = (q, k) (39)

(a/a)2 = 8$G/3%(1& %/%crit) (40)

(a/a)2 = 8$G/3(%& %1(v))(%2(v)& %)/%crit (41)

" > 0 (42)

O(je) =!

e!!

KF (je)

dje

(43)

Gravity + Particle

The momentum of the particle is group valued

•Is it really a Feynman graph evaluation? •Do we recover QFT in the GN 0 limit ?• Then, what are the quantum gravity corrections ?

I!(")

Ge = !f!e

g f

Carroll, Matschull,Bais,Muller, Schroers ...

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)"

v

(21)

2

6 Laurent Freidel


!Xe = !e! (1.9)





Z!,T,j0 =!

{je}

"

e

dje

"

e!T

!je,j0e

(dj0e)2

"

t

#je1 je2 je3

je3 je5 je6

$. (1.10)






I"(e) =%D""Dx"Dp" eiS! . (1.11)

where

S"(x", p","") =12

!

e!"

%

ed# tr

&peet #

"e

2(p2

e # µ2e)

'. (1.12)


!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)"

t

(21)

2

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)

$ %& '

"

t

(21)

O" ="

e!"

KF (je)dje

(22)

KF (G) =!

d"ei!

“P 2(G)"( sin !m

! )2”

=i

P 2(G)"(

sin "m"

)2 " i$(23)

Pi(G) # tr(G%i)2i&

&2P 2(G) $ 1 (24)

K(G) =#

j!NdjKF (j)'j(G) djKF (j) =

!dG K(G)'j(G) (25)

2

!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)

$ %& '

"

t

(21)

2

!"(Ge) Ge = xeh!exe!1on shell

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

3

QG correction: GN expansionplanar and M =S3

After some work...

!

I!(") =

6 Laurent Freidel


!Xe = !e! (1.9)





Z!,T,j0 =!

{je}

"

e

dje

"

e!T

!je,j0e

(dj0e)2

"

t

#je1 je2 je3

je3 je5 je6

$. (1.10)






I"(e) =%D""Dx"Dp" eiS! . (1.11)

where

S"(x", p","") =12

!

e!"

%

ed# tr

&peet #

"e

2(p2

e # µ2e)

'. (1.12)


!! = !

"

I!(") =! "

e

dXe

"

f

dgf

"

e!"


I!(") =! "

f

dgf

"

e!"

KF (Ge)"

e/!"

#(Ge) =#

je

"

e

dje

"

e!"

KF (je)"

t

(21)

2

QG correction: GN expansionplanar and M =S3!

with

cyclically order product of group valued momenta meeting at the vertex v

!P(G) =1

2i!Tr(G!")

we can get rid of the triangulation dependence

!X ! X = Xi!i

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

3

3

m ! sin(!m)/! (31)

S3 " SU(2) (32)

X " "P (33)

P 2 # 1/!2 (34)

GN ! 0 (35)

g = # + h (36)


|x& y|2 (37)

s = (!, je, iv) (38)

g = (q, k) (39)

(a/a)2 = 8$G/3%(1& %/%crit) (40)

(a/a)2 = 8$G/3(%& %1(v))(%2(v)& %)/%crit (41)

" > 0 (42)

O(je) =!

e!!

KF (je)

dje

(43)

KF (g) =

"dTeiT (P 2(g)" sin !m

! ) (44)

Gv =&!!

e#v

gv (45)

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

AS(#0, #1) =!

jf ,ie

"

f

Af (jf )"

e

Ae(jf , ie)"

v

Av(jf , ie) (0.40)

SPlebanski =


Bi #Bj = $!ij (0.42)#

Dg (0.43)

I!(#) =

# "

v!"

dXv

# "

e!"

dge

"

e!"

