new spin foam : modeling quantum space time · 2009. 7. 30. · perimeter institute mg12 paris...

Spin foam :Modeling quantum space time

Laurent Freidel

Perimeter Institute

MG12 Paris2009

Background independent Gravity

There has been a wealth of new developments in the recent years in the covariant formulation of loop quantum gravity (spin foam formulation)

Diffeomorphism symmetry and background independence

•To review briefly the general principles of Loop quantum gravity

The key points behind these approaches are:

Aim of this talk:

Non perturbative dynamics

•To present some of the new developments in spin foam models:The construction of a discrete path integral for non perturbative quantum gravity

•To present the remarkable convergence between the two approaches

The Gauge principle

•It determines the dynamics of the gravitational field and coupling to matter•Redundancy of the description no obs. def. of spacetime points

Background independent GravityOne of the basic problems we want to address is: How can we describe the elementary degrees of freedom of quantum gravity in a context where there is no background spacetime?

The key is the symmetry: Diffeomorphism invariance plays a key role.

In quantum gravity the nature of short distance quantum spacetime geometry cannot be postulated, it should be determined dynamically

What is needed is a radical change of our concept of space and time

One of the basic element is that the UV divergences that one encounters in quantizing GR are of a different nature : perturbation theory breaks down at the place where we do not know how to define spacetime

These problems can be addressed in the context of Loop quantum Gravity and Spin foams: No transplanckian dof.

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

Quantum Gravity

At the quantum level the description of gravity is in term of states and operators and the dynamics is encoded in constraint equations

In usual description states are functional of a spacelike metric

One wish to rigously define the path integral

The perturbative expansion around asymptoticaly flat configuration: graviton propagator and scattering amplitude.

No definition is yet available in WdW theory: Spin foam, AdS/CFT ?

5

BIJ ! XIJf

Xf

M !M = !

"M(") =!

g|!Dg eiSM (g)

Boundary state

5

BIJ ! XIJf

Xf

M !M = !

"M(") =!

g|!=!

Dg eiSM (g)

The dynamics = Symmetry = operatorial constraints : Space diffeo constraints and time reparametrisation: Wheeler de Witt eq.

Operators are needed to extract physical information from these states

Canonical approach

Covariant approach

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

The dynamics is encoded in Transition amplitudes

5

BIJ ! XIJf

Xf

M !M = !

"M(") =!

g|!Dg eiSM (g)

5

BIJ ! XIJf

Xf

M !M = !

"M(") =!

g|!Dg eiSM (g)

Given

LQG and spin foam

The expression of the gauge principe and diffeomorphism symmetry leads to the construction of new type of boundary states: spin network states LQG.

The challenge is to formulate the dynamics in terms of these states and give a definition of the transition amplitude with these boundary states Spin foam model

In order to resolve the difficulty with the usual metric formulation one uses new variables (gauge fields)

A fundamental geometrical cutoff is incorporated in these states.

Loop Quantum gravity Spin Foam

Covariant Canonical Dynamics

Geometry has two functions:

A metric or a frame field to give us rods and clocks

A medium which allows parallel transport from one point to another a connection

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

Loop Quantum Gravity

We can use the unifying language of gauge theory that appears in the description of Electromagnetism, the strong and weak interaction + principle of general covariance

These principles leads to a unique proposal for a set of variables that describe the geometry of space at the quantum level

The gauge principle:

•

•

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

Eai

The gauge principle

At the canonical level one chose a spacelike slice S of spacetime and we can formulate the hamiltonian dynamics of gravity as an SU(2) gauge theory

Magnetic Field: Connection allowing parallel transport

Conjugate variables: SU(2)

Ashtekar-Barbero connection: Extrinsic curvature

Electric field: pull back of the frame field on S

Spin connection Immirzi parameter

Wave functions are gauge invariant functional of A

3

Kab = Kaieib

{Eai (x), Aj

b(y)} = !" #ab #

ji #

(3)(x! y)

"" = 1

$ = 1

#F (A) = F (A)

tµ = Nn + Nµ

S =

!dt

!

!

Eai LtA

ia !H(A, E, N)

H(A, E, N) = NaCa + NC + AitGi

Ca = F iabE

bi

C =!!$

"F ij

abEai Eb

j

Gi = !%aEai

!(ya) =

! "

A

dXAeiS(XA,ya)

%aS & Pa(X, y)

'PA

#S = P a%yaS + PA%XAS = &abPaPb ! V (y)

%XAPA = 0

(!!y + V (y)) !(y) = 0

Aia = "i

a !Kia

", $

[Eai (x), Aj

b(y)] = i!" #ba#

ji #

3(x! y)

'(A)

he(A) = (!exp

#! 1

0

A(t)dt

$) SU(2)

A(t) & i$aAaµe

µ

%th(t) = A(t)h(t)

h(0) = 0

A * gA = gAg!1 + dgg!1

he * g(1)heg!1(0)

