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A 2nd quantized (Fock space) formulation of LQG
(and what is can be useful for)
Daniele Oriti
!
Albert Einstein Institute
!
!
LQG workshop - Tux, Austria
10/02/2014
Plan of the talk
•
Part I: from LQG to GFT
!
•
LQG states as “many-particle states”
•
Second quantization of spin networks and Group Field Theory
!
•
•
!
kinematics: states, observables and GFT fields/observables
dynamics: canonical projector and GFT action
!
!
!
•
Part II: What is the 2nd quantised (QFT) formalism good for?
•
•
!
relating LQG and Spin Foam models
!
dealing with continuum limit (many d.o.f.) at dynamical level
defining full LQG dynamics via QFT methods
•
continuum phase structure and LQG vacua
•
GFT condensates
•
extracting eﬀective continuum dynamics
•
• (Quantum) Cosmology as GFT condensate hydrodynamics
Introduction
Reisenberger,Rovelli, ’00
we already know:
GFT <---> Spin Foams
- actually: GFT = Spin Foams
L. Freidel, ’06
DO, ’06, ’11
A. Baratin, DO, ’11
!
GFT is often presented as the 2nd quantized version of LQG
!
!
we show (Part I):
(DO, 1310.7786 [gr-qc])
!
•
this is true in a precise sense: reformulation of LQG as GFT
•
very general correspondence (both kinematical and dynamical)
•
do not need to pass through Spin Foams (LQG/SF correspondence obtained via GFT)
!
!
!
the reformulation provides powerful new tools to address LQG open issues (Part II)
Part I
!
2nd quantized LQG:
!
From Loop Quantum Gravity to Group Field Theory
DO,1310.7786 [gr-qc]
The (kinematical) Hilbert space(s) of LQG
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
!
quantum states: cylindrical functions of holonomies (fluxes) along links (surfaces) for graphs (+surfaces)
1
G14
H 3
G12
(G1 , ..., GE )
= (V, E)
plus gauge invariance at vertices
G13
4
Gi 2 SU (2)
2
G24
G34
G23
H1 =
3
[
H
not a Hilbert space
T. Thiemann, ’01
A. Ashtekar, J. Lewandowski, ‘04
M
•
huge
• diﬀerent graphs ~ orthogonal states
•
prominence to graphs
H
H2 = lim
S
H
⇡
=L
2
A¯
• based on kinematical continuum limit (states eﬀectively defined on “infinitely refined graph”)
• equivalence classes of graphs
• diﬀerent graphs ~ orthogonal states
The (kinematical) Hilbert space(s) of LQG
algebra of observables: holonomy-flux algebra for paths (+ dual surfaces)
!
quantum states: cylindrical functions of holonomies (fluxes) along links (surfaces) for graphs (+surfaces)
[
modified class of quantum states: • extended: closed + open graphs
•
restricted: d-valent graphs
˜d
can generalise
H ˜d
plus gauge invariance at d-valent vertices
W. Kaminski, M. Kieselowski, J. Lewandowski, ’09
DO, J. Ryan, J. Thuerigen, ‘14
M
Hdext =
V
HV
turned into Hilbert space by:
!
• considering Hilbert space of states for V “open vertices”
1
g13
1
g1
g2
•
1
1
g2
3
g4
g2
g2
4
4
2
g14
2
g32
g23
g33
3
1
g3
•
! g V , ..., g V = ' ~g 1 ; ....; ~g V
HV 3 ' g11 , ..., gd1 ; ....;
d
! 1
embedding
!
summing over V
H ˜dV ⇢ H
V
vertex is a node with d outgoing open links, and can be thought as dual to a polyhedron with d faces .
j
ab
1 1
1
V
V
V
(g
)
=
(g
,
g
,
.
.
.
,
g
;
.
.
.
;
g
,
g
,
.
.
.
,
g
),
),
where
the group
elements Gdescribing
2
G
are
assigned
to
each
link
e
:=
[(i
a)
(j
b)]
2
E(
)
of
the
tates
are given
by wavefunctions
V
vertices
or
their
dual
polyhedra,
of
the
type
1
2
d
1
2
d
i
ij
led by two pairs of indices:
the first
pair identifies
the
pair of
vertices (ij) connected, while the
j
1
1
1
V
V
V
(gi ) =where
(g1 ,each
g2 , . open
. . , gdlink
; . . .outgoing
; g1 , g2 ,from
. . . , each
gd ) ,vertex is associated a group element of
(7)the
the outgoing edges (ab) of each
vertex
together
form
theand
link.
assume
the gauge
or SL(2,
C) inglued
quantum
gravity to
GFT
models,
G =We
SU(2)
in standard
LQG), wi
1
j
j
ab
in
V
:
(g
)
=
(g
V elements
ofthe
G. The
set G
of (G
such=functions
(restricting
=
({
G
}),
for
any
V
group
elements
to
vertices.
j of
i
i associated
i
i j )a for
ij j
open link outgoing
from each vertex is associated
group
element
the
group
SU(2),
Spin(4),
2
d·V
V
be
turned
into
the
Hilbert
space
L
(G
/G
) by defining the be
innerobtained
product via
lent
graph with
V GFT
vertices
(specified
E( )),inastandard
cylindrical
function
bythe
in quantum
gravity
models,
and G by
= SU(2)
LQG),
with gauge can
invariance
at vertices
some right/left-invariant
V
2measure
dV in the
V non-compact
1 case. V
j
y=wavefunction
, embedding
V ⇢
HThe
' Lfunctions
G /G
' to
~g square-integrable
, ..., ~g
(gi j ) for V elements
setH
of such
(restricting
ones) can
j of G.
˜ d3
The ˜Hilbert
space for these functions, H
, includes as a special class of states the usu
d
d·V
V
nto the Hilbert space L2 (G
/G
) by defining
the
inner graphs
product. via
theis Haar
measure
on({1,
the. .group,
Z
associated
to closed
d-valent
There
a relation
E( ) ⇢
. , V } ⇥ or
{1, .
Y
ab
a case.
ab b ab
a b
1
left-invariant
measure
in the d↵
non-compact
E(
))
which
specifies
the
connectivity
of
such
a
graph:
({Gab
})
=
({g
↵
;
g
↵
})
=
({g
(g
)
})
, if [(i a) (j b)] 2 E( ), there
(8) is
ij
ij
i ij
j ij
i
j
Spin network functions as “many-particles” states
˜ includes
ert space for these functions,
a special
class
of states
the
cylindrical
of LQG,
link atasthe
i-th node
to the
b-th link
at usual
the j-th
node, withfunctions
source i and
target j
e2E( ) GHd ,a-th
1
ab
2 G are assigned to each link
of the
({G
where
Gab
o closed d-valent graphs . There
is aform
relation
E(
)⇢
({1, the
. . . ,group
V } ⇥elements
{1, . . . , d})
ij }),
ij 2(satisfying
1 g [(i a) (i a)] 62
g
a ),bof
graph.
These
are
labelled
by two
pairs
indices:
the
first
pair
identifies
of ve
g13 the
g11edge
h specifies
the
connectivity
of
such
a
graph:
if
[(i
a)
(j
b)]
2
E(
there
is
a
directed
the
1g connecting
ach
edge
in
is
associated
with
two
group
elements
g
,
g
2
G.
The
integrals
over
thepair
↵’s
i
j
wave function second pair identifies the outgoing edges (ab) of each vertex glued
to form
g2 together
the
i-th
node
to
the
b-th
link
at
the
j-th
node,
with
source
i
and
target
j.
Cylindrical
functions
are
then
for
closed
graph
1
k vertices corresponding to the
function
,
pairwise
along
common
links,
thus
forming
the
spin
1 net vertices
ab
abspin
wave
function
for
many
open
invariance ab ({Gij }) =
({ i Gij j }), for any V group elements i associated
to
ab
g the
({G
}),
where
the
group
elements
G
2
G
are
assigned
to
each
link
e
:=
[(i
a)
(j
b)]
2
E(
)
of
g
ij
ij
y the closed
graph . The same
construction
can graph
be phrased
in theg(specified
flux
representation
and 1in fu
3
Given
a closed
d-valent
with V vertices
by E( )), a cylindrical
G14
G while gthe
4
same
in other
fluxes,ofspins
se are
labelled
bybasis:
two pairs
indices:
the first of
pair
identifies
the pair
of vertices
(ij)
connected,
2
group-averaging
any
wavefunction
,
g
n.
g
4
2
2
g2
identifies the outgoing edges (ab) of each vertex glued together to form
the link.
We assume theggauge
Z
4
2
G
13
Y 4
1
ab
ab
ab
abvertices.
a ab b ab
a2
({Gij }) =
({
G
}),
for
any
V
group
elements
associated
to
the
⇧
({G
})
=
d↵
({g
↵
;
g
↵
})
=
({g
i spin
iij
⇥4
ij ⇥jnetwork vertices:d
ij
i ij
j ij
i (g
“gluing”
of
˜
d
1
d
1
gG
the subspace
ofXsingle-particle
(single-vertex)
states,
i.e.
elements
of Hd with
V
=be
1. obtained
A general
g 2
....,
X
=
,
.....
=
(X
+
X
)
⇤
⇥
X
,
X
,
....
˜ ....,
ij
e2E(
)
i
i
j
i
j
closed d-valent
graph
with
V
vertices
(specified
by
E(
)),
a
cylindrical
function
can
by
G
24
1
imposition of symmetry, identification
g4
g3
2
(ij)
Eelements of Hv ,
eging
decomposed
intospecific
products
ofcombination
of variables,
linear
of ⇤
any
wavefunction
,
a b
⌅
in such a way that each edge in
is associated with two group
g 2elements gi , gj 2
1
d
03 )dV Z/(closure)
1 V)
g3
⌅ 1 , ..., X
⌅V
⌅
g
⇥˜ X
L2 ((R
X
=
(X
,
...,
X
34
i
g G23along comm
to the G
function
,
pairwise
i
i)
YVglue open spin network vertices corresponding
1
3
ab
ab
a ab b ab
a b
1
Y
X
g
g
({Giji}) function
= @ network
d↵
({g
gj ↵closed
}) =V ({g. iThe
(gj )same
}) construction
, 3
(8) in
represented
by
can3 be phrased
~ 1in
...~
1; the
i ↵ij
ij
V
Aij
everyi cylindrical
is contained
new
Hilbert
spacegraph
3
1
1
1
2
1
1
3
4
2
12
2
4
2
2
1
4
3
2
2
3
3
3
(gI ) = hgI | i =
hg |~ i · · · hg |~ i ,
1
3
3
(9)
V
I
) G
⇧e2E( the
⌃I 1
spin representation.
i=1 ~ i
3
⌦
⇥
3
d
d 1
1
⌥
a
b
(....,
j
,
...,
I
,
....)
=
⇥
....,
j
,
m
,
j
,
m
, Ij , ....
d
1
d
1
˜
ij
i
mi ,mjtwo
ji ,jH
i ofg
isingle-particle
j , ...,
way that each edge in
is associated
group
elements
G. IiThe
integrals states,
over the
↵’s
j v the subspace
Let uswith
denote
by
(single-vertex)
i.e. elemen
i , jgj 2
d
1
(ij) E mi ,mj
pin network vertices ⇤corresponding
tocan
thebe
function
,decomposed
pairwise
commonoflinks,
thusof forming
the spin
Any LQG state
written
terms ofalong
“many-vertices”
states
V -particle
state
can
into products
elements
Hv ,
⌅ bein
d
1
⇤jV same
⇤ji can
resented by the closed
. ...,
The
phrased
in0(m
the1i , flux
representation
and in
⇥ ˜ (⇤j1graph
,m
⇤ 1 , I1 ),
,m
⇤ V , Iconstruction
= (ji1be
, ...,
jid ) m
⇤i=
...,
m
V)
i)
V
Yeach
X GFT quantum (or spin
t attention to the simplicial case, in which d equals the spacetime dimension, and
i
i
~ 1 ...~ V
1
V
presentation.
