Loop quantum gravity with AL vacuum. â¢based on dual graphs: carry excitations (spin networks = functional of holonomies labelled by spins). â¢refining ...
A new vacuum for loop quantum gravity Bianca Dittrich, Perimeter Institute Second EFI Winter School, Tux, Feb 2014
BD, Sebastian Steinhaus, Time evolution as refining, coarse graining and entangling arXiv: 1311.7565[gr-qc]
BD, Marc Geiller, A new vacuum for loop quantum gravity arXiv: 1401.6441[gr-qc] and to appear
Overview Motivation. How to construct continuum physical theory with refining time evolution. As an exercise construct BF vacuum: dualize the Ashtekar-Lewandowski construction. Need to dualize everything!
BF refinement and BF cylindrical consistent observables. Holonomies and integrated fluxes
BF measure and cylindrically consistent inner product. Compactification /discretization of excitations via inductive limit construction. Remarks on diffeomorphism symmetry and (full) dynamics. Simplicity constraints again
i s u l c n o C
k o o l t u o d n a on
How to construct (continuum) physical vacuum. [Bahr: cylindrical consistent path integral measure] [BD,12: dynamical cylindrical consistency] [BD, Steinhaus 13: Refining by time evolution]
What is vacuum? coarse representation of vacuum
(Hartle-Hawking) wave function associated to closed boundary of space time, given by amplitude for basic space time building block [Oeckl: generalized boundary proposal] [Conrady, Rovelli et al 03] ... [Hoehn: discrete generalized boundary proposal 14]
finer representation of vacuum
Refined vacuum wave function by evolving i.e. with refining Pachner moves. [BD, Steinhaus 13: Refining by time evolution]
adding degreesDof−freedom in hin Refining: D − 1 hypersurface: 1 Pachner (interpolating) vacuum state within D − 1 hypersurface: D − 1 Pachner Same refining onto can be applied to atriangle general state: tetrahedron single
ng of tetrahedron onto single triangle
−→
:
ctive:
:ctive:
−→
general state
ΣΣk k
[BD, Hoehn, Steinhaus, ...]
fits nicely the heuristics of tensor network renormalization [BD 12]
refined state
Σk+1 Σk+1 −→ −→
[BD, Steinhaus 13: Refining by time evolution, Read this!]
Lesson: think about refining with Pachner moves.
Can we use this to construct continuum limit in the same way standard LQG is based on (dual) graph refinements?
As we will see this makes sense for topological theories and allows the construction of the continuum theory via inductive/projective limit used in LQG.
This limit does not only describe the topological theory but also excitations! In fact we obtain a representation (of gauge invariant projection) of the (modified) holonomy flux algebra. Thus the space of `refining Dirac observables’ = cylindrically consistent observables is (unexpectedly) large.
Excitations and vacuum (general) (discretized)
vacuum
topological field
state
theory
(dynamics)
determine observables stable under refining time evolution
continuum excitations
Hilbert space ‘inductive limit’ construction
for excitations
From AL to BF: Dualize everything! Ashtekar Lewandowski
`BF’
Hilbert space
Hilbert space
cylindrical functions over A
cylindrical functions over E
Main lesson: dualize everything! (new insight: BF refining needs triangulation) [Gambini, Griego, Pullin 97, Bobienski, Lewandowski, Mroczek 01: A two-surface quantization of Lorentzian gravity] [Bianchi 09 ]
LQG as theory of curvature defects.
[LOST-F 05/04]
[Freidel, Geiller, Ziprick 11]
Uniqueness theorem for AL vacuum.
LQG continuum phase space with BF gauge fixing.
Can there be another vacuum?
[Baratin, Oriti & Baratin, BD, Oriti, Tambornini 10 ]
Non-commutative flux representation of LQG. [ BD, Guedes, Oriti 12]
LQG in terms of E-bar instead of A-bar? Problem: Usual refining is not consistent with E-bar!
= vacua Loop quantum gravity
ˆ = I2 − cyl C Zb1 ,b2 [ψb1 , ψb2 ]
cyl Z∞,∞ [ιb1 ∞ (ψb1 ), ιb2 ,∞ (ψb2 )]
geometric variables:
Cˆ = I2 −
connection
{A , E} -= δ Ashtekar Lewandowski vacuum (90’s) ψvac (A) ≡ 1
{A , E} = δ
,
ψvac (A) ≡ 1
E≡0
peaked on degenerate (spatial) geometry maximal uncertainty in connection
excitations: spin network states supported on graphs
(representation) labels for edges
[Koslowski: vacuum with shifted E]
flux: spatial geometry
condense
,
E≡0
BF (topological) theory vacuum
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
shift connection to homogeneous curvature?