Eai

2

(je, Zv)

ze = je + i!e

z1 z2 z3 z4

T !G = GC

S2j z

!j = 2j ln(1 + |z|2)

!j, "|j, z" = (1 + "z)2j = e!j(z,w)

SL(2, C)

|#$, Z"Xi = jiNi N2 = 1

Hv =!

i

jiNi = 0

{Na, N b} =1

j%ab

cNc

Z Z "

"(z) = !"|z" " |z"!(z, z) " = &&! , &"(z) = 0

||"||2 =

"

P

dzdz#

"e#1! !(z,z)!"|z"!z|""

#(Re(z))#(Im(z))

j1 j2 j3 j4

$ia $

1

2%i

jk"jka

g(vv!) % kvg(vv!)k#1v! ge % kvgek

#1v! e = (vv")

Kia $ "i0

a

dei & %ijk$

j ' ek = 0

% =1

'(

"

"

Ai ' Ei

Eai =

1

2%ijke

jae

kb %

abc

Aia = $i

a + (Kia

)

det(()(ab = Eai Ebi

Ei =1

2%ijke

j ' ek

SU(2) index

The gauge principle

Wilson Line: Given a loop in a SU(2) representation one can integrate the connection along the loop (holonomy) and construct a simple gauge invariant functional.

This are eigenstates of the electric (frame) field They represent singular configuration of the E field

In order to have a complete set of observables one needs to introduced more than loops: Spin networks

Fundamental excitations are one dimensionalContinuum arise as a coarse grained limit

Graphs labelled by edge spins and invariant maps on vertices

A fundamental discretness: quantisation of the spin labels

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

Geometrical OperatorsWhat emerge from this is a a picture in which Quantum Geometry is a discrete geometry

Geometrical objects such as the Area of a surface Sthe volume of a region V

Are now observables with finite and discrete spectraspin networks are eigenstates

A(S) = sum over intersection of S with spin network

10-33 cm

7

153F903957A7C27ED077364961!ab

!(!)

H!(!) = 0

H!M(!) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1 From BH counting

Ashtekar, Rovelli, Smolin, Lewandowski ...

Diffeo invarianceA fundamental problem in QCD about the continuum limit:states supported on infinitesimally distinct graphs are orthogonal

Resolved by use of diffeo symmetry: Only the diffeomorphism class of the graph is relevant.

A theorem stating that this is a unique representation of the algebra of kinematical observables. (Remarkably strong result!)

•The basic result is that the knowledge quantum gravity amplitude for all possible “discrete” spin networks states is equivalent to the definition of the full theory.

Lewandowski, Okolov, Salman,Thiemann, ...

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j) = 0 ?

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

The DynamicsHow can we formulate a fully non perturbative quantum dynamics in the context of QFT?

Spin foam is a framework whose aim is to define this entity in the context where boundary states are labeled by spin networks.

One should learn from the one successful example: QCDThe only QFT that we know to define fully from first principle.The only way known is to regulate the theory on a space time lattice define the dynamics respecting the gauge principle and take the continuum limit.

Extremely successful route.

But the lattice in QCD is fixed and external: Regular and it is carrying an external cutoff.The continuum limit is controlled by sending the cut-off to zero.

This is not suitable for GR where one should have a formulation without preferred coordinates

More suitable lattice formulation for gravity?

Regge gravityRegge calculus (‘61): gravity without coordinates


Label this triangulation with edge lengths and assign an action:

is the Hamilton-Jacobi functional of Einstein gravity evaluated on solutions

But

• It is based on classical considerations

• It is a second order formulation:

• There is no canonical (Hamiltonian) formulation,

• No expression of the gauge principle,

• No UV cutoff (the length spectra is continuus)

• Strong ambiguities about the measure and higher derivatives corrections

• No relationship with LQG nor reference to the Immirzi parameter

Take a spacetime triangulation (no need for it to be regular): It represent a manifold structure without the gravitational field.

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

SRegge ="

f

Af (#)#f (#) $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

"

eiSregge + c.c

New Spin foam modelsRecently following a effort led by several groups, all these questions have been successfully addressed.


It is now possible to propose a discrete first order formulation of quantum gravity amplitudes which

•Possess SU(2) spin networks as boundary states: same states of the canonical LQG approach (Totally independent derivation)

It brings new tools (geometrical coherent states) and a new and sharp interpretation of the spin network basis

It provides a purely algebraic formulation of Quantum gravity. And show a deep interplay between pure algebra and geometry. (NC geometry)

•Possess the right semi-Classical limit

•Incorporate the Immirzi parameter

It allows first computations of n-point gravitons amplitudes

•Fixes some important ambiguities in the functional integral measure

Rovelli, Engle, Pereira, Livine, Freidel, Krasnov, Speziale, ...

•Admit a canonical interpretation

4d gravity: constraint BFAn important ingredient: 4d gravity can be written as a constrained SO(4) topological BF theory. Plebanski, Ooguri, Reisenberger, Freidel ,

Krasnov, De Pietri, Perez, Baez ...

