Download Coarse graining and renormalization: the bottom up approach

Coarse graining and
renormalization:
the bottom
up approach
Bianca Dittrich,
Perimeter Institute, Canada
Asymptotic Safety Seminar,
April 2014
Raussendorf
[Picture by R. Raussendorf]
Copyright: The M. C. Escher Company B. V.
1
arXiv:1209.4539
[gr-qc]
B. Dittrich, J. Hnybida,
arXiv:1312.5646 [gr-qc]
Literature
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Bahr, B. Dittrich,
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“Quantum
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044048 (18pp)
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M. (2013)
Martin-Benito,
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[gr-qc]
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[45] “On the path integral measure for 4D discrete gravity,”
Conceptual.
[32] “On
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B. Dittrich,
W. of
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S. Steinhaus,
[41] “Time
evolution
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Geiller,
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theories from
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the discrete
tofield
the continuous
– towards
a cylindrically
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Publication List – Bianca Dittrich
B. Dittrich, J. Hnybida,
[30] “How to construct di↵eomorphism symmetry on the lattice,”
arXiv:1312.5646
[gr-qc]
Refereed
(Latest)
Coarse
graining
resultsapproach
for spin to
foams
/ spin
nets. arXiv:hep-th/0006199. R. Oeckl, “RenorB. Dittrich,
[6]
F.
Markopoulou,
“An
algebraic
coarse
graining,”
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[6] F. Markopoulou,
“An
algebraic
approach to coarse graining,” arXiv:hep-th/0006199. R. Oeckl, “RenorPoS
(QGQGS
2011)
012
(23pp)
malization
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discrete
models
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malization
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“Coarse
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of
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intertwiners,”
J. A. Zapata, “Loop quantisation from a lattice gauge theory perspective,” Class. Quant. Grav. 21 (2004)
B. Dittrich,
M.
Martin-Benito,
S.
Steinhaus,
J. A. Zapata,
“LoopM.
quantisation
from
a lattice
gauge theory perspective,” Class. Quant. Grav. 21 (2004)
B.
Dittrich,
Martin-Benito,
E.
Schnetter,
[45]
“On
the
path
integral
measure
for
discretestructure
gravity,”of spin foams,”
L115-L122,
[arXiv:gr-qc/0401109].
[29] “Towards
computational insights into the4D
large–scale
arXiv:1312.0905[gr-qc]
L115-L122,
[arXiv:gr-qc/0401109].
New
J. Phys. W.
15 (2013)
103004
(38pp)
B.
Dittrich,
Kaminski,
S.
Steinhaus,
B.
Dittrich,
F.C.
Eckert,
[7] B. Bahr, arXiv:1306.2987[gr-qc]
B. Dittrich, F. Hellmann and W. Kaminski, “Holonomy Spin Foam Models: Definition and Coarse
[7] B. Bahr,
B.
Dittrich,
F. Hellmann
and
W. (10pp)
Kaminski,
“Holonomy
Spin
Foam Models: Definition and Coarse
J.
Phys.
Conf.
Ser.
360
(2012)
012004
to
appear
[41]
“Time
evolution
as
refining,
coarse
graining
and
entangling,”
New
representation
for
LQG.
Graining,” Phys. Rev. D 87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. KaminGraining,”
Rev.S.D[gr-qc]
87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. KaminB.Phys.
Dittrich,
Steinhaus,
[38] arXiv:1111.0967
“Bubble
divergences
andBoundary
gauge symmetries
in spin
foams,”
ski, “Holonomy
Spin
Foam
Models:
Hilbert
spaces
and Time
Evolution Operators,” Class. Quant.
[44] arXiv:1311.7565[gr-qc]
“A new
for loop
quantum
gravity,”
ski, “Holonomy
Spinvacuum
Foam Models:
Boundary
Hilbert
spaces and Time Evolution Operators,” Class. Quant.
V.
Bonzom,
B. Dittrich,
[28]
integral
measure
and triangulation
Grav.
30“Path
(2013)
085005
[arXiv:1209.4539
[gr-qc]]. independence in discrete gravity,”
B.
Dittrich,
M.
Geiller,
Grav. 30Phys.
(2013)
085005
[arXiv:1209.4539
[gr-qc]].
Rev. S.
D88
(2013) 124021 (23pp)
B.
Dittrich,
Steinhaus
[8] B. Dittrich,
M. Martı́n-Benito
and
E. theories
Schnetter,from
“Coarse
graining ofdynamics,”
spin net models: dynamics of interarXiv:1401.6441[gr-qc]
[40] Phys.
“Topological
lattice
field
intertwiner
[8] B. Dittrich,
M.
Martı́n-Benito
and
E.
Schnetter,
“Coarse
graining
of
spin net models: dynamics of interarXiv:1304.6632
[gr-qc]
Rev. D85 (2012) 044032 (21pp)
twiners,”B.
New
J.
Phys.
15
(2013)
103004
[arXiv:1306.2987
[gr-qc]].
Dittrich,
W.[gr-qc]
Kaminski,
twiners,”arXiv:1110.6866
New
J. Phys.
15
(2013) 103004 [arXiv:1306.2987 [gr-qc]].
[43]
“Ising
model
from
intertwiners,”
[37] “Dirac’s
discrete hypersurface
deformation
algebras,”
[9] B. Dittrich,
M. Martı́n-Benito,
S.
Steinhaus,
“The refinement
limit of quantum group spin net models,” to
arXiv:1311.1798[gr-qc]
[9]Tensor
B. Dittrich,
M.
Martı́n-Benito,
S.
Steinhaus,
“The
refinement
limit
of quantum
[27]network
“Coarse
graining
methods
for spin
net and(plus
spin foam
models,”
B.Bonzom,
Dittrich,
Hnybida,
method
in condensed
matter.
many
many
more) group spin net models,” to
V.
B.J.Dittrich,
appear
appear B.
Dittrich,
F.C.Grav.
Eckert,
Martin-Benito
arXiv:1312.5646
[gr-qc]
Class.
Quant.