Spin network functions as “many-particles” states
embedding
H ˜dV ⇢ HV
with standard Haar measure
LQG kinematical scalar product for given graph (with V vertices) is restriction of scalar
product for V open-vertices states
same pattern of “gluings”
⇧
|
⇥= ⇤
=
,
⌥
⇤
(ij)⇥E
i⇥V
⌦
⌦
dGij
d⇧gi
⌅
⌃
(ij⇥E)
(..., Gij , ....)
⌦
d
ij d⇥ij ⌅ ˜
(..., Gij , ...) =
⇥
...., gid ij , gj1 ij , ....
⌅˜
⇥
...., gid ⇥ij , gj1 ⇥ij , ....
= ⌅ ˜ |⌅ ˜ ⇥
this shows embedding of Hilbert space of given graph into new Hilbert space
however, generic cylindrical functions for diﬀerent graphs
are embedded diﬀerently in new Hilbert space:
Hdext =
M
V
HV
• states associated to diﬀerent graphs with same number of vertices are NOT orthogonal
•
states associated to graphs with diﬀerent number of vertices ARE orthogonal •
prominence to number of vertices, not to graph structure
•
no cylindrical consistency, no projective limit
Spin network functions as “many-particles” states
ext
Hd
=
M
V
H
V
HV 3 ' g11 , ..., gd1 ; ....; g1V , ..., gdV = ' ~g 1 ; ....; ~g V
with standard Haar measure
require also symmetry under relabelling of vertices (permutations of vertices)
each state can be decomposed in products of “single-particle” (vertices) basis states:
|⇥ ˜ =
⇥ ⇥˜ 1 ...⇥ V
⇥i
⇥
⇧ i = ⇧ji , m
⇧ i, I
⇧i
⇥
⇧i = X
⇥
⇥
g|⇥ ˜ ⇥ =
|⇤ 1 .....|⇤ V
⇤ ⇤ (⇧g ) = ⇥⇧gi |⇧
⇥i ⇤ =
⇤ ⇤ (⇧g ) = ⇥⇧gi |⇧
⇥i ⇤ =
d
⌃
a=1
d
⌃
⇥ ⇥˜ 1 ...⇥ V
⇥i
⇥
i V
⇤gi |⇤ i ⇥
ja
j1 ...jd ;I
Dm
(g
)C
a
n1 ..nd
a na
a=1
Ega (Xa ) ⌅
⇥
⇤
⇧
a
Xa
⌅
related by unitary transformation
(NC Fourier transform, Peter-Weyl decomposition)
2nd quantized reformulation: kinematics
standard procedure for writing same states in 2nd quantized form
Hdext =
result:
M
V
HV ' F(Hd ) =
M⇣
V
symmetry under vertex relabeling
˜
(⇥g1 , ..., ⇥gi , ..., ⇥gj , ..., ⇥gV ) =
(1)
(V )
Hd ⌦ · · · Hd
⌘
(only a justification,
not a proof:
assumption!)
bosonic statistics
˜
(⇥g1 , ..., ⇥gj , ..., ⇥gi , ..., ⇥gV )
sketch of procedure (1):
•
•
•
define ordering of links in each vertex (e.g. 1 < 2 < ... < d)
define ordering (e.g lexicographic) for single-vertex labels (e.g. (j,m,I))
count how many times each set of labels appears and label states by these numbers:
n1
⇥ ⇥˜ 1 ...⇥ V
•
•
⇥i
⇥)
!
na = V
normalize states re-write generic wave function in “occupation number”
basis:
⇤
⇤ ⇥˜ 1 ...⇥ V
⇤ ˜ (⌅gi ) =
(a
na
⇥ ⇧⌅ ⇤ ⇥ ⇧⌅
⇤ !
⇤ 1 ...⇤ 1 ... ⇤ a ...⇤ a .... !
= ⇥˜
!= C (n1 , ..., na , ...)
⇥
i⇥V
=
⌅gi |⌅ i ⇥ =
C˜ (n1 , .., na , ..)
{na }
C˜ (n1 , .., na , ..) ⇥{na } (⌅gi ) =
n1 !...n !
V!
a=1
⇥
{⇥ i |na } i⇥V
⌅gi |⌅ i ⇥ =
C˜ (n1 , .., na , ..) g|n1 , ..., na , ...⇥
2nd quantized reformulation: kinematics
sketch of procedure (2):
•
na =
send number of vertices to infinity - no constraint on occupation numbers
!
a
•
orthonormal basis (in LQG scalar product): •
define creation/annihilation operators:
c ⇥ , c†⇥
⇥
⇥
= |n1 .....|n
|n1 , ..., na , ..., n
!
=
⇥ ,⇥
c ⇥ |n ⇥ ⇥ =
⇤
1⇥
c†⇥ , c†⇥
⇥
=0
⇤
†
c ⇥ |n ⇥ ⇥ = n ⇥ + 1|n ⇥ + 1⇥
[c ⇥ , c ⇥ ] =
n ⇥ |n ⇥
n⇥ =
N ⇥ |n ⇥ ⇥ = c†⇥ c ⇥ |n ⇥ ⇥ = n ⇥ |n ⇥ ⇥
all quantum states generated from Fock vacuum | 0 >
(“no-space” state)
can define conjugate bosonic field operators:
⇥(g
ˆ 1 , .., gd )
⇥(⇤
ˆ g) =
cˆ⇥
g)
⇥ (⇤
⇥ˆ† (g1 , .., gd )
cˆ†⇥
⇥ˆ† (⇤g ) =
⇥
⇥
⇥ = ⇥j, m,
⇥ I
⇥
or
⇥
⇥= X
⇥
g)
⇥ (⇤
Spin networks in 2nd quantization
Motivation
Fock vacuum: “no-space” (“emptiest”) state | 0 >
~ AL LQG vacuum
Motivation
(this is the(III)
natural background independent, diﬀeo-invariant vacuum state)
and
up fieldsingle
theory
we consider
defined
in terms
of a field
field “quantum”:
spinisnetwork
vertex
or tetrahedron
'(g1 , g2 , g3 , g4 ) $ '(B1 , B2 , B3 , B4 ) ! C
(“building
blockofofspace:
space”)
ng an elementary building
block
g1⇥
†
ˆ (g1, g2, g3, g4)|⇤⌅ = |
⇥
⇥
⇥⌅
⌅
g4
g2
⌅
⌅
⌅
•
⌅
⇥ ⇥ ⇥ ⌅⌅ g 3 ⌅
⇤
⇥g4⇥
⇤⇤
⇤
⌅
⇥
⇤⇤
g3
g1
⇥
⌅
⌅
⇤
⌅⇥⇤⇤⇥⇥
⇥
⌅
g2
⌅
nts gi specify the geometry of the elementary tetrahedron by
quantum state: arbitrary collection of spin network vertices (including glued ones) or
mies ofgeneric
the gravitational
SO(4) connection along links dual to
tetrahedra (including glued ones)
Riemannian signature – the extension to SL(2, C) is doable but
)
SO(4) gauge transformations on the vertex:
Quantization of Systems with Constraints
Two dynamical models for full LQG
Outlook and Work in Progress
Hamiltonian formulation of GR
Relational Formalism: Observables & Evolution
Basis of Hkin
Spin network functions
[Ashtekar, Isham, Lewandowski, Rovelli, Smolin ’90]
j15
j14
j19
j18
j22
j17
j16
j20
j8
j21
j12
j13
j11
j23
j10
j2
1 , g2 , g3 , g4 ) = (g1 h, g2 h, g3 h, g4 h) ,
j1
j3
j6
j7
j5
j4
Kristina Giesel
Dynamics of LQG
j9
h ⇥ SO(4).
Homogeneous cosmologies in discrete quantum gravity
5 / 16
2nd quantized reformulation: kinematics - observables
any LQG operator can be written in 2nd quantized form
ˆ = O\
O
(E, A)
“2-body” operator
(acts on single-vertex, does not create new vertices)
⌃2
O
⌃2 |⇤ ⇤ = O2 (⇤ , ⇤ )
⇥⇤ |O
⌅
⇤
⇥
†
⌃2 ⇥,
O
⇧ ⇥
⇧ =
O2 (⇤ , ⇤ ) cˆ†⇥ cˆ⇥ = d⇤g d⇤g ⇥
⇧† (⇤g ) O2 (⇤g , ⇤g ) ⇥(⇤
⇧g)
⇥ ,⇥
“(n+m)-body” operator
(acts on spin network with n vertices, gives spin network with m vertices)
On+m
⇥⇤ 1 , ...., ⇤ m |On+m |⇤ 1 , ..., ⇤ n ⇤ = On+m (⇤ 1 , ..., ⇤ m , ⇤ 1 , ..., ⇤ n )
⇤
⇥
On+m ⇥,
ˆ ⇥ˆ† = d⇤g1 ...d⇤gm d⇤g1 ...d⇤gn ⇥
⌅† (⇤g1 )...⇥
⌅† (⇤gm )On+m (⇤g1 , ..., ⇤gm , ⇤g1 , ..., ⇤gn ) ⇥(⇤
⌅ g1 )...⇥(⇤
⌅ gn )
basic field operators and the set of observables as functions of them define
the quantum kinematics of the corresponding GFT
2nd quantized reformulation: dynamics
can use general correspondence for operators to rewrite also any dynamical quantum
equation for LQG states in 2nd quantized form
assume quantum dynamics is encoded in “physical projector equation”:
P| ⇥=| ⇥
Hdext
| ⇥
or:
⇣
Fˆ | i = Pˆ
⌘
Iˆ | i = 0
projector operator will in general decompose into 2-body, 3-body, ...., n-body operators (weighted by
(coupling) constants), i.e. will have non-zero “matrix elements” involving 2, 3, ..., n spin network vertices
P ⇤=| ⇤
⇥
⇤
2 P2 + 3 P3 + .... | ⇤ = | ⇤
⇥⇤
⇥1 , ...., ⇥
⇤ m |Pn+m |⇤
⇥1 , ..., ⇥
⇤ n ⇤ = Pn+m (⇤
⇥1 , ..., ⇥
⇤ m, ⇥
⇤ 1 , ..., ⇥
⇤ n)
the same quantum dynamics can be expressed in 2nd quantized form (using general map for operators):
⌅
⇤
⇥
⌥ †
⌥ †
⌥
†
⌃
⇧
=
⇥1 , ..., ⇥
⌅ m, ⇥
⌅ 1 , ..., ⇥
⌅ n ) cˆ⇥ 1 ...ˆ
c⇥ n |
cˆ⇥ cˆ⇥ |
cˆ⇥ 1 ...ˆ
c ⇥ m Pn+m (⌅
n+m
n,m/n+m=2
⇥
⌥
n,m/n+m=2
n+m
{⇥ ,⇥ }
⇥
⇥
d⌅g1 ...d⌅gm d⌅g1 ...d⌅gn ⇤† (⌅g1 )...⇤† (⌅gm )Pn+m (⌅g1 , ..., ⌅gm , ⌅g1 , ..., ⌅gn ) ⇤(⌅g1 )...⇤(⌅gn ) |
=
d⌅g ⇤(⌅g )⇤(⌅g ) |
=
equation 36 contribute. It would be the obvious option in a strict
which
states
quantum
are only language,
those solving
st choice we
makephysical
is to phrase
theofquestion
in aspacetime
quantum statistical
that i
not the only
in partition
a generalized
theory
that
on 14
the
text of a definition
of a possible
quantum option
statistical
function
for the
theory
. one
2nd
quantized
reformulation:
dynamics
starting
point
is of course
the
operator
equation
36.
Knowing
this, once wouldand
like othe
to d
discrete
context
(where
the
very notion
of diﬀeomorphisms
he quantum
theory,hand
in turn
definedininits
terms
of a statistical
density
ρ.