This family is labeled by the Immirzi parameter

B needs to satisfy simplicity constraints.There exists in fact a one parameter family of simple B field

1

I. REMARKS

review by abhay and jerzy Check the immirzi parametrisation (dowload Artem L paper)Check the Lost papers. downlaod them Check the coe! for volume and area (prper ref)Check the BH entropy paper

S =1

2G

!!IJKLeI ! eJ !RKL(A) (1)

S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

S =1

2G

!!IJKLeI ! eJ !RKL +

1

2"eI ! eJ !RIJ +

#

3!IJKLeI ! eJ ! eK ! eL (4)

ggab = tr(EaEb) (5)

{Eai (x), Aj

b(y)} = "G$ab $

ji $(x, y) (6)

" (7)

" = 0 (8)

" =" (9)

" = 1, i (10)

"(A) (11)

S(e, A) = SP (e, A) +1

2"%

!eI ! eJ !RIJ(A) (12)

% = 8&G (13)

"(A) = "(A!) (14)

{FirstPage}Towards an Hamiltonian formulation of spin foam model

Laurent Freidel and Bianca Dittrich

a Perimeter Institute for Theoretical Physics, Waterloo, N2L 2Y5, Canada.(Dated: July 2007)

I. THE MODELS

In this section we revised some of the new developpements concerning the constructionof spin foam models. One of the main feature of the recent developpement concerning theconstruction of a spin foam model is the fact that it is now possible to construct spin foammodels including the insertion of a non zero Immirzi parameter. This is essential becauseone of the main claim of Loop quantum gravity, the fact that the spectra of geometricaloperator is discrete is valid only when the value of the immirzi parameter is non zero. It wasargued in [? ] that a non-trivial value of this parameter implies a compactification of thephase space of Riemannian gravity explaining in a di!erent way why all the spectra appearto be discrete. Indeed The spectra of the area is quantized in the form

A = !"!

j(j + 1) (1)

where j label an SU(2) spin and ! = 8#G. This formula obviously become ill defined if" = 0 or " = !, these values of " shoud be understood as limiting value for the immirziparameter and no conclusion can be drawn for the loop quantum gravity discreteness. Infact one expect a continuus spectra in the limit " " 0. Note that this discreteness statementis valid for the kinematical area operator, wether the discreteness survives in a Lorentziantheory for physical operators is not established and under debate ([? ],[? ]).

At the classical level this parameter is present in the classical action as shown by Hotlz [?], and, even though does not modify the equations of motion of the theory, it does modifyits symplectic structure. The action for gravity can be written as

SG,! =1

!

"BIJ

! (e) #R(A)IJ (2)

where A is an SO(4) connection and B!(e) Lie algebra valued two-form field BIJ of the BFformulation. It depends on the gravity frame field which are one-forms eI and on the Immrziparameter ":

BIJ! (e) =

1

!

#1

2$IJ

KL +1

"%[IK%J ]

L

$eK # eL = $(e # e)IJ +

1

"eI # eJ . (3)

A straightforward but important remark made by (Engle) is the fact that we have theidentity

$B!!1(e) = "B!(e). (4) {star}

This identity is important for the following and in key order to interpret properly the previousworks [? ]. Suppose first that we take the limit " " ! while keepping ! fixed then the

{FirstPage}Towards an Hamiltonian formulation of spin foam model

Laurent Freidel and Bianca Dittrich

a Perimeter Institute for Theoretical Physics, Waterloo, N2L 2Y5, Canada.(Dated: July 2007)

I. THE MODELS

In this section we revised some of the new developpements concerning the constructionof spin foam models. One of the main feature of the recent developpement concerning theconstruction of a spin foam model is the fact that it is now possible to construct spin foammodels including the insertion of a non zero Immirzi parameter. This is essential becauseone of the main claim of Loop quantum gravity, the fact that the spectra of geometricaloperator is discrete is valid only when the value of the immirzi parameter is non zero. It wasargued in [? ] that a non-trivial value of this parameter implies a compactification of thephase space of Riemannian gravity explaining in a di!erent way why all the spectra appearto be discrete. Indeed The spectra of the area is quantized in the form

A = !"!

j(j + 1) (1)

where j label an SU(2) spin and ! = 8#G. This formula obviously become ill defined if" = 0 or " = !, these values of " shoud be understood as limiting value for the immirziparameter and no conclusion can be drawn for the loop quantum gravity discreteness. Infact one expect a continuus spectra in the limit " " 0. Note that this discreteness statementis valid for the kinematical area operator, wether the discreteness survives in a Lorentziantheory for physical operators is not established and under debate ([? ],[? ]).

At the classical level this parameter is present in the classical action as shown by Hotlz [?], and, even though does not modify the equations of motion of the theory, it does modifyits symplectic structure. The action for gravity can be written as

SG,! =1

!