30 M.
(2013)
205013 (35pp)
[10] M. Levin,
C. J.P.Phys.
Nave,
“Tensor
renormalization
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New
14
(2012)
025008
(43pp),
[10] M. Levin,arXiv:1304.5983
C. P. Nave, “Tensor
[gr-qc] renormalization group approach to 2D classical lattice models,” Phys. Rev.
Refereed
Lett.
99 (2007)
120601,group
[arXiv:cond-mat/0611687
“Quantum
spin net models: [cond-mat.stat-mech]].
refinement limit and relation to spin foams,”
Lett.[42]
99 arXiv:1109.4927
(2007)
120601,[gr-qc]
[arXiv:cond-mat/0611687
[cond-mat.stat-mech]].
[36]
“Constraint
analysis
for variational
discrete systems,”
B.
Dittrich,
M.
Martin-Benito,
S. Steinhaus,
[11] Z. -C.
Gu
and
X. -G.simplicial
Wen,
“Tensor-Entanglement-Filtering
Renormalization Approach and Symmetry Pro[26]
“Canonical
gravity,”
[11] Z. -C. GuB.and
X.
-G.
Wen,
“Tensor-Entanglement-Filtering
Renormalization Approach and Symmetry ProDittrich,
P.A.
Höhn,
arXiv:1312.0905[gr-qc]
tected
Topological
Order,”
Phys.
Rev.
B
80
(2009)
155131
[arXiv:0903.1069
[cond-mat.str-el]].
[39]Topological
“Coarse
graining
of spin
net
models:
dynamics
of intertwiners,”
B.
Dittrich, Order,”
P.A.
Höhn,
tected
Phys.
Rev.
B
80
(2009)
155131
[arXiv:0903.1069
[cond-mat.str-el]].
J.
Math.
Phys.
54
(2013)
093505
(43pp)
Class.
Quant.
Grav.
29
(2012)
115009
(52pp),
B. Dittrich,
M. discrete
Martin-Benito,
Schnetter,Towards a cylindrically consistent dynamics,” New J.
[12] B. Dittrich,
“From the
to the E.
continuous:
[41]arXiv:1108.1974
“Time
evolution
as
refining,
coarse graining
entangling,”
arXiv:1303.4294
[math-ph]
[12] B. Dittrich,
“From
the
discrete
to
the
continuous:
Towards aand
cylindrically
consistent dynamics,” New J.
[gr-qc]
J. Phys.
(2013) 103004 (38pp)
Phys. 14New
(2012)
12300415
[arXiv:1205.6127
[gr-qc]].
Dittrich,
S. Steinhaus,
Phys. 14 B.
(2012)
123004
[arXiv:1205.6127 [gr-qc]].
2
Bottom-up design
causal sets
group field theory
tensor models
(causal) dynamical
triangulations
loop quantum
gravity / spin foams
emergent gravity
string theory
Your input of
“fundamental excitation/
fundamental theory”
continuum/
refinement limit:
extract low energy/
large scale physics
KEY
PROBLEM
?
Renormalization
tools
(auxiliary
discrete?)
building blocks
3
Overview
Motivation.
Spin foams and space time atoms.
Coarse graining without a scale.
The importance of refining states. Tensor network coarse graining algorithms.
Application to spin foams / spin nets.
Classifying (symmetry protected) phases.
New representations and vacua in loop quantum gravity
Expanding the theory around different vacua corresponding to different ways of refinement.
4
Why loop quantum gravity / spin foams?
(My personal short list of reasons)
•“minimal assumptions”: theories result from quantizing gravity (as a geometric theory), taking
background independence seriously
[gives you a higher probability to obtain a quantum theory of gravity]
•choice of (connection) variables: originally motivated by “convenience”,
•allowed first rigorous quantization of gravity (kinematical)
•so far leads to the most advanced inclusion of diffeomorphism symmetry
[kinematics: for instance LOST uniqueness theorem, but also BD, Geiller 14]
[conceptually very important in particular for coarse graining / renormalization, key role in regaining gravity]
[dynamics: we know what we are looking for! [for instance in the formulation BD, Steinhaus 13]]
•(in particular spin foams): deep connection to topological field theories,
(because there is) no Wick rotation involved
[topological field theories are fixed points of renormalization flow]
[Wick rotation leads to action unbounded from below: prevents useful continuum limit]
5
(not
aBF
fixed
point)
• Barrett
Crane
analogue
model:
=1
a b=
bcoupling):
=
con=aquotient
1= b = cgrou
• a
high
temp
(strong
•point)
a
=
"
(not
a
fixed
a
=
1,
b
=
c
=
0Crane
•
Barrett
Crane
analogue
model:
•
Barrett
analo
=
1,
b
=
c
=
0
%
•
BF
on
•
BF
quotient
on
quotient
group
Z
group
=
SZ
%
2
a
=
"
b
BF Crane
%=
{α! }
{Adt! (α! )}
a =aa1,="=
b•b=
c=
==
01%model:
high
temp
(strong
coupling):
a
b
c
=
c
a
=
b
=
0,
c
=
1
a
b
c
• Barrett
analogue
•
Barrett
Cra
b
(not
fixed
point)
=a 1,
bb=
=
0=1,a10,
a cbc=
b"=c=b=c 1= 0
(not a afixed
=
=abbac0,
==
apoint)
"=
a
(not
a
fixed
point)
=fixed
c = 1(not
Barrett
Crane
=a(not
b= b a•c
a = 1, b = c = 0a
aanalogue
=point)
1, b mod
=m
c
%
point)
a
fixed
•
Barrett
Crane
analogue
a "=• bBarrett Crane
•
high
temp
(strong
c
atemp
= analogue
1, (strong
btemp
= c =(strong
0 a fixed
model:
(not a fixed
(not
coupling):
coup
aa=
a =point)
b "=c•b high
•=
Cran
•=•b
Barrett
analogue
1,high
b c=
cCrane
=
0Barrett
a
=
b
c
=
1
a = b baca=
01,bab =
0.1
×
a0c==
a=b0.1
fixed
point)
b cc=
×
a1
=0
b(not
ccc=
a==
a0.1
b1=
|L|
" −1 • Barrett
a==
aanalogue
1,
b×
=apoint)
0 1, b = c =
(not
aa=
fixed
Crane
model:
b
=
0
!
