Twistors in spin networks and loop quantum gravity - …people.maths.ox.ac.uk/lmason/New...

s(l)t(l)

τs τt

Nl Nl

jl

ξl

NlNl

Simone Speziale Oxford, 3 Jan. 2017

Twistors in spin networks and loop quantum gravity

X12

X34

X24

X13X14

23

21

12

32

14 24

34

41

31

13

43

42

X32

Loopquantumgravity:abackgroundindependentapproachtoquantumgravitybasedontheuseofrealAshtekarvariablesandspinnetworks

• Mainresults:spaceisquantised,geometricoperatorshavediscretespectraandnon-commutingproperties;applicationstocosmologyandblackholes

• Mainopenquestions:recoveringsmoothgeometriesfollowingEinstein’sequationsinthesemiclassicallimit;systematicallycomputetransitionamplitudes

Each`Feynmandiagram’ofthetheorycorrespondstoatruncationtoafinitenumberofdegreesoffreedom:thistruncationdescribestheevolutionofafuzzyquantumgeometry.

Inrecentyearsmanypropertiesofthisfuzzyquantumgeometryhavebeenclarified,andtwistorshaveemergedasnaturaltoolstoprovideacovariantdescriptionofthegeometry,toansweropenquestions,andtoleadtonewresearchdirections.

Itisarichresearcharea,andIwillonlypresentsomeisolatedresults,morecloselyrelatedtotwistors,withoutattemptingduetothelimitedtimeacomprehensiveandcoherentoverview.

1. SU(2)spinnetworksrepresentacollectionoffuzzypolyhedragluedtogetheralongfaces

2.Theirspacetimeembedding,basedonSU(2)SL(2,C),isnaturallydescribedbytwistors�!

3.UsingtwistorstoconstructSU(2,2)spinnetworks

Intro and overview

I. Loop Quantum Gravity

• number of quanta • momenta • helicites

• number of quanta and their relations • volumes of regions • areas of interconnecting surfaces

Fuzzy spinning particles

dynamics described by Feynman diagrams

Fuzzy linked polyhedra

dynamics described by spin foams

Quantum field theory Loop quantum gravity

Spin networks and quantum geometry

QFT LQG

F = �nHn H = �

�H�

|n, pi, hii ! quanta of fields |�, je, ivi ! quanta of space

Spin networks are eigenstates of geometric operators such as surface areas

• spins 7! quanta of area

• intertwiners 7! quanta of volumes

geometric operators turn out to have discrete spectrawith minimal excitations proportional to the Planck length

Key result

Speziale — LQG and polyhedra Motivations and overview 8/38

diagrams can be organised in PT or EFT

diagram organisation not yet established! hands-on approach for the moment: compute in a given truncation, then change truncation

Quanta of space in loop quantum gravity

Classical limit of a spin network

Onthefullgraph:

L2[SU(2), dµHaar] ! T ⇤SU(2)

H� ! T ⇤SU(2)L//~Cn

theclassicalphasespaceofaspinningtop

turnsouttodescribetheintrinsicandextrinsicgeometryofadiscretisedspace-likehypersurface

Onasinglelink:

Onasinglenode: Invh⌦i Vji

i! SKM acertainphasespacecorrespondingto

shapesofpolyhedra

H� = L2[SU(2)L/SU(2)N , dµHaar] = �jl

⇣⌦n Inv

⇥⌦l2n Vjl

⇤⌘|�, jl, ini 2

(aclassicalcounterparttoPenrose’sspin-geometrytheorem)

II. SU(2) singlets and polyhedra (Freidel-Krasnov-Livine ’09, Bianchi-Dona-S ’10)

Itfollowsfromthreetheorems:

• Kapovich-Millson:reductionbytheclosureconstraintgivesasymplecticmanifoldwithactionanglevariablesthediagonalsanddihedralanglesofthepolygon

• Guillermin-Stenberg:(forcompactorbits)quantisationcommuteswithreduction

• Minkowski:aclosedsumofvectorsinR3definesauniquebounded,convexpolyhedron

OA1n1

A2n2

A3n3

A4n4

SU(2)singletsandpolyhedra

⇥ni=1S

2ji �! ⌦iVji

# ~C = 0 #

SKM �! Invh⌦i Vji

i|j1, j2, . . . ; j12, j123, . . .i

j1

j2j3

j12j123

. . .

SKM :=�⇥i S

2ji

�//~C

dimSKM = 2(n� 3) virtual spins = n� 3

Polyhedrareconstruction

Geometry of polyhedra

Explicit reconstruction procedure: Xl fiÑ edge lengths, volume, adjacency matrix

For F ° 4 there are many di↵erent combinatorial structures, or classes

F “ 6

Dominant:

Codimension 1:

Codimension 2:

Codimension 3:

‚ The classes are all connected by 2-2 Pachner moves�@

�@Ø

@� @

�(they are all tessellations of the 2-sphere)

It is the configuration of normals to determine the class

‚ The phase space SF can be mapped in regions corresponding to di↵erent classes.

´ Dominant classes have all 3-valent vertices.[maximal n. of vertices, V “ 2pF ´ 2q, and edges, E “ 3pF ´ 2q]´ Subdominant classes are special configurationswith lesser edges and vertices, and spanmeasure zero subspaces.[lowest-dimensional class for maximal number of triangular faces] 3d slice of S6, cuboids blue

Speziale — Twistors and LQG From loop quantum gravity to twisted geometries 25/33

(Bianchi-Donà and S ’10)

AdaptinganalgorithmbyLasserre,numericalexplicitreconstruction

Bianchi, Doná, and Spezialenumerical reconstruction ’11

HMH, Han, KamiÒski, Riello curvedtetrahedron reconstruction ’16

O4O3O2O1 = 1l, Oi œ SO(3) ≈∆

Bianchi and HMH Bohr-Sommerfeldtetrahedral volume spectrum ’11

0 2 4 6 8 100

2

4

6

8

10

12

14

HMH described full phase diagramof pentahedral adjacencies ’13

1 2 3 4 51

2

3

4

5

2

Minkowskitheorem:aconvexflatpolyhedronatfixedareashas2(n-3)degreesoffreedom,andcanbeuniquelyreconstructedfromthenormals

Noticethatapolyhedronhasingeneralmanypossibleadjacencyclasses:theadjacencymatrixandvalenceofeachfaceisuniquelydeterminedbythenormals

{ ~Ni}��X

i

ji ~Ni = 0

• Guillermin-Stenberg:(forcompactorbits)quantisationcommuteswithreduction,andsymplecticreductionisobtaineddividingbytheorbitsofthecomplexifiedgroup

4. Spinors 63

the Regge action. This is an encouraging result for the model, however the limiting procedure is delicate tohandle on a full triangulation, and it has been argued that only flat solutions are compatible with the saddlepoint equations of the large spin limit [93]. This has started a debate in the literature on whether non-trivialcurvature is properly accounted for, and if not, what is the problem with the model or the limiting procedure.See [94, 29, 44] for some references. Furthermore, the result relies on the special geometric properties of the 4-simplex, and it is not clear how to interpret the most general spin foams that are not dual to a triangulations[84, 95]. These partial results show the importance of improving our understanding of the dynamics alreadyat the classical level. Actions for twisted geometries and their relations to the Regge action on-shell ofthe shape matching conditions have been studied in [51, 88, 96, 75, 97, 98], but there is as of yet no clearconsensus on the meaning of curvature and dynamics away from the shape matching subspace, nor on theo↵-shell role of the torsion a priori kinematically present in the theory [99]. An interesting development inthis sense is the alternative interpretation of the same phase space in terms of discrete geometries with torsionalong the edges proposed in [100]. Studying the dynamics of the non-shape-matched configurations is alsoimportant to understand whether they admit a continuum limit reproducing general relativity, or whetherthe latter property can be satisfied only by the Regge configurations. Lastly, spinors have also found manyapplications in the study of the quantum dynamics of spin foam models, e.g. [101, 102, 103, 104, 105, 106].

4.1 Polyhedra as a Grassmannian

2(n � 3) variables parametrizing SKM can be given by KM variables, or as we have seen by angles �ij up totheir linear dependencies, or by the scale-less triple products wijk, up to their linear dependencies encodedin Plucker relations.

Alternative to the KM parametrization, another simple one that already solves the linear dependenciesis the one in terms of cross ratios. This was pointed out in (Krasnov-Livine-Freidel 09), and it is easy toprove: first, we use GS theorem:

SKM ' (CP 1)n//~C ' (CP 1)n/SL(2,C) (3.111)

Then, we recall that there is a bijection (⇣1, . . . ⇣n) = (g, Z1, . . . Zn�3) where g 2 SL(2,C) and

Zi =(⇣1 � ⇣2)(⇣3 � ⇣i)

(⇣1 � ⇣3)(⇣2 � ⇣i)(3.112) {Zdef}

are the cross ratios (or the alternative (⇣1 � ⇣3)(⇣2 � ⇣i)/(⇣2 � ⇣3)(⇣1 � ⇣i)). (Notice that while the crossratios are perfectly rotational invariant quantities, the reconstruction of the polyhedron from them isn’t;one fixes three normals ⇣i arbitrarily, typically 1, 0, 1; then there is a unique boost that closes the orientedtetrahedron (1, 0, 1, Z1); or alternatively in real terms, one fixes one normal parallel to z, one to lies in the(xy) plane, then its longitude as well as the two components of the third normal are derived from the closurerelation. All this just to stress that I do not see my projective geometry in the use of these cross ratios asvariables).