ˆ
on LQG
the other
includes
covariant
‘histories’,
i.e.operator
dynamica
b) correlations) for
F
partition function (and correlations) of GFT is then obtained from partition function δ(
(and
be to define
analoguespin
of networks,
the
microcanonical
ensemble:procedure.
ρˆ = Zm Of
with:
theancontinuum)
a canonical
quantization
course, an ex
recast in 2nd quantised languagem
for the dynamics which includes topology change for space and/or s
!
One
may
expect
from
general
arguments,
and
we
will
indeed
con
"
first candidate (“microcanonical ensemble”): Zm =
⟨s| δ(F )|s⟩
,|s> is arbitrary basis
corresponds to a quantum LQG dynamics
of
this
generalized
type.
s
statistical
amounts
to a contribute
choice of
a density
operator
oftheory)
the ca
only statesterms
solving dynamical
constraint
(natural
from continuum
canonical
e s denotes an arbitrary complete basis of states in the Hilbert (Fock) space of the qu
choice would mean that one defines a (quantum statistical) dynamics in which only stat
In It
thewould
same
spirit,
can option
weight
states with
m
!continuum
general context
(abstract
structures,
noone
continuum,
topologydiﬀerently
change, …) quantum
b
n more
36 contribute.
be
the
obvious
in
a
strict
canonical
qua
−F
suggest more general ansatz (“canonical ensemble”):
Z
=
⟨s|e
|s⟩
c
physical states
of quantum
spacetime
are to
only
those solving the
canonical
constraint
network
vertices,
and move
a grandcanonical
ensemble,
defined
by th
s
only possible option in a generalized theory that on the one hand is defined
in a more
e context
(where
theparameter
very notion
of diﬀeomorphisms
and other continuous
symmetries
or, introducing
a new
weighting
diﬀerently quantum states
!
where
we
have
put
any
“inverse
temperature”
constant
β
=
b−
b ) 1, fo
−
F
µ
N
(
with
diﬀerent
numbers
of
vertices
(“grandcanonical
ensemble”):
other hand includes in its covariant ‘histories’, i.e. dynamical
Zg = processes,
⟨s|e spacetimes
|s⟩ t
non-zeroquantization
weight to procedure.
quantum states
notansolving
the
constraint
eq
tinuum) aacanonical
Of course,
example
of
the
latter
is
a
p
s
compared
to generic
(and
so
that
such so
dynamicssuch
whichsolutions
includes topology
change
for space
spacetime.
thisstates
is and/or
the expression
leading
mostonly
directly
to GFTs confirm
and determines
Spin
Foam
where
the arguments,
sign
the and
‘chemical
qua
onefrom
reproduces
the of
microcanonical
ensemble).
One
also
imag
may expect
general
we willpotential’µ
indeed
it could
inmodels
thewhether
following,
verticesLQG
are
favored.
wetype.
work This
in this
general context.
onds to a imaginary
quantum
dynamics
this
generalized
generalized
quantu
constant
β In
=ofthe
iT
, following,
giving
such
weights
the
appearance
of
tool: 2nd
quantised
coherent
states ————->>>>
that
corresponds
to
existing
group
field
theories,
clarifying
in
the
proce
e− Fb
15
. However,
in this
this rewriting
may be type
probably
more
cal terms T
amounts
to a choice
of acontext,
density operator
of the canonical
: ρˆc =
correspondence between the corresponding spin foam amplitudes andZcc
proceed is the field theory analogue16 of the standard procedure of constructing the coherent state pat
quantum mechanics [53]. Indeed, the 2nd quantized formulation of 36 is already set up to make this p
most convenient one.
We introduce then
a 2nd quantized basis of eigenstates of the annihilation operator, that is the GFT q
P
R
†
ϕ
b
c
g ϕ(⃗
g )ϕ(⃗
b g )†
χ
⃗
χ
⃗ |0⟩ = e d⃗
operator, |ϕ⟩ = e χ⃗
|0⟩ satisfying:
2nd quantized reformulation: dynamics
†
c
"
|ϕ⟩
=
ϕ
|ϕ⟩
⟨ϕ|
c
"
= ϕχ⃗ ⟨ϕ|
χ
⃗
χ
⃗
χ
⃗
basis of 2nd quantized coherent states:
#
or equivalently
ϕ(⃗
g )|ϕ⟩ = ϕ(⃗g ) |ϕ⟩
⟨ϕ|ϕ
"† (⃗g ) = ϕ(⃗g ) ⟨ϕ|
$
$
!
− |ϕ|2
2
I = DϕDϕ e
|ϕ⟩⟨ϕ|
|ϕ| ≡ d⃗g ϕ(⃗g ) ϕ(⃗g ) =
ϕχ⃗ ϕχ⃗
X
Z
χ
⃗
b µN
b)
F
(
ef f (',')
gives:
The functions
classical
fields, as we are going to see17 . Th
hs|eϕ can be understood
|si ⌘ as the
D'D'
Zg = ϕ and
e SGFT
integration DϕDϕs is the (formally defined) functional measure for fields on Gd entering the GFT path
bpartition
terms of this basis of states (and using the bosonic statistics of the GFT states),h'|
theF
function
|'i
Sef f (', ') = S (', ') + O(~) =
+ O(~)
where the quantum corrected action is:
h'|'i
$
!
b − µN
b)
b
b
− (F
− |ϕ|2
Zg =
⟨s|e
|s⟩ =
DϕDϕ e
⟨ϕ| e− (F − µN ) |ϕ⟩
.
this is the GFT partition function with classical GFT action:
s
⌃
⇥
†
S ⇥, ⇥It is=cleard⇤gthat
⇥† (⇤gwe
) ⇥(⇤
g ) here the GFT path integral with a quantum amplitude
have
⇤⌃
⌅
⇥
⇧
†
†
d⇤g1 ...d⇤gm d⇤g1 ...d⇤
g
g
)...⇥
(⇤g Nb) Vn+m (⇤g1−
, ...,
⇤g , ⇤g , ..., ⇤gn ) ⇥(⇤g1 )...⇥(⇤gn )
2⇥ (⇤
n+m
1
b
n
− (F − µm
− |ϕ|
Sef fm 1
)
e
⟨ϕ| e
|ϕ⟩ ≡ e
n,m/n+m=2
g1S, ..., ⇤g(ϕ,
⇤g1 , ...,
⇤gn )task
= Pn+m
(⇤g1 , ..., ⇤gthe
gpartition
gn ) fun
n+m (⇤
m , ϕ).
m, ⇤
1 , ..., ⇤
that can be expressed in terms of an eﬀectiveVaction
The
of relating
ef f
quantum
theory,term
defined
some
specific
operator
F" to the GFT path integral of specific models t
the GFTLQG
interaction
is thebySpin
Foam
vertex
amplitude
task of understanding the form of the eﬀective action Sef f .
E. Alesci, K. Noui, F. Sardelli, ’08
For generic operators F", this is a very diﬃcult task [53], of course. The general strategy for solv
quantum corrections give new interaction terms or renormalisation of existing ones
ever, is clear: one should a) expand the exponential operator function of F" in power series of polyno
2nd quantized reformulation: dynamics - 3d example
test construction in “known” example: 3d quantum gravity (euclidean)
Hamiltonian and diﬀeo constraints impose flatness of gravity holonomy
general matrix elements of projector operator:
K. Noui, A. Perez, ’04
|P⇥|
⇥
⇥=
|
f ⇤˜
,
(Hf ) |
⇥
⇥
(independent) closed loops
such action decomposes into an action on 2, 4, 6,... spin network vertices
(glued to form closed graphs, because of gauge invariance of P - graphs formed by an odd number of spin net vertices do not arise)
this in turn should give possible GFT interaction terms
Vn+m (g1 , ..., gm , g1 , ..., gn ) = Pn+m (g1 , ..., gm , g1 , ..., gn )
E. Alesci, K. Noui, F. Sardelli, ’08
we expect these to give rise to the known Boulatov GFT model for 3d QG
(NB: because matrix elements are real, do not expect any distinction between using the GFT field and its conjugate)
2nd quantized reformulation: dynamics - 3d example
indeed....
using gauge invariance of GFT fields (i.e. of spin net vertices)
gives the identity kernel
g 1'
g1
g2
g6'
g4'
g3
g5
g5
g2
g6
=
g5'
g6'
⇤
⇤
[dgi dgi ] ⇥123 ⇥3 56 ⇥5 4 6 ⇥62 1
[dgi dgi ] ⇥123 ⇥3 56 ⇥5 4 6 ⇥62 1
g2'
g4'
g4
⇥
1
G4 G6 G5
g3 g3
1
⇥
1
⇥
= ... =
1
g4 g4
Gi = gi gi
1
⇥
g5 g5
1
⇥
g6 g6
1
⇥
⇥ijk = ⇥ (gi , gj , gk )
exactly usual tetrahedral interaction term of Boulatov GFT
g3'
g3
g 1'
1
g6
g5'
g1
⇥
[dgi dgi ] ⇥123 ⇥3 45 ⇥5 2 6 ⇥6 4 1
G3 G5 G2
G2 G6 G1
⇤
⇥
⇥
1
1
=
[dgi dgi ] ⇥123 ⇥3 45 ⇥5 2 6 ⇥6 4 1
g1 g1
g2 g2
1
g4
g3'
g2'
⇤
already shown to correspond to GFT diﬀeos - even clearer in flux variable
(A. Baratin,F. Girelli, DO, ’11)
(G2 G1 ) (G6 G5 ) (G3 G5 G4 G2 ) = ... =
g1 g1
1
⇥
g2 g2
1
⇥
g3 g3
1
⇥
g4 g4
Gi = gi gi
1
1
⇥
g5 g5
1
⇥
g6 g6
1
⇥
⇥ijk = ⇥ (gi , gj , gk )
so-called “pillow” interaction term, also considered in Boulatov GFT
L. Freidel, D. Louapre, ’02
can then compute diagrams of order 6,8,.... - GFT action will in general contain infinite number of interactions
Part II:
!
What for?
extended discussion
Relating LQG and Spin Foams via GFT
LQG can be reformulated in 2nd quantized form to give a GFT - kinematical and dynamical correspondence
insights:
•
•
•
!
direct route LQG <—> GFT (SF fully defined via perturbative expansion of GFT)
!
SF vertex is elementary matrix element of projector operator
!
E. Alesci, K. Noui, F. Sardelli, ’08
SF partition function (transition amplitude) contains more than canonical projector
equations (scalar product)
L. Freidel, ’06; T. Thiemann, A. Zipfel, ‘13
key issues:
•
!
class of diagrams to be summed over? subsector of canonical dynamics?
!
(L. Freidel, ’06)
•
choice of quantum statistics? relation to diﬀeomorphisms and to GFT symmetries?
•
criteria for restricting GFT interactions (matrix elements of canonical projector P)
•
•
(B. Bahr, T. Thiemann, ’07) (A. Baratin,F. Girelli, DO, ’11)
!
!
(V. Bonzom, R. Gurau, V. Rivasseau, ’12)
exact relation between Hilbert spaces (physical meaning of graph structures)
!
role and significance of open spin networks?
• more rigorous meaning to canonical partition function, ensembles and thermodynamic potentials
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
(superpositions of “many-vertices” states, refinement as creation of new vertices, etc)
1. making sense of quantum dynamics and LQG partition function (correlations)
2.
understanding LQG phase structure
3.
extracting eﬀective continuum dynamics
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
1.making sense of quantum dynamics and LQG partition function (correlations)
•
approximate tools for computing quantum dynamics (transition amplitudes) around appropriate
vacuum state (spin foam perturbative expansion)
•
control quantum corrections and interactions of many quantum LQG degrees of freedom,
compute eﬀective dynamics at diﬀerent scales (# LQG d.o.f.):
GFT (perturbative) renormalization
alternative: spin foam (lattice)
refinement/coarse graining (B. Bahr, B. Dittrich, ’09, ’10; B. Bahr, B.