"BIJ

! (e) #R(A)IJ (2)

where A is an SO(4) connection and B!(e) Lie algebra valued two-form field BIJ of the BFformulation. It depends on the gravity frame field which are one-forms eI and on the Immrziparameter ":

BIJ! (e) =

1

!

#1

2$IJ

KL +1

"%[IK%J ]

L

$eK # eL = $(e # e)IJ +

1

"eI # eJ . (3)

A straightforward but important remark made by (Engle) is the fact that we have theidentity

$B!!1(e) = "B!(e). (4) {star}

This identity is important for the following and in key order to interpret properly the previousworks [? ]. Suppose first that we take the limit " " ! while keepping ! fixed then the

This parameter has no classical effect in pure gravity

At the quantum level: • Leads to inequivalent quantisation (like a theta term) • Compactification of the phase space Discrete spectra of geometrical operators.

1

I. REMARKS


S =1

2G


1

2"eI ! eJ !RIJ +

#


"ggab = tr(EaEb) (2)

{Eai (x), Aj

b(y)} = "G$ab $

ji $(x, y) (3)

" (4)

" = 0 (5)

" =# (6)

" = 1, i (7)

"(A) (8)

S(e, A) = SP (e, A) +1

2"%

!eI ! eJ !RIJ(A) (9)

% = 8&G (10)

"(A) = "(A!) (11)

h!(A) = Pexp

!

!

A (12)

ES,f =

!

S

Eafa (13)

[ES,f , h!] ="

!"S

h!1fh!2 (14)

3

X = (X+, X!)

X = (X+,!X!)

Xf = (Xf ,!ueXfu!1e )

Xf = (Xf , ueXfu!1e )

ue " SU(2) # S3

j+ = j!

n!ef = uen+ef

1j+ $ 1j!

Gj =!

k

dk Cjk j k

Tj = d2j Cjk j 2j

!

f"t

Xf± ·

"$f

#due|jf , uenf%

$= 0. (9)

A & !%

j(j + 1)

Av(jf , ie) =!

i+t ,i!t

15jSO(4)(jf , jf , i+t , i!t )

&

t

di+tdi!e

f iei+e i!e

(jf ). (10)

jf jf1 jf4 ie i+e i!e 2jf

k(j) = 2j k(j) =!

k

Ckj k k(jf )

S(e, A) =

#1

2"IJ

KLeI ' eJ 'RKL(A) +1

!

#eI ' eJ 'RIJ(A) (11)

ggab = tr(EaEb) (12)

{Eai (x), Aj

b(y)} = !G#ab #

ji #(x, y) (13)

! (14)

! = 0 (15)

! = ( (16)

Yang-Mills SU(N) BF +

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

BF theory quantisationThe strategy is now clear:

One use the fundamental fact that we know how to quantise BF theory exactly using spin foam models

Given an exact quantization of BF theory one needs to implement the Simplicity constraint as a restriction on the discrete integration measure.

SO(4) versus SU(2),Non locality and on commutativity of the constraints (second class)Lack of geometrical interpretation (area variables not edge lenght)...

There has been earlier attempts: Reisenberger, Barrett-Crane

For many years a wealth of technical and conceptual difficulties were addressed but only recently resolved :

New developpements: localisation of the constraints, coherent states...

BF theory quantisation

Independent of the triangulation and finite after gauge fixing

The state sum quantization of any gauge theory is characterised by a choice of face and vertex amplitude

1

0 < ! < 1

1 < ! < !! > 1

! = 1

e " e =!

1# !2(B! # ! $B!) =

1

1# !2($B!!1 # !B!!1)

X = $(e " e)

X = (e " e)

XIJnJ = 0

! = 0

! = !B = e " e

B = $(e " e)

AIJ =!

1# !2

!XIJ # !XIJ

"

Xf =#(1 + !)Xf , (1# !)ueXfu

!1e

$

Xf =#(1 + !)Xf ,#(! # 1)ueXfu

!1e

$

Af = !(Xf , ueXfu!1e )

Xf · Xf = 0 Xf · Xf " = 0%

f"e

Xf = 0

BCj =

&dndn|j, n% & |j, n%'j, n|& 'j, n|

0

XIJf =

(Xj+n+ ,#Xj!,n!)

(X,#ueXu!1e )

(X, ueXu!1e )

XIJf = V !1u[I

e uJ ]e"

uIe XIJ

f ueI = 0

uIXIJ # !uIX

IJ = 0

f e v

1

0 < ! < 1

1 < ! < !! > 1

! = 1

e " e =!

1# !2(B! # ! $B!) =

1

1# !2($B!!1 # !B!!1)

X = $(e " e)

X = (e " e)

XIJnJ = 0

! = 0

! = !B = e " e

B = $(e " e)

AIJ =!

1# !2

!XIJ # !XIJ

"

Xf =#(1 + !)Xf , (1# !)ueXfu

!1e

$

Xf =#(1 + !)Xf ,#(! # 1)ueXfu

!1e

$


Xf · Xf = 0 Xf · Xf " = 0%

f"e

Xf = 0

BCj =

&dndn|j, n% & |j, n%'j, n|& 'j, n|

0

XIJf =

(Xj+n+ ,#Xj!,n!)