The same cross ratios (3.112) can also be understood as solutions to a Plucker embedding of SKM as aprojective variety. Indeed, (Etera-Laurent 09) show that

GrC(2, n) ' PC2n/SL(2,C) (3.113) {EL}

which I will prove below. For the time being let us observe that on the LHS we have a Grassmannian, whichis known to be a projective variety in P (Cn ^ Cn), defined by the quadratic Plucker relations. Let me callFij = z0

i z1j � z0

j z1i the coordinates for Cn ^ Cn, and P their set of Pluckers FijFkl � FikFjl + FilFjk = 0.

Then

Zi =F12F3i

F13F2i, (3.114) {ZF}

as I will also prove below. Then, on the RHS of (3.113) we can use

PC2n/SL(2,C) ' PCn ⇥ SKM (3.115) {totti}

which trivially follows from PC2n ' PCn ⇥ (CP 1)n. This shows that PCn ⇥ SKM is a projective variety(which I will later review how it gets identified with the space of ‘framed’ polyhedra at varying shapes, areas

• Parametrizationintermsofcross-ratios:

• PhasespaceofshapesasaGrassmannian(Freidel-Krasnov-Livine’09)

4. Spinors 63





SKM ' (CP 1)n//~C ' (CP 1)n/SL(2,C) (3.111)


Zi =(⇣1 � ⇣2)(⇣3 � ⇣i)

(⇣1 � ⇣3)(⇣2 � ⇣i)(3.112) {Zdef}



GrC(2, n) ' PC2n/SL(2,C) (3.113) {EL}


i z1j � z0


Then

Zi =F12F3i

F13F2i, (3.114) {ZF}




(andidentificationofthePluckerembedding

ofSKMinaprojectivespace)

4. Spinors 63





SKM ' (CP 1)n//~C ' (CP 1)n/SL(2,C) (3.111)


Zi =(⇣1 � ⇣2)(⇣3 � ⇣i)

(⇣1 � ⇣3)(⇣2 � ⇣i)(3.112) {Zdef}



GrC(2, n) ' PC2n/SL(2,C) (3.113) {EL}


i z1j � z0


Then

Zi =F12F3i

F13F2i, (3.114) {ZF}




Polyhedraandprojectivegeometry

AnalyticaladjacencymatrixofpentahedronfromDesargues’crossratios?(wipHaggardandS’)

Many alternative parametrization possible: KM variables, or normals,

or dihedral angles, or triple products, or cross-ratios

SKM :=�⇥i S

2ji

�//~C

dimSKM = 2(n� 3)

OA1n1

A2n2

A3n3

A4n4

III. Twisted geometries and twistors (Freidel and S ’10, S and Wieland ’12)

s(l)t(l)

τs τt

Nl Nl

jl

Aspinnetworkisthenacollectionofpolyhedragluedtogether

TheassociatedintrinsicgeometryisageneralisationofRegge’s:afaceseenfromadjacentpolyhedrahasauniquearea,butcanhavedifferentshapes

Whereistheinformationontheextrinsicgeometry?

1.EmbedtheSU(2)spinnetworksinSL(2,C)(thisγ-dependentembeddingisthediscreteequivalentoftheuseofrealAshtekar-Barberoconnection)

2.DefinetheextrinsiccurvatureàlaReggeviatheboost`dihedralangle’between4dnormalstothepolyhedra

Twistedgeometries

Geometry on the graph

Each classical holonomy-flux configurationon a fixed graph describesa collection of polyhedra and their embedding

‚ each polyhedron is locally flat: curvature emerges at the faces, as in Regge calculus

‚ extrinsic geometry encoded in the parallel transport g between adjacent polyhedra

the structure defines a piecewise discrete metric, in general discontinuous:two neighbouring polyhedra share a face with same area but di↵erent shape

‚ shape-matching conditions can be written explicitlyDittrich and S ’08

This subcase describes ordinary Regge calculusj

k l

i

shape-matchingskinematical LQG ›Ñ Regge geometries

(twisted geometries)

Speziale — Twistors and LQG From loop quantum gravity to twisted geometries 26/33

Twistorshaveprovedcrucialtoaddressthesetwosteps

Theresultisasymplecticreductionfromflattwistorstotwistedgeometries

Ai = �!0i � 1

2✏ijk!

jk(E)

Twistorstotwistedgeometries

Takeacollectionofflattwistors,eachassociatedtoahalf-linkoftheorientedgraph

Imposingthefollowingsetofconstraintsgivesasymplecticreductiontothephasespaceofloopgravity:

• oneachlink,helicityanddilatationmatching:definesthephasespaceT*SL(2,C)

• oneachhalf-link,simplicityconstraintsselectingaspace-likeplane:reducestoAshtekar-BarberoT*SU(2)

• oneachnode,aclosureconditionfortheSU(2)generators

LQG phase space: smearing

‚ Di↵eomorphism covariance requires a distributional smearing of the Poisson algebra

A on 1d paths E on 2d surfacesg “ exp

≥lA X “

≥l˚ E

‚ Poisson structures:

continuous ùñ on each link :

tAiapxq, Aj

bpyqu “ 0 tgAB , gC

Du “ 0

tAiapxq, Eb

j pyqu “ �ij�ba�

p3qpx, yq tXi, gu “ ´⌧ig

tEai pxq, Eb

j pyqu “ 0 tXi, Xju “ ✏ijkXk

Non-commutativity introduced by the discretization (same happens in LGT)

‚ LQG phase space on a fixed graph:

P� “ T˚SUp2q

L

‚ Quantization gives the holonomy-flux algebra on

H� “ L2rSUp2q

Ls

In the same sense as pL2rRs, rq, ps “ i~q is a quantization of pTR, tq, pu “ 1q

Speziale — Twistors and LQG Motivations and goals 9/30

Zl

Zl

T ⇤SU(2)L//~CnT2L//{Cl, Sl, ~Cn} '

Conversely,asimpleSchroedinger-likequantisationoftwistorsgivesSU(2)spinnetworks

(seenasembeddedinunitarySL(2,C)irrepsoftheprincipalseries,)⇢ = �k, k = j

TwistorspacecarriesarepresentationofSL(2,C),whichisHamiltonianwrt

takes the form of a Lie derivative of the group character, and when interpreted in terms of spinors, it acts asa boost in mixing the source and target spinors. In spite of its simplicity, its action on holonomies and fluxesis completely non-linear, and has no resemblance with usual dilatations. This is unavoidable, as the discretephase space has compact directions, so there is no usual meaning of linear dilatations. On the other hand, thegenerator has the property of preserving the symplectic structure, and all transformed quantities recover theexpected behaviour in the continuum limit.

All our results, computing the action of SU(2, 2) dilatations, the relation between the di↵erent boost dihe-dral angles in the literature and proving the equivalence of di↵erent secondary simplicity constraints, as wellas computing the finite action of the discrete holonomy-flux dilatation, rely heavily on the use of twistorialformalism, and are a demonstration of its usefulness to loop quantum gravity.

2 Twistors and twisted geometries

A twistor can be described as a pair of spinors, Z↵ = (!A

, i⇡

A

) 2 C2�C2⇤ =: T. It has a dual, Z↵

= (�i⇡

A

, !

A),which defines a pseudo-Hermitian norm of signature (+ +��),

Z

↵

Z

↵

= 2Im(⇡A

!

A), (1)

preserved by SU(2,2) transformations. It is well-known [8] that these transformations can be realized by Hamil-tonian vector fields, if we equip the space with canonical Poisson brackets,

�Z

↵

, Z

�

= i�

↵

�

. (2)

In Penrose’s abstract index convention, the generators of SU(2,2) can be written as

J

IJ = !

(A⇡

B)✏

AB + cc, P

I = i⇡

A

⇡

A

, C

I = i!

A

!

A

, D = Re(⇡A

!

A), (3)

and their finite action is realized via the exponential map,

exp(!IJ

M

IJ) . f := exp(!IJ

{M IJ

, ·})f. (4)

Half the twistor norm, s := Im(⇡A

!

A), is called helicity of the twistor,1 since

W

I = �sP

I

, (5)

where W

I := 1/2✏IJKL

P

J

J

KL

is the Pauli-Lubanski vector. The complex scalar ⇡A

!

A = D + is, for which wealso use the index-free notation [⇡|!i, is invariant under the Lorentz subalgebra.