a
=
b
c
a
=
1/2
1
3/2
0b3/2 c
1
(not
fixed
point)
0% a0.1
× a1/2
{α! }
%
%
% BF
%ba =
a "=
b
%
(not
a fixed
(not
a
fixed
point)
Crane
analogue
= •1, Barrett
ba="=cb1/2
=
0
1c
3/2model:
0 p
a
=
"
b
a
=
b
1/2
1
3/2
0
bfixed
=
0
b%= 0(not0.1
×=
aa1,
=point)
b=0.1
c=×0a
a
b
c
!
a
b
=
0
0.1
×
a
%
%
a=
1/2
1point)
3/2
0analo
=b
01(notc0.1
ab
b×
×Crane
a
•0
Barrett
1/2b
3/2
aCrane
=
b analogue
cmode
==
c00.1
a afixed
•
Barrett
•
Barrett
Crane
analogue
1/2
1
3/2
0
b = 0 0.1 × a
1b 0=0c 0.1
a 3/2
=b1,=
=3/2
0× a
a = b 1/2
c a =11, ba1/2
=
=
c
1,
=
b
0
=
c
=
0
× aa 1/2
1/2
1
3/2b =b 00= 0 0.10.1
1
(not
fixed point)
×
a
a
=
b
c
−1
fixed
fixed
point) 0
|L|b = 0 0.1(not
" 3/2
1apoint)
×1/2
a a (not
!
!−1
b
=
0
0.1%0× a
b
=
0
0.1
×
a
1/2
1
3/2
%
%
%1/2
%
{α! α! }
1
3/2
0c
a =1b 1/2
1/2
3/2 1
b
=
0
0.1
×
a
a
=
b
a
=
c
b
c
!
%
%a
%
0
We keep the1/2ib! = 01 0.1 ×3/2
discretization / quantization/
1/2
1
3/2
0
b = 0 0.1 × a
reformulation into generalized
b=0 b=
0.10 × a0.1 × a
1/2
1
3/2
1/2
1/2
1
1
3/2
3/2
0
0
lattice gauge theory
%!
%
χ
:= R
η
What are spin foams?
R{α!−1 α! }
!
"
%!
δ(g )
years of !
RCe ∼ EF
research:
Reisenberger,
Rovelli, BarrettCrane, Barbieri,
Freidel-Krasnov,
Engle-PereiraRovelli, Livine, ...
"
!
δ(g ) d
=
˙
" # " $
g µ = (χ δ) α= α δ̃(α
, )
exp(iS(geom))"
Dgeom "
R
δ(g )
δ(g ) d
%
geom labels
" #
=
δ α
g
%
exp(iS(geom))
Dgeom
geom labels
δ(g α ).
(0.175)
A( geom labels)
!
(0
exp(iS(geom)) Dgeom
A( geom labels)
29
%
28
28
(0.176)
A( geom labels)
with local (i.e. factorizing) amplitude:
geom labels
29
$ (0
α ,
A( geom labels) =
"
fund building blocks
Af bb ( geom labels)
6
ψ{j} (A) = ψ{j} (A) ·!ηAL (A)
χ{α! }
exp(iS(geom))
Dgeom
"
%
η Dgeom
=
˙
:=exp(iS(geom))
R
δ(g% α% ).
labels)
(0.175) (0.173)
"
(0.175)
Space time atoms: relation to gravity action
µ(ψ{j} ) = δ{j},
µ(ψ{j} ) = δ{j},
{Ad
)} BF
%t! (α!A(
geom
(0.176)
exp(iS(geom))
Dgeom
(0.175) (0.176)
%
A( =
geom% labels)
geom
labels "
A( geom
labels)
A
% labels) "
f bb ( geom labels)
%
A(
geom
(0.176)
geom
ηBFlabels
=
˙ geom
χ{α! } := R{Adt! (α
δ(glabels)
(0.173) (0.176) (0.174)
A(
% α% ).
! )} (χ
δ̃(α
µBF
) =
"
geom
labels)
(0.176)
%)
{αA(
"
!}
fund
building
blocks
%
geom labels%
geom
η =
˙
:= labels
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A( geomχ{α
labels)
="
t! (α! )} BF %
!}
" δ(g%A
geom
labels
labels)
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%
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µBF labels)
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) =geom
fund
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% )building blocks
=
A
(
geom
labels)
f
bb
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"
" labels)
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# " −1 $
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labels)
= "
(0.177)
f bb
|L|
%
geom
labels
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!
δ̃(α
)
(0.174)
µ
(χ
)
=
j1 A(j2jgeom
j
j
j
j
R
!−1
δ(g
)
δ(g
)
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g
=
δ
α
α
,
labels)% %
(0.177)
j 4
j4 {α
j=
%BF fund
%A building
% % blocks
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bb ( geom
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5 !5 jα6! } 6
buildingf blocks
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geom
labels)
=
A
(
geom
labels)
(0.177)
!
"
"
"
f bb
%
#
$
%
%
%
−1
fund building
|L|
"
j1 j1j2 j2 j3jR
jj4!−1
jj}55 jδ(g
δ(g%! ) dblocks
g% building
=
δ blocks
α" % α% ,
3 {α
4 α
6j6 % )
!
fund
!
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" # labels)
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="
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(0.177)
RCe ∼ EF
$
%
%!
% f bb
29
|L|
" −1
j3 j4 j5 j6
δ(g )
δ(g ! ) d g =
δ α % α% ,
j10 ji1R1Ce j∼2 EF
i5j3 j4 j5 jR6 {α!−1
fund% building %blocks %
! α! }
29
!%
%!
%
sum over orientations of
29
j1 j2 j3 R
j4Ce ∼j5EF j6
exp(iS(geom)) Dgeom
(0.175)
!
