twistorial structure of loop quantum gravity transition...

Twistorial structure of loop quantum gravity transition amplitudes

Simone Speziale

Centre de Physique Theorique de Luminy, Marseille, France

ILQG 13-11-2012mainly based on work with Wolfgang Wieland 1207.6348

Outline and goals

Classical Theory

‚ Description of the covariant phase space in terms of twistorsalgebra of the area matching and simplicity constraints

‚ Smearing the connection-tetrad algebra to the holonomy-flux algebra before or aftersolving the simplicity constraints is equivalent: T˚SUp2q with AB holonomy

‚ Notion of simple twistors solving the simplicity constraints

Quantum theory

‚ Hilbert space represented via homogeneous functions on spinor space(instead of cylindrical functions)

‚ Dynamics as integrals in twistor space

‚ Embedding of the Regge data of the EPRL asymptotics in the initial phase space

Work and ideas shared with a number of collaboratorsL. Freidel, E. Livine, W. Wieland, B. Dittrich, C. Rovelli, M. Dupuis, J. Tambornino, . . .

This talk: w Wolfgang Wieland 1207.6348 PRD12 w Miklos Langvik (to appear)

Speziale — Twistors and LQG 2/24

Smearing and simplicity constraints

Phase spaces:

SmearingSLp2,Cq variables ùñ T˚SLp2,Cq

ó Solve simplicity constraints ó

(primary and secondary)

Ashtekar-Barbero variables ùñ T˚SUp2q

Does the diagram commute?

LQG path: ë spin foam path: the other one

In the continuum:

‚ primary: simple bivectors, unique metric structure

‚ secondary: embedding of AB variables in covariant phase space: Γ “ ΓpEq

In the discrete:

‚ primary: simple twistors, unique (twisted) geometry

‚ secondary: non-trivial embedding of T˚SUp2q in T˚SLp2,Cq. Discrete ΓpEq ?


Smearing and simplicity constraints

Phase spaces:

SmearingSLp2,Cq variables ùñ T˚SLp2,Cq

ó Solve simplicity constraints ó

(primary and secondary)

Ashtekar-Barbero variables ùñ T˚SUp2q

Strategy: do symplectic reduction of the primary constraints, and think of thesecondary constraints as providing a non trivial gauge-fixing section.

ñ allows to show commutativity of the procedure, while postponingknowning explicitly the form and solution of the secondary constraints

Technique: twistorial tools: parametrize the phase spaces and the constraints interms of twistors

ñ allows to embed the non-linear HF phase space into a linear one,explicit calculations, plus direct link with variables used in the twistedgeometry parametrization


Panorama

twistors Ñ covariant ðñ covariantC-area matching twisted geometry loop gravity

SLp2,CqÓ simplicity constraints Ó Ó

SU(2) spinors Ñ “open” twisted geometry ðñ loop gravityarea matching SUp2q

Ó closure Ó Gauss law

“closed” twisted geometries ðñ gauge-inv.(polyhedra) loop gravity

Ó shape matching

Regge calculus


LQG and twisted geometries

On a fixed graph HΓ “ L2rSUp2qLs PΓ “ T˚SUp2qL

represents a truncation of the theory to a finite number of degrees of freedom w Rovelli, PRD 10

These can be interpreted as discrete geometries, called twisted geometries w Freidel, PRD 10, . . .

X, gùñ

Npζq, j, ξ, Npζq

hol-flux area-angles

Each classical holonomy-flux configuration on a fixed graph can be visualized as acollection of adjacent polyhedra with extrinsic curvature between them

‚ Each polyhedron is locally flat: curvature emerges at the hinges, as in Regge calculus

‚ They induce a discontinuous discrete metric: two neighbouring polyhedra areattached by faces with same area but different shape

‚ ξ: extrinsic geometry

‚ Realizes a link suggested by Immirzi, Smolin et al., with the subtlelty of discontinuity: Reggecalculus is too “rigid” to capture the kinematical degrees of freedom of a spin network

‚ The matching with Regge geometries becomes exact in 2+1 dimensions

The picture can be embedded in a SLp2,Cq formalism described by simple twistors


Twistors

‚ A pair of spinors, ω and π, chiral under Lorentz transformations,

Z “

ˆ

ωA

π 9A

˙

P T :“ C2ˆ C2˚

Twistor “complex helicity”: πAωA“ rπ|ωy SLp2,Cq-invariant

Twistor norm (helicity) s :“ Im`

rπ|ωy˘

SUp2, 2q invariant

‚ physical picture as a massless particle with a certain spin and momentum

‚ geometric picture via incidence relation ωA “ iXA 9Aπ 9A§ X a light ray in Minkowski space for a null twistor s “ 0§ X a congruence of light rays for a generic twistor