The representation of the conformal algebra constructed in this way is not the most general one, since thealgebra has 15 dimensions, whereas the carrying space has only 8 dimensions. In fact, the generators (3) arenot independent: P I and C

I are null and related to D and the Lorentz generators by

2P [IC

J] = (D + s?) JIJ

, P · C = �(D2 + s

2). (6)

Accordingly, the three Casimir operators of su(2, 2) (see the Appendix for definitions) are not independent:they are all proportional to the only conformally invariant quantity in , the helicity:

C

(2) = 6 s2, C

(3) = �6 s3, C

(4) = 6 s4. (7)

Hence, the representation of the algebra on twistor space is special, and furthermore reducible, with irreducibleslabelled by s. In this sense, Penrose refers to twistors as the spinors of the conformal group.

Irreducible representations with independent real values of the Casimirs can be obtained working withthe larger carrying space 2 made of pairs of twistors (Z, Z), with generators constructed by linearity. Thisspace has now 4 conformally invariant quantities, given by the two helicities plus the pseudo-hermitian productZ

↵

Z

↵, and the Casimirs are independent functions of these four variables. 2 has been considered in the twistor

1In the twistor literature, the helicity is usually given by the real part. The di↵erence comes from the extra i used here in thedefinition of the twistor. In this way we can match Dirac’s conventions for a bi-spinor, and bridge more easily with the notationused in loop quantum gravity. Notice also that we use metric conventions with mostly plus, thus we map vectors to anti-hermitianmatrices, see Appendix A.

3





, i⇡

A


= (�i⇡

A

, !


Z

↵

Z

↵

= 2Im(⇡A

!

A), (1)


�Z

↵

, Z

�

= i�

↵

�

. (2)


J

IJ = !

(A⇡

B)✏

AB + cc, P

I = i⇡

A

⇡

A

, C

I = i!

A

!

A

, D = Re(⇡A

!

A), (3)


exp(!IJ

M

IJ) . f := exp(!IJ

{M IJ

, ·})f. (4)


!


W

I = �sP

I

, (5)

where W

I := 1/2✏IJKL

P

J

J

KL


!




2P [IC

J] = (D + s?) JIJ

, P · C = �(D2 + s

2). (6)


C

(2) = 6 s2, C

(3) = �6 s3, C

(4) = 6 s4. (7)



↵

Z



3

Step1:T*SL(2,C)fromtwistors

Torepresentthecovariantholonomy-fluxalgebraofLQG,weseekarepresentationofthesymplecticmanifoldT*SL(2,C)(SandWieland’12)

T2?//C ' T ⇤SL(2,C)

C = ⇡A!A � ⇡A!

A :matcheshelicityanddilatationofthetwotwistors

BreaksSU(2,2)toSL(2,C)

Conn

ectio

nswi

thpr

ojec

tive

geom

etry

were

alrea

dyhi

ghlig

hted

inni

cepa

pers

byFr

eidel,

Kras

nov,

&Li

vine;

[e.g

.10

05.2

090]

Usin

gsp

inor

sthe

ysh

owed

expl

icitly

the

isom

orph

ism:

Fixe

d to

tal a

rea

Fixe

d to

tal p

hase

Spin

or’s

norm

s: ar

eas

Spin

or’s

phas

es: a

fram

ing

of e

ach

face

(rel

ated

to e

xtrin

sic g

eom

etry

)

A pr

ojec

tive

varie

ty: e

mbe

dded

in

via

spin

or b

iline

ars

and

Plüc

ker r

elat

ions

Que

stio

ns:

Wha

tist

hepr

ecise

relat

ionbt

wnth

eco

mpl

exGr

assm

anni

anpr

ojec

tivity

and

the

real

proj

ectiv

ityof

the

adjac

encie

s?Ca

nwe

analy

ticall

yre

cons

truct

the

adjac

ency

usin

gpr

ojec

tive

geom

etry,

eithe

rthe

com

plex

one

ofth

eF

’s,or

the

real

one

ofth

eW

ijk’s?

27

T2? :2twistors,eachwith(Z↵, Z↵) ⇡A!

A 6= 0

In terms of this decomposition, the Poisson brackets (A.1) read

{Li, Lj} = �✏ijkLk, {Li,Kj} = �✏ijkKk, {Ki,Kj} = ✏ijkLk,

{P I , JJK} = �⌘IJPK + ⌘IKP J , {CI , JJK} = �⌘IJCK + ⌘IKCJ , (A.4)

{CI , P J} = ⌘IJD � JIJ , {D,P I} = �P I , {D,CI} = CI .

In the main text we refer to the three Casimir invariants of su(2, 2). These are given by

C(2) = Mab

Mab = 2C1 � 2D2 � 4P · C, (A.5)

C(3) =1

8✏abcdef

MabM cdMef = 3DC2 � 6W · C, (A.6)

C(4) = Mab

M bcMcd

Mda

= 2C12 + C2

2 + 2D4 + 8JIJ

JJKP ICK

+ 8DJIJ

P ICJ + 8D2P · C + 4P 2C2 + 4(P · C)2, (A.7)

where

C1 =1

2JIJ

JIJ = L2 �K2 = 2(⇧2 + ⇧2), C2 =1

2(?J)

IJ

JIJ = 2K · L = 2i(⇧2 � ⇧2) (A.8)

are the Lorentz Casimirs, and

WI

=1

2✏IJKL

MJKPL (A.9)

is the Pauli-Lubanski vector, which satisfies WI

= �sPI

for a massless particle. In deriving C(4), we used theidentities

JIJ

JJKP ICK

= �1

2✏IJKL

JIJWKCL � 1

2JIJ

JIJP · C, JIJ

JJKJKL

JLI = 2C12 + C2

2. (A.10)

Casimir evaluations: with one twistor

C(2) = 6s2, C(3) = �6s3, C(4) = 6s4 (A.11)

With two twistors:

C(2) = 4(s2 + s2 + |⇢|2) + 2(s� s)2, (A.12)

C(3) = �6(s+ s)[(s� s)2 + |⇢|2], (A.13)

C(4) = 2(s+ s)2 + 4[(s� s)2 + |⇢|2]2 (A.14)

B SU(2) case revisited

C Formularium: to be removed

Twistor’s definitions:

Z↵ =

✓!A

i⇡A

◆, {Z↵, Z

�

} = i�↵�

, {⇡A

,!B} = �BA

, {!A,⇡B} = ✏AB (C.15)

and

⌘↵�

=

✓0 2

2 0

◆, Z

↵

= ⌘↵�

Z � = (�i⇡A

,!A) (C.16)

The i factor allows to see the twistor as a Dirac bi-spinor in the chiral basis, and it is used in the LQGliterature, as well as the bar over ⇡ to keep track of its anti-selfdual nature: ⇡A

LQG

= i⇡A

Penrose

. As a consequence,dilatation generators and helicity are inverted:

⇡! := ⇡A

!A = D + is, s =1

2Za

Za. (C.17)

2

Thegrittydetails



2P [IC

J] = (D + s?) JIJ

, P

I

C

I

= �(D2 + s

2), (6)

where ? is the Hodge-star. Accordingly, the three Casimir operators of su(2, 2) (see the Appendix for definitions)are not independent: they are all proportional to the only conformally invariant quantity in , the helicity:

C(2) = 6 s2, C(3) = �6 s3, C(4) = 6 s4. (7)



↵

Z

↵, and the Casimirs are independent functions of these four variables. 2 has been considered in the twistorliterature [11,12], among other things in relation to ambitwistor theory and to build representations of massiveparticles. But as shown in [13], 2 has another interesting property that links it to loop quantum gravity: itcontains T

⇤SL(2,C) as a symplectic submanifold, obtained by Hamiltonian reduction with respect to the firstclass constraint

C := [⇡|!i � [⇡|!i = 0, (8)

referred to as (complex) area-matching condition. Explicitly, we take

J

IJ = !

(A⇡

B)✏

AB + cc, h

A

B

=!

A

⇡

B

� ⇡

A

!

Bp[⇡|!i

p[⇡|!i

, J

IJ = �!

(A⇡

B)✏

AB + cc, (9)

and following the conventions of [13], we take opposite signs for the twistor brackets2 :

{⇡A

,!

B} = �

B

A

= �{⇡A

, !

B} (10)

It is easy to check that (9) are C-invariant. When the constraint (8) is satisfied, the generators are relatedby the adjoint action of h and thus (9) span a 12-d submanifold of 2, for which one recovers the Poissonbrackets of T ⇤SL(2,C) with J and J acting respectively as left-invariant and right-invariant vectors fields onthe group manifold, parametrized by h as a left-handed group element. This symplectic reduction holds provided[⇡|!i 6= 0, which we exclude from our analysis from now on, and is 2-to-1: there is a Z2 symmetry associatedwith the simultaneous sign flip of the spinors. See [13–15] for more details.