Dittrich, F. Hellmann, W. Kaminski, ‘12)
•
!
!
(V. Bonzom, J. Ben Geloun, ’11;
A. Riello, ’13; J. Ben Geloun, ’12;
S. Carrozza, DO, V. Rivasseau,
’12, ’13; S. Carrozza, ‘14)
give non-perturbative meaning to full partition function, control the full sum over spin foams:
constructive GFT (and summability)
(L. Freidel, D. Louapre, ’03; J. Magnen, K. Noui, V. Rivasseau, M. Smerlak, ‘09)
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
AL vacuum
AL h0|ES |0iAL
AL ES << 1
|0iAL
= 0
8S
AL AS >> 1
totally degenerate geometry (emptiest state)
connection highly fluctuating
unique diﬀeo invariant
J. Lewandowski, A. Okolow, H. Sahlmannm T. Thiemann’06
C. Fleischack, ‘06
physical vacuum with non-degenerate space-time and geometry and GR as eﬀective dynamics?
LQG condensate vacuum (condensate of spin networks)
in canonical LQG context:
T. Koslowski, 0709.3465 [gr-qc]
in covariant SF/GFT context:
DO, 0710.3276 [gr-qc]
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
AL vacuum
AL h0|ES |0iAL
AL ES
<< 1
|0iAL
= 0
AL AS
KS vacuum
KS h0|ES |0iKS
KS ES
<< 1
>> 1
|0iKS
= ES
KS AS
8S
totally degenerate geometry (emptiest state)
connection highly fluctuating
diﬀeo invariant
8S
>> 1
non-degenerate geometry (triad condensate)
connection highly fluctuating
diﬀeo covariant
T. Koslowski, H. Sahlmann, 1109.4688 [gr-qc]
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
AL vacuum
AL h0|ES |0iAL
AL ES
<< 1
|0iAL
= 0
AL AS
8S
>> 1
DG vacuum (or BF vacuum)
|0iDG
DG h0|F (A)|0iDG
DG A
<< 1
DG ES
= 0
>> 1
B. Dittrich, M. Geiller, 1401.6441 [gr-qc]
KS vacuum
KS h0|ES |0iKS
KS ES
<< 1
|0iKS
= ES
KS AS
8S
>> 1
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
AL vacuum
DG vacuum (or BF vacuum)
?
KS vacuum
phase transitions
?
?
The idea of “geometrogenesis”:
continuum spacetime and geometry from GFT
•
GFT is QG analogue of QFT for atoms in condensed matter system
!
!
• continuum spacetime (with GR-like dynamics) emerges from collective behaviour of large numbers
of GFT building blocks (e.g. spin nets, simplices), possibly only in one phase of microscopic system !
•
continuum spacetime as a peculiar quantum fluid
•
•
more specific hypothesis: continuum spacetime is GFT condensate
GR-like dynamics from GFT hydrodynamics
•
even more specific suggestion: phase transition leading to spacetime and geometry (GFT
condensation) is what replaces Big Bang singularity (geometrogenesis)
cosmological evolution as relaxation towards (simple) condensate state
exact GFT condensate state to correspond to highly symmetric spacetime
•
•
!
!
!
!
!
(...., Hu ’95, Volovik ’10,...., Oriti ’07, ’11, ’13, Rivasseau ’11, ’12, Sindoni ’11)
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
(DO, L. Sindoni, 1010.5149 [gr-qc];
other simple candidates for LQG physical vacuum: GFT condensates
S. Gielen, DO, L. Sindoni, 1303.3576 [gr-qc], 1311.1238 [gr-qc]
all GFT quanta have the same (gauge invariant) “wave function”, i.e. are in the same quantum state
Bi(1) , ...., Bi(N )
N
Y
1
=
(Bi (m))
N ! m=1
•
such states can be expressed in 2nd quantized language one can consider superpositions of states of arbitrary N
continuum geometric interpretation: homogeneous (anisotropic) quantum geometries
1 to
4
4In addition
†
†
1 termines
metrics
into
the
states.
1 states.
ˆ such
1evolution
4termines
4 the
†
†of
1
4
4
†
†
⇤
:=
d
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
(17)
1
ˆ
ahedra,
specified
by
data
metrics
into
the
evolution
of
such
In
addition
to
ˆ
I
I
I
1
4
4
d
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
(18)
⇤
:=
I
†
†
⌅
:=
d
g
d
h
⌅(g
h
)
⌃
ˆ
(g
)
⌃
ˆ
(h
),
(17)
edra,
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ˆ
I
I
I
ˆ
depends
on2ond the
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we
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I
I
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)
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),
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o
depends
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we
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is
inIinvariance
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gical
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de2
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heevolution
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such
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†
†
†
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where
due
to
(1)
and
[
⇧
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),
⇧
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)] =
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function
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here
due
to
(1)
and
[
⇧
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(g
),
⇧
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(h
)]
=
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I
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ng
that
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corresponding
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ling
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⇤
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corresponding
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(8)
that
where
due
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(1)
and
[
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(g
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)]
0
the
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nvariance
(1),
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due
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and
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re
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ng
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ariance
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⌅k,
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.
.
.
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⇤(g
=
⇤(kg
k
)
for
all
k,
k
in
n
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
⇥
k,
m
=
1,
.
.
.
,
N.
(15)
I
I
s,
the
leftthe
state
only
depends
on
gauge-invariant
data.
⇥
⇥
elds,
the
leftthe
state
only
depends
on
gauge-invariant
data.
I
I
der
right
multiplication
of
all
group
elements,
can
be
taken
to
satisfy
⌅(g
)
=
⌅(kg
k
)
for
all
k,
k
in
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
I
I
be
taken
I group
I on gauge-invariant data.
ds,
the
leftthe state
depends
right
multiplication
of all
elements,
1
1 only
1
SU(2)
and
⇤(g
)
=
⇤(g
).
⇤been
is
aimfunction
1
)
invariant
quantities
g
,
we
similarly
obtain
if
we
require
⌅(g
k)
= ⌅(gon
forthe
allgaugek ⇥ SU(
U(2)
and
⇤(g
)
=
⇤(g
).
⇤
is
a
function
on
the
gaugenner
prodAssuming
that
the
simplicity
constraints
been
imI= ⌅(g have
trresponding
inner
prodAssuming
that
the
simplicity
constraints
have
corresponding
to
invariance
under
(8)
so
that
& function
& )on
!
"
I
I
SU(2)
and
⌅(g
)
).
⌅
is
a
the
gaugeI
2)
and
⇤(g
)
=
⇤(g
).
⇤
is
a
function
I
2)
and
on
the
gaugenner prodAssuming
that(8)
thesosimplicity
constraints
II to invariance
under
that
Ihave been imI and
⇥ tetrahedron.
4
4
rinsic
geometric
data
does
two
simple
choices
of
quantum
GFT
condensate
states
invariant
configuration
space
of
a
single
out
loss
of
generality
⌅(k
g& ) = ⌅(g& ) for all k
ue
up
to
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
nly
depends
on
gauge-invariant
data.
sic
geometric
data
and
does
up
to
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
variant
configuration
space
of
a
single
tetrahedron.
4
invariant
configuration
space
of
a single
tetrahedron.
two
simple
of
quantum
GFTcondensate
condensate
states
riant
space
ofdata.
a choices
single
tetrahedron.
two
simple
choices
of
quantum
GFT
states
up
toconfiguration
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
ydding
depends
on
gauge-invariant
riant
configuration
information
apart
from
(homogeneous
continuum
quantum
spacetimes)
ghat
that
the
simplicity
constraints
have
been
imWe
then
consider
two
types
ofcandidate
statesforfor
mbedded
tetrathis
additional
symmetry
under
the
action
of
SU(2).
It candidatestates
gedded
=
g(x
)(e
(x
),
e
(x
))
,
(14)
because
of
(1).
dded
tetrathis
additional
symmetry
under
the
action
SU(2).
It
ng
information
apart
from
We
then
consider
two
types
of
candidate
states
for
#
!
#
"
#
!
"
(#)
(homogeneous
continuum
quantum
spacetimes)
(homogeneous
continuum
quantum
spacetimes)
We
then
consider
two
types
of
the
simplicity
constraints
have
been
imtetrathis
additional
symmetry
under
the
action
of
SU(2).
It
e
then
consider
two
types
of
candidate
states
for
We
then
4
very
natural
notion
of
spatial
by
(6),
⇧
is
a
field
on
SU(2)
and
we
tor
fields,
can
be
imposed
on
arequire
state
created
by
4
macroscopic,
homogeneous
configurations
of
tetrahedra:
Acreated
second
possibility
is condensate
toofcondensate
use
a two-particle
single-particle
condensate
rroscopic,
fields,
can
be
imposed
on
a
one-particle
state
by
natural
notion
of
spatial
acroscopic,
homogeneous
configurations
of
tetrahedra:
two-particle
dipole
(6),
⇧
is
a
field
on
SU(2)
and
we
require
single-particle
condensate
macroscopic,
homogeneous
configurations
tetrahedra:
ry
fields,
can
be
imposed
on
a
one-particle
state
created
by
homogeneous
configurationsone-particle
two-particle
dipole
single-particle
condensate
roscopic,
of
tetrahedra:
two-particle dipole condensate
⌅
(Gross-Pitaevskii
approximation)
nal
symmetry
under
the
action
of
SU(2).
It
e
context.
⌅
(Bogoliubov
approximation)
⇥
⇤
are
the
metric
components
in
the
frame
which automatically
has
the
required gauge inv
⇥⇤ ⇤
(Gross-Pitaevskii
⌅⇥ It
context.
#)
(Gross-Pitaevskii
approximation)
al symmetry
under approximation)
the action of SU(2).
(Bogoliubov
approximation)
⇤
(Bogoliubov
approximation)
⇥
4
†
(14)
4
†)
ˆ |0⇧ .
osed
on
a
one-particle
state
created
by
(14)
⌅
ˆwill
:=
dˆ|0⌦
g|0⇧
⌅(g
ˆ(19)
(g
(17)
4⇤
†⇧
|⌅⇧
:=
exp
(ˆ
⌅
)
|0⇧
,
|⇤⇧
:=
exp
.
(19)
(14)
mpatible
with
spatial
homoI:=
I )(ˆ
|⌅⇧
exp
⌅
)
|0⇧
,
|⇤⇧
:=
exp
⇤
(18)
s
frame
a
homogeneous
metric
be
one
ˆ
⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(g
)
(17)
ˆ
ed on|⌅⌦
a
one-particle
state
created
by
I
I
:=
exp
(ˆ
⌅
)
|0⌦
,
|⇤⌦
:=
exp
⇤
.
⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(g
)
(17)
patible
with
spatial
homo|⇧⌦ I:= expI (ˆ
⇧ ) |0⌦ , |⌅⌦ := exp ⌅ |0⌦ .
(18)
|⌅⇧ ⌅
|0⇧
|⇤⇧
|0⇧
⌅
⌅with
mpatible
spatial
isotropy
nt
coe⇤cients.
We can
then say that a disned
by
pushatible
with
spatial
isotropy
4
†
1
dcorresponds
by ⌅ˆpush:=
g
⌅(g
)
⇧
ˆ
(g
)
(17)
4 d 2to
†simplest
1
4with4
†
†
I
I
ˆ
if
we
require
⌅(g
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
⇧
the
case
of
single-particle
conI
I
⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(g
)
(17)
2
⇤
:=
d
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h& ),
2)
and
g
=
a
for
some
a.
|⌅⇧
corresponds
to
the
simplest
case
of
single-particle
contry
of
N
tetrahedra,
specified
by
the
data
orresponds
to
the
simplest
case
of
single-particle
conI
I
if
we
require
⌅(g
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
with& &
&
ij a ij for
ij some a.
corresponds
elds
on
G.