3D
29
exp(iS(geom))
Dgeom
%
!
(0.175)
29
%
exp(iS(geom))
29A( geomDgeom
labels)
A( geom
geom
labels
labels)
!
space time atoms
(0.175)
(0.176)
(0.176)
%
labels
A(j)
exp(iS
)
+
exp(−iS
)
discr
grav
discr
grav
A( geom
geom
labels)∼A(
=geom
A
(
geom
labels)
(0.177)
f
bb
(0.176)
j>>1 " labels)
"
4D
A( labels)
geom labels)
=labels
A
( geom labels)
(0.177)
geom
fund geom
building
f bbblocks
A( geom
=
labels)
(0.177)
! Af bb ( !
fund building blocks
A( geom labels)
= building
(0.177)
fund
A( geom
labels) blocks
= " Af bb ( geom
Af bblabels)
( geom labels)
(0.177)
j j j j j56 j6labels) fund
= buildingfund
(0.177)
f bb ( geom labels)
blocks
building A
blocks
j1 j1j21 jj23 2jj43 j3j54 j6j45A(jgeom
j1
j10
j10
i1
j2 j j3 j j4 jj1j5jj2 j6jj3 jj4 j5 j6
i 1 i2 3 4 5 6
1
i5
5
j10
i1
i5
fund building blocks
Large jψ(j)
(semi-classical)
limit
for single building
blocks
∼
cos(S
)
discr grav
gives 29
discrete j>>1
GR action (for flat building blocks)!
29
29
[ Ponzano-Regge, Barrett et al, Conrady-Freidel, ... ]
ψ(j)
∼
j>>1
cos(Sdiscr grav )
7
Many space time atoms?
Is there a phase describing smooth space time?
Do we get General Relativity at “large scales”?
What are the phases of spin foam theories
(and in loop quantum gravity)?
8
Spin foams as lattice theories
•there is an underlying (auxiliary) lattice: refinement limit with respect to this lattice
•use tools developed in condensed matter, in particular tensor network renormalization.
[Cirac,Levin, Nave, Gu, Wen,Verstraete,Vidal, ...]
[‘Emergent gravity’]
•Can deal with complex amplitudes: (we do not Wick rotate!)
•However many conceptional differences:
•result should be independent on choice of lattice (discretization independence)
•diffeomorphism symmetry should emerge (at fixed points of renormalization flow)
•(well supported) conjecture: discrete notion of diffeomorphism symmetry equivalent to
discretization independence - should emerge in refinement limit
[BD 08, Bahr, BD 09, Bahr, BD, Steinhaus 11, BD 12] .
•There is no lattice scale, instead notion of scale included in dynamical variables.
•Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once.
9
Coarse graining without a scale
10
Fixed point model encodes all scales!
•There is no lattice scale, instead notion of scale included in dynamical variables.
•Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once.
View point:
•Scale encoded in boundary state
- not in renormalization step.
•The initial model is an approximation (via discretization), which we hope to be valid in a small
curvature regime
[determined by choice of boundary states and number of building blocks].
•Models obtained by coarse graining improve this approximation and increase domain of validity.
•Refinement limit corresponds to a ``perfect discretization’’.
11
Fixed point as a continuum theory
1 .0
1 .0
0 .5 a 3
If we do not have a lattice scale, how do we
know that we reached the continuum limit?
0 .0
1 .0
0 .5
a2
0 .0
Answer:
diagram for k = 8 with ↵0 = 0. The coloured dots indicate to which fixed point the respective
w to: The
green
dots show
the factorizing models,
green for
= 1 (area that
starts at the
Fixed
points
of renormalization
flow lighter
correspond
toJ continuum
limit.
) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex (0,1,0)). Analogue BF theory is
tarts at the(a)
vertex
(0,0,1)).
The so-called
‘mixed’ (discretization
fixed point is orange
(area that starts
at thecorrespond
“local”
amplitudes:
topological
independent)
theories
•
to phases
(b) “non-local” amplitudes (infinite bond dimension) correspond to phase transitions
a2
1 .0
0 .8
0 .6
Ê
Ê
Ê
ÊÊ
Ê Ê Ê
Ê
0 .2
0 .0
0 .0
Ê Ê
Ê
Ê
0 .4
Ê Ê
Phase space: (some) parameters in initial
model determine end points of coarse
graining flow encoded in different colours
Ê
Ê
Ê
Ê
ÊÊ
Ê
Ê
0 .2
0 .4
0 .6
0 .8
1 .0
a1
of the phase diagram for k = 8 with ↵0 = ↵3 = 0. The colouring is the same as in the previous
the right corner corresponds to models flowing to the factorizing fixed point with J = 1, the upper
ponds to models that flow to the factorizing fixed point with J = 2, the models at the bottom left
12
Fixed point as a continuum theory
Fixed point model still looks discrete. How to reconstruct continuum theory?
Answer (here a short summary, expanded upon in the next slides):
•Build continuum theory as inductive limit: standard technique of loop quantum gravity
[Ashtekar,
Isham, Lewandowski 92+]
•Based on embedding (or refining) coarser (boundary) states into Hilbert space describing finer
boundary states
• In this way any coarse (discrete looking) state can be understood as a state of the continuum
Hilbert space.
•(Cylindrically) consistency requirements on observables, amplitudes, inner product.
• Surprisingly (condensed matter) tensor network renormalization methods are well adapted to this
technique and can provide embedding maps - determined by dynamics of the system. [BD 12]
•Allows to formulate continuum theory of spin foams based on discrete boundary states
[BD, Steinhaus 13]
13
Inductive limit techniques and
Tensor network coarse graining algorithms.
14
Boundary Hilbert spaces and transition
amplitudes
A
A
A
A
A
A
(local) amplitudes depending on
(boundary) variables, represented by blue edges
A
A
!
A
A
A
A
summing over (bulk) variables in path integral
ONB
A
A
A
A
boundary Hilbert space (tensor product over local sites)
This represents a transition amplitude built from local amplitudes.
The boundary
Hilbert
T
T space has two components.