KF (ge)"

v!"

etr(XvGv) (0.44)

3

3

m → sin(κm)/κ (31)

S3 ∼ SU(2) (32)

X ∼ ∂P (33)

P 2 ≤ 1/κ2 (34)

GN → 0 (35)

g = η + h (36)

〈hab(x)hcd(y)〉 ∼ P abcd

|x− y|2 (37)

s = (Γ, je, iv) (38)

g = (q, k) (39)

(a/a)2 = 8πG/3ρ(1− ρ/ρcrit) (40)

(a/a)2 = 8πG/3(ρ− ρ1(v))(ρ2(v)− ρ)/ρcrit (41)

Λ > 0 (42)

O(je) =!

e!!

KF (je)

dje

(43)

KF (g) =


! ) (44)

Gv =−→!

e#v

ge (45)

Λ (46)

A -Product!X ! X = Xi!iIdea: non commutative structure on R3

!

ei

2!Tr(Xg1) ! ei

2!Tr(Xg2) = ei

2!Tr(Xg1g2)

Properties: • New Fourier transform R3! S3

f (X) =Z

dgei

2!Tr(Xg) f (g)

• is dual to convolution product on su(2)!• 0-Bessel function replace , it has finite width

!(x)!

• generalize Fourier theory to curved momentum space L.F, S. majid

[xi,x j] = i!"i jkxknon commutative spacetime

Algebra of derivation of a curved momentum space

QG amplitude as Feynman diagram

K!(G) =Z dT

2"eiT

!P2(G)!sin2 #m

#2

"

with

is a Feynman diagram of ...

ei

2! Tr(XvGv)! !"e"v

e±i2! Tr(XvGe)

!

!e=v

U ve Pe + [!v

e , Pe] = 0 (0.32)

S =!

e

tr(XeGe) (0.33)

!Xe =!

v!e

U ve "v ! [!v

e , "v] = 0 (0.34)

Xe = X0e e " T (0.35)

Z!,T

"

e!T

!je,j0e

dj2e

(0.36)

(! + m2)kF (x) = i!(x) (0.37)

Re(KF)(Ge) = (0.38)

" = #m (0.39)

AS(#0, #1) =!

jf ,ie

"

f

Af (jf )"

e

Ae(jf , ie)"

v

Av(jf , ie) (0.40)

SPlebanski =


Bi #Bj = $!ij (0.42)#

Dg (0.43)

I!(#) =

# "

v!"

d3Xv

# "

e!"

dge

"

e!"

KF (ge)"

v!"

etr(XvGv) (0.44)

I!(#) =

# "

v!"

d3Xv

# "

e!"

dge

"

e!"

KF (ge)"

v!"

$%e!ve

tr(Xvge)%

(0.45)

3

A non commutative field theory

• Perturbative expansion of massive coupled to 3d gravity !"3 !

"

#|v"|

S"I$(")

is the perturbative expansion of a non commutative field theory

S =1

8!"3

Zd3x

!12(#i$!#i$)(x)! 1

2sin2 m"

"2 ($!$)(x)+%3!

($!$!$)(x)

"

Particles behave as if they propagate on a non commutative space time

G = ei!"p·"# = 1 + i!("p · "#) + O(!2) (1)

G1G2 = ei!"p·"# = 1 + i!("p · "#) + O(!2) (2)12!

Tr(XG) = i"p · "X + O(!) (3)

eiSeff ($) =!

DgeiS($,g)+ i

lpS(g) (4)

$j(h%) =sin dj%

sin %(5)

$+j (h%) =

eidj%

2i sin %(6)

Xe gf Ge =!""

f!e

gf (7)

X = Xi#i S =#

e"!

tr(XeGe) (8)

! "

e

dXe

"

f

dgf eitr(XeGe) (9)

Z!(!o,!i) =! "

e

dXe

"

f

dgf eitr(XeGe)&"o(gf )&#"i(gf ) (10)

!i,o!o !i &"i,o(gf ) (11)

=#

j"Ndj$j(G) (12)