I |⇧⌦ corresponds
I
and
g
=
to
the
simplest
case
of
single-particle
conij
⇥
⇥
s
on
G.
dsnsation with gauge
2k ⇥ SU(2)
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
⇥
⇥
invariance
imposed
by
hand;
|⇤⇧
I
I
with
gauge
invariance
imposedbybyhand;
hand;|⌅⌦|⇤⇧
mpatible
with
spatial
homogeneity
ifdensation
ation
with
invariance
imposed
|⇤⌦
out)number
loss
of generality
⌅(k
gby
=hand;
⌅(ggauge
for all
k ⇥ SU(2)
metric
at
a gauge
finite
number
ofSU(2);
now
reads
sation
I ) with
I )|⇤⇧
densation
invariance
imposed
tric
at
a
finite
of
re
⌅(g
k)
=
⌅(g
for
all
k
⇥
withow
reads
I = ⌅(g
Ifor
because
of
(1).
⌅(g
k)
)
all
k
⇥
SU(2);
withtomatically
has
the
right
gauge
invariance.
I
I
†
†
automatically
has
the
right
gauge
invariance.
matically
has
the
right
gauge
invariance.
because
of
(1).
⇥
⇥
ically
meaningless.
Our
interomatically
where
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h& )] = 0 the f
automatically
has
the
right
gauge
invariance.
&
lly
meaningless.
Our
intergenerality
⌅(k
⌅(g
) for
all
⇥ gI ) =
⇥ k ⇥ SU(2)
Ifor
A
second
possibility
is
to
use
a
two-particle
operator
nerality
⌅(k
g
)
=
⌅(g
)
all
k
⇥
SU(2)
,
(15)
I ⌅k,
I= GFT
Let
us
consider
generic
GFT
models
infour
four
dimen⇥
A
second
possibility
is
to
use
a
two-particle
operator
=
g
m
1,
.
.
.
,
N.
(15)
et
us
consider
generic
models
in
dimenLet
us
consider
generic
GFT
models
in
four
dimen(15)
ormation
given
by
knowing
the
!
"
(#)
!
"
($
)
et
us
can
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
) for a
(1).
Let
us
consider
generic
GFT
models
in
four
dimen&
&
mation
given
by
knowing
the
which
automatically
has
the
required
gauge
invariance:
).
ons,
whose
actions
consist
of
a kinetic
term
and
an
in-and
1
which
automatically
has
the
required
gauge
invariance:
s,
whose
actions
consist
of
a
kinetic
term
and
an
insions,
whose
actions
consist
of
a
kinetic
term
an
in-on t
mpling
of
an
underlying
contins,
whose
SU(2)
⇤(g
)
=
⇤(g
).
⇤ isand
a and
function
possibility
is
to
use
a
two-particle
operator
sions,
whose
actions
consist
of
a
kinetic
term
an
inin
the
frame
&
ing
of
an
underlying
contin&
possibility
is
to
use
a
two-particle
operator
the
frame
the
frame
raction
quintic
(but
otherwise
general)
in
the
field
⇧:
n
only
uses
intrinsic
geometric
data
and
does
ction
quintic
(but
otherwise
general)
in
the
field
⇧:
teraction
quintic
(but
otherwise
general)
in
the
field
⇧:
matically
has
the
required
gauge
invariance:
⌅
s
are
distributed
in
a
region
of
ction
quintic
invariant
configuration
space
of
a
single
tetrah
c
will
be
one
teraction
quintic
(but
otherwise
general)
in
the
field
⌃:
are
distributed
in
a
region
of
atically
has
the
required
gauge
invariance:
⌅
will
be
one
⌅1
⌅
will
be
one
4
4 from ⌅1
†
†
⌅
on
any
embedding
information
apart
ˆ
1
⌅
pect
to
a
background
metric),
We
then
consider
y
that
a
disd
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(18)two types of candidate
⇤
:=
1
4
4
†
†
1
1background
I⌅1I)⇧
I † (h (h
I ),
ˆ
ct
to
a
metric),
4
4
†
1
⇥
4
4
⇥
⇥
that
a
disd
g
d
h
⇤(g
h
ˆ
(g
)
⇧
ˆ
),
(18)
⇤
:=
ˆ
1
⌅
ˆ
I
I
I
1
44ad very
44d⇥⇥ gnatural
2, gId,⇥⇥ g)⇧(g
I⇥V
that
a
disg
d
h
⇤(g
h
)5⇧ˆ[⇧]
(g
)⇧ˆ⇥4 (h
(18)
⇤
:=
ˆ
⇥⇥ )spatial
1/3
S[⇧]
=
)⇧(g
)
+
g
⇧(g
)
K(g
⇥ configurations of te
4I4(20)
⇥ I ), homogeneous
⇥
1
I
f⇧]
G.
It
is
notion
of
⌅1by
I
ˆ
⇧]
=
+
⇥V
[⇧]
(20)
d
g
d
g
⇧(g
)
K(g
I
I
I
2
1/3
⇥
4
⇥
the
data
I
I
5
rs
up
to
N
/L.
In
this
sense
macroscopic,
=
)⇧(g
)
+
⇥V
[⇧]
(20)
d
g
d
g
⇧(g
)
K(g
,
g
)⇧(g
)+
⇥V
(19)
S[⇧]
=
d
g
d
g
⇧(g
)
K(g
,
g
I
I
ˆ
I
I
5
1
5 [⇧](19)
I
Inaturally
4N
4 /L. In this
†sense
† I
2
up
to
simplest
I (18)
2
I
I
•
)⌃(g
)
+
⇥V
[⌃]
S[⌃]
=
d
g
d
g
⌃(g
)
K(g
,
g
y
the
data
5
I
I
simplest
2
gauge
invariant
•
I
I
d
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
1
•
4
4
†
†
simplest
ytyinthe
•2data
Icontext.
Iˆ (h ),I
† 2 2 †
I)⇧
the
discrete
d
g
d
h
⇤(g
h
ˆ
(g
)
⇧
(18)
if
I
I
I
⇥ ⇤
eity
is,
at
any
N
,
an
approxiwhere
due
to
(1)
and
[
⇧
ˆ
(g
),†⇧
ˆ (h
)] =
0 the
function
⇤account some correlations
2
†
I
I⇧
I=
yififis, at any N where
, an approxitakes into
•
due
to
(1)
and
[
⇧
ˆ
(g
),
ˆ
(h
)]
0
the
function
⇤
†
†
I
I
ading
to
the
quantum
equation
of
motion
ˆ |0⇧
where
due
to (1)spatial
and
[⇧ˆ homo(g⇤(g
ˆ) (h
)] = ⇥I0k ⇥the
function
⇤⇥(ˆ
ing
to
the
quantum
equation
of
motion
I ), I⇧
I⇤(kg
geometry
compatible
with
|⌅⇧
:=
exp
⌅
)
|0⇧
,
|⇤⇧
:=
exp
⇤
can
be
taken
to
satisfy
=
)
for
all
k,
k
in
⇥
ing
to
the
quantum
equation
of
motion
rto
continuous
geometries.
† can be
† taken to satisfy
leading
quantum
equation
motion
ontinuous
geometries.
⇤(g
= the
⇤(kg
all
k, k⇥ of
in of
leading
to
the
quantum
motion
I⇤)to
I k⇥ ) forequation
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
function
†
†
N.
(16)
I
I
1
can
beI )]Consider
taken
to
satisfy
⇤(kg
) for on
allthe
k, kanisotropic
in
⌅[⇧ˆassame
(1)
(gvariable:
⇧ˆSU(2)
(h
=
0 ⇥the
function
⇤case):
geometric
variables
(in
SU(2)
for
homogeneous
geometries I ) ⇤=data
Ik
(16)
•and
addition
compatible
with
spatial
isotropy
⌅⌅(16)
I ),
1⇤(g
and
⇤(g
)
=
⇤(g
).
is
a
function
gaugeI
⇥
ˆ
ink
of
N
I
SU(2)
and
⇤(g
)
=
⇤(g
).
⇤
is
a
function
on
the
gaugeˆ
V
k
ofto
Nsatisfy
as 4variable:
Consider
en
⇤(g
) non-perturbative
=
⇤(kg
k for
)⇥2I for
all
k,
in
⌅a(infinite
ˆ
V
Iˆ=
I⇥ 1k5
4
⇥truly
⇥I kI⇥⇤(g
5
quantum
states
QG on
dofs,
superposition
ofˆgraphs)
⌅
V
•
⇥
⇥
⇥
to
satisfy
⇤(g
)
⇤(kg
)
all
k,
k
in
SU(2)
and
)
=
⇤(g
).
⇤
is
function
the
gauge5
I
ˆ
I
invariant
configuration
space
of
a
single
tetrahedron.
U(2)
or
Hom(2)
and
g
=
a
for
some
a.
4
⇥
⇥
⇥
|⌅⇧
corresponds
to
the
simplest
case of single-pa
d
g
K(g
,
)
⇧(g
ˆ
)
+
⇥
=
0
.
(21)
1
I
ˆ
V5
!
"
!
"
ˆ
I
d
g
K(g
,
g
)
⇧(g
ˆ
)
+
⇥
=
0
.
(21)
I
I
data
and
does
ose
geometry
is
approximated
I
V
4
⇥
⇥
⇥
invariant
configuration
space
of
a
single
tetrahedron.
d
g
K(g
,
g
)
⇧(g
ˆ
)
+
⇥
=
0
.
(21)
I
I
⇤(g
)
=
⇤(g
).
⇤
is
a
function
on
the
gauge1
5
ˆ
e
geometry
is
approximated
I
support
perturbations
sampling
I•
4atdany
⇥ gˆ K(g
⇥, g scale
⇥ N
I
I on
ta
and
does
⇧(g
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I). ⇤ invariant
)
⇧(g
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)+
⇥ invariance
=
0 .imposed
(20)by
I ) types
⇧(g
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)
gsaI )and
=
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the
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g
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,
g
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I
I+
⇧(g
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)
We
then
consider
two
of
candidate
states
for
I
I
densation
with
gauge
I
about
the
metric
at
a
finite
number
of
I
I
does
ngapart
apart
from
⇧(g
ˆI ) I )
We
consider
twoquantized
types of
candidate
N
leading
to
di⇥er2nd
coherent
states
states for ⌃(g
onfiguration
space
of
a then
single
tetrahedron.
• increase,
ˆ
N
increase,
leading
to
di⇥erfrom
figuration
space
ofmacroscopic,
a
single
tetrahedron.
We meaningless.
then
consider
two
types
of
candidate
states
for
homogeneous
configurations
of
tetrahedra:
automatically
has
the right gauge invariance.
general
physically
Our
intercan
be
studied
using
BEC
techniques
apart
from
nce
|⌅⇧
is
an
eigenstate
of
⇧(g
ˆ
),
when
(21)
acts
on
|⌅⇧
of
spatial
•
macroscopic,
homogeneous
configurations
of
tetrahedra:
eon
|⌅⌦
is
an
eigenstate
of
⇧(g
ˆ
),
when
(21)
acts
on
|⌅⌦
consider
types
offor
states(21)
for acts on |⌅⇧
reach
each
If
(15)
holds
for
I ),I when
|⌅⇧spatial
eigenstate
ofcandidate
⇧(g
ˆallall
of
Nis.Nan
If.two
(15)
holds
QuantumGFT
GFTcondensates
condensates
Quantum
(14)
(14)
obtained by
by pushpushobtained
ctor
fields on
on G.