A
T
A
A
A
T
This A
represents a one-component boundary Hilbert space.
A
A
ψ
A
A!
ψ
A!
A!
[generalized boundary formalism: Oeckl 03]
15
Embedding Hilbert spaces
coarser boundary (with four sites) with
attached boundary Hilbert space
(tensor product over four sites)
AAA
AAA
ψψψ
finer boundary (16 sites) with
attached boundary Hilbert space
ψψψ
(local) embedding map
AAA
AAA
AA!A! !
AAA
AAA
ψψ2ψ22
AA!A! !
ψψ1ψ11
ψψ1ψ11
ψψ2ψ22
AAA AAA
ψψ1ψ11
AA!A! !
embedding map identifies coarse states with states
in finer boundary Hilbert space
16
Good embedding maps?
What are the embedding maps good for?
Answer:
•efficiency!
•do computation on the most coarse grained level possible
•can represent continuum theory by discrete (boundary) data
➡ To be really efficient, embedding maps have to be adjusted to dynamics of the system:
➡Coarse Hilbert space should represent ``most typical states”
(state describing smooth geometry / low energy)
Whereas initial boundary states may rather describe fundamental excitations, and one expects a
large number of these to describe smooth geometry.
17
projections ⇡b0 b : Cb0 ! Cb on the configuration spaces at our di
in which the
states are functions
A
A on the configuration
cb ∼ cb!
A quantum
A
A!
Ainjections: A
of
the
projections
to
define
A family of functions {Sb }b∈B is cylindrically consistent on the inductive fa
A
A
A
A
ψ
(Cb , ιb ), if
ψ2
ψ
ψ1
ψ
0
0
◆
:
H
!
H
b
bb
b
•embedding maps:
! (c))
Sb = ι∗bb! Sb! , i.e.
Sb (c)
= Sbb0 ! (ιbb
∀cb0∈(cCb0b) = b (
where
b 7!
•order boundaries into∗coarser and finer ψ
ψ
ψ1
ψ2
!
pullback
of
ι
.
This
just
implements
that
the
value
the fu
where ιbb! is the
The reader willbbnote that in the classical case we alsoofused
in
not depend onInthe
on which
to evaluate.
therepresentative
case we discussed
hereone
we chooses
can however
obtain projection
A A
A A graining was !based on
variables. I.e.
in the caseA that coarse
A
A A
A A
a refinement which doubled the number of fields, { i } to { i
Ab : 1H
C ) .and define the projection
b !→
transformation
=
(
+
ij
ij
i
j
•(transition) amplitudes encode dynamics:
2
Amplitudes encode the dynamics of the system
!
!
1
2
1
1
2
!
A
[transition amplitudes if there are two boundary components]
A
ψ =( ,
⇡ ( ,
, 2 , . . .)
(0.147) 1
2 , . . .) .
A(x1 , x2 , x3 , x4 ) =
!
a(x1 , x2 , x3 , x4 , xbulk )
path integral
•(transition) amplitudes for instance
ιbb! : ψdefined
!→ ψb!by where
ψ (φ , γ , φ , . . .) = xψb (φ
.
1 , φ2 , . . .) class
b
Such
a
projection
satisfies
⇡
◆
=
Id
where
here
the
◆
b
(blue edges represent sum over variables)
b
where x are boundary data
for the configuration spaces.
A
A
ψ(x , x , x , x )
ψ(x1A
, x! as
x4 )
2 , x3 ,linear
A measure (which
herewavewill
be just understood
fun
is a boundary
function
amplitudes labelled by boundaries b is a boundary wave function
•in fact we have a family of (transition)
Hilbert spaces can
defined
byA providing a representation on
A is be
an (anti-)linear
A functional
A is an (anti-)linear functional
on bdry Hilbert space H ,
Such a family of measures
{µ}
is
cylindrically
consistent
ψ
b 15
on1bdry
Hilbert space H1if
,
ψ
!2
ψ
A(x1 , x2 , x3 , x4 ) =b0 b
1 12
!
a(x1 , x2 , x3 , x4 , xbulk )
xbulk
12 2
b! 0 1 class
b
b
bb0
where x are boundary data
1
2
3
bulk
4
1
A(ψ) =
A(xi )ψ̄(xi )
xi
(0.148)
!
A(ψ) =
A(xi )ψ̄(xi )
b0 bbx0i b
µb ( b ) = µ (◆ ( )) .
b and
b’.
•Cylindrical consistency conditions relate amplitudes for different boundariesdefines
(transition) amplitudes
defines (transition) amplitudes
State sum models associate amplitudes to space time regions with boundary (data)
State sum models associate amplitudes to space time regions w
Amplitude for a ‘larger’ region
ψ1 for a ‘larger’ region
glued from amplitudesψof2 smaller regions,
Amplitude
acts on ‘refined’ bdry Hilbert space H2
glued from amplitudes of smaller regions,
acts on ‘refined’ bdry Hilbert space H2
The Ashtekar–Lewandowki measure [?, ?] for the kinematical
gravity satisfies such consistency relations. One way to define
inner product) would be via the path integrals acting as functi
kinematical Hilbert spaces (technically on some dense subspac
18
Sb = ι"bb! Sb!
i.e.
Sb (c) = Sb! (ιbb! (c))
∀c ∈ Cb
Cylindrically
consistent
amplitudes
A
A
ψ
ψ
Ab (ψb ) = Ab! (ιbb! (ψb ))
A
A
Computing the amplitude for
a coarse state should give the same result, as
d variables
Ab
Ab
state to a finer state
•embedding coarse
A
A •using
A the amplitude map for finer states. ψb
A
ιbb! (ψbψ
)
ψ
ψb
ction
A
A
ιbb! (ψb )
A
A
ψb )
itude
over
ables
bb!
Ab
Ab!
Ab
ψb
ψb
ψb
A
A
A
A
(ψb )
ιbb! (ψb )
Ab
ψb
Ab!