The symplectic manifold T

⇤SL(2,C) is the building block of projected spin networks [16], which appear asboundary states of covariant spin foam models [17, 18]. There, a copy of T ⇤SL(2,C) decorates each link of anoriented graph, and the orientation is used to associate unambiguously Z to the source node, and Z to thetarget. Each twistor is then subject to the primary simplicity constraints, which reduce the covariant boundarystates to those relevant for the SU(2) spin networks of loop quantum gravity. The constraints are deducedfrom a discretisation of the gravitational action in Ashtekar variables, and relate the tetrad to the bivector B

which is the field canonically conjugated to the connection. To represent the constraints at the discrete level,we introduce a time direction N

I(n) on each node n of the graph. In their linear form introduced in [19] (seealso [8, 20, 21]) the primary simplicity constraints read

N

I

B

IJ = 0, (11)

for all bivectors B associated with links sharing the node n, and imply simplicity of B, namely ✏

IJKL

B

IJ

B

KL =0. The bivector B is related to the Lorentz generators via B = ( ��?)J , where � 2 R is the Immirzi parameter.See [8, 22, 23] for details, and [24] for extensions to the case of a null hypersurface.

It is customary the fix the time gauge N

I = (1, 0, 0, 0), but the construction extends to an arbitrarygauge [22]. In this gauge, we identify the left-handed generators with 3-vectors ⇧i, whose real and imaginaryparts are rotations L

i and boosts K

i, via ⇧i = (Li + iK

i)/2. The normal N I is conserved by the canonical

2This is convenient for the geometric interpretation of the theory. Switching to the alternative conventions with equal-signbrackets also used in the literature is straightforward, via the map !A 7! ⇡A, ⇡A 7! �!A.

4

4

The functions ⇧, ⇧e

and h are related by

⇧

e

= �⇡!

⇡e

!e

h⇧h�1, (11)

and span 14 out of 16 dimensions of T2. They obey the Poisson brackets�

⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h, (12a)�

hA

B

, hC

D

= � 2C

(⇡!)(⇡e

!e

)

�

✏AC

⇧

BD

+ ✏BD

⇧

e

AC

�

. (12b)

On the constraint hypersurface C = 0, two key properties hold: Firstly, the adjoint representation relates thefluxes, ⇧

e

= �h⇧h�1. Secondly, the components of the holonomy commute. Hence, we recover the Poissonalgebra of T ⇤

SL(2,C) with ⇧ and ⇧

e

as the (chiral) left- and right-invariant Hamiltonian vector fields on thegroup manifold:

�

⇧

i

,⇧j

= �✏ij

k

⇧

k

,�

⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h,�

hA

B

, hC

D

�

�

C=0= 0. (13)

In fact, we can easily see that this procedure amounts exactly to a symplectic reduction T ⇤SL(2,C) '

SL(2,C) ⇥ sl(2,C) ' T2//C. On the C = 0 constraint hypersurface, the Hamiltonian vector field XC

= {C, ·}generates the orbits

exp (zXC

+ zXC

) : (!,⇡,!e

,⇡e

) 7! (e

z!, e�z⇡, ez!e

, e�z⇡e

), z 2 C. (14)

The functions ⇧,⇧e

and h are invariant under such gauge transformations, and thus span the space obtained bysymplectic reduction.

Let us add two more remarks to complete the analysis. First, the map between twistors and holonomy-fluxvariables is 2–to–1, since exchanging spinors as

(!,⇡,!e

,⇡e

) 7! (⇡,!,⇡e

,!e

) (15)

leaves both holonomy (10) and flux (2) unchanged. Hence, to arrive at the reduced phase space, we alsohave to divide out this residual Z2 symmetry (15). Second, because of the restriction ⇡! 6= 0, the spinorialparametrization cannot cover the submanifold of (h,⇧) : ⇧

AB

⇧

AB

= �Tr⇧

2= 0, and what we truly find

is T ⇤SL(2,C) removed from all its null configurations. The complete isomorphism could be defined through

a suitable treatment of the degenerate configurations, see e.g. the analogue situation in the SU(2) case [30].However, below we will identify ⇧ with the Plebanski field smeared over 2-dimensional surfaces in a t = const.slice of initial data. Be this hypersurface space-like, the restriction is automatically fulfilled, and has no physicalconsequence for the following.

B. Twistors from the LQG action

The interest in the phase space of SL(2,C) holonomies and fluxes comes from loop quantum gravity. We workin the first-order tetrad formalism, and start from the often-called Holst action for general relativity. In termsof chiral variables, it is

SHolst[A, e] =~`P

2

� + i

i�

Z

M

⌃

A

B

(e) ^ FB

A

(A) + cc., (16)

where `P =

p

8⇡~GN/c3 is the Planck length, � > 0 is the Barbero–Immirzi parameter, and “cc.” denotescomplex conjugation of everything preceding. See e.g. [31–34] for the case with boundary terms. The action (16)is a non-analytic functional of the left-handed sl(2,C) connection A, and the four soldering forms e transformingin the irreducible (

12 ,

12 ) representation of SL(2,C). Curvature FA

B

and Plebanski 2-form ⌃

A

B

(e) are uniquelydetermined by the equations

FA

B

(A) = dAA

B

+AA

C

ÂC

B

, ⌃

A

B

(e) = eAC ^ eBC

=

i

2

⌧ABi

✓

i

2

✏ilm

el ^ em + e0 ^ ei◆

, (17)

andanadditionalsymmetry:Z2

T ⇤G ' g⇥G 3 (⇧, h) ⇥ = Tr(⇧dhh�1)

generatorsasleft-invariantvectorfields,theiradjointright-invariant⇧ := Adh(⇧)4



⇧

e

= �⇡!

⇡e

!e

h⇧h�1, (11)


⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h, (12a)�

hA

B

, hC

D

= � 2C

(⇡!)(⇡e

!e

)

�

✏AC

⇧

BD

+ ✏BD

⇧

e

AC

�

. (12b)


e



e


�

⇧

i

,⇧j

= �✏ij

k

⇧

k

,�

⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h,�

hA

B

, hC

D

�

�

C=0= 0. (13)




exp (zXC

+ zXC

) : (!,⇡,!e

,⇡e

) 7! (e

z!, e�z⇡, ez!e

, e�z⇡e

), z 2 C. (14)




(!,⇡,!e

,⇡e

) 7! (⇡,!,⇡e

,!e

) (15)


AB

⇧

AB

= �Tr⇧






SHolst[A, e] =~`P

2

� + i

i�

Z

M

⌃

A

B

(e) ^ FB

A

(A) + cc., (16)

where `P =

p


12 ,


B


A

B


FA

B

(A) = dAA

B

+AA

C

ÂC

B

, ⌃

A

B

(e) = eAC ^ eBC

=

i

2

⌧ABi

✓

i

2

✏ilm

el ^ em + e0 ^ ei◆

, (17)

4



⇧

e

= �⇡!

⇡e

!e

h⇧h�1, (11)


⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h, (12a)�

hA

B

, hC

D

= � 2C

(⇡!)(⇡e

!e

)

�

✏AC

⇧

BD

+ ✏BD

⇧

e

AC

�

. (12b)


e



e


�

⇧

i

,⇧j

= �✏ij

k

⇧

k

,�

⇧

i

, h

= �h⌧i

,�

⇧

e

i

, h

= ⌧i

h,�

hA

B

, hC

D

�

�

C=0= 0. (13)




exp (zXC

+ zXC

) : (!,⇡,!e

,⇡e

) 7! (e

z!, e�z⇡, ez!e

, e�z⇡e

), z 2 C. (14)




(!,⇡,!e

,⇡e

) 7! (⇡,!,⇡e

,!e

) (15)


AB

⇧

AB

= �Tr⇧






SHolst[A, e] =~`P

2

� + i

i�

Z

M

⌃

A

B

(e) ^ FB

A

(A) + cc., (16)

where `P =

p


12 ,


B


A

B


FA

B

(A) = dAA

B

+AA

C

ÂC

B

, ⌃

A

B

(e) = eAC ^ eBC

=

i

2

⌧ABi

✓

i

2

✏ilm

el ^ em + e0 ^ ei◆

, (17)

• thesequantitiesareinvariantundertheCorbits

• theyfurthersatisfy

C = ⇡A!A � ⇡A!

A

Take2twistors,eachwith(Z↵, Z↵) ⇡A!A 6= 0ToconstructT*SL(2,C), and

γ-nulltwistorswithincidenceconstrainedtoatime-likedirection

Remark:canbeeasilyextendedtotime-likeandnullhypersurfacesNulltwistedgeometriescharacterisedbythe2dspace-likedataonly





, i⇡

A


= (�i⇡

A

, !


Z

↵

Z

↵

= 2Im(⇡A

!

A), (1)


�Z

↵

, Z

�

= i�

↵

�

. (2)


J

IJ = !

(A⇡

B)✏

AB + cc, P

I = i⇡

A

⇡

A

, C

I = i!

A

!

A

, D = Re(⇡A

!

A), (3)


exp(!IJ

M

IJ) . f := exp(!IJ

{M IJ

, ·})f. (4)


!