(X,#ueXu!1e )

(X, ueXu!1e )

XIJf = V !1u[I

e uJ ]e"

uIe XIJ

f ueI = 0

uIXIJ # !uIX

IJ = 0

f e v

1

0 < ! < 1

1 < ! < !! > 1

! = 1

e " e =!

1# !2(B! # ! $B!) =

1

1# !2($B!!1 # !B!!1)

X = $(e " e)

X = (e " e)

XIJnJ = 0

! = 0

! = !B = e " e

B = $(e " e)

AIJ =!

1# !2

!XIJ # !XIJ

"

Xf =#(1 + !)Xf , (1# !)ueXfu

!1e

$

Xf =#(1 + !)Xf ,#(! # 1)ueXfu

!1e

$


Xf · Xf = 0 Xf · Xf " = 0%

f"e

Xf = 0

BCj =

&dndn|j, n% & |j, n%'j, n|& 'j, n|

0

XIJf =

(Xj+n+ ,#Xj!,n!)

(X,#ueXu!1e )

(X, ueXu!1e )

XIJf = V !1u[I

e uJ ]e"

uIe XIJ

f ueI = 0

uIXIJ # !uIX

IJ = 0

f e v

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

We chose a 2d cell complex S that bounds a graph and reproduce to topology of M: dual triangulation

vertex faceedge

1

I. REMARKS


S =1

2G


S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

!IJKLBIJµ!BKL

"# " !µ!"# (4)

BIJ = ±eI ! eJ BIJ = ±!IJKLeK ! eL (5)

S =1

2G


1

2"eI ! eJ !RIJ +

#


Jf ie

ggab = tr(EaEb) (7)

{Eai (x), Aj

b(y)} = "G$ab $

ji $(x, y) (8)

" (9)

" = 0 (10)

" =# (11)

" = 1, i (12)

"(A) (13)

1

I. REMARKS


S =1

2G


S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

!IJKLBIJµ!BKL

"# " !µ!"# (4)


S =1

2G


1

2"eI ! eJ !RIJ +

#


Jf ie

ggab = tr(EaEb) (7)

{Eai (x), Aj

b(y)} = "G$ab $

ji $(x, y) (8)

" (9)

" = 0 (10)

" =# (11)

" = 1, i (12)

"(A) (13)

2

S =!

f

Tr(XfGf ) (7)

Xf Gf ="

e!f

Ge

Ge1 · · · Gen

Jf

Jf = (j+f , j"f )

Ge = (g+e , g"e )

ieXf = XIJ

f JIJ

XIJf = V n[I

e nJ ]e! = !IJ

KLAKLf

= AIJf

f = (ee#)

V nIe AIJ

f

XIJ = (1/2)!IJKLXKL

XIJf XfIJ = 0

XIJf Xf !IJ = 0

!

f$e

XIJf = 0

XIJf neI = 0

XIJf neI = 0

X = (X+, X")

X = (X+,!X")

Xf = (Xf ,!ueXfu"1e )

Xf = (Xf , ueXfu"1e )

ue " SU(2) # S3

ggab = tr(EaEb) (8)

{Eai (x), Aj

b(y)} = "G#ab #

ji #(x, y) (9)

" (10)

7

Gv =!"!

e!v

ge (56)

! (57)

S(!) ! (58)

L2(G!) = #je ($vH!"v) (59)

Hj1,···,jn = (Vj1 $ · · · Vjn)SU(2) (60)

BIJ =

"%+

1

!

#"(e)IJ , "(e)IJ = eI & eJ (61)

B± =1' !

!(62)

Ie = (i+, i") Vi+ $ Vi! = #kVk (63)

Is there a preferred choice for the vertex amplitude for gravity?

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Spin Foam models: constrained BF

1

I. REMARKS


S =1

2G


S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

!IJKLBIJµ!BKL

"# " !µ!"# (4)


S =1

2G


1

2"eI ! eJ !RIJ +

#


Z =

! "

e

dGe

"

f

dXfeiS(Xf ,Ge)

=

! "

e

dGe

"

f

$(Gf )

$(G) =#

J

dJ%J(G)

Z =

!DADBei

RTr(B!F (A))

#

Jf ,Ie

"

f

dJf

"

v

$Jf1 · · · Jf10

Ie1 · · · Ie5

%

& '( )

S =#

f

Tr(XfGf ) (7)

Xf Gf ="

e"f

Ge

vertex amplitude

State sum model formulation of quantum gravity amplitude incorporating

Rovelli, Engle, Pereira, L.F Krasnov, Livine Speziale.

Sum over discrete geometry containing no transplanckian degree of freedom

represents the area of the face in Planck unit: It is discretised (fondamental cut-off) and represent a sum over “area” geometry

represent the “shape” of the tetrahedra with given area

6

A(S)! = !"!

e!S"!