What a vacuum needs to deliver a) The vacuum state itself.
b) The vacuum defines how to refine an arbitrary state. c) The vacuum is cyclic: obtain entire Hilbert space by applying cylindrically consistent observables to the vacuum.
d) Via vacuum measure defines inner product on Hilbert state.
Need in particular define refinement and find cylindrically consistent observables with respect to this refinement.
Loop quantum gravity with AL vacuum
[ ... Ashtekar, Isham, Lewandowski 93]
•based on dual graphs: carry excitations (spin networks = functional of holonomies labelled by spins) •refining operations on this graph: matches composition of holonomies •cylindrically consistent holonomy and flux observables: commute with this refining •allows the construction of a cylindrically consistent measure and inner product and the definition of the continuum Hilbert space via a so-called inductive limit Class. Quantum Grav. 30 (2013) 055008
m Grav. 30 (2013) 055008
B Dittrich et al
Figure 1. The three elementary moves on graphs.
Figure 2. Consistency conditions for fluxes across surfaces associated with the three elementary moves on graphs.
Refinement operation on holonomies and fluxes.
ck of these defines the elementary injections for Hγ : add := p∗add : He → He,e# ;
B Dittrich et a
f (g) $→ (add · f )(g, g# ) = f (g)
[Thiemann 00, QSD7]
[ BD, Guedes, Oriti 12]
! = Hom(P, ! agrees with the Pontryagin dual E = A U(1)) = Hom(Hom(P, U(1)), C). The
BF refinement and BF cylindrically consistent observables [ BD, Geiller 14]
Change to triangulation for refinement is essential! Exercise: come up with a refinement rule in dual for BF vacuum plus excitations!
Set-up for BF vacuum
More (regular) structure than just dual graph! [Bonzom, Smerlak 12: needed for BF quantization] [Gurau: colored triangulation]
•simplicial version of LQG, see also [Thiemann 00, QSD7] •instead of embedding dual graph, embed triangulation (vertices) •refining given by operation on triangulation (and not on dual graph) •classical phase space: projective limit of phase spaces as in [Thiemann 00, QSD7] •however change AL embedding to BF embedding
•gauge group: discrete or Lie group
Set-up •manifold with auxiliary metric •set of embedded triangulations •embedded vertices: carry coordinate labels •edges: geodesics with respect to auxiliary metric (replaces piecewise linear) •triangles, tetrahedra:
given by minimal surfaces
•dual complex (for instance barycentric, however details do not matter) with a root node (fixing a reference frame)
•refining operations given by refining Alexander moves (alternative: set of refining Pachner moves)
•equips the set of triangulations with a partial (directed)* order
Alexander moves, d=2 subdividing (sub) simplices
In d=2:
subdividing a triangle
subdividing an edge
Alexander moves, d=3 subdividing (sub) simplices
In d=3:
subdividing a tetrahedron:
subdividing an edge:
subdividing a triangle:
(there are infinitely many of those moves)
)≡0
,
!
Phase spaces
!
Xπ
!
F (A) ≡now 0 fix triangulation and a root node •Ffor (A) ≡ 0
!
dT e−S(T ) −S(T )
dT e
point )separating functions •specify a set of−S(T h dT e (0.168) γ ) we consider phase space functions invariant under gauge invariance: •gauge dT e−S(T (0.168) transformations ! ! at all nodes except at the root ) ) −S(T dTdT e−S(T e
-closed holonomies with source at root hγ
(0.168) (0.168)
Xπ
hγ
,
hγ
hγ hγ -integrated (simplicial) fluxes transported to the root
(0.169)
(0.169) (0.169)
(0.169)
(can understand these as vector fields acting on functions of holonomies) k k {X , g } = g T Xd=2: , πXπ, , d=3: XXσσXσ e (tree),(0.170) π X e root (+transport π to + treee on σ )(0.170) (0.170)
X
,
Xσ
,
X
(0.170)
Xσ
-Poisson brackets deducible from basic (standard) Poisson brackets:
k {Xe , ge }
= ge T
k
k l {Xe , Xe }
= f
[Thiemann 00, QSD7]
klm
m Xe(0.171)
!
simplicial fluxes in the case where G = SU(2) and in −S(T ) (2+1) dimensions. Let e∗ ∈ ∆∗ be the oriented edge dual to e ∈ ∆. We assume that the pair (e, e∗ ) is positively oriented. Given the continuum E = Eai τi dxa (with an almost canonical choice of triad parallelfield transport): (where τi is a basis of su(2) and a = 1, 2 a spatial one– form index), for one edge:the fluxes are defined by ! " ∗ # ∗ a Xe = he∗ (t),e(0) Ea e (t) (e˙ ) (t)h−1 dt, (7) e∗ (t),e(0)
dT e Integrated simplicial fluxes, d=2
!
action o γ!.