‚ can provide notion of non-linear graviton

We will see that twistors can be thought of as non-linear gravitons in a completelydifferent way than Penrose’s original one, as classical counterparts of LQG’s quantumgeometry


Representation of the Lorentz algebra

‚ C2 carries a representation of the Lorentz algebra (the celestial sphere)

‚ Similarly, there is a representation of the Lorentz algebra on twistor space T

ΠAB“

1

2pL` iKqAB “

1

2ωpAπBq

‚ We are interested in a representation of a larger Poissonian algebra, the one ofT˚SLp2,Cq: the holonomy-flux algebra of (covariant) loop quantum gravity

‚ This can be achieved on T2Q pZ, Zq,

ΠAB“

1

2ωpAπBq, ΠAB

“1

2ωpAπBq, hAB “

ωAπB ´ πAωB

?πω?πω

imposing a complex first-class constraint called complex area matching

C “ rπ|ωy ´ rπ|ωy “ 0,

C8{{C – T˚SLp2,Cq

That is, both Lorentz generators and holonomies can be expressed as simple functions ona space of two twistors, provided they have the same complex helicity


Twistor Poisson brackets

The symplectic structure simplifies in the larger space T2:

T2 C “ 0 T˚SLp2,CqtωA, ωBu “ 0 th, hu “ 0tπA, ω

Bu “ δBA ùñ tΠi, hu “ τ ih

tπA, πBu “ 0 tΠi,Πju “ εijkΠk

In particular, momenta commute.

16 twistors T2 Ñ T˚SLp2,Cq 12C-area matching C

Ó simplicity constraints Ó

8 simple twistors Ñ T˚SUp2q 6SU(2) spinors area matching C

Note: the complete system of constraints is reducible


Simplicity constraints

Gauge-fixed approach Pick a time direction nI (typically time gauge, nI “ p1, 0, 0, 0q)(cfr. with gauge-invariant approach e.g. Bianca and Jimmy)

‚ Allows to define SU(2) norm }ω}2 :“ δA 9AωAω

9A and xω|πy :“ δA 9Aω9AπA

‚ Primary simplicity constraints: matching of left and right geometries

K ` γL “ 0 ô Π “ eiθΠ, γ “ cotθ

2

and the same for tilded variables: Π “ eiθ ¯Π

‚ In terms of spinors:`

rπ|ωy “ R` iI˘

F1 “ R´ γI “ 0, F2 “ xω|πy “ 0, F2 “ 0

‚ On each link, pM,Fi, Fiq form a second class system: tF2, F2u ff 0which is further reducible: M and F1 imply F1.

3 indep. 1st class constraints`

C,D :“ F1 ` F1

˘

aut`

Cred P R, F1, F1

˘

4 2st class constraints F2, F2


Counting

3 indep. 1st class constraints`

C,D :“ F1 ` F1

˘

aut`

Cred P R, F1, F1

˘

4 2st class constraints F2, F2

16 twistors T2 Ñ T˚SLp2,Cq 12C-area matching C


8 simple twistors Ñ T˚SUp2q 6SU(2) spinors area matching C

Counting:

Right-Down: T2 C T˚SLp2,Cq D,F2, F2 T˚SUp2q

16 -2x2 12 -1x2-4 6

Down-Right: T2 F C4 Cred T˚SUp2q

16 -2x2-4 8 -1x2 6


Solution of the simplicity constraints

Focus first on half-link: pω, πq,`

F1 “ R´ γI, F2 “ xω|πy˘

‚ F1 “ 0 ñ rπ|ωy “ pγ ` iqj, j P R

‚ orbits: tF1, }ω}u “1

1`γ2 }ω}

‚ convenient to introduce a F1-gauge-invariant spinor

zA “?

2JωA

}ω}iγ`1, }z} :“

?2J, tF1, z

Au “ 0

constraint surface orbitspωA, πAq ÝÑ pωA, jq ÝÑ zA

Fi “ 0 }ω}

Symplectic reduction: C4{{F – C2

‚ Solutions parametrized by a SU(2) spinor zA

(transforms linearly under rotations but not under boosts)

‚ Simple twistors (on the 5d constraint surface): Z “ pωA, jq “ pzA, }ω}q


Area matching and T ˚SUp2q

On the link:16 pωA, πA, ωA, πAq P T2 Ñ pΠ, hq 12

C-area matching C


8 pzA, zAq P C4 Ñ pX, gq 6area matching Cred

Cred “ }z} ´ }z} C4{{Cred “ T˚SUp2q Q pX, gq Freidel PRD10

Symplectic reduction by the linear primary simplicity constraints selects an SUp2qholonomy-flux algebra from the covariant SLp2,Cq one