"je(je + 1)

! ! H!,je

dim(H!,je) = 1

Conformal factor: Paneitz 1983 Riegert, Fradkin Tseylin 1984 jack shaposhnikov 1998 mazurMottolla 1998

H!j = (Vj1 " Vj2 " Vj3)SU(2) = 1

V (R) =

#

R

#" =

#

R

$%%%#abc

3!Ea

i EbjE

ck#

ijk%%%

limN#$

$%%%#abc

3!Ei(Sa)Ej(Sb)Ek(Sc)#ijk

%%%!

i

J iav = 0

lim"#0

Ei(S")! = 0

S" #

"eDje(ge) ! Vje " Vje

iv ! (Vje1" · · ·" Vjen

)SU(2)

v ! ei

!!,je,iv(ge) = $"viv|"e Dje(ge)%"

!(ge) =!

je,iv

Cje,iv!!,je,iv(ge)

$!!,je,iv |!!,j!e,i!v% =

&

e

$je,je!dje

&

v

$iv|i%v%

V (R)!! =!

v

'(() 1

48

%%%%%!

e1,e2,e3

#(e1, e2, e3)#ijk&e1i &

e2j &

e3k

%%%%%!!

&eif(ge) =

d

dtf(ei#itge)|t=0

!

a

&eai !(ge) = 0

= 0

(1 + ")B& = (1' ")B+

jf

j±f = (1 ± ")jf

1

I. REMARKS


S =1

2G


S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

!IJKLBIJµ!BKL

"# " !µ!"# (4)


S =1

2G


1

2"eI ! eJ !RIJ +

#


Z =

! "

e

dGe

"

f

dXfeiS(Xf ,Ge)

=

! "

e

dGe

"

f

$(Gf )

$(G) =#

J

dJ%J(G)

Z =

!DADBei

RTr(B!F (A))

#

Jf ,Ie

"

f

dJf

"

v

$Jf1 · · · Jf10

Ie1 · · · Ie5

%

& '( )

S =#

f

Tr(XfGf ) (7)

Xf Gf ="

e"f

Ge

Sum over internal Su(2) repand intertwiner

6

A(S)! = !"!

e!S"!

"je(je + 1)

! ! H!,je

dim(H!,je) = 1


H!j = (Vj1 " Vj2 " Vj3)SU(2) = 1

V (R) =

#

R

#" =

#

R

$%%%#abc

3!Ea

i EbjE

ck#

ijk%%%

limN#$

$%%%#abc


%%%!

i

J iav = 0

lim"#0

Ei(S")! = 0

S" #


iv ! (Vje1" · · ·" Vjen

)SU(2)

v ! ei


!(ge) =!

je,iv

Cje,iv!!,je,iv(ge)

$!!,je,iv |!!,j!e,i!v% =

&

e

$je,je!dje

&

v

$iv|i%v%

V (R)!! =!

v

'(() 1

48

%%%%%!

e1,e2,e3

#(e1, e2, e3)#ijk&e1i &

e2j &

e3k

%%%%%!!

&eif(ge) =

d

dtf(ei#itge)|t=0

!

a

&eai !(ge) = 0

= 0

(1 + ")B& = (1' ")B+

jf

j±f = (1 ± ")jf

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

!

The weight is expressed in a pure algebraic form

6

A(S)! = !"!

e!S"!

"je(je + 1)

! ! H!,je

dim(H!,je) = 1


H!j = (Vj1 " Vj2 " Vj3)SU(2) = 1

V (R) =

#

R

#" =

#

R

$%%%#abc

3!Ea

i EbjE

ck#

ijk%%%

limN#$

$%%%#abc


%%%!

i

J iav = 0

lim"#0

Ei(S")! = 0

S" #


iv ! (Vje1" · · ·" Vjen

)SU(2)

v ! ei


!(ge) =!

je,iv

Cje,iv!!,je,iv(ge)

$!!,je,iv |!!,j!e,i!v% =

&

e

$je,je!dje

&

v

$iv|i%v%

V (R)!! =!

v

'(() 1

48

%%%%%!

e1,e2,e3

#(e1, e2, e3)#ijk&e1i &

e2j &

e3k

%%%%%!!

&eif(ge) =

d

dtf(ei#itge)|t=0

!

a

&eai !(ge) = 0

= 0

(1 + ")B& = (1' ")B+

jf

j±f = (1 ± ")jf

eiS NC geometry

1

I. REMARKS


S =1

2G


S =

!BIJ ! F IJ(A) (2)

BIJ =1

2G!IJKLeK ! eL (3)

!IJKLBIJµ!BKL

"# " !µ!"# (4)


S =1

2G


1

2"eI ! eJ !RIJ +

#


Z =

! "

e

dGe

"

f

dXfeiS(Xf ,Ge)

=

! "

e

dGe

"

f

$(Gf )

$(G) =#

J

dJ%J(G)

Z =

!DADBei

RTr(B!F (A))

S =#

f

Tr(XfGf ) (7)

Xf Gf ="

e"f

Ge

Ge1 · · · Gen

Jf ie

ggab = tr(EaEb) (8)

1

0 < ! < 1

1 < ! < !! > 1

! = 1

e " e =!