Z! =! "

f

dgf

"

e

'(Ge) =#

{je}

"

e

dje

! "

f

dgf

"

e

$je(Ge) (13)

GF (x, y; e) =! x(1)=y

x(0)=xDpDxD( e

i

!pie

it + ((p2 !m2)

$ %& ' (14)

GF (x, y; e) =! x(1)=y

x(0)=xDpDxD( eiSP (p,x,&) (15)

SP (p, x, () =!

pieit + ((p2 !m2) (16)

eit # ei

µxµ (17)

I"(g) =! "

v

dxv

"

e

GF (xse , xte ; g) =! "

e

DxeD(e eiP

e SP (pe,xe,&e) (18)

( > 0 (

I" =!

Dg eiS(g)I"(g) =!

DgDpeD(e eiS(g)eiP


! = 4)G

1

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

mass renormalisation

PropertiesIn momentum space

Renormalisation of the massno deformation of dispersion relation

but momenta cut-off

Deformed conservation law at vertices

0=!P1!!P2!!P3

=!i

!Pi""(!P1#!P2 + ...)+O("2)

Symmetric under deformed action of the Poincare group:

Lorentz part undeformed, the action on one particle state is isomorphic to Poincare, the action of translation is deformed on multiparticle states

!

Satisfies the principles of Doubly Special Relativity

12

Zdg

!P2(g)! sin2 !m

!2

"#"(g)#"(g!1)+

#3!

Zdg1dg2dg3 $(g1g2g3)#"(g1)#"(g2)#"(g3)

• The mass gets renormalised

• The momentum space acquire a curvature proportional to the Planck mass.

In the euclidean case the momentum space is a 3 sphere

addition of momenta is deformed. All momenta are bounded

• By duality the position space is non commutative

there is a minimal length scale accessible to the theory Snyder NC 1947!

• The action of the Poincare group and diffeomorphism group is undeformed on one particle state. It is deformed on multiparticle states in order to preserve the existence of a maximal energy scale.

Effective geometry from quantum gravity

G = ei!"p·"# = 1 + i!("p · "#) + O(!2) (1)

G1G2 = ei!"p·"# = 1 + i!("p · "#) + O(!2) (2)12!

Tr(XG) = i"p · "X + O(!) (3)

eiSeff ($) =!

DgeiS($,g)+ i

lpS(g) (4)

$j(h%) =sin dj%

sin %(5)

$+j (h%) =

eidj%

2i sin %(6)

Xe gf Ge =!""

f!e

gf (7)

X = Xi#i S =#

e"!

tr(XeGe) (8)

! "

e

dXe

"

f

dgf eitr(XeGe) (9)

Z!(!o,!i) =! "

e

dXe

"

f

dgf eitr(XeGe)&"o(gf )&#"i(gf ) (10)

!i,o!o !i &"i,o(gf ) (11)

=#

j"Ndj$j(G) (12)

Z! =! "

f

dgf

"

e

'(Ge) =#

{je}

"

e

dje

! "

f

dgf

"

e

$je(Ge) (13)

GF (x, y; e) =! x(1)=y

x(0)=xDpDxD( e

i

!pie

it + ((p2 !m2)

$ %& ' (14)

GF (x, y; e) =! x(1)=y

x(0)=xDpDxD( eiSP (p,x,&) (15)

SP (p, x, () =!

pieit + ((p2 !m2) (16)

eit # ei

µxµ (17)

I"(g) =! "

v

dxv

"

e

GF (xse , xte ; g) =! "

e

DxeD(e eiP


( > 0 (

I" =!

Dg eiS(g)I"(g) =!

DgDpeD(e eiS(g)eiP


! = 4)G

1

The effect of quantum gravity is fourfold

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

X " $P (23)

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

S3 " SU(2) (23)

X " $P (24)

By duality it is related to a minimal length scale or discrete spectra of time intervals

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

m" sin($m)/$ (23)

S3 # SU(2) (24)

X # %P (25)

P 2 $ 1

$2(26)

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

m" sin($m)/$ (23)

S3 # SU(2) (24)

X # %P (25)

P 2 $ 1/$2 (26)

3

g1

g1

g1

g1

g2

g2

g3

g!1g!