e∗
h
[Husain 91] [Thiemann 00, QSD7] [Freidel, Louapre 04 ]
where t ∈ [0, 1] is a parametrization of the edge e∗ such γ that and ine∗the intersect at t = 1/2, and he∗ (t),e(0) is the for ae path triangulation: ∗ parallel transport from the point e (t) to the source vernce it ∗ tex e(0) of e. This holonomy starts at t ∈ e , goes along ariant ∗ ∗ e until it reaches the point e ∩ e , and then goes along Since e until its source e(0). x two The advantage of using these fluxes is that under a wave gauge transformation or an orientation reversal of the edge they transform σ path π as " #−1 (6)parallel transport Xe #→ g(e(0))Xe g(e(0) , Xe−1 = −ge Xe ge−1 . (8) . These properties and definitions can be generalized to root other (Lie) groups, such as SL(2, C) [23]. The action of the fluxes leaves the space of functions vector fromasource to target vertex path. moves Interpretation: ψ : G|E| → C over fixed vertex dual graph (with |E| of edges) ators. In (2+1) k allow open paths! invariant. is defined as gravity This closedaction pathsk give Dirac observables - but here we hoose, e" i # e ie
X
,
X
{X , g } = g T
he holonomy loop that starts at the root and goes along the tree to the source node nce frame in which the first edge is expressed in), than goes along ⇡1 ( as near as of the path ⇡1 ) to the source node of X⇡2 and then returns via (the inverse edges t ⇢. replaces composition of holonomies in AL embedding the composition rule
Composition of integrated simplicial fluxes, d=2 X ⇡2
X ⇡1
= `⇡21 ⇡1 X⇡2 `⇡2
⇡1
+ X ⇡1
.
(1)
n gauge variance of holonomies and fluxes
obably just consider holonomies and fluxes transported to the root, i.e. via a tree. ransport can be generated by one system of fluxes based on a given tree, together holonomy observables. going to discuss projective limit of phase spaces, we can select a set of observize the phase space (with the other observables obtained by finite polynomials/ uld even reduce the set of fluxes to fluxes associated to leaves of a tree. 4
Integrated simplicial fluxes, d=3 (need a surface tree for parallel transport):
black arrows: elementary fluxes blue: piece of a surface red: bonsai tree for piece of surface parallel transport (dashed red) takes place in tetrahedra `below’ the surface
For composition of integrated fluxes need to specify a `bridge’ edge (which connects the two surface trees). Choice does not matter for BF refinement.
Continuum phase space Can be defined as a projected limit of phase spaces associated to fixed triangulations.
[Thiemann 00, QSD7: for AL embedding]
The discussion needed for that is exactly the same as for cylindrically consistency of the quantum observables.
AL embedding
BF embedding
holonomies
compose
stay constant
fluxes
stay constant
compose
Quantum theory
Coincides with 1
3 Pachner moves
Refining for8.2(holonomy) wave functions Subdividing an edge 8.2.1
Holonomies
We glue this dual complex
need to make notation consistent throughout - could think of some way to haviour of root, see remarks below - should really dospatial that! hypersurface: (but is there to the triangle refinement?) The refinement in the triangulation and dual is depicted in over figurered ??. edges. This refine integrate from gluing the dual complex ⌥e , in figure ??, to ⌥. This dual complex ⌥e has four in Impose flat holonomies. – thus the amplitude is given by four delta functions, imposing flatness. This leads to the following refinement/embedding
=
Ee5 (g10 , · · · , g80 , · · · ) Z (g2 1 g20 g80 g10 1 g1 ) (g3 g30 1 g70 1 g40 g4 1 ) (g5 g4 g40 1 g60 1 g10 1 g1 ) (g70 g50 1 g80 g60 ) (g1 , · · · , g5 , · · · ) dg1 · · · dg5
.