Thus, computation for coarse states give already continuum results.
ations,
Cylindrically consistent amplitudes define the continuum limit of the theory.
ly subparts
A A
erator
A
A
ψ(x1 , x2 , x3 , x4 )
ιbb! (ψb )
is a boundary wave function
Ab
[BD 12, BD, Steinhaus 13]
19
What are good choices for the embedding maps
(for the basis of boundary states)?
20
on to continuum
Hamilton’s
field actually
exact!function evaluated on ‘edge wise’ linear
The same procedure for squares with refined boundary data will in
Motivation: transfer
operator
technique
this approximation.
A
A
A
A
The same procedure for squares with refined boundary data will in general give a correction to
this approximation.
uares with refined boundary data will in general
to
γ233 ,a. correction
.. = 0
set give
AA
A
AA
AA
A
set γ233 , . . . = 0
A
A A
AA A
AA A
AA
AA
!
!
!
ONB
T
T
A=
T
T
A
A
TN
A
T
A
A
T
A
A = TN
A
A
insert id =
A
A
T
A
A AA
T
21
T
A AA
ONB |ψ"!ψ|
insert id =
T
T
!
A
A
A!
!
A!
A!
Truncate by restricting ONB
to the eigenvectors of T with the
(in mod)
A χ largest
A
A! eigenvalues. A!
AA
AA
A
A
!
A
ONB |ψ"!ψ|
21
T
A
Transition amplitude between
two states !ψ|A|ψ2 "
Transition
betweenA
A
A amplitude
A
two states !ψ1 |A|ψ2 "
T
AA
ONB
A A
ONB
A
AA
A
T
Transition amplitude between
two statesT !ψ|A|ψ2 "
T
insert id =
!
ONB |ψ"!ψ|
A = TN
A
A
A
21
A
A
A
Here coarse states
correspond to low energy states.
Results (typically) in non-local embedding maps
(example:
Fourier transform for free theories).
A
Explicit diagonalization difficult for large systems.
A!
A!
A!
A!
21
A = TN
Localized embedding
maps
!
Truncate by restricting
A A AA A AA A AA A
A
A
A
A
A A AA A AA A AA A
A
A
A
A
A
A A AA A AA A AA A
= id
A
A
A
A
!
!
!
ONB
!ONB ONB
TN
A = TN
A
ONB
!
toTruncate
the eigenvectors
of T with the
by restricting
ONB
!
!
!
χ largest
(in
A AA AA AA A Ato
of T with the
A eigenvectors
A mod) eigenvalues.
A A AA AA
T
amplitudethe
function
ONB
A
T
A
A
A
A
χ largest (in mod) eigenvalues.
Expect good approximation if ψ1 , ψ2
A areAin span
A of these
A
eigenvectors.
amplitude
A
A function
A
A=
T
A
T
N
C2 good approximation ifAψ
Expect
=1 ,Tψ
2
C2
effective
C
areamplitude
in2 span of these eigenvectors.
T
!
But: explicit diagonalization
of T by
difficult.
Truncate
restricting
T T
T T T
!
amplitude
function
Truncate
by restricting
T
Tamplitude ONB
effective
ON
= id
to the eigenvectors of T with t
Localize
truncations,
But: explicit
diagonalization
of T difficult.
Determined by (generalized)
χ largest (in mod) eigenvalue
only subparts
to the eigenvectors of T with
A AA A AA A AA diagonalize
A A
A the
EV-decomposition.
ofA transfer
operator by
χ largest (inTmod)
eigenvalues.A
A ! A
Determined
(generalized)
Expect
good approximation if ψ
Ttruncations,
T T T Localize
TA
A
A
A!
A!
are in span of these eigenvector
effective
amplitude
T
T only subparts
EV-decomposition.
diagonalize
C2
a
C2
c
blocking
of transfer operatorA
C2
b
A
A
A
But: explicit diagonalization of T d
Localize truncations,
iteration procedure
22
diagonalize only subparts
of transfer operator
A A iteration
AA A Aprocedure
A! A! A!
A
A
d
f
e
A! A! AA! ! A! A!
A!
A!
A!
determine embedding maps
H
blocking
H Determined by (generalized)
embedding
H
EV-decomposition.
= id
embedding
blocking
embedding map after 3 iterations
embedding
A
A A AA A AA A AA A
A
A
A
A= id = id
= id
= id
22
22
Relation to vacuum
Here coarse states correspond to low energy states / states with few excitations from vacuum.
Embedding maps depend on the dynamics of the system.
However in General Relativity:
What is vacuum?
Hamiltonian is a constraint and hence zero (and we do not Wick rotate).
Transfer matrix should be a projector if diffeomorphism symmetry is realized
(with eigenvalues one and zero).
NB: In 4D diffeomorphism symmetry is broken, eigenvalues take other values too.
So we can hope that even for GR, unphysical states (eigenvalue zero) are truncated away.
There is however a further mechanism that differentiates states even in GR: radial evolution.
[BD, Steinhaus 13: Time evolution as refining, coarse graining and entangling]
23
Radial time evolution and coarse states
•
•
•
•
•
•
•
•
•
•
igureConsider
6: Illustration
of radial
evolution
in tensor
networks:
adding eight
additional
tensor (in
a “radial
time time
evolution”
from
a ``smaller”
to By
a “larger”
Hilbert
space.
ray) we perform one time evolution step. The boundary data grows exponentially from 4 to 48 .
(We can easily construct such evolutions for systems based on simplicial discretizations, such as
spin foams).
The radial time evolution can be split into steps with time evolution operators
Z R2map:
This time evolution itself defines an embedding
T (R1 , R2 ) = exp(
Hr dr) .
(6.1)
The image of the time evolution
defines coarser
states
in
the
larger
Hilbert
space.
R1
his is a refining time evolution in the sense that the Hilbert space H2 at R2 will have more kinematical
egrees of
freedom than
thedefinition
Hilbert space
H1 to
at states
R1 . The
main idea coarse
is to truncate
theexcitations.
degrees of freedom
Conjecture:
This
leads
describing
grained
H2 . Ideally one would
performtime
a singular
valuecan
decomposition
T (R1 , R
would givemaps.
a maximal
2 ). This
(Radial)
evolution
be used toofdefine
useful
embedding
umber of dim(H1 ) singular values. Hence T (R1 , R2 ) will have a non–trivial co–image in H2 , which can
[BD, Steinhaus 13]
e projected out.