1# !2(B! # ! $B!) =

1

1# !2($B!!1 # !B!!1)

X = $(e " e)

X = (e " e)

XIJnJ = 0

! = 0

! = !B = e " e

B = $(e " e)

AIJ =!

1# !2

!XIJ # !XIJ

"

Xf =#(1 + !)Xf , (1# !)ueXfu

!1e

$

Xf =#(1 + !)Xf ,#(! # 1)ueXfu

!1e

$


Xf · Xf = 0 Xf · Xf " = 0%

f"e

Xf = 0

BCj =

&dndn|j, n% & |j, n%'j, n|& 'j, n|

0

XIJf =

(Xj+n+ ,#Xj!,n!)

(X,#ueXu!1e )

(X, ueXu!1e )

XIJf = V !1u[I

e uJ ]e"

uIe XIJ

f ueI = 0

uIXIJ # !uIX

IJ = 0

f e v Ie

7

153F903957A7C27ED077364961qab

!(q)

H!(q) = 0

H!M(q) = 0

qab = !ijEai Eb

j

Aia

Eai

A(S) = "#2P

!j(j + 1)

" < 1

!M(q)

q

H!(", j)?

=?0 ?

SRegge ="

f

Af (#)#f (#) $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j!f , i!e )

$

e

f iei+i!

ie

"

eiSregge + c.c

Canonical Structure

1

0 < ! < 1

1 < ! < !! = 1

e " e =!

1# !2(B! # ! $B!) =

1

1# !2($B!!1 # !B!!1)

X = $(e " e)

X = (e " e)

XIJnJ = 0

! = 0

! = !B = e " e

B = $(e " e)

AIJ =!

1# !2

!XIJ # !XIJ

"

Xf =#(1 + !)Xf , (1# !)ueXfu

!1e

$

Xf =#(1 + !)Xf ,#(! # 1)ueXfu

!1e

$


BCj =

%dndn|j, n% & |j, n%'j, n|& 'j, n|

(Xj+n+ ,#Xj!,n!)

A ( !

&j +

1

2

'

"

! < 1

The boundary states arising in the model are 4-valent SU(2) spin networks

Moreover these states for a complete set for the spin foam amplitudes

The sum being over all SU(2) coloring of the graph

2

G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

G!j =

!dn|j+, n! " |j!, n!#j, n|" #j, n|

j+ + j!

A =!

2(j+ + j!) =

1

2(j+ $ j!)

j+f1

j+f4

j!f1j!f4

j+f1

+ j!f1

k(j+, j!

+ $

j+

j!=

! + 1

! $ 1

2j!"

m=0

Cmj+,j!

j+ $ j! + m

Cmj+,j! =

(2j+)!(2j!)!

(2j+ + m)!(2j! + 1$m)!

dj+dj!G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

! %&!

Cmj+,j! ' e!"m

" = ln

#! + 1

! $ 1

$

( e!m2

j+, j! >> m

Z!(!, ke, iv)

(!, ke, iv)

Z(!, !"") ="

ke,iv

Z(!, !"ke,iv)Z(!"

ke,iv , !"")

!"

Given the spin foam model S we can slice it and obtain boundary state amplitudes for spin foam with boundaries.

2

G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

G!j =

!dn|j+, n! " |j!, n!#j, n|" #j, n|

j+ + j!

A =!

2(j+ + j!) =

1

2(j+ $ j!)

j+f1

j+f4

j!f1j!f4

j+f1

+ j!f1

k(j+, j!

+ $

j+

j!=

! + 1

! $ 1

2j!"

m=0

Cmj+,j!

j+ $ j! + m

Cmj+,j! =

(2j+)!(2j!)!

(2j+ + m)!(2j! + 1$m)!

dj+dj!G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

! %&!

Cmj+,j! ' e!"m

" = ln

#! + 1

! $ 1

$

( e!m2

j+, j! >> m

Z!S(!, ke, iv)

(!, ke, iv)

ZS#S"(!, !"") ="

ke,iv

ZS(!, !"ke,iv)ZS"(!

"ke,iv , !

"")

!"

2

G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

G!j =

!dn|j+, n! " |j!, n!#j, n|" #j, n|

j+ + j!

A =!

2(j+ + j!) =

1

2(j+ $ j!)

j+f1

j+f4

j!f1j!f4

j+f1

+ j!f1

k(j+, j!

+ $

j+

j!=

! + 1

! $ 1

2j!"

m=0

Cmj+,j!

j+ $ j! + m

Cmj+,j! =

(2j+)!(2j!)!