W

I = �sP

I

, (5)

where W

I := 1/2✏IJKL

P

J

J

KL


!




2P [IC

J] = (D + s?) JIJ

, P · C = �(D2 + s

2). (6)


C

(2) = 6 s2, C

(3) = �6 s3, C

(4) = 6 s4. (7)



↵

Z



3

Using theconstraintsread !A =||!||2

(� � i)j�AA⇡A

Step2:SimplicityconstraintsandconstrainedincidencerelationSU(2) subgroup of the Lorentz group, and allows us to define the Hermitian scalar product between spinors,

h!|⇡i := �

AA

⇡

A

!

A

. We will also use the short-hand notation k!k2 := h!|!i for the norm.The constraints (11) read K

i + �L

i = 0, and are equivalent to imposing the matching of left-handed andright-handed sectors up to a phase,

⇧i = �e

i✓⇧i

, � = cot✓

2. (12)

Using (3), the constraints have simple spinorial equivalents [13, 14],

F1 = D � �s = 0, F2 = h!|⇡i = 0. (13)

The constraint F1, real and Lorentz-invariant, is solved posing

[⇡|!i = (� + i)j, j 2 R (14)

and fixes the relative global phase between the spinors. F2, which explicitly depends on the chosen timedirection, implies that one spinor is proportional to the parity transform of the other, ⇡

A

/ �

AA

!

A. Puttingthe two conditions together, the constraints can be solved expressing one spinor in terms of the other and thenorm j:

⇡

A

=(� + i)j

k!k2 �

AA

!

A

. (15)

There is an alternative parametrization of the solution space, motivated by the canonical structure of theconstraints: while F2 is second class, F1 is first class. A good coordinate among its orbits is the SU(2) normk!k2, and the F1-gauge-invariant spinor is

z

A :=p2j

!

A

k!ki�+1, kzk =

p2j. (16)

Here j 6= 0 by assumption, and we have restricted to j > 0 using the Z2 symmetry of the symplectic reductionby C. See [13] for details.

The solution space is thus a five-dimensional manifold 0�

, dependent on the Immirzi parameter (via ✓) andthe choice of time gauge, that can be parametrized by one of the two Lorentzian spinors, plus the norm j; oralternatively, by the reduced spinor zA, plus the norm k!k :

0�

= {!A

, j} = {zA, k!k}. (17)

The two norms j and k!k play complementary roles in the geometric interpretation of loop quantum gravity :j is the area of the face dual to the link, while ||!|| is related to the extrinsic curvature, as we will recall below.We refer to twistors satisfying the simplicity constraints (13), or equivalently (15), as simple twistors.3

On a link, we impose the same set of constraints on both source and target twistors,

z

A

kzk =!

A

k!ki�+1,

z

A

kzk =!

A

k!ki�+1, (18)

with induced Poisson brackets {zA, zB} = i�

AB = �{zA, ¯zB}, and reduced area matching condition C =kzk2 � kzk2 = 0. The latter implies kzk2 = kzk2 = 2j, and reduces this C4 symplectic space to T

⇤SU(2), asshown in [25], parametrized by

~

X =1

2hz|~�|zi, g =

|zihz|+ |z][z|kzkkzk ,

~

X = �1

2hz|~�|zi, (19)

where ~� is the vector of Pauli matrices and X

2 = X

2 = j

2. Again, X and X are related on-shell of C by theadjoint action of g, and act as right-invariant and left-invariant vector fields.

That is, twistors constrained by the area matching and primary simplicity constraints describe the phasespace T ⇤SU(2) of loop quantum gravity. When SU(2) gauge-invariance holds, by means of the closure conditions

3The term simple twistors is also used in the twistor literature, but in reference to a bi-twistor being simple in the same senseof a simple bivector as defined above.

5

namelythematchinguptoaphaseofleftandrightsu(2)algebras

Considerthefollowingconstraint

NI( � �?)JIJ = 0 ,

) Selectsa2dspace-likeplane(whichwillactasthepolyhedron’sface)

• Oneofthese3isaCasimirequation:

SU(2) subgroup of the Lorentz group, and allows us to define the Hermitian scalar product between spinors,

h!|⇡i := �

AA

⇡

A

!

A

. We will also use the short-hand notation k!k2 := h!|!i for the norm.The constraints (11) read K

i + �L

i = 0, and are equivalent to imposing the matching of left-handed andright-handed sectors up to a phase,

⇧i = �e

i✓⇧i

, � = cot✓

2. (12)

Using (3), the constraints have simple spinorial equivalents [13, 14],

F1 = D � �s = 0, F2 = h!|⇡i = 0. (13)

The constraint F1, real and Lorentz-invariant, is solved posing

[⇡|!i = (� + i)j, j 2 R (14)

and fixes the relative global phase between the spinors. F2, which explicitly depends on the chosen timedirection, implies that one spinor is proportional to the parity transform of the other, ⇡

A

/ �

AA

!

A. Puttingthe two conditions together, the constraints can be solved expressing one spinor in terms of the other and thenorm j:

⇡

A

=(� + i)j

k!k2 �

AA

!

A

. (15)

There is an alternative parametrization of the solution space, motivated by the canonical structure of theconstraints: while F2 is second class, F1 is first class. A good coordinate among its orbits is the SU(2) normk!k2, and the F1-gauge-invariant spinor is

z

A :=p2j

!

A

k!ki�+1, kzk =

p2j. (16)

Here j 6= 0 by assumption, and we have restricted to j > 0 using the Z2 symmetry of the symplectic reductionby C. See [13] for details.

The solution space is thus a five-dimensional manifold 0�

, dependent on the Immirzi parameter (via ✓) andthe choice of time gauge, that can be parametrized by one of the two Lorentzian spinors, plus the norm j; oralternatively, by the reduced spinor zA, plus the norm k!k :

0�

= {!A

, j} = {zA, k!k}. (17)

The two norms j and k!k play complementary roles in the geometric interpretation of loop quantum gravity :j is the area of the face dual to the link, while ||!|| is related to the extrinsic curvature, as we will recall below.We refer to twistors satisfying the simplicity constraints (13), or equivalently (15), as simple twistors.3

On a link, we impose the same set of constraints on both source and target twistors,

z

A

kzk =!

A

k!ki�+1,

z

A

kzk =!

A

k!ki�+1, (18)

with induced Poisson brackets {zA, zB} = i�

AB = �{zA, ¯zB}, and reduced area matching condition C =kzk2 � kzk2 = 0. The latter implies kzk2 = kzk2 = 2j, and reduces this C4 symplectic space to T

⇤SU(2), asshown in [25], parametrized by

~

X =1

2hz|~�|zi, g =

|zihz|+ |z][z|kzkkzk ,

~

X = �1

2hz|~�|zi, (19)

where ~� is the vector of Pauli matrices and X

2 = X

2 = j

2. Again, X and X are related on-shell of C by theadjoint action of g, and act as right-invariant and left-invariant vector fields.

That is, twistors constrained by the area matching and primary simplicity constraints describe the phasespace T ⇤SU(2) of loop quantum gravity. When SU(2) gauge-invariance holds, by means of the closure conditions

3The term simple twistors is also used in the twistor literature, but in reference to a bi-twistor being simple in the same senseof a simple bivector as defined above.

5

• TherestalignsthenullpolesofthespinorstoadduptoN4 Research Report

Results. Using twistors, the classical formalism can be extended in a natural way to null hypersurfaces,with the Euclidean polyhedra replaced by null polyhedra with space-like faces, and SU(2) by the littlegroup ISO(2). The main di↵erence is that a larger symmetry group appears, due to the fact that on anull hypersurface, the isometry group is larger: the primary simplicity constraints present in the formalismbecome all first class, and removing the gauge orbits during the symplectic reduction selects only the helicitysubgroup of the little group. This fact has an appealing counterpart in particle theory: as well-known, therepresentations of massless particles only depend on the spin quantum number, the translations beingredundant gauges. In our setting, the gauge orbits have the geometric interpretation of shifts along the nulldirection of the hypersurface.

As a consequence, information on the shapes of the polyhedra is lost,

NRNL

S0

NR

NL

Figure 2: Spacetime foliation bynull hypersurfaces

and the result is a much simpler, abelian geometric picture. It can bedescribed by an Euclidean singular structure on the 2-dimensional space-like surface defined by a foliation of space-time by null hypersurfaces. Thisgeometric structure is naturally decomposed into a conformal metric andscale factors, forming locally conjugate pairs. Proper action-angle variableson the gauge-invariant phase space are described by the eigenvectors of theLaplacian of the dual graph. Finally, we quantised the phase space and itsalgebra using Dirac’s algorithm, obtaining a notion of spin networks for nullhypersurfaces. Such spin networks are labelled by SO(2) quantum numbers, and are embedded non-triviallyin the unitary, infinite-dimensional irreducible representations of the Lorentz group.