Such a scheme might
be indeed
worthwhile
to 14,
investigate
(for
[also
BD, Hoehn
11,13, Hoehn
BD, Hoehn,further
Jacobson
wip]statistical systems), in order to
btain an intuition about the truncations. One would expect that the reorganization of the degrees of
eedom via the singular value decomposition will be highly non–local, as we have seen for the example
section
Thethis
timedefinition
evolution would
considered
exactlyto
corresponds
T (R1 , R2 ). maps.
NB:2.1.
Again
leadthere
in general
non-localtoembedding
Even such a classical investigation might be worthwhile: the radial evolution as a refining time
volution will result in post constraints. Below we will introduce coarse graining time evolution steps,
hat will implement a truncation, and should classically lead to pre–constraints. This suggest that for a
ood choice of truncation, the pre–constraints should match to the post–constraints.
24
Tensor network renormalization methods
... rather using local embedding maps (to keep algorithms efficient)
but different versions are possible.
A
A
A!
A
A
Contract initial amplitudes (sum over bulk variables).
Obtain “effective amplitude” with more boundary
variables.
bare/initial amplitude
depending on four variables
A
A
A A
A
A
A
A
A
A!
A
A A
A! A!
A A
Find an approximation (embedding map) that would
32
minimize the error as compared to full summation
(dotted
lines). For instance using singular value
A!
decomposition, keeping only the largest ones.
Leads to field redefinition, and ordering of fields into
more and less relevant.
A A
32
Use embedding maps to define coarse grained
32 with the same (as initial) number of
amplitude
boundary variables.
25
Tensor network renormalization methods
Iteration. Summation over bulk variables are truncated
using the embedding maps.
Associated embedding maps for boundary Hilbert
space.
26
ψ
ψ
C2
A
Fixed points
give cylindrical consistent amplitudes
A
A
A
C2
A
c
0.8
b
0.6
ψ
ψ
d
A
ω(g1 g2−1 )
g1
g
A2
A
A
A
H
0.2
H
f
e
A
0.4
A
Gauss ω̃(k)
0
Ab
10
15
A
20
25
!
AA
k
A
5
H
Ab
A
A
A
!
30
A
A
A
A
A
ONB
A
A
Ab
Ab
A
A
ψb
ψ
ψ
ψbfor cylindrically consistent amplitudes b
Condition
ψb
T
22
T
A
Ab
b)
ψb
T
Ab!
Ab
A
T
ψb
ψb
A
ψb
Ab
A
ψb
satisfied for
Ab!
ιbb! (ψb )
A
Ab
A
A!
A!
A
25
Ab!
ψb
A
A
Ab!
A!
A
A
ιbb! (ψb )
C2
C2
C2
=
ψ(x1 , x2 , x3 , x4 )Ab
ιbb! (ψb )
is a boundary wave function
ιbb (ψb )
Ab
!
A is an (anti-)linear functional
C2
C2 27
Fixed points corresponding to topological field
theories or interacting field theories
•method works particularly well for convergence to a “gapped phase”
(lead to topological field theories)
•finitely many ground states separated from excited states: determine bond dimension
(number of non-vanishing singular values)
•phase transition: usually involve ‘infinite bond dimension’
•expect to find interacting theories / propagating degrees of freedom
•truncation can be nevertheless quite good, and provides a method to determine phase transition
(and sometimes even to calculate critical exponents)
• However argument about radial time evolution suggest that better truncations (near phase
transitions) might be possible if non-local embedding maps are incorporated.
• Might even not need infinite bond dimension.
28
Applications: (analogue) spin foam models
•4D spin foam models are very very complicated
•devised analogue models capturing the essential dynamical ingredients of spin foams
[similar to 4D lattice gauge and 2D spin system duality]
•these spin nets can actually be interpreted as spin foams based on very special
discretizations
•unlike spin foams the spin net models are non-trivial in 2D
•investigated the phase diagrams for such models (with quantum group structure)
•these phase diagrams have a very rich structure
(The details of this and the adaption of the tensor network methods
take easily up another talk.)
29
Applications: (analogue) spin foam models
6.2.2
Lattice gauge theory/
Ising like systems
Phase diagram for k = 8
For the quantum group with k = 8, we discuss the linear combintation of four fixe
with P
a maximal (even) spin 1  J  4, where we neglect J = 0 as argued above
that J ↵J = 1, we have three free parameters. In figure 9 we show the full p
coloured points indicating the fixed point they flow to. In figure 10 we show the
Spin nets
coupling
0 .0
a1
0 .5
confining phase
‘no space’ phase
1 .0
1 .0
0 .5 a 3
deconfining phase
(topological phase)
topological phase
(gives 3D gravity!)
0 .0
0 .5
a2
0 .0
1 .0
Figure 9: Phase diagram for k = 8 with ↵0 = 0. The coloured dots indicate to
initial models flow to: The green dots show the factorizing models, lighter green
vertex (↵1 , ↵2 , ↵3 ) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex
blue (area that starts at the[ BD,
vertex
(0,0,1)). The
so-called
fixed point i
Martin-Benito,
Schnetter
2013,‘mixed’
BD, Martin-Benito,
vertex (0,0,0)).
Steinhaus 2013]
Much richer phase structure!
Interpretation: different phases describe uncoupled
a2
space time atoms
(green) and coupled space time
1 .0
atoms (orange,blue).
0 .8
0 .6
Ê
Ê
Ê
ÊÊ
Ê
Ê Ê
30
Towards spin foam coarse graining
•So far encouraging results for ‘spin foam analogue’ models.
Conclusions:
•Relevant parameters related to (SU(2)) intertwiners (also appearing in spin/anyon chains)
leading to rich phase space structure. This is expected from the gravity dynamics.