or stick to 0 notation ... Solving the delta functions andg4 . The remaining inte We can solve the first three delta functions say for g1 , g2 and gThus a gauge averaging the fixing nodes A the and `old’ B in nodes: the coarser dual graph. For 5 implement gauge at we obtain for the refinedat wave function function we can therefore evaluate the integral (2) by gauge fixing g3 = g30 and g5 choice of gauge fixing we can of a possible root at A or B unde Ee5determine ( )(g10 , · ·the · , gbehaviour 0, · · · ) 8 figure ??. 1
=
(g70 g50 g80 g60 ) (g1 , · · · , g5 , · · · )˛˛ g
1 ˛ 1 = g10 g80 ˛ ˛ g2 = g20 ˛ ˛g =g 0 3
3
˛ ˛ g = g 1g ˛ 4 70 40 ˛ ˛g =g 0 5
5
Determines refining map for By constructions gauge invariant functions f (h ) of closed holonomies h (adapte 5 holonomies. ) will be cylindrically consistent with respect to this refining
wecylindrically obtain for the refinedwith waverespect function )Thus will be consistent to this refining
The root
Ee5 ( )(g · · · ), g8)0 , ·=· · ) f (hE( ) )Ee ( ) . 0 , (h Ee 1(f = (g70 g50 1 g80 g60 ) (g1 , · · · , g5 , · · · )˛˛ g = g 0 g 01 ˛ 1 ˛ by ˛ g path To this end we have to describe to replace path 8 Gauge fixing determines behaviour of root, in case ithow coincides with ana old node. ˛ 1 a1 refined ˛ 4 = g70R ˛g =g ˛
be obtained by lifting a given path to a path 0 under gluing comp ˛ 2 of20the dual ˛g = g 0 5 5 ˛g =g 0 3 3 is not unique, however the di↵erent choices lead to the going same through result. the Thus we can Thus not only holonomies description give examples and region write some explicit formula again this is similar to are cylindrically consistent, but also By constructions gauge invariant functions f (h ) of closed holonomies h (adap flux in the AL embedding can let start holonomies at vertex A, remark that this also holonomies startingto at this the root. ) will be cylindrically consistent with respect refining
new root
g2 g5 g3 ! g20 g50 g30
Ee (f (h ) )
=
f (hE( ) )Ee ( ) ...
.
To this end we have to describe how to replace a path by a refined pat Note that also holonomies starting or ending at the 0 node A or B are cylindricall be by holonomy lifting a given path tothe a path undernodes gluingA0ofand theBdual cw 0 endobtained the refined has to start at corresponding and old root isreplacement not unique,rules however the di↵erent lead result.with 0 Thus we Moreover gaugefixing). actiontoThus atthe rootasame commutes (5) (which involve a choices gauge function coming from description give at examples write some explicit again this we is similar gauge covariant node A and will refining: be gauge covariant at formula node R(A). Indeed have flux in the AL embedding can let start holonomies at vertex A, remark that this =
Ee GA h
E(A) Gh Ee (
)
g2 g5 g3 ! g20 g50 g30 ˛˛ 11 ˛ (3) { edgeref1 )˛˛ ˛gg ==gg0 g0 g0where (3) { edgeref A ˛ G denotes the action of the gauge group at the node A, for instance ... ˛ 11 11 880 ˛ ˛g h = g 1 1 =g ˛˛ ˛˛ g 0 0 gg 00 4 7 4 7 4 1 A ˛ ˛gg2 ==gg20 0 ˛˛ 4 (g , . . . , g ) = (g h, g h, g , g , h g5 ) cylindr . G 1 5 1 2 3 4 ˛˛ 2 ˛ ˛ggthat 2 Note ==also gg0 0holonomiesh starting or ending at the node A or B are 55 55 ˛ ˛gg ==gg0 end the refined 0 0 holonomy has to start at the corresponding nodes A and B a 0 3 3 3
3
we could make up some convention for gauge fixing / behaviour of r 0 replacement (which involve a gauge fixing). a function could makerules that(5) deponent on orientation of the Thus new dual edges... coming need gauge covariant at node A will gauge formulas! (or just invert thebe edge 80 ) covariant at node R(A). Indeed we ha
dholonomies holonomieshh (adapted (adaptedto tothe thetriangulation triangulation A
E(A)