The results, obtained in with PhD student Mingyi Zhang, have been published in Phys. Rev. D [5],and have been presented at various international conferences. The result that the kinematical quanta arecarried by a co-dimension two surface could point towards a holographic structure of the theory, a factwhich would have far reaching consequences. However, many non-trivial steps are needed before theseresults can be extended to include the complete dynamics of the system. First of all, our analysis needs tobe complemented with a continuum canonical analysis of the Plebanski action for general relativity on anull hypersurface. This motivated my next piece of research.

1.6 Spin foam dynamics

Description of the problem. The current state of the art for the dynamics of LQG uses the Engle-Pereira-Rovelli-Livine spin foam model to implement the Hamiltonian constraint and reduce the kinematical scalarproduct among spin network states to a physical one. The model is supported by a number of results.Explicit evaluation of transition amplitudes is on the other hand hindered by the sheer complexity of themodel. This is especially true for Lorentzian signature (the model exists for both Lorentzian and Euclideansignatures), where the definition of the vertex amplitude includes unbounded integrals, due to the non-compact nature of the Lorentz group. The field has focused for a number of years on technical and formaldevelopments, and partial studies of the semiclassical limit. It is my opinion that the most urgent questionin this approach is to show how to do calculations, and compute. I have recently obtained two importantresults in this direction. First, I have shown how the model can be recasted in a way in which the vertexamplitudes are purely SU(2). All the non-compacteness goes into integrals over boosts associated with edges.In this way, the model resembles a lattice gauge theory model, with the plaquette weight decomposed intoedge weights. Secondly, the edge integrals can be related in a precise way to Clebsch-Gordan coe�cients forSL(2,C). Using and extending results in the relevant literature, I have been able to derive a number of finiteand asymptotic properties, and more importantly, set up the ground for systematic numerical evaluations.

2 General relativity on null hypersurfaces

A central feature of general relativity is the presence of non-linear constraints among the initial data on aCauchy surface, which prevent an explicit description of the physical degrees of freedom, and complicate thedynamics. Taming them is a crucial part of the theory, be it for formal studies and quantisation attempts, orfor numerical relativity and applications to astrophysics and gravitational waves. An interesting approach

Theareascomefromthevaluesofthehelicities:importanceofworkingwithfulltwistorspacenotsimplytheprojectiveone

SimplicityconstraintsfromthePlebanskiaction

ThesimplicityconstraintscomefromwritingGRasaBFtheory(Plebanskiaction)

Remarks:• :self-dualformulation• Forreal,twoUrbantkemetrics;theconstraintsmakethemcoincide• IntheHamiltonianformulation,theconstraintsleadtosecondaryconstraintswhichimposepartofthe

torsion-lesscondition

� = ±i�

10 Chapter 2. General relativity in connection variables and loop quantum gravity

1.2 Plebanski action.

Anticipating on the canonical analysis to appear below, the structure of the action (2.1) shows that thepull-back !IJ

a of the connection on a spatial slice will be conjugated to a wedge product of two tetrad fields,more precisely to the bi-vector

BaIJ :=1

2✏abceI

beJc . (2.30)

Recall that a bivector BIJ which is the exterior product of two vectors is called simple, and satisfies thesimplicity constraint ✏IJKLBIJBKL = 0. Notice that for each component a, the bi-vector in the internalindices is of a special form, called simple Hence one way to do the canonical analysis is to start from BFand add so called simplicity constraints reducing the B field to the exterior product of two tetrads. Thisequivalent reformulation of the theory is also interesting because it gives a direct link to TQFTs.

Consider the following action,

S(B, !,�) =

Z( +

1

�?)BIJ ^ FIJ(!) � 1

2

✓�IJKL +

⇤

6✏IJKL

◆BIJ ^ BKL, (2.31){S}

where F (!) is the curvature of an so(4) connection !, and BIJ a 2-form with values in the algebra. Thisaction can be called non-chiral, to distinguish it from the original Plebanski action, which is identical butuses g = su2. ⇤ is the cosmological constant, and the field �IJKL a Lagrange multiplier, symmetric underexchange of the first and the second pair, and antysimmetric within each pair. In addition, we impose on� the constraint ✏IJKL�IJKL = 0. It has therefore the same symmetries of the Riemann tensor, and 20components. Its variation gives the following equations, known as simplicity constraints,

BIJ ^ BKL =1

12✏IJKL hB ^ ?Bi. (2.32){simpl}

Solutions satisfying the non-degeneracy condition hB ^ ?Bi 6= 0 can be divided in two sectors [12],

BIJ = ±(1/2)✏IJKLeK ^ eL, BIJ = ± eI ^ eJ . (2.33){Bsols}

We will review the derivation below. In both cases, the 20 constraints (4.12) reduce the initial 36 independentcomponents of BIJ

µ⌫ down to 16, parametrized by a tetrad eIµ, and the action reduces to (2.1). � gets further

correctly identified provided we choose the first of the four solutions in (2.33). All other 3 correspond tosimple redefinitions of �.

It is also useful to review the on-shell meaning of the Lagrange multiplier. To that end, let us look atthe other field equations. These read

Self-duality and metricity

It is intriguing to unravel the mechanism behaind this mechanism. Indeed, the key to the formalism is thefact that the 2-forms B can always be used to introduce a metric, regardless of any constraints. The roleof the constraints is rather to single out this metric, namely to freeze the remaining components of the2-forms which do not enter the definition of the metric. To be more specific, given an su2-valued 2-formBi

µ⌫ , i = 1, 2, 3, a metric can be defined through the well-known Urbantke formula [13, 14],

pgU gU

µ⌫ =1

12✏ijk ✏↵��Bi

µ↵Bj��Bk

�⌫ . (2.34){gU}

Notice in this formula the completely antisymmetric tensor ✏ijk, the unique singlet in the tensor product ofthree adjoint representations of SU(2). The B field needs to be complex for this metric to have Lorentziansignature, while a real field yields Euclidean signature. The same mechanism can be applied to both theoriginal Plebanski actions for general relativity and the modified theories. In the original case, the actiongives quadratic field equations, some of which are the “metricity constraints” which freeze the remainingcomponents of Bi not captured by (2.34), and the rest reduce to the Einstein equations for (2.34). If weconsider a modification where the constraints are removed, not only we get new field equations, but the extra





BaIJ :=1

2✏abceI

beJc . (2.30)



S(B, !,�) =

Z( +

1

�?)BIJ ^ FIJ(!) � 1

2

✓�IJKL +

⇤

6✏IJKL

◆BIJ ^ BKL, (2.31){S}


BIJ ^ BKL =1











pgU gU

µ⌫ =1

12✏ijk ✏↵��Bi

µ↵Bj��Bk

�⌫ . (2.34){gU}


Lagrangemultipliersatisfiesandleadsto20equationsimposingthesimplicityofB:





BaIJ :=1

2✏abceI

beJc . (2.30)



S(B, !,�) =

Z( +

1

�?)BIJ ^ FIJ(!) � 1

2

✓�IJKL +

⇤

6✏IJKL

◆BIJ ^ BKL, (2.31){S}


BIJ ^ BKL =1











pgU gU

µ⌫ =1

12✏ijk ✏↵��Bi

µ↵Bj��Bk

�⌫ . (2.34){gU}


"IJKL�IJKL = 0

NnI( � �?)JIJl = 0

s(l)t(l)

τs τt

Nl Nl

jl

Discretisation:onenormalpernode,onegeneratorperlink

constraints:

canonicalconjugatevariabletotheconnection:Jsuchthat

NnIBIJl = 0

BIJ = ( � �?)JIJl

Summary

Spinnetworks (Flat)Twistors

Twistedgeometries

Reggegeometries

Twistorsprovidepowerfultoolstostudythesemiclassicalgeometryofloopquantumgravity,andpermitanelegantcovariantdescriptionoftheclassicallimitofspinnetworks

IV. SU(2,2) spin networks and a self-dual octahedron (wip Maciej Dunajski, Miklos Langvik and S)

AsmentionedintheRoadtoReality,PenroseoriginallyintendedtoextendSU(2)spinnetworkstoSU(2,2),asawayofintroducinganotionofscaleanddistancesviathePoincaregenerators,whilekeepingsemi-simplicityofthegroup

ButhowwoulddoSU(2,2)spinnetworkrelatetotheSU(2)onesofLQG?Aretheassociatednotionsofscalesanddistancescomparable?

CanweuseSU(2,2)spinnetsastoolstoinvestigatequestionsasbreakingofconformalsymmetryinLQG?