•Need to implement a weak version of discretization independence to uncover these rich phase
spaces (escape the two lattice gauge theory phases).
•Positive indication for finding a geometric phase in spin foams.
•Are now looking at spin foam analogue and actual spin foam models in higher dimensions.
•Need to extend tensor network tools (applicable to lattice gauge theories) to this end. [wip]
31
Each phase corresponds to a toplogical field theory.
Defines embedding maps and associated vacua.
New representations (Hilbert spaces) for
Loop quantum gravity.
32
Loop quantum gravity
•succeeded in constructing a quantum representation of geometric operators
[Ashtekar, Lewandowski, Isham, Rovelli, Smolin, Corichi, ... 90’s]
•spatial geometric operators (area, volume operators) have discrete eigenvalues
•spin networks based on (finite) graphs give a basis in the (non-separable) LQG Hilbert space
•despite this (discrete) basis the theory is defined in the continuum: via embedding maps
•dynamics based on Hamiltonian constraints [Thiemann ’96]
So far the theory is based on the
Ashtekar-Lewandowski representation
based on a (AL) vacuum describing
•
•
zero volume spatial geometry
maximal uncertainty in conjugated variable.
33
What is vacuum?
•Ashtekar-Lewandowski representation /vacuum corresponds to confining phase for lattice
gauge theories (i.e. embedding maps coincide).
Question: Can we base the representation on a different vacuum
corresponding to one of the other phases?
34
Loop quantum gravity vacua
cyl
Ĉ =
I2,−
Z∞,∞
[ιb1 ∞ (ψb1 ), ιb2 ,∞ (ψb2 )] = Zbcyl
[ψ
b1 ψb2 ]
1 ,b2
geometric variables:
Ĉ = I2 −
{A , E} = δ
connection
ψvac (A) ≡ 1
Ashtekar
Lewandowski
representation
{A
, E}- =
δ
(90’s)
ψvac (A) ≡ 1
flux: spatial geometry
,
E≡0
peaked on degenerate (spatial) geometry
maximal uncertainty in connection
excitations:
spin network states supported on graphs
(representation)
labels for edges
,
[BD, Geiller 2014,
BD, Geiller to appear]
E≡0
BF (topological) theory representation
ψvac (EGauss ) ≡ 1 ,
F (A) ≡ 0
peaked on flat connections
maximal uncertainty
in spatial geometry
28
excitations:
flux states supported on (d-1) D-surfaces
(group) labels
for faces
35
We applied this to BF (topological field
theory), corresponding to unconfined
phase. [BD, Geiller 14]
But construction can be generalized to other
topological field theories:
36
Excitations and vacuum
(discretized)
vacuum
topological field
state
theory
(dynamics)
determine observables stable
under refining time evolution
(non-trivial step)
continuum
excitations
Hilbert space
‘inductive limit’
construction
for excitations
37
Advantages
[BD, Geiller 2014,
BD, Geiller to appear]
•This new construction allows to expand loop quantum gravity around different vacua
corresponding to the different phases (fixed points) for spin foams.
• Ashtekar-Lewandowski representation: peaked on completely degenerate spatial geometry,
maximal fluctuation in conjugated variable: difficult to built up states corresponding to smooth
geometry
•New representation [BD, Geiller 2014] : peaked on flat connection, maximal fluctuation in spatial
geometry. Corresponds to the physical state in (2+1)D gravity and in general to condensate states
with respect to Ashtekar-Lewandowski representation.
•Facilitates construction of states corresponding to smooth geometries and should be useful for
discussions of i.e. black hole entropy in loop quantum gravity [Sahlmann 2011].
•New representation much easier to interpret geometrically: new handle on the dynamics of the
theory. [need substitute of Thiemanns (1996) quantization of Hamiltonian constraints]
38
coloured points indicating the fixed point they flow to. In figure 10 we show the inter
0 .0
Phase transitions?
a1
0 .5
1 .0
1 .0
0 .5 a 3
0 .0
1 .0
0 .5
a2
0 .0
to expand
quantum
around
different vacua
•This new construction allowsFigure
9: Phaseloop
diagram
for k =gravity
8 with ↵
0 = 0. The coloured dots indicate to whic
initial
models
flow to:
The green
the factorizing models, lighter green for J
corresponding to the different
phases
(fixed
points)
for dots
spinshow
foams.
vertex (↵1 , ↵2 , ↵3 ) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex (0,1,0
blue (area that starts at the vertex (0,0,1)). The so-called ‘mixed’ fixed point is ora
vertex (0,0,0)).
• What about phase transitions (‘non-trivial’ fixed points)?
/ propagating degrees of freedom.
• Instead of topological theory expect conformal theory
a
2
1 .0
• Expect such fixed point amplitudes to be non-local
0 .8
Ê
non-local
embedding maps
Ê
Ê
ÊÊ
0 .6
Ê Ê
0 .4
Ê Ê
Ê
(also
given ÊbyÊ time evolution?)
Ê
Ê
Ê
Ê
Ê
Ê
symmetry
there?
Ê
• Do we regain triangulation independence / diffeomorphism
0 .2
0 .0
0 .0
ÊÊ
Ê
Ê
0 .2
0 .4
0 .6
0 .8
1 .0
a1
39
Summary
Coarse graining is about refining boundary states.
Tensor network methods construct embeddings of coarse states,
representing “low energy/coarse degrees of freedom”,
into continuum Hilbert space.
40
Summary and Outlook:
Quantum Space Time Engineering
•We are on a good way to understand the ‘many body physics’ of spin foams and loop quantum
gravity.
•Corresponds to the ‘continuum limit’ of these models.
•In the path integral approach (spin foams): mapping out the phase diagrams
•connections to condensed matter physics / (new?) topological field theories and phases
•challenge: going to higher dimensions
•challenge: understanding symmetries of phase transition fixed points
•In the canonical approach (loop quantum gravity): expanding around different vacua
•facilitates investigation/construction of physical states describing extended/smooth geometries
•each vacuum comes with its own set of excitations: investigate dynamics of these.
41
Thank you!
42