transported to root.Note Need tofollowing: slightly redefine composition of fluxes flux as component the With X = Xk T denoting transported root.kwe Need totoslightly redefine composition of fluxes as k undo this 1 i rted to roottoand have transport. {X , g} = gT . We then have for a transported flux (h Xh) rted we have undo thiswe transport. ith Xto= root Xk T kand denoting fluxto components have the basic Poisson brackets k 1 ith Xfor=aXtransported flux we have the basic1Poisson brackets k T denoting have flux (h components Xh)i k i i 1 {[h T X h] , g} = g hT h k have for a transported flux (h 1 Xh)i {[h 1 T k Xk h]i , g} = g hT i h 1 . (9) {[h 1 T k Xk h]Ifi , we g} exponentiate = g hT i h 1 .the flow of the transported (9) flux, we will e flow of the transported h exp(↵i T iflux, )h 1we . will obtain a right translation by H = e flow of the transported flux, we will obtain a right translation by H = 1 i Exponentiated fluxes act= by R exp(↵ {[h Xh] , ·}) i h exp(↵i 1 i i 1 exp(↵i {[h Xh] , ·}) = Rh exp(↵i T )h (10) right translations. 1 i exp(↵i {[h Xh] , ·}) = Rh exp(↵i T i )h 1 (10) i i abuse of notation we identity With a slight ↵ = exp(↵ T ). Define: i ation we identity ↵ = exp(↵i T ). i ation we exponentiated identity ↵ Let = exp(↵ ). us denote the action of↵ .the exponentiated (and integrated) i Tintegrated) on of the (and flux by X n of the exponentiated (and integrated) flux by X↵ . Weflux will(example): need exponentiated fluxes. tency of To show cylindrical consistency of flux (example): tency of flux (example):
Integrated (exponentiated) fluxes: action
6 6
6
{Xargue = ingegeneral) T (The order does not matter ... need to check and for that e , ge } Could make commutative diagram. Need the following formulas
Integrated (exponentiated)X fluxes: consistency (· · · , g , · · · ) = (· · · , ↵ g , · · · ) ↵ e5 1
1
5
5
kGauss klm k lconstraints l klm mm Basically follows {X from{X and the old new , X } = f X , eXe }e = f↵ X e e e ↵ us assume the root is atis thebased node A and the fluxA. Xe which is base t root is at the node A root and consider theLetflux X which at consider this node 1 fact that we add a flat (Gauss-closed) piece of eE ( )(g 0, · · · , g 0, · · · ) 5 e 5 1 8 Under refining we have 1 ˛ = geometry. (g 0 g 0 1 g 0 g 0 ) (g ⇣, · · · ,⌘g , · · · )˛˛ 5
⇣
⌘π
7
E(π)
5
8
6
Ee51 X↵ e
1
5=
g 1= g 0↵g 0 1 81 X e ˛6 g2 = g20 5 ˛ ˛ g↵ = g 0 X which
˛ 1 Xh↵h e˛ 1
1
˛ g = g 1g ˛ 4 70 40 ˛ ˛g =g 0
1 ↵ Thus abased vectoratpointing from one vertex to the ↵ consider the h↵h ↵ s at the node A and flux X which is this node A. 5 5 1 π Ee5 XE(π) = X X (11) 1 the node A and consider the flux e5h 1 1 with 1 Let us assume the root is at is based at this n 3 3 = g g g . e5 e6 e56 7next e 5 vertex stays invariant after subdividing (in a
⇣
Ee5 X↵ e
1
⌘
5
0
0
0
5
1
Under refining we(The haveorder does not matter ... need to check and argue for that in general) Could make commutative Need the following formulas flat manner). ⇣ diagram. ⌘ ↵ h↵h 1 ↵ ↵E h↵h 1 ↵ X = X X 1 · · · , g 0 , · ·1 · ) E (X )(g , e 1 0 5 e ↵ 1 e e e5 1 5 Xe 1 Xe 1 15 1 (· · · 8, g5 ,6 ·(11) e5 X · · ) = (· · · , ↵ g5 , · · · ) e
=
6 atter ... need to check and argue5 for1 that in general) (g7 g5 1 g8 g6 ) (g1 , · · · , ↵ 1 g5 , · · · )˛˛ g = g g with h = g6 g7 g5 . = ˛ ive diagram. Need the following formulas ↵ order does notismatter check A. and argue for that in˛˛ general) at the node A and consider the (The flux X which based...atneed thistonode g =g 5
5
0
0
0
0
0
0
e5 1
0
10 80
1
20 ˛ 2 ˛g =g 0
Ee ( the )(g1following , · · · , g8 , formulas ···) Could make in commutative diagram. Need ter ... need to check and argue for that general) 3 3 ↵ 1 1 Xe 1⇣Need (· ·⌘·the , gfollowing (· · · , ↵ g5 , · X· ↵· ) =(· · ·(g, g75g,5· · ·g)8 g6=) (g1(·, ·· ·· ·, ,↵g5 ,1·g·5·,)·˛˛˛·g· )=(12) g g 5, · · · ) = diagram. formulas ˛ 1 5 e
X↵ e5 1
Ee5 X↵ e
5
1
= Xh↵h e 1
(· · · , g5 , · · · )
6
=
0
5
0
0
0
0
0
As the shifts cancel in the 5delta function, we have X↵ (11) 1 e 1
5
(· · · , ↵
1
g5 , · · · )
(12)
1
1
10 80
˛ g2 = g20 ˛ ˛g =g 0 3
3
1
1
˛ ˛ g = g 1g ˛ 4 70 4 ˛ ˛g =g 0 5
5
˛ ˛ g = g 1g ˛ 4 70 40 ˛ ˛g =g 0 5
5