Motivatedbythesequestions,oneisledtolookforatwistorialembeddingofT*SU(2,2)

Thislineofthoughtthenblossomedintwistortheory…Andconversely,LQGsolvedtheissueofdistancesviatheAshtekarvariables,whichendowtheSU(2)spinswithquantaofareasandvolumes

Somemotivations…

Theideaistogeneralisethepreviousconstruction:takethefundamentalrepresentationofSU(2,2)onC4,withtwoindependentbasisand,andagroupelementmappingonebasistotheother

Z↵i Z↵

i

G↵�

Theconstraintsmustbeconformallyinvariantquantities:

• helicities

• incidences

• `volumes’

In the following, we will denote the abstract set of 15 su(2, 2) generators by antisymmetric matrices Mab,(a, b) = 0, . . . 5. The action is Hamiltonian with respect to the canonical SU(2, 2)-invariant Poisson structure

{Z↵, Z�

} = i�↵�

. (4) {ZZ}

Our goal is to extend this representation to the larger Poisson algebra carried by the symplectic manifoldT ⇤SU(2, 2). Let us fix some notations and provide explicit expressions, which will be useful references for laterresults. Using the isomorphism between a Lie algebra and the (co-)tangent vector space at the origin of itsLie group manifold, T ⇤SU(2, 2) ' SU(2, 2) ⇥ su(2, 2) can be parametrized by a unitary 4x4 matrix G↵

�

, andalgebra generators M↵

�

in the fundamental irrep. Using (2) and its inverse with upstair indices, we write thecondition for unitarity as follows,

G�1↵�

= G†↵�

⌘ ⌘↵↵G�

↵

⌘��

(5) {Gunitary}

We denote by �ab↵

�

= ��ba↵

�

, (a, b) = 0, . . . 5, a basis of su(2, 2) in the fundamental 4x4 representation (whichcan be written from tensor products of identities and Pauli matrices, see (C.45) for explicit formulas); so thatM↵

�

=P

a<b

Mab �ab↵

�

, and the representation (3) can be written as a bilinear in the twistors,

Mab = Z↵

�ab↵

�

Z� . (6) {MZ}

If we take M to be left-invariant vector fields, the right-invariant ones, called them M , are obtained throughthe adjoint action, i.e.

M↵

�

= �(GMG�1)↵�

(7) {Ad}

Finally, all cotangent bundles of a Lie group carry a natural symplectic structure, with potential given by theinner product between the Maurer-Cartan form and the algebra generators, resulting in the following Poissonbrackets, (see e.g. [?])

{G↵

�

, G�

�

} = 0, {Mab, G↵

�

} = i(G�ab)↵�

, {Mab, G↵

�

} = �i(�abG)↵�

, (8a)

{Mab, M cd} = 0, {Mab,M cd} = �fabcd

ef

Mef , {Mab, M cd} = �fabcd

ef

Mef , (8b)

where fabcd

ef

the constant structures (see (??)), and the actions in the first line are the identification ofleft-invariant (resp. right-) as right derivatives (resp. left derivatives).

Notice that the Poisson algebra is not diagonal; this reflects the non-linear nature of a non-abelian Lie group.Our goal is to find a suitable complex space CN with diagonal symplectic structure, such that T ⇤SU(2, 2) is asubmanifold in a symplectic sense. The canonical CN variables play then the role of local Darboux coordinatesfor T ⇤SU(2, 2). It is natural to take CN to be a twistor space, since the algebra generators are already givenby (3). The task is to find a parametrization of the group element in terms of twistors, in such a way that thetwistor’s Poisson algebra (4) induces the right Poisson algebra of T ⇤SU(2, 2). To do so, we will need multiplecopies of the twistor space , and impose a number of constraints among the twistors. A simple countingargument shows that we will need at least 4 independent twistors (T ⇤SU(2, 2) has 30 real dimensions), and thestrategy we use will lead us to start with 8 twistors, and impose a total of 5 first class constraints, and 24 secondclass constraints. Whether a simpler reduction exists we don’t know; but we will explain below why we do notthink so.

Clearly, the constraints will have to be conformal invariant functions of the twistors. These are of threetypes:

• the helicitiessi

:= 12 Zi

· Zi

, (9)

• the incidences⇢ij

:= Zi

· Zj

. (10)

• Finally, with four or more twistors we can also consider the ‘volumes’ (S: how should we call theseinvariants?)

Z :=1

4!✏ijkl✏

↵��

Z↵

i

Z�

j

Z�

k

Z�

l

. (11)

Notice that the norm of Z is not independent from the helicities and the incidences; however the phaseis. S: the first statement is easy to prove writing ZZ and expanding ✏✏ = �� + . . .. Thesecond I dont have a direct proof, I just tried and could not see how the phase could be alsorewritten in terms of incidences.

2


{Z↵, Z�

} = i�↵�

. (4) {ZZ}


�


�


G�1↵�

= G†↵�

⌘ ⌘↵↵G�

↵

⌘��

(5) {Gunitary}


�

= ��ba↵

�


�

=P

a<b

Mab �ab↵

�


Mab = Z↵

�ab↵

�

Z� . (6) {MZ}


M↵

�

= �(GMG�1)↵�

(7) {Ad}


{G↵

�

, G�

�

} = 0, {Mab, G↵

�

} = i(G�ab)↵�

, {Mab, G↵

�

} = �i(�abG)↵�

, (8a)


ef


ef

Mef , (8b)

where fabcd

ef





:= 12 Zi

· Zi

, (9)


:= Zi

· Zj

. (10)


Z :=1

4!✏ijkl✏

↵��

Z↵

i

Z�

j

Z�

k

Z�

l

. (11)


2


{Z↵, Z�

} = i�↵�

. (4) {ZZ}


�


�


G�1↵�

= G†↵�

⌘ ⌘↵↵G�

↵

⌘��

(5) {Gunitary}


�

= ��ba↵

�


�

=P

a<b

Mab �ab↵

�


Mab = Z↵

�ab↵

�

Z� . (6) {MZ}


M↵

�

= �(GMG�1)↵�

(7) {Ad}


{G↵

�

, G�

�

} = 0, {Mab, G↵

�

} = i(G�ab)↵�

, {Mab, G↵

�

} = �i(�abG)↵�

, (8a)


ef


ef

Mef , (8b)

where fabcd

ef





:= 12 Zi

· Zi

, (9)


:= Zi

· Zj

. (10)


Z :=1

4!✏ijkl✏

↵��

Z↵

i

Z�

j

Z�

k

Z�

l

. (11)


2

and

⌘↵�

=

✓0 2

2 0

◆, Z

↵

= ⌘↵�

Z � = (�i⇡A

,!A) (C.16)

The i factor allows to see the twistor as a Dirac bi-spinor in the chiral basis, and it is used in the LQGliterature, as well as the bar over ⇡ to keep track of its anti-selfdual nature: ⇡A

LQG

= i⇡A

Penrose

. As a consequence,dilatation generators and helicity are inverted:

⇡! := ⇡A

!A = D + is, s =1

2Za

Za. (C.17)

The other important di↵erence is that we work with (� + ++), so we map world vectors to anti-hermitianmatrices via

XAA =ip2�AA

I

XI =ip2

✓X0 +X3 X1 � iX2

X1 + iX2 X0 �X3

◆, (C.18)

with�AA

I

= (�AA,�A

i

B

�BA), �I

AA

= (��AA

,��AB

�B

i

A

). (C.19)

Finally, the incidence relation is unchanged,

!A = iXAA⇡A

, (C.20)

since the two i’s inside Z↵ and the antihermitian matrices map compensate one another.With these conventions, the generators are

JIJ = !(A⇡B)✏AB + cc, P I = i⇡A⇡A, CI = i!A!A, D = Re(⇡A

!A), (C.21)

C.1 Incidences

Defining the bi-twistors

R↵�

ij

:= 2Z [↵i

Z�]j

= ⇡i

⇡j

✓� 1

2r2✏AB �rA

B

rA

B �✏AB

◆, R

ij↵�

:=1

2✏↵��

R↵�

ij

= ⇡i

⇡j

�✏

AB

�rA

B

rAB

� 12r

2✏AB

!

(C.22)(note: R(r) = �R

Penrose

(�irPenrose

)) and

rAA

ij

= � i

⇡i

⇡j

(!A

i

⇡A

j

� !A

j

⇡A

i

) 2 MC, r2 = �2!i

!j

⇡i

⇡j

(C.23)

is the intersection point of the two ↵ planes (notice this is the same formula as with Penrose’s conventions, thetwo extra i’s cancel out each other); we have

Rij↵�

R↵�

kl

= ⇡i

⇡j

⇡k

⇡l

(rij

� rkl

)2 (C.24)

and

Z =1

48✏ijkl⇡

i

⇡j

⇡k

⇡l

(rij

� rkl

)2, Zi

↵

=1

6Z✏ijklR

kl↵�

Z�

j

. (C.25)

Other properties of r:

C.2 Symplectic structures and dual basis

C4 and twistors. Poisson brackets:

{Z↵

i

, Zj�

} = i�ij

�↵�

, {Z↵

i

, ¯Zj�

} = �i�ij

�↵�

(C.26) {AppZZ}

{si

, Z↵

j

} = �i�ij

Z↵

i

, {si

, Z↵

j

} = i�ij

Z↵

i

, e{�(s�s),·}Z↵Z�

= Z↵Z�

(C.27)

Assume that the four twistors are not parallel. Then

Z :=1

4!✏ijkl✏

↵��

Z↵

i

Z�

j

Z�

k

Z�

l

6= 0, (C.28)

9

T8?//C ' T ⇤SU(2, 2)

• :non-zerohelicities,eachquadruplehas si = (+ +��), Z 6= 0T8?