G.
tor fields
metric now
now reads
reads
metric
⌅
ˆ :=
⌅
d 4g ⌅(gI )⇧ˆ† (gI )
(17)
iflimit
we require
require
⌅(gI k) (at
= ⌅(g
withwe
I ) for all k ⇥ SU(2);
Continuumifout
of
LQG
dynamical
level)
out loss
loss of
of generality ⌅(k ⇥ g ) = ⌅(g ) for all k ⇥ ⇥ SU(2)
I
I
because of
of (1).
because
(i.e. GFT reformulation
LQG) useful toisaddress
of large numbers
of LQG d.o.f.s,
A second
secondof possibility
operator
A
to usephysics
a two-particle
(xmmQFT
)),, methods(15)
(15)
(x
))
i.e. many and refined graphs (continuum limit)
which automatically
automatically has the required gauge invariance:
which
ion 2.
ofin
gijthe
under
the
adjoint
action of
Cosmological dynamics. — The GFT dynamics de
nents
frame
nents
in
the
frame
understanding
LQG
phase structure
ion of gphysically
adjoint
actioninto
of⌅
Cosmological
dynamics.
— The
GFTIndynamics
de
3 such
ij under the
nsforms
distinct
metrics
termines
the
evolution
of
states.
addition
t
metric
metric will
will be
be one
one
1on 4the
nsforms
physically
distinct
metrics
into
termines
the
states.
In
addition
t
1evolution
4gauge
† geometric
†ofwesuch
what
is
the
LQG
continuum
phase
structure?
what
is
the
physical,
LQG
phase?
ˆ
ˆ
notion
of
homogeneity
also
depends
invariance
(1),
require
that
the
state
is
in
en say
that
aa disd g d h ⇤(gI hI )⇧ˆ (gI )⇧ˆ (hI ),
⇤⇤ :=
hen
say
that
dis(18)
:=
notion
of of
homogeneity
also depends
theThe
gauge
invariance
(1),
that1010.5149
the state
is in
nt
action
Cosmological
dynamics.
GFT
dynamics
de-we require
2on —
(DO,
L.
Sindoni,
[gr-qc];
variant
under
right
multiplication
of
all
group
elements
cified by
ecified
by the
the data
data
variant
under
multiplication
of allDO,
group
elements
metrics
into issues
termines
the evolution
of such
states.
In†right
addition
toto invariance
S.
Gielen,
L.
Sindoni,
tha
†
th
of
those
by
recalling
that
the
g
⇤
g
h,
corresponding
under
(8)
so
other
simple candidates
for
LQG
physical
vacuum:
GFT
condensates
I
I
geneity
if
ogeneity
if
where
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
function
⇤
where
due the
I
th
of those
recalling
that
gIthat
h,I corresponding
under
(8) so
tha
depends
on issues
theby
gauge
invariance
(1), we grequire
the state
is in-to invariance
I ⇤
1303.3576 [gr-qc],
1311.1238
[gr-qc]
a natural basis of vector can
fields,
the
left-to satisfy
the state
only
⇥on gauge-invariant
be
taken
⇤(g
=depends
⇤(kg
can
be
taken
) for
all k, k ⇥⇥ in data.
I )group
a natural basis of
vectorunder
fields,
the
leftthe state
only
depends
gauge-invariant
data.
I k on
variant
right
multiplication
of
all
elements,
.
.
.
,
N.
(16)
, . . .all
, N.
1 function”,
fields.
Fixing
aa (16)
G-invariant
inner
prodAssuming
that
the
simplicity
constraints
have
been
im
GFT
quanta
(spin
neth,
vertices)
have
theIIsame
“wave
i.e.
are
in the
same
quantum
state
SU(2)
and
⇤(g
)
=
⇤(g
).
⇤
is
a
function
on
the
gaugeSU(2)
and
fields.
Fixing
inner
prodAssuming
that
the
simplicity
constraints
have
been
im
ing that
the
gG-invariant
⇤
g
corresponding
to
invariance
under
(8)
so
that
I
I
I
4
gebra
ggleftthis
basis
is
unique
up
to
the
plemented
by
(6),
⇧
is
aatetrahedron.
field
on
SU(2)
and
we
requir
4condensate
two
simple
choices
of
quantum
GFT
stat
invariant
configuration
space
of
a
single
gebra
this
basis
is
unique
up
to
the
plemented
by
(6),
⇧
is
field
on
SU(2)
and
we
requir
lds,
the
the
state
only
depends
on
gauge-invariant
data.
invariant
configuration
N
etric
data
and
does
Y
etric
data
and does
1
We
now
demand
that
the
embedded
tetrathis
additional
symmetry
under
the
action
of
SU(2).
(homogeneous
continuum
quantum
spacetimes)
such
states
can
be
expressed
in 2nd
quantized
language
II
inner
prodAssuming
that
the
simplicity
constraints
have
been
imWe
now
demand
that
the
embedded
tetrathis
additional
symmetry
under
the
action
of
SU(2).
We
then
consider
two
types
of
candidate
We
then
states
for
B
,
....,
B
=
(B
(m))
i
i(1)
i(N )
mation
apart
from
mation
apart
from
4 consider
N
!
one
can
superpositions
of
states
of arbitrary
N
left-invariant
vector
fields,
can
be
imposed
on
a
one-particle
state
created
by
uewith
up the
to
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
with
the
left-invariant
vector
fields,
can
be
imposed
on
a
one-particle
state
created
by
macroscopic,
homogeneous
configurations
of
tetrahedra:
m=1
single-particle
condensate
macroscopic,
two-particle
ll notion
notion of
of spatial
spatial
⌅
⌅⇥ It
(Gross-Pitaevskii
bedded tetrathis additional symmetry
under approximation)
the action of SU(2).
(Bogoliubo
⇤
4
†
vvi(m)
= e (xcan
), be imposed on|⌅⌦
4 g ⌅(g )⇧
† (g )
(14)
ˆ
⌅
ˆ
:=
d
ˆ
(17
m ),condensate
i(m) = eii (xm
or fields,
a(14)
one-particle
by
I
II )
:= exp (ˆ
⌅ ) state
|0⌦
, created
|⇤⌦
:=
|0⌦
single-particle
⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(17
|⌅⇧
|0⇧
|⇤⇧
exp
⇤
|0⇧
.
(19)
⇥(g
ˆ
)|
=
(g
I | (g
I
I
ith
spatial
homowith spatial homo⌅
he
fields
on M obtained by push- 4
†
(14)
h vector
spatial
isotropy
th
spatial
isotropy
\
⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(gI ) case
(17)
|⌅⌦
corresponds
to
the
I simplest
if
we
require
⌅(g
k)
=
⌅(g
)
for
all
k ⇥
with
h
|
'(g
i = (g
|⌅⇧
corresponds
of
single-particle
conI
I
I )|SU(2);
I)
22 left-invariant vector fields on G.
s
of
⇥ by hand; |⇤⌦
⇥
= aa ijij for
for some
some a.
a.
densation
with
gauge
invariance
imposed
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
k
⇥ SU(2
densation
|⇤⇧
I
I
ned
byofpush(13)
the
physical
metric
now
reads
of
newnumber
IRREP of
observables
algebra,
inequivalent
wrt
Fock
vacuum
(AL withvacuum)
if
we
require
⌅(g
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
aldsfinite
finite
number
of
I
I
automatically
has
the
right
gauge
invariance.
because
of
(1).
automatically
onsymmetry
G.
⇥ BF vacuum)
breaking
U(1)
symmetry &
BF⇥ gdiﬀeos/translations
(not
(A. Baratin,F. Girelli, DO, ’11)
ngless.
Our
interout
loss
of
generality
⌅(k
)
=
⌅(g
)
for
all
k
⇥
SU(2)
ingless.
Our
interI
I
Let
us
consider
generic
GFT
models
in
four
dimenA
second
possibility
is
to
use
a
two-particle operato
=
g(x
)(e
(x
),
e
(x
))
,
(15)
reads
usfunction - more general than constant triad field
mparameter:
m condensate
m
ii m
jj m Let
))now
order
wave
of (1).
ven
the
venby
by knowing
knowingbecause
the
which
automatically
hasterm
the required
gauge invariance:
sions, whose actions
consist
of a kinetic
and an in-
Quantum GFT condensates
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
2. understanding LQG phase structure
what is the LQG continuum phase structure? what is the physical, geometric LQG phase?
AL vacuum
DG vacuum (or BF vacuum)
?
KS vacuum
phase transitions
GFT condensate
?
issue is to prove dynamically the choice of vacuum and the phase transitions
V. Bonzom, R. Gurau, A. Riello, V. Rivasseau, ’11;
experience and results in tensor models and GFTs
A. Baratin, S. Carrozza, DO, J. Ryan, M. Smerlak, ‘13
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
3.extracting eﬀective continuum dynamics
general strategy: change vacuum
• obtain eﬀective GFT or spin foam amplitudes around new vacuum
•
write approximate SD equations in new vacuum
•
approximate techniques, e.g. mean field theory
applied in simple models for:
•
conditions on non-perturbative vacuum (DO, L. Sindoni, 1010.5149 [gr-qc])
•
eﬀective spin foam dynamics (DO, L. Sindoni, 1010.5149 [gr-qc]; E. Livine, DO, J. Ryan, 1104.5509 [gr-qc])
• eﬀective dynamics of simple fluctuations around new vacuum (W. Fairbairn, E. Livine, gr-qc/0702125)
most recently: cosmology from full QG (via GFT formalism) ——>
(Quantum) Cosmology from GFT
S. Gielen, DO, L. Sindoni, arXiv:1303.3576 [gr-qc], arXiv:1311.1238 [gr-qc]
problem 1: identify quantum states in fundamental theory with continuum spacetime interpretation
many results in LQG (weaves, coherent states, statistical geometry, approximate symmetric states,....)
Quantum GFT condensates are continuum homogeneous spacetimes
described by single collective wave function (depending on homogeneous anisotropic geometric data)
similar constructions in LQG (Alesci, Cianfrani) and LQC (Bojowald, Wilson-Ewing, .....)
problem 2:
extract from fundamental theory an eﬀective macroscopic dynamics for such states
following procedures of standard BEC
QG (GFT) analogue of Gross-Pitaevskii hydrodynamic equation in BECs is
non-linear and non-local extension of quantum cosmology equation for collective wave function
similar equations obtained in non-linear extension of LQC (Bojowald et al. ’12)
Quantum GFT condensates
Quantum GFT condensates
2. . , N.
pends
on
the gauge (15)
invariance (1),
we require
I
I that the state is in⇥
⇥
⇥
⇥satisfy
m
=
1,
.
⇥
⇥
atural
basis
of
vector
fields,
the
leftthe
state
only
depends
on
gauge-invariant
data.
can
be
taken
to
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
taken
to
satisfy
(
g
)
=
(
kg
k
)
for
all
k,
k
in
I
I
he
leftthe
state
only
depends
gauge-invariant
data.
I group
I on
can
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
ht
multiplication
of
all
elements,
tural
basis
of
vector
fields,
the
leftthe
state
only
depends
on
gauge-invariant
data.
I
I
variant
under
right
multiplication
of
all
group
elements,
†
†
†= 0 the† function ⇤
1
1[)]
,rnd
N.
(16)
1
nd
[
⇧
ˆ
(g
),
⇧
ˆ
(h
eponding
due
to
(1)
and
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
function
⇤
ds.
Fixing
a
G-invariant
inner
prodAssuming
that
the
constraints
have
been
im- w
SU(2)
and
⇤(g
)
=
⇤(g
).the
⇤been
issimplicity
aimfunction
onforthe
gaugeI
I
riant
quantities
g
,
we
similarly
obtain
if
we
require
⌅(g
k)
=
⌅(g
)
all
k
⇥
SU(2);
I
I
(
g
)
=
(
g
).
is
a
function
on
the
gaugeI ). ⇤
Assuming
that
the
simplicity
constraints
have
under
(8)
so
that
&
&
!
"
SU(2)
and
⇤(g
)
=
⇤(g
is
a
function
on
the
gaugeI
s. prodFixing
inner
prodAssuming
that
simplicity
constraints
have
been
imthat
theI toainvariance
gG-invariant
⇤
g
h,
corresponding
to
invariance
under
(8)
so
that
I
I
I
I
I
⇥
⇥⇥
4
⇥4
⇥ontetrahedron.