EeX (h↵h )(g · ,↵g↵801,(E · · ·e)5 ) 1 10 , · · X 5 e e Ee5 (Xe5 1 )(g10 , · · · , g80 , · · · ) 6 ↵ to check and argue for that in general) 1 Eere5... (Xneed )(g , · · · , g , · · · ) = (g70 g50 g801g60 ) (g5 1 , · · · , g5 , · · · )˛˛ g = g 0 g˛ 01 ˛ 1 0 0 1 following 8 formulas e5 Need the = (g7= g800gg60 10 )g 0 g(g0 )1 , (g · · ·, ·,·g˛˛· 5,1,↵· ·1·1g)˛,8·g· · ˛˛)=˛˛gg4g==0 ggg70011gg401˛ ˛ 0 g 0 (g diagram. 5 7 5 8 6 1 ˛ g2 = g205˛ 1 ˛ ˛ 11 810 80 ˛ g˛ g ==gg 11g ↵ ˛ ˛ ˛g = g 0 ˛ ˛˛4 4 7700 4400 ˛ 1 1 5g = g 5 0 (X )(g ˛g =g 0 ˛g = ˛g ˛ 1 0, · · · , g 0, · · · ) ˛ ˛{g = 0 2 2 1 ˛ 2 g , · · · ) (13) (gXe7↵50 g50(·g· ·810, gg6,0 ·)· · )(g 3 3 ˛ 2 ˛ 8 ,··· ,↵ ˛ g 5=edgere ↵g501 g 0 ˛ 1 ˛ g = g g = g g 1= 5 5 ˛ g = g3 0 30 (· · · , ↵ 5g5 , · · · ) ˛ 1 1 ˛ g = g 1(12) 5 10 80 e1 3 3 g 1 5 ˛ 1 ˛ 4 ˛ 70 40 g , · · · ) (13) { edgeref1b} g = g 0 g 0 g 0 g 0 ) (g , · · · , ↵ ˛ ˛ ˛ ˛ 0 1 5 8 6 5 the shifts cancel in the delta function, we have 2 2 0 1 ˛ g1 = g10 ˛gAs ˛ ˛ g = g 8 0 g ↵0 1g 0)(g ˛ ˛ gg4Ee=5 (X ˛ 7e 4 510 , · · ·5, g80 , · · · ) g = ˛ 0 8.3 2˛ g =2 gmove 0 5
(X↵ e5 1
)(g10 , · · · , g80 , · · · )
2 ˛ 2 ˛g =g 0
3
˛ g 3= g
the1 delta function, we have
1 e0 gdelta , ↵ have g , · · · )˛ 0 ) (g , · · ·we 0 g 0 gfunction,
1
5
0
1 5 1 1, · · · , ↵ (g g g g ) (g g , · · · )˛˛ g = g 0 g 01 0 0 0 0 h↵h ↵ 3 3 7 5 8 6X 1 1 X 1 (E 5 ) ˛ 1 1 8 e5 Should we mention that? Just give ethe result? e5 ˛ 6 g =g
=
=
(g70 g50 1 g(13) g ) 80 60
˛ ˛ g = g 1g ˛ 4 70 40 ˛ ˛ 2 0 2 ˛g =g 0 ˛ 5 5 ˛↵g 1=gg , 0· · · )˛˛ (g , · · · , g = g 0 g 01 ˛ 3 {1 edgeref1b} 53
1
Same mechanism in (3+1)D
Subdividing a tetrahedron. (Gluing a 4-simplex)
Subdividing a triangle.
5. Measure
The
k l klm m π E(π) Xe } = f Xe measure: dualize {X ALe ,measure
{X
The AL measure can be characterized by ations of the positive linear functional µAL ηAL (A) ≡ 1 π E(π) network basis, which itself can be generated Excitations via AL-vacuum plication of holonomy Spin network basis observables ψ{j} ({g}) holonomy observables state ηAL ({g }) ≡ ψ{j}e(A) = 1. ψ{j}The (A) · holonomies ηAL (A) ηAL (A) vacuum ≡ 1 with {j} denoting the set of representations la µ(ψ{j} ) = δ{j},operators and lead t edges, are multiplication ψ{j} (A) network = ψ{j} (A)basis. · ηAL (A) More precisely, the functional fined on this basis as µAL (ψ{j} ) = δ∅,{j} , wh µ(ψ{j} ) = δ{j}, vanishing if and only if all representation labe AL measure ial. To construct the new measure we will pr
leaves $ define and are inl one–to–one correspondence w ental cycles of Γ. We the cycle c associ! k klm m ∗ {X , X } = f X π (1) e e e the cycle c! peaked on fundamental cycles of Γ. We define the leave $ by choosing the same starting vertex BFinvacuum and atedexcitations to the leave $ by choosing the same startin ven the entation cycle as for the leave (edge) $. !E(π) for the and orientation for the cycle as for the leave ( δ(g ), First consider fixed triangulation. Gauge fix with maximal tree. ! rom $, all other edges in the cycle are elements ! ) Apart from $, all other edges in the cycle are e dtree. g! is We the will denote t of the tree. We will denote the holonomy associate to asso Leafsthe are in one-to-one correspondence with holonomy fundamental ηAL (A) ≡ 1 cycles c! by . the cycle C . A BF vacuum state that is pe ! e c! by C! . A BF vacuum state that is peaked on a locally and(does globally flatonconnection Γ isfluxes. give BF vacuum not depend tree): constant! infor (Gauss-) ! A) ≡ 1globally flux yrated and flat connection for Γ is given in the holonomy representation by η! ˙ BF = ! ! δ(C! ) = ansported my representation = the ˙ δ(g! ), and ψ{j} (A) = ψ{j} (A)where ·by ηALη= ˙(A) denotes gauge–fixed BF !) = ! δ(C !expression group element associated and to theg!leave $. = by denotes expression is the groupthe gauge–fixed excitations bythe action of exponentiated A) =) =ψ{j} (A) · ηAL (A) WeObtain can basis nowof consider exponentiated integra ψement δassociated {j} {j}, to the leave $. the expo(integrated) fluxes: observables associated to the leaves $ and tran holonomy from root "flux an now consider the exponentiated integrated e t is the to the χ root r.:=These act as right translations b ! to source of leaf R η = ˙ δ(g α ). = δ{j}, % % {α! } {Adt! (α! )} BF elements Adt (h! ).