2

! !(g1g2g3)! Km(g1)

! !(g1g2g!1"1g!

2"1) !(g2g

!2"1)

FIG. 1: Feynman rules for particles propagation in thePonzano-Regge model.

where ! is a coupling constant. |v!| is the number ofvertices of ! and S! is the symmetry factor of the graph.Remarkably, this sum can be obtained from the pertur-bative expansion of a non-commutative field theory givenexplicitly by:

S =

!d3x

8"#3

"1

2($i% & $i%)(x) !

1

2

sin2 m#

#2(% & %)(x)

+!

3!(% & % & %)(x)

#(19)

where the field % is in C!(R3). Its momentum has sup-port in the ball of radius #!1. We can write this actionin momentum space

S(%) =1

2

!dg

$P 2(g) !

sin2 #m

#2

%&%(g)&%(g!1) (20)

+!

3!

!dg1dg2dg3 '(g1g2g3) &%(g1)&%(g2)&%(g3).

This is our e"ective field theory describing the dynamicsof the matter field after integrating out the gravitationalsector. This non-commutative field theory action is sym-metric under a #-deformed action of the Poincare group.Calling # the generators of Lorentz transformations andT"a the generators of translations, the action of these gen-erators on one-particle states is undeformed:

# · &%(g) = &%(#g#!1) = &%(# · P (g)), (21)

T"a · &%(g) = ei"P (g)·"a &%(g). (22)

The non-trivial deformation of the Poincare group ap-pears at the level of multi-particle states and only theaction of the translations is deformed :

# · &%(P1)&%(P2) = &%(# · P1)&%(# · P2), (23)

T"a · &%(P1)&%(P2) = ei"P1""P2·"a &%(P1)&%(P2). (24)

It is straightforward to derive the Feynman rules fromthe action (21) (see fig.1). The e"ective Feynman prop-agator is the group Fourier transform of &K#(g),

Km(X) = i

!dg

e12!

tr(Xg)

P 2(g) !'

sin !m!

(2 . (25)

The e"ect of quantum gravity is two-fold. First the massgets renormalized m " sin #m/#. Then the momentum

space is no longer the flat space but the homogeneouslycurved space S3 # SO(3). This reflects that the momen-tum is bounded |P | < 1/#.At the interaction vertex the momentum addition be-comes non-linear with a conservation rule P1$P2$P3 = 0which implies a non-conservation of momentum P1+P2+P3 %= 0. Intuitively, part of the energy involved in acollision process is absorbed by the gravitational field:gravitational e"ects can not be ignored at high energy.This e"ect, which is stronger at high momenta and fornon-collinear momenta, prevents the total momenta frombeing larger than the Planck energy.A last subtlety of the Feynman rules is the evaluation ofnon-planar diagrams. A careful analysis of I! shows thatwe have a non-trivial braiding: for each crossing of twoedges, we associate a weight '(g1g2g#1

!1g#2!1) '(g2g#2

!1)(see fig.1). This reflects a non-trivial statistics where theFourier modes of the fields obey the exchange relation:

&%(g1)&%(g2) = &%(g2)&%(g!12 g1g2) (26)

which is naturally determined by our choice of star pro-duct. Indeed, let us look at the product of two identicalfields:

% & % (X) =

!dg1dg2 e

12! tr(Xg1g2) &%(g1)&%(g2), (27)

Under change of variables (g1, g2) " (g2, g!12 g1g2), the

star product reads

%&% (X) =

!dg1dg2 e

12! tr(Xg1g2)&%(g2)&%(g!1

2 g1g2). (28)

The identification of the Fourier modes of %&% (X) leadsto the exchange relation (26). This braiding was firstproposed in [10] for two particles coupled to 3d QG andthen computed in the Ponzano-Regge model in [5]. It isencoded into a braiding matrix

R · &%(g1)&%(g2) = &%(g2)&%(g!12 g1g2). (29)

This is the R matrix of the #-deformation of the Poincaregroup [10]. Such field theories with non-trivial braidedstatistics are simply called braided non-commutative fieldtheories and were first introduced in [11].