⇥
geometric
data
and
does
tisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
bra
g
this
basis
is
unique
up
to
the
plemented
by
(6),
⇧
is
field
SU(2)
and
we
require
e
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
4
two
simple
choices
of
quantum
GFT
condensate
states
invariant
configuration
space
of
a
single
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
k
⇥ SU
I
I
I
I
to
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
pends
on
gauge-invariant
data.
nt
configuration
space
of
a
single
tetrahedron.
two
simple
choices
of
quantum
GFT
condensate
states
&
&
ra
g
this
basis
is
unique
up
to
the
plemented
by
(6),
⇧
is
a
field
on
SU(2)
and
we
require
the
leftthe
state
only
depends
on
gauge-invariant
data.
invariant
configuration
space
of
a
single
tetrahedron.
c⇤(g
data
1 and does
1(homogeneous
information
apart
from
continuum
quantum
demand
that
the
embedded
tetrathis
additional
symmetry
theaction
action
ofSU(2).
SU(2).ItIt
).
⇤
is
a
function
on
gauge)now
and
⇤(g
)
=
⇤(g
).
⇤that
is,symmetry
athe
function
on
the
gaugethe
simplicity
constraints
have
been
imWe
then
consider
types
ofunder
candidate
states
for
d
tetrathis
additional
under
the
action
of
SU(2).
Itunder
=
g(x
)(e
(x
),
e
(x
))
(14)
because
ofspacetimes)
(1).
(homogeneous
continuum
quantum
spacetimes)
I
hen
consider
two
types
of
candidate
states
for two
!
# the
#
er prodAssuming
the
simplicity
constraints
have
been
imI #
I"
ow
demand
that
embedded
tetrathis
additional
symmetry
the
of
We
then
consider
two
types
of
candidate
states
for
ion
apart
from
4
natural
notion
of
spatial
4 created
,
⇧
is
a
field
on
SU(2)
and
we
ith
the
left-invariant
vector
fields,
can
be
imposed
on
aone-particle
one-particle
state
created
tion
space
of
a
single
tetrahedron.
ant
configuration
space
of
asingle-particle
single
tetrahedron.
elds,
can
be
imposed
on
arequire
state
by
macroscopic,
homogeneous
of tetrahedra:
A
second
possibility
is tocondensate
use
acreated
two-particle
oper
condensate
copic,
homogeneous
configurations
of
tetrahedra:
ph single-particle
to
plemented
by
(6),
⇧
isone-particle
ahomogeneous
field
on condensate
SU(2)
andtwo-particle
we
require
dipole
thethe
left-invariant
vector
fields,
can
be
imposed
on
aconfigurations
state
byby
macroscopic,
configurations
of
tetrahedra:
two-particle
dipo
otion
of
spatial
⌅
⌅
(Gross-Pitaevskii
approximation)
ymmetry
under
the
action
of
SU(2).
It
text.
(Bogoliubov
⌅⇥ It
then
consider
two
types
of
candidate
states
forof SU(2).
er
twometric
types
ofadditional
candidate
states
for
components
in
the
frame
(Gross-Pitaevskii
which
automatically
hasapproximation)
the
required gauge
invarian
⇥ ⇤
edthe
tetrathis
symmetry
under approximation)
the
action
(Bogoliubov
ap
⇤
follow
closely
procedure
used
in
real
BECs
4
†
(14)
4
†
v
=
e
(x
),
(14)
4
†ˆused
ni(m)
aaone-particle
state
created
by
i spatial
vible
(x
:=
dˆone
g|0of
⌅(g
)state
⇧
ˆexp
(g
)(ˆ
|i(m)
:=
= eexp
(m),
ˆ )be
|0 imposed
, metric
|configurations
=⌅ˆwill
exp
.⌅tetrahedra:
(19)
ˆ:=
iconfigurations
m
with
homo⌅ˆby
:=
dgprocedure
gexp
⌅(g
(g
(17)
closely
BECs (17)
I:=
Icreated
|⌅⇧
⌅ ) follow
|0⇧
,:=
|⇤⇧⇤(17)
⇤ˆ)ˆ⇧
|0⇧
fields,
can
on
a(14)
one-particle
oscopic,
homogeneous
me
homogeneous
be
geneous
of: tetrahedra:
⌅
ˆ
d
⌅(g
)
(g
I⇧
|⌅⇧
:=
exp
(ˆ
)
|0⇧
,
|⇤⇧
:=
exp
|0⇧
.
(19)
I
I )I.)in real(18)
single-particle
GFT
condensate:
dipole
GFT
condensate:
⌅
spatial
homo⌅ a dis⌅
⇥ ⇤
ble
with
spatial
isotropy
⇤cients.
We
can
then
say
that
⇥
⇤
y
push4
†
vector
fields
onwe
M
obtained
byof
push1
ector
fields
on
M
obtained
by
push4 ˆ ) for †all k ˆ⇥ SU(2);
(14)
⌅
ˆspatial
:= |⌅⌦
d:=
g ⌅(g
)
⇧
ˆ
(g
)
(17)
1
4with4
†
†
2to
isotropy
I
I
ˆ
if
require
⌅(g
k)
=
⌅(g
esponds
the
simplest
case
single-particle
con⌅
ˆ
:=
d
g
⌅(g
)
⇧
ˆ
(g
)
(17)
exp
(ˆ
⌅
)
|0⌦
,
|⇤⌦
:=
exp
⇤
|0⌦
.
(19)
I
I
I
I
⇤
:=
d
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h&with),withdnfleft-invariant
g⌅
=
a
for
some
a.
(ˆ
)
|0⇧
,
|⇤⇧
:=
exp
⇤
|0⇧
.
(18)
|⌅⇧
corresponds
to
the
simplest
case
of
single-particle
conif
we
require
⌅(g
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
if
we
require
⌅(g
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
N
tetrahedra,
specified
by
the
data
|⌅⇧
corresponds
to
the
simplest
case
of
single-particle
con& &
&
ij
ij vector
II
II
G.
left-invariant
vector
fields
on
G.
fields
on
G.
⇥
⇥
2k ⇥ SU(2)
⇥ ⇥
⇥ ⇥
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
for
some
a.
on
with
gauge
invariance
imposed
by
hand;
|
I
I
ij
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
k
⇥⇥SU(2)
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
k|⇤⇧
SU(2)
densation
with
gauge
invariance
imposed
by
hand;
le
with
spatial
homogeneity
if
densation
with
gauge
invariance
imposed
by
hand;
|⇤⇧
cby
at
a
finite
number
of
reads
I
I
I
I
push)nite
of
the
physical
metric
now
reads
3)
of
the
physical
metric
now
reads
k)
=
⌅(g
)
for
all
k
⇥
SU(2);
withI
Ito
because
of
(1).⌅(g
rresponds
the
simplest
case
of=single-particle
conifcase
we
require
k)
⌅(g
)the
forright
all of
kof
⇥(1).
SU(2);
with- invariance.
number
ofOur
tically
has
the
right
gauge
Ibecause
†
†
(1).
the
simplest
of
single-particle
conbecause
automatically
has
the
right
gauge
⇥
⇥Iinvariance.
automatically
has
gauge
invariance.
on
G.
meaningless.
interwhere
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h& )]
= 0 (real
the functi
&
ality
⌅(k
g
)
=
⌅(g
)
for
all
k
⇥
SU(2)
⇥
⇥
I
I
microscopic
quantum
GFT
dynamics
obtained
(first
approximation)
from
GFT
action
fields)
A
second
possibility
is
to
use
a
two-particle
operator
out
loss
of
generality
⌅(k
g
)
=
⌅(g
)
for
all
k
⇥
SU(2)
(15)
tion
with
gauge
invariance
imposed
by
hand;
|⇤⌦
less.
Our
interI
I
us
consider
generic
G
F
T
models
in
four
dimen⇥
A
second
possibility
is
to
use
a
two-particle
operator
g(x
)(e
(x
))
,. , N.
(15)
=
g!
⌅k,
m
=
1,
. ))
.the
quantum
GFT
dynamics
obtained
(first
approximation)
from
GFT
(real
fields)
invariance
byus
hand;
|⇤⇧
Acan
second
possibility
ismodels
to
use
a) action
two-particle
operator
wuge
reads
Let
us consider
generic
GFT
in
four
dimenion
given
by
knowing
Let
consider
generic
GFT
models
in
four
dimenm
mm
=
g(x
)(e
(xmm),
),eejimposed
(15)(15)
"
($
) i (x
be
taken
to
satisfy
⇤(g
=
⇤(kg
k
)
for
all k, k
m
imicroscopic
j (x
&
&
which
automatically
has
the
required
gauge
invariance:
because
of
(1).
matically
hasthe
the
right
invariance.
by
knowing
whose
actions
consist
ofgauge
a whose
kinetic
term
and
an
in1required
with
extra
approximations
required
for
consistent
continuum
geometric
which
automatically
has
the
required
gauge
invariance:
which
automatically
has
the
gauge
invariance:
right
gauge
invariance.
sions,
whose
actions
consist
of
a
kinetic
term
and
an
in-on the ga
gethe
of
an
underlying
continsions,
actions
consist
of
a
kinetic
term
and
an
inSU(2)
and
⇤(g
)
=
⇤(g
).
⇤
is
a
function
bility
is
to
use
a
two-particle
operator
frame
&
&
more
precisely,
from
truncation
of
SD
equations
for
GFT
model
A
second
possibility
is
to
use
a
two-particle
operator
(15)
metric
components
in
the
frame
interpretation:
GFT
quanta
“small
enough”
and “flat enough”:
us
consider
generic
GFT
models
in
four
dimenderlying
continthe
metric
components
in
the
frame
n
quintic
(but
otherwise
general)
in
the
field
:
yhe
uses
intrinsic
geometric
data
and
does
generic
GFT
models
in
four
dimenteraction
quintic
(but
otherwise
general)
inof⇧:
the
field ⇧:
ally
has
the
required
gauge
invariance:
⌅
distributed
in
a
region
of
teraction
quintic
(but
otherwise
general) in
the
field
invariant
configuration
space
a single
tetrahedron
be
one
which
automatically
has
the
required
gauge
invariance:
⌅
˜
a
homogeneous
metric
will
be
one
1 of
⌅
whose
actions
consist
a
kinetic
term
an
inV
ed
in
a
region
of
e
a
homogeneous
metric
will
be
one
1and
4
4
†
†
ny
embedding
information
apart
from
ˆ
1
⌅
ns
consist
of
a
kinetic
term
and
an
in⌅
a
background
metric),
We
then
consider
two
types
of
candidate
states
he
frame
to
a
disd
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
(18)
⇤
:=
1
˜
4
4
†
†
1 iId] K(g
1
I I
I [dg
ˆ ˆ:=
,
g
)
⇤(g
ˆ
)
+
⇥
=
0
1
4
4
†
†
⇥
4
4
⇥
⇥
ients.
We
can
then
say
that
a
disg
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
(18)
⇤
i
S.
Gielen,
DO,
L.
Sindoni,
1
ˆ
1
I )⇧
I ),
i⇥ I I h
2gin⌅I that
tion
quintic
(but
general)
I i⇥ )⇧
⇤cients.