• Requiredconstraints:

The geometric meaning of these functions, both in twistor space and (complexified) Minkowski space, will bediscussed in the next Section.

Let us consider 8, and split the 8 twistors into two sets Z↵

i

, i = 1, . . . 4 and Z↵

i

. We posite the brackets

{Z↵

i

, Zj�

} = i�ij

�↵�

, {Z↵

i

, ¯Zj�

} = �i�ij

�↵�

. (12)

The advantage of an opposite sign on the second bracket will become clear later on. With each set we can definegenerators Mab and Mab, extending (6) by linearity, which we will want to represent left-invariant and right-invariant vector fields. The strategy we use is a generalisation of one previously used to construct T ⇤SU(2) andT ⇤SL(2,C), see Appendix ?? for a brief review. SU(2, 2) is the group of linear transformations in C4 preservingthe norm (2). Therefore we can write a corresponding group element as a dyadic representation demanding thatboth quadruples form an orthogonal basis. To that end, they should not be parallel, Z↵

i

6= �Z↵

j

, in particular

both Z and Z should not vanish; second, we restrict each quadruple to have two positive helicities and two withnegative helicities. Furthermore, we exclude zero helicities. As well as equal helicities as degenerate. Theserestrictions do not a↵ect the dimensionality of the space. We denote 8

?

the sector of 8 satisfying them. Then,we need to impose the orthogonality, which in twistor terms is the incidence of twistors.

[remove: The required constraints are of three types: first, the incidences of the twistors, which guaranteethe unitarity of the matrix element. Second, the helicity matching constraints, which guarantee that M andM are related by adjoint action Third, the unit determinant condition. Surprisingly, the Poisson algebra worksdireclty.2 ]

We will do so in two steps:

Proposition 1. The symplectic reduction of

8?

by the following constraints

Hi

= si

� si

= 0, (4 real, first class) (13a)

⇢ij

= 0, (6 complex, second class) (13b)

⇢ij

= 0, (6 complex, second class) (13c)

describes a symplectic space of 32 real dimensions isomorphic to T ⇤U(2, 2), parametrized by

G↵

�

:=X

i

Z↵

i

Zi�p

si

psi

, Mab =X

i

Zi↵

�ab↵

�

Z�

i

, Mab = �X

i

¯Zi↵

�ab↵

�

Z�

i

, (14) {GM}

U =X

i

si

, U = �X

i

si

, (15)

with G↵

�

2 SU(2, 2), Mab

and Mab

respectively left-invariant and right-invariant vector fields generating the

su(2, 2) algebra, and U and U are the center of u(2, 2).

Proposition 2. Further reduction by the additional constraint

� = argZ � arg Z = 0 (16)

imposes detG = 1, makes U (and thus U) pure gauges, and describes a symplectic space of 30 real dimensions

isomorphic to T ⇤SU(2, 2), parametrized by (14) above.

See Appendix C for detailed demonstration, I still need to type it here in a short and clear form.

3 Self-dual octahedron

The various constraints that we have considered have interesting geometric meanings. Let us begin with theincidences.

2This is di↵erent from the previous cases, where the equivalent of the helicity constraints played the role of both relating Mand M, but also closing the Jacobi identity for the reduced algebra. The origin of this di↵erence is in the di↵erent structure of thegroup element. There di↵erent dyads did not commute a priori. Here they do.

3

T*SU(2,2)fromtwistors

The geometric meaning of these functions, both in twistor space and (complexified) Minkowski space, will bediscussed in the next Section.

Let us consider 8, and split the 8 twistors into two sets Z↵

i

, i = 1, . . . 4 and Z↵

i

. We posite the brackets

{Z↵

i

, Zj�

} = i�ij

�↵�

, {Z↵

i

, ¯Zj�

} = �i�ij

�↵�

. (12)

The advantage of an opposite sign on the second bracket will become clear later on. With each set we can definegenerators Mab and Mab, extending (6) by linearity, which we will want to represent left-invariant and right-invariant vector fields. The strategy we use is a generalisation of one previously used to construct T ⇤SU(2) andT ⇤SL(2,C), see Appendix ?? for a brief review. SU(2, 2) is the group of linear transformations in C4 preservingthe norm (2). Therefore we can write a corresponding group element as a dyadic representation demanding thatboth quadruples form an orthogonal basis. To that end, they should not be parallel, Z↵

i

6= �Z↵

j

, in particular

both Z and Z should not vanish; second, we restrict each quadruple to have two positive helicities and two withnegative helicities. Furthermore, we exclude zero helicities. As well as equal helicities as degenerate. Theserestrictions do not a↵ect the dimensionality of the space. We denote 8

?

the sector of 8 satisfying them. Then,we need to impose the orthogonality, which in twistor terms is the incidence of twistors.

[remove: The required constraints are of three types: first, the incidences of the twistors, which guaranteethe unitarity of the matrix element. Second, the helicity matching constraints, which guarantee that M andM are related by adjoint action Third, the unit determinant condition. Surprisingly, the Poisson algebra worksdireclty.2 ]

We will do so in two steps:

Proposition 1. The symplectic reduction of

8?

by the following constraints

Hi

= si

� si

= 0, (4 real, first class) (13a)

⇢ij

= 0, (6 complex, second class) (13b)

⇢ij

= 0, (6 complex, second class) (13c)

describes a symplectic space of 32 real dimensions isomorphic to T ⇤U(2, 2), parametrized by

G↵

�

:=X

i

Z↵

i

Zi�p

si

psi

, Mab =X

i

Zi↵

�ab↵

�

Z�

i

, Mab = �X

i

¯Zi↵

�ab↵

�

Z�

i

, (14) {GM}

U =X

i

si

, U = �X

i

si

, (15)

with G↵

�

2 SU(2, 2), Mab

and Mab

respectively left-invariant and right-invariant vector fields generating the

su(2, 2) algebra, and U and U are the center of u(2, 2).

Proposition 2. Further reduction by the additional constraint

� = argZ � arg Z = 0 (16)

imposes detG = 1, makes U (and thus U) pure gauges, and describes a symplectic space of 30 real dimensions

isomorphic to T ⇤SU(2, 2), parametrized by (14) above.

See Appendix C for detailed demonstration, I still need to type it here in a short and clear form.

3 Self-dual octahedron

The various constraints that we have considered have interesting geometric meanings. Let us begin with theincidences.

2This is di↵erent from the previous cases, where the equivalent of the helicity constraints played the role of both relating Mand M, but also closing the Jacobi identity for the reduced algebra. The origin of this di↵erence is in the di↵erent structure of thegroup element. There di↵erent dyads did not commute a priori. Here they do.

3

Thesymplecticreduction:

Prooftoappear,letmejustgiveacountingargument:8x8-(4x2-12x2)-1x2=30

Oneconstraintsurfaceandtheself-dualoctahedron

TWELVE COMPLEX NULL RAYS 3

X12

X34

X24

X13X14

23

21

12

32

14 24

34

41

31

13

43

42

X32

Fig. 4. The octahedral configuration of twelve complex null rays in MC.

Maciej Dunajski, Cambridge, February 27th, 2015.

6

RecallthatgenerateaU(4)algebrasi, ⇢ij

Theconfigurationsofoctahedrashouldbespannedbytheorbitsofthegenerators⇢ij

4(non-null)twistorsmutuallyincident: ⇢ij = Zi · Zj = 0

6complexequationsin

Constraintsurfaceisten-dimensionalLetuslookattheprojectivetwistorspaceandtheassociatedsix-dimensionalconstraintsurfaceincomplexifiedMinkowskispace

T4 ' C16

4αand4βplanesmutuallyintersectingalongnullcomplexlines,withverticescoincidingwiththeintersectionsofpairsofαplanes

AfungeneralisationoftheproblemsolvedbyTodalongtimeagoofthesymplecticreductiontoSouriauphasespace

Self-dualinthesensethat Xij =1

2"ijklXkl

mixingthetwistors

Conclusions

Therelationisforthemomentrestrictedtoflattwistors,andafinitenumberofthem!

Stillmuchmoretoexplore,conformalspinnetworks,curvedtwistors,hyperbolictwistedgeometries,continuumlimit…

andiftheextensiontocurvedtwistorsworks,couldtheγ-embeddingprovidenewcuestothegooglyproblem?

Twistorsprovidepowerfultoolstostudythesemiclassicalgeometryofloopquantumgravity,andpermitanelegantcovariantdescriptionoftheclassicallimitofspinnetworks

Twistors in spin networks and loop quantum gravity - …people.maths.ox.ac.uk/lmason/New...

Documents

Transcript of Twistors in spin networks and loop quantum gravity - …people.maths.ox.ac.uk/lmason/New...