$ and We will therefore denote th bles associated to the leaves transported from the % "R{Ad (h )} , where nentiated flux observables by r.have These act as Excitations right translations by group doot we R{Ad η bytogroup = ˙theelements δ(g:% αpath are labelled % ). going f n integrated flux t! (α! )} BF holonomy associated unique scuum Adt!vector (h ). We will therefore denote the expo! acement in is sourceCan vertex of the leave $% to the root r, and group average at root. path. −1 d flux observables by R , where t is the ! defined Ad (h) = ghg . The action on the vacu {Ad (h )} g t! ! !
t!
!
µ(ψ{j} ) = δ{j}, Measure
{Xek , ge } = ge T k
Basis of excitations labelled by group elements.
χ{α! } := R{Ad η = ˙ BF (α )} t ! k ! l klm m {Xe , Xe } = f
π
Define measure in the same way as the AL measure:
"
µBF (χ{α! } ) =
δ(g% )
" %!
"
δ(g%! ) d
"
Bohr compactification of the dual to the group
%
δ(g% α% ).
ψ{j} (A) = ψ{j}
(heuristically similar to SU(2) quantum group)
(0.173)
leads to compactification of Z) (A) (For · η G=U(1) (A) AL
µ(ψ{j} ) = δ{j}, ˜ δ(α% )
(0.174) χ{α := R{Ad η = ˙ Choice as! }group delta: t! (α! )} BF
g%
%
Choice as formal Kronecker Delta:
%
|L|
Xe
δ(g% α% ).
E(π)
ηAL (A) ≡ 1
:= R{Adt! (α! )} ηBF = ˙
"
δ(g% α% ).
%
Same inner product on fixed" graph
" # −1 $ µAL (χ{α! } ) " = δ α % αas% with , Haar measure: %
"
!
R{α!−1 α! } !
" %
δ(g% )
" %!
˜ %) δ(α
=
%
|L|
δ(g%! ) d
g%
" # −1 $ = δ α" % α% , %
Shows independence of choice of tree.
Can now attempt to construct the continuum limit as an inductive limit of Hilbert spaces in the same way as in standard LQG. Need to make sure that inner product is cylindrically consistent, i.e. does not depend on the choice of triangulation it is computed on. For the Bohr compactification this is the case.
Bohr compactification of the dual to a group [Soltan 06] Bohr compactifications of quantum groups C^* algebraic framework Dual of a group formulated as a quantum group
Useful for C^* algebraic construction of Hilbert space.
measure and the inner product. To cure these diencies for the inner product we can choose a (heat el) regularization for the delta functions, and divide nner product of two states by the norm of a reference we choose group we need to modify product e (in Ifthis case the BFdelta, vacuum). That is, the we inner define a to make it cylindrically consistent. some regulated delta function: ifiedWith inner product as group (see also [14])
Compactification of excitations
!ψ1 , ψ2 "ε , !ψ1 , ψ2 " = lim ε→0 !ηBF , ηBF "ε !
[Bahr, Fleischhack, hopefully to appear very soon] with pure BF, dual refinements (23)
re ε indicates regulator. Heuristicallythe equivalent to a Bohr compactification. Need exponentiated fluxes. et us also mention a notion of spatial diffeomorphism riance for the vacuum. Consider a triangle subdid byLesson: a 1–3Inductive move but with theconstruction inner vertex Hilbert space putsplaced discreteat topology on excitations. different positions leading to two states ψ and ψ2 . in triangulations with group labels. For AL: dual graphs with discrete labels. For BF: 1(d-2) objects ure 1 shows that there exists a common refinement indeed the two different ways to obtain this refinet starting with triangle via ψ1 and ψ2 lead tocontinuum the [Okolow 13] theConstructs (inductive Hilbert space) limit e state. Thus the inner product identifies the two for non-compact configuration spaces R^N es which differ by a vertex translation. This nicely via a projective limit of density matrices (i.e. functionals). cts the notion of vertex translations as a diffeomorm symmetry [27] (see also for acompactification similar mechaResults also[28] in a Bohr / almost periodic functions. m for the physical inner product of (2 + 1) gravity). he AL projective limit construction [4] leads to the
Deformations/ Generalizations Bohr compactification is expected to lead to appearance of quantum group.