We
then
a
disground
g
d
ˆ
(g
ˆ
(h
(18)
⇤
:=
1/3
=but
)
(
g
)
d metric),
g d can
gIn
(gotherwise
Ksay
(
,
g
⇥ h ⇤(g
4+ in4Vthe
⇥ [ ]field
⇥d ˆ
4 (20)
4⇧:
⇥
Itbedata
isotherwise
a
very
natural
notion
of
spatial
I
I(19)
I )sense
5
ˆ
I
I
general)
the
field
⇧:
2
I
l4to
one
⇤(g
ˆ
)
he
N
/L.
this
macroscopic,
homogeneous
configurations
of
tetrahe
S[⇧]
=
)⇧(g
)
+
⇥V
[⇧]
(20)
d
g
d
g
⇧(g
)
K(g
,
g
)⇧(g
)
+
⇥V
[⇧]
S[⇧]
=
d
g
d
g
⇧(g
)
K(g
,
g
i
I
5 I
1
5
4
†
†
simplest
II I
2I II
2tetrahedra,
•
NN
tetrahedra,
specified
by
the
data
1303.3576
[gr-qc],
1311.1238
[gr-qc]
1
⌅
g
d
h
⇤(g
h
)
⇧
ˆ
(g
)
⇧
ˆ
(h
),
(18)
1
4
4
†
†
simplest
I
I
I
2
2
•
ˆ
specified
by
the
data
L.
thisNsense
† I h †)⇧
I
e, at
discrete
context.
at
aIn
disg d [h⇧
⇤(g
ˆ I(g)]I )=⇧ˆ 0(hthe
(18)
⇤ :=to (1)d and
⇥function
⇤ functions
I ), function
any
,
an
approxiwhere
due
ˆ
(g
ˆ
(h
1
† + ⇤SD equations
†
I⇧
I ),
with
spatial
if
⇥
4 homogeneity
4 dynamics
⇥
⇥
effective
for
dipole
condensate
extracted
from
this
for
n-point
where
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
⇤
2
ˆ
†
†
I
I
]tinuous
=
)⇧(g
)
+
⇥V
[⇧]
(20)
d
g
d
g
⇧(g
)
K(g
,
g
⇥
⇥
to
quantum
equation
of
motion
with
spatial
homogeneity
if
ˆ
N
an
approxi⇥
4
⇥, the
⇥
I
I
5
where
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
function
⇤
the
data
I
I
metry
compatible
with
spatial
homoˆ
|⌅⇧
:=
exp
(ˆ
⌅
)
,
|⇤⇧
exp
⇤
|0⇧
.
I|0⇧
I :=
Z
can
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
)
for
all
k,
k
in
geometries.
⇥
⇥
leading
to
the
quantum
equation
of
motion
I
I
)⇧(g
)
+
⇥V
[⇧]
(19)
g
⇧(g
)
K(g
,
g
†
†
leading
to
the
quantum
equation
of
motion
5
I
I
2
˜⇥ for all k, k ⇥in
taken to satisfy ⇤(gI ) = ⇤(kgI k )V
when
GFTfunction
condensate
state,
and
[⇧ˆ applied
(gI ), ⇧ˆto(hI(coherent)
(16)
I )] =I0 the
†1 ⇤ can
† be
eometries.
0
0
0
can
be
taken
to
satisfy
⇤(g
)
=
⇤(kg
k
all k,
k0 in
where
due
to
(1)
and
[
⇧
ˆ
(g
),
⇧
ˆ
(h
)]
=
0
the
function
⇤
compatible
spatial
isotropy
and
⇤(g
⇤(g
).I ⇤ is asystem
function
on
the gaugegftion
⌅k,
mSU(2)
=equation
1,Consider
.with
. . , N.
(16)
I )+
I ) for
˜
I
1 ) (g
I) = ⌅
ij(k)
⇥for
⇥
ˆ
N
as
variable:
of
equations
I
[dg
]
K(g
,
g
|
=
it
gives
“wave
function”:
and ⇥ ⇤(gIi) = ⇤(g
the gaugei I⇥ i).
satisfy
⇤(g
==⇤(kg
k. ,)⇥N.
for allofk,V(16)
k5 in SU(2)
⌅= ⇤(kg
gij(k)
⌅k,
1,
1 ⇤ isiˆa function on '⌘
Iˆ)m
I.equation
ˆ
4Consider
⇥ quantum
⇥ .taken
2
ng
to
the
motion
V
can
be
to
satisfy
⇤(g
)
k
)
for
all
k,
k
in
iable:
'(g
and
⇤(g⇥I ) = ⇤(g
).simplest
⇤V
is5 a function
the gaugeinvariant
of
5 Ithe
or
Hom(2)
gfunction
=
for some
corresponds
to
case
ofi )on
single-particle
dequation
(isg Ia, of
)motion
ˆ (configuration
gaI )on
+"the
=Iˆinvariant
0SU(2)
. ,ag4|⌅⇧
(21)
I
1g isKand
4space
⇥ a.
⇥single
⇥ tetrahedron.
!
"
!
I
nd
does
eometry
approximated
⇥
⇥
ntum
=
⇤(g
).
⇤
gaugeconfiguration
space
of
a
single
tetrahedron.
d
g
K(g
)
⇧(g
ˆ
)
+
⇥
=
0
.
(21)
(16)
ˆ II ,on
Iad function
ˆ (two
g I1)). types
I
Ig candidate
K(g
gIwith
)the
⇧(g
ˆstates
)+
⇥ invariance
=
0 .imposed
(20)by hand;
ses
intrinsic
geometric
data
and
does
SU(2)
and
⇤(g
)
=
⇤(g
⇤
is
gauge⌅
Igauge
s
approximated
We
then
consider
of
for
I
invariant
configuration
space
of
a
single
tetrahedron.
densation
⇧(g
ˆ
)
I
⇥(g
ˆ
)|
=
(g
)
|
for
odd-order
GFT
interactions,
eqn
from
kinetic
term
decouples
separate
equations
out
the
metric
at
a
finite
number
of
since:
I
rt from
ˆ5 I We then consider two
Itetrahedron.
uses
intrinsic
geometric
data
and
does
⇧(g
ˆ
)
ncrease,
leading
to
di⇥erration
space
of
a
single
types
of
candidate states for
I
V
embedding
information
apart
from
4
⇥
⇥
⇥
invariant
configuration
space
of
a
single
tetrahedron.
ˆ meaningless.
macroscopic,
homogeneous
configurations
of tetrahedra:
ˆ)⇧(g
eading
to
differWe
then
consider
types
of candidate
states for
automatically
hastwo
the configurations
right gauge
invariance.
d
g
K(g
,
g
ˆ
)
+
⇥
=
0
.
(21)
and
ral
physically
Our
interisdoes
an
eigenstate
of
ˆ
(
g
),
when
(21)
acts
on
|
I
spatial
V
I
I
I
der
two
types
of
candidate
states
for
embedding
information
apart
from
5
h
N
.
If
(15)
holds
for
all
macroscopic,
homogeneous
of
tetrahedra:
⇥
⇥
Since
|⌅⇧
is two
an
eigenstate
of ⇧(g
ˆ⇥ ⇤Ihomogeneous
),constraint-like
when
(21)
acts
on
|⌅⇧
⇧(g
ˆ1(20)
) ismacroscopic,
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very
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notion
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⇥+
⇥consider
⇥Since
Hamiltonian
eqn
for
collective
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We
then
of
candidate
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for
Itypes
(g
g
)
⇧(g
ˆ
)
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=
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.
˜
sday,
7,
2013
|⌅⇧
an
eigenstate
of
⇧(g
ˆ
),
when
(20)
acts
on
|⌅⇧
I
I
art
from
16)
holds
for
all
configurations
of
tetrahedra:
Let
us
consider
generic
GFT
models
in
four
dim
I
[dg
]
K(g
,
g
)
(g
g
˜
)
=
0
w
the
information
given
by
knowing
the
mes
a
non-linear
equation
for
:
⇥
⇤
mogeneous
configurations
ofextension
tetrahedra:
i
t
is
a
very
natural
notion
of
spatial
Thursday,
March
7,
2013
non-linear
and
non-local
of
quantum
cosmology-like
equation
for
“collective
wave
function
N
,
the
spatial
geometry
⇧(g
ˆ
)
i
i
i
i
ˆ
I
discrete
context.
|⌅⇧
:=
expit(ˆ
⌅a
) |0⇧
, configurations
|⇤⇧
:=
exp ⇤of tetrahedra:
|0⇧
. ⌅: (19)
+
non-linear
equations
coming
from higher-order
correlato
it⇥becomes
non-linear
equation
foractions
macroscopic,
homogeneous
ˆ
⇥
⇤
fdiscrete
spatial
becomes
a
non-linear
equation
for
⌅:
⇤
sions,
whose
consist
of
a
kinetic
term
and an
geometry
homo|⌅⇧
:=
exp
(ˆ
⌅
)
|0⇧
,
|⇤⇧
:=
exp
⇤
|0⇧
.
(19)
spatial
as
a
sampling
of
an
underlying
contin|⌅⌦
is
an
eigenstate
of
⇧(g
ˆ
),
when
(21)
acts
on
|⌅⌦
!
I
context.
uracy.
⇥
⇤
⌅
try4(ˆ
compatible
with
spatial
V.5homoˆ |0⌦ .
ˆ(20)
⇥) |0⇧
⇥ ),
⇥
|⌅⌦
:=
exp
(ˆ
⌅
)
|0⌦
,
|⇤⌦
:=
exp
⇤
(19)
xp
⌅
,
|⇤⇧
:=
exp
⇤
|0⇧
(19)
ˆ
⌅
nstate
of
⇧(g
ˆ
when
acts
on
|⌅⇧
sotropy
teraction
quintic
(but
otherwise
general)
in
the
field
V
the
points
are
distributed
in
a
region
of
QG
(GFT)
analogue
of
Gross-Pitaevskii
hydrodynamic
equation
in
BECs
ˆ
I
d
g
K
(
g
,
g
)
(
g
)
+
=
0
.
(22)
comes
a
non-linear
equation
for
⌅:
5
I
4
⇥
⇥
⇥
etry
compatible
with
spatial
homo|⌅⇧
:=
exp
(ˆ
⌅
)
|0⇧
,
|⇤⇧
:=
exp
⇤
|0⇧
.
(19)
|⌅⇧
corresponds
to
the
simplest
case
of
single-particle
conI
I
ˆ
can
identify awith
quantum
V=5 0 .case of(22)
n
compatible
spatial isotropy
d
g
K(g
,
g
)⌅(g
)
+
⇥
(
g
)
⇥
=
4
⇥
⇥
⇥
I
l
homoI
|⌅⇧
corresponds
to
the
simplest
I
I
ˆ
ome
a.
⌅with
neararespect
equation
⌅:
tify
quantum
d g K(g , g )⌅(g⌅ ) + ⇥
=single-particle
0.
(21) contofor
a background
metric),
Effective
cosmological
dynamics
from
GFT
Effective
cosmological
dynamics
from
GFT
Effective cosmological dynamics from GFT
derivation of (quantum) cosmological equations from GFT quantum dynamics very general
it rests on:
!
•
•
•
continuum homogeneous spacetime ~ GFT condensate
good encoding of discrete geometry in GFT states
2nd quantized GFT formalism
•
•
•
general features:
quantum cosmology-like equations emerging as hydrodynamics for GFT condensate
non-linear
non-local (on “mini-superspace”)
derivation of (quantum) cosmology from fundamental QG formalism!
exact form of equations depends on specific model considered
if GFT dynamics involves Laplacian kinetic term, then FRW equation is contained in eﬀective
cosmological dynamics for GFT condensate, with QG corrections
S. Gielen, DO, L. Sindoni, arXiv:1303.3576 [gr-qc], arXiv:1311.1238 [gr-qc]
Continuum limit of LQG (at dynamical level)
QFT methods (i.e. GFT reformulation of LQG) useful to address physics of large numbers of LQG d.o.f.s,
i.e. many and refined graphs (continuum limit)
1. making sense of quantum dynamics and LQG partition function (correlations)
2.
understanding LQG phase structure
3.
extracting eﬀective continuum dynamics
beside the formal role in linking canonical LQG and covariant Spin Foam models
(and in giving a complete definition of the latter)
GFT (2nd quantized LQG formalism) key for further developments!
Thank you for your attention!