Finally, the &-product induces a non-commutativity ofspace-time and a deformation of phase space:

[Xi, Xj ] = i#(ijkXk,

[Xi, Pj ] = i)

1 ! #2P 2 'ij ! i#(ijkPk. (30)

This non-commutativity reflects the fact that momen-tum space is curved. Indeed the coordinates X are re-alized as right invariant derivations on momentum spaceand derivations of a curved manifold do not commute.Moreover, this non-commutativity being related to hav-ing bounded momenta implies the existence of a minimal

Reconciling QM and GR

Quantum Mechanics: Heisenberg duality/democracy between space and momentum space

2+1 QG restore the duality by curving momentum space

Gravity: space is singled out and curved

momentum space : 3sphere of radius Planck mass

geometrical interpretation of non commutativity

Xi : right invariant derivative on S3

NC geometry a better approximation ....

Is it possible to extend this strategy to 4d gravity ? The exact coupling is not yet known. Two key results:

Matter can be described as a topological gravitational defect

Usual Feynman diagram in flat space can be effectively written in a purely combinatorial manner as some expectation of certain observable in a topological spin foam model.

This gives a new background perspective on field theory and gives the form of the limit quantum gravity spin foam model

New point of view on possible dimensionfull deformation of field theories.

Baratin, L.F, Starodubtsev, Kowalski-Glikman

2

!Vol!!,je,iv =

"8!"l3P

#

v!R"!

vje,iv

$!!,je,iv (15)

F (A) ! E (16)

!Area(H) = A (17)

H = tr([E, E]F (A)) (18)

#

i

Hj(i) (19)

%Dg e

ilP

S(g)(20)

KF (j) ! kF (lP dj) (21)


k (22)

m " sin($m)/$ (23)

S3 # SU(2) (24)

X # %P (25)

P 2 $ 1/$2 (26)

GN " 0 (27)

%hab(x)hcd(y)& # P abcd

|x' y|2 (28)

s = (", je, iv) (29)

g = (q, k) (30)

(a/a)2 = 8!G/3&(1' &/&crit) (31)

(a/a)2 = 8!G/3(&' &1(v))(&2(v)' &)/&crit (32)

4d ?

Conclusion Shown that a Non Commutative Braided QFT gives the effective

description of matter field coupled to 2+1QG

Generalisations: Lorentzian gravity: more involved but natural implementation of the Feynman propagator, S3 ADS3Non trivial cosmological constant: both x and p space are curved and non-commutative

Feynman diagrams = e.v of observables in a topological spin foam model.

Unitarity, conserved charges, renormalisability?

Extension to 4D ? New and background independent spin foam perspective on usual field theory in any dimensions

AdS/CFT: is it possible to do a similar computation? What is the effective quantum geometry

QG affects in a non trivial ( dependent) the density of state of FT

A way to tame down the cc problem?

3

m ! sin(!m)/! (31)

S3 " SU(2) (32)

X " "P (33)

P 2 # 1/!2 (34)

GN ! 0 (35)

g = # + h (36)


|x& y|2 (37)

s = (!, je, iv) (38)

g = (q, k) (39)

(a/a)2 = 8$G/3%(1& %/%crit) (40)

(a/a)2 = 8$G/3(%& %1(v))(%2(v)& %)/%crit (41)

" > 0 (42)

O(je) =!

e!!

KF (je)

dje

(43)

KF (g) =


! ) (44)

Gv =&!!

e#v

gv (45)

" (46)

ef fectiv e f ield theory fr om spin f oam models :3d e...

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