(2j+ + m)!(2j! + 1$m)!

dj+dj!G!j =

!dn|j+, n! " |j!, n!#j+, n|" #j!, n|

! %&!

Cmj+,j! ' e!"m

" = ln

#! + 1

! $ 1

$

( e!m2

j+, j! >> m

Z!S(!, ke, iv)

(!, ke, iv)

ZS#S"(!, !"") ="

ke,iv

ZS(!, !"ke,iv)ZS"(!

"ke,iv , !

"")

!"

An independent, purely path integral, derivation of the fact that SU(2) spin network states can be used to label pure gravity amplitudes

Semi-Classical Limit

F. Conrady L.F ; Barrett, Fairbain et al.

Two keys results about the asymptotics

Since the semiclassical limit of the spin foam amplitude is controlled by the large spin limit

4

ZBF =

! "

e

dge

"

f

!(Gf )

ZBF =#

Jf

"

Jf

dJf

! "

e

dGe

"

f

"Jf(Gf )

1j = dj

!

SU(2)

dn |j, n!"j, n|

=$X(j+

f ,n+ef ), X(j!f ,n!ef )

%

S =#

f,v

S (Xf (v), Gf (v)) Xf (v) Gf (v)

ZBF =

! "

e

dGe

"

(v,f)

dXf (v) eP

v,f S(Xf (v),Ge)

XIJf ueJ = 0

k = j+ # j!

A $ #(j+ # j!) = j+ + j!

Cjk = !k,j+!j!

n+L = n!L n+

R = n!R nL %= nR

G!j =

!dnLdnR|j+, nL! & |j!, nL!"j+, nR|& "j!, nR|

A!v(jf , le, kef ) '

#

i+ev ,i!ev

15j(j!+f , j!!

f , i+e , i!e )"

e"v

di+edi!e

f lei+e ,i!e

(j!+f , j!!

f , kef ) .

jf $ Area(f)/$2p

$p ( 0

4

ZBF =

! "

e

dge

"

f

!(Gf )

ZBF =#

Jf

"

Jf

dJf

! "

e

dGe

"

f

"Jf(Gf )

1j = dj

!

SU(2)

dn |j, n!"j, n|

=$X(j+

f ,n+ef ), X(j!f ,n!ef )

%

S =#

f,v

S (Xf (v), Gf (v)) Xf (v) Gf (v)

ZBF =

! "

e

dGe

"

(v,f)

dXf (v) eP

v,f S(Xf (v),Ge)

XIJf ueJ = 0

k = j+ # j!

A $ #(j+ # j!) = j+ + j!

Cjk = !k,j+!j!

n+L = n!L n+

R = n!R nL %= nR

G!j =

!dnLdnR|j+, nL! & |j!, nL!"j+, nR|& "j!, nR|

A!v(jf , le, kef ) '

#

i+ev ,i!ev

15j(j!+f , j!!

f , i+e , i!e )"

e"v

di+edi!e

f lei+e ,i!e

(j!+f , j!!

f , kef ) .

jf $ Area(f)/$2p

$p ( 0

7

153F903957A7C27ED077364961!ab

!(!)

H!(q) = 0

H!M(q) = 0

!ab = "ijEai Eb

j

Aia

Eai

A(S) = !#2P

!j(j + 1)

! < 1

!M(q)

q

H!(", j)?

=?0 ?

S! ="

f

Af (#)#f $

S! ="

S4S

"

f

Af#f $

gY M

#!BIJ "BIJ

fki+i!

Z! ="

jf ,ie

djf

$

v

Av(jf , ie)

Av(jf , ie) ="

i+i!

15j(j+f , i+e )15j(j+

f , i+e )$

e

f iei+i!

ie

!

eiSregge + c.c•The vertex amplitude is either exponentially suppressed or asymptotic to when boundary labels allow the reconstruction of a metric

•A similar results generalised to an arbitrary triangulation where we sum over all intertwinners labels (non degeneracy condition)

The dependence on the Immirzi parameter disappear in the limit!

Highly non trivial results!

Non trivial consistency check

Graviton propagator


Full control of the flat space sector of the theory at the quantum level in the coarse grained limit

First checks that the perturbative sector around these quantum flat space configurations can be computed

Graviton scattering computation program.

Provide an approximation scheme for QG amplitudes.

Spin foam ansatz: the possibility that for a given boundary state the exact coarse grained amplitude is captured by a finite internal triangulation.

Rovelli, Bianchi, Speziale....

Conclusion Spin foam models provides a purely algebraic framework allowing us to deal

covariantly with the dynamics of loop quantum gravity

This gives an independent derivation that the boundary states of quantum gravity are SU(2) spin network states

These amplitudes incorporate the Immirzi parameter and have the propersemi-classical limit.

A program of study of the renormalisation property under refinement is underway

A lorentzian version is available

A program of extracting physical prediction from this model (graviton scattering...) is under developpement

new spin foam : modeling quantum space time · 2009. 7. 30. · perimeter institute mg12 paris...

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