Dualising the Koslowski shift (introduction of background triad): leads to shift of background connection. A homogeneous curvature requires flux dependent background connection, which requires change in (group) measure to keep holonomies as unitary operators. Thus one expects a deformation of the symmetry group. Derivation of quantum group?
Can we use non-commutative flux representation and compactify this space?
2
µ(ψ{j} ) = δ{j},
(Spatial) Diffeomorphism symmetry 2 2
Common refinement.
2
Identifies BF vacuum on different triangulations as one (spatial diffeomorphism invariant state).
Example of refinement via 1–3 and 2–2 moves. The of refinement via 1–3isand 2–2 than moves.the Theleftmost one, and st triangulation finer ulation iscommon finer thanrefinement the leftmost ∆ one, and nts the of the triangulations on Example of refinement and 2–2 moves. The mmon refinement ∆ of via the 1–3 triangulations on Finite (spatial) diffeomorphisms: and on the bottom. triangulation is finer than the leftmost one, and he bottom.
s the common refinement ∆ of the triangulations on nd on the bottom. (Right shift
!
R{α!−1
RCe ∼ EF
with holonomy around vertex.)
refinement that we are going to useNeeds is exponentiated flux. (Diffeos and exponentiated fluxes not notion of refinement that we are going to use is r moves. Pachner moves for triangulated otion of refinement that we are going to use is weakly continuous.) onsurfaces Pachner moves. Pachner moves for triangulated consist of the so–called 1–3, 3– Pachner moves. Pachner moves forThe triangulated ment via 1–3 and 2–2 moves. mensional surfaces of the so–called 1–3, 3– s, and two such triangulated are1–3, ensional surfaces consistconsist of thesurfaces so–called 3– Might explain `finite action interpretation’ of Hamiltonian. finer than the leftmost one, and hic if one can be transformed into the 2–2 moves, and two such triangulated –2 moves, and two such triangulated surfaces aresurfaces are nement ∆ of thecan triangulations onwe number of these moves. Since here eomorphic if one be transformed into the meomorphic if one can be transformed into the due to simplicity In (3+1)d: complications .refining operations, we will consider onlyhere we a finite number of these moves. Since by a finite number of these moves. Since here we [Zapata 96, BD & Ryan 08] constraints, but doable. s the 2–2 Pachner moves (i.e. discard the ested in refining operations, we will consider only erested in refining operations, we will consider only
!
On Hamiltonian dynamics and simplicity constraints Good news! Some Hamiltonian constraints are already there! Dual regularization mechanism to ThiemannHamiltonian (with adjusted ordering).
•non-graph changing (interpretation as tent move) •for `flat or homogeneous sector’ (stacked spheres) and in (2+1)D free of discretization anomalies
[classical: BD & Ryan 08, Bonzom & Dittrich 13 quantum: Barrett , Crane 96, Bonzom 11, Bonzom, Freidel 11, ]
•`graph’ changing: Pachner or Alexander moves: spin foam dynamics [Example: BD, Steinhaus 13] •dynamics can be understood as first refining and then imposing dynamics •could impose dynamics by gluing spin foam amplitudes [Alesci, Rovelli 10]
On Hamiltonian dynamics and simplicity constraints Coming back to “spin foam amplitudes give the physical vacuum” (Why are we not already with a very physical vacuum? This BF vacuum has constant distribution in twisted geoemtries.)
[Speziale, Freidel 10]
Can we get (Regge) physical vacuum with (almost) constant distribution in Regge like geometries, and (some) suppression of non-Regge geometries? This however is a non-local problem (Area constraints are non-local). Tautological claim : Imposition of simplicity constraints making everyone happy equivalent to Continuum limit, i.e. with construction of physical vacuum. We therefore need coarse graining and refining ... ... coming in the next ILQGS talk.
Conclusions and outlook A new Hilbert space for Loop Quantum Gravity! Lots of interesting mathematical structures to fill in. Very near to spin foam dynamics. Facilitate extraction of low energy physics, cosmology etc. Another view on quantum geometry and simplicity constraints. Many generalizations possible. Quantum deformations of SU(2)? Does it allow SL(2,C) Hilbert space, supporting self dual variables?
LARGER PROGRAM: Apply this to (non-trivial) fixed points of renormalization flow!