‚ Flux: X “ L “ ´K{γ‚ Holonomy X Ø h „ AγAB Ø A

Recall that Π “ ´hΠh´1. From this plus Fi “ Fi “ 0, it follows that

ph:hqABωB “ e´ΞωA, ph:hqABπ

B “ eΞπA

Using this one proves that the reduced g is the holonomy of the Ashtekar-Barberoconnection, a non-trivial and γ-dependent mixing of real and imaginary parts oflnh “ Γ` iK:

g „ eRe lnheγIm lnh ` opΞq


Geometric interpretation

Properties of the reduction:

‚ On C “ 0 surface, 1st class diagonal simplicity constraint D “ F1 ` F1

‚ Each orbit of D spanned by

Ξ :“ 2 ln

ˆ

}ω}

}ω}

˙

‚ In the time gauge,nIΛphq

IJnJ “ ´ cosh Ξ

The dihedral angle parametrizes the orbits of the diagonal simplicity constraint

‚ Symplectic reduction eliminates the dependence on Ξ, which is the coordinate of theorbits of the diagonal simplicity constraint D

‚ Secondary constraints: provide a non-trivial gauge-fixing of the orbits and thusnon-trivial embedding of T˚SUp2q in T˚SLp2,Cq, precisely as in the continuum


Geometric interpretation: covariant twisted geometries

‚ gauge-fixed approach pick a time direction nI (typically nI “ p1, 0, 0, 0q)

nIΛphqIJnJ “ ´ cosh Ξ, Ξ :“ 2 ln

ˆ

}ω}

}ω}

˙

pΠ, hq ÞÑ´

rπ|ωy, ζ, ζ, α, α, ξ,Ξ¯

given by

ζ “ ´ω1

ω0, α “

e2i argpω0q

rπ|ωyF2, ξ :“ 2 argpω0

q ´ 2 argpω0q ` γΞ.

reduction to twisted geometries:

Π “ ´ i2rπ|ωynpζqTατ3T

´1α n´1

pζq R´ γI “ α “ α “ 0 X “ jnpζqτ3n´1pζq

ÝÑ

h “ npζqTαep´ξ`pγ´iqΞqτ3T´1

α n´1pζq Ξ “ 0 g “ npζqe´ξτ3n´1

pζq

A “ Γ` iK “ AγAB ´ pγ ´ iqK


More on the brackets and the role of γ

Symplectic potential of T˚SLp2,Cq: pΠ, hq ÞÑ´

rπ|ωy “ R` iI, ζ, ζ, α, α, ξ,Ξ¯

Θ “ ´2Re´

TrrΠh´1dhs¯

“

“ pR´ γIqdΞ` Idξ ´iI

1` |ζ|2pζdζ ´ ζdζq ´

1

1` |ζ|2pαrπ|ωydζ ` αxω|πsdζq

´iI

1` |ζ|2pζd

¯ζ ´

¯ζdζq ´

1

1` |ζ|2prπ|ωyαd

¯ζ ` xω|πs ¯αdζq.

“Abelian” sector, described by the pR, I, ξ,Ξq, with Poisson brackets

tR,Ξu “ 1, tR, ξu “ γ, tI,Ξu “ 0, tI, ξu “ 1.

On the constraint surface, we recover the twisted geometry brackets, in particular

tj, ξu “ 1

where ξ :“ 2 argpω0q ´ 2 argpω0

q ` γΞ encodes the extrinsic curvature(But to be able to extract it I need to know explicitly the solution of the secondaryconstraints, i.e. the embedding in the covariant phase space)


Geometry of a simple twistor

‚ Incidence relation, ωA “ iXA 9Aπ 9A, X PM iff null twistor, s “ 0

‚ Simple twistors (Fi “ 0q:

ωA “1

rnA

9Aeiθ2 π 9A, r :“

ja

1` γ2

}ω}2

1. Not null, but isomorphic to null twistors:

Z ÞÑ Zγ “ pωA, eiθ{2π 9Aq, spZγq “ 0

Isomorphism depends on γ, reduces to the identity for γ “ 8

2. F2 “ xω|πy “ 0 aligns the spinors’ null poles to the time normal

`Ipωq `1

r2kIpπq “

?2}ω}2nI

A simple twistor is a γ-null twistor with a time-like direction picked up

The projective twistor space PN does not depend on the “scale” jwhereas it depends on |xω|πy|


Twistor networks

Zsl

Ztl

Zsm

Graph Γ decorated with a twistor on each half-link,

pT2qL{{CL – T˚SLp2,CqL

Quantization gives covariant spin networks ΨΓ,pρl,klq,In rGls,

with the covariant holonomy-flux algebra

‚ The standard Penrose interpretation is a collection of spinning massless particles

‚ A new interpretation in terms of geometries can be achieved thanks to thegauge-invariance at the nodes (Ψrh´1

splqGl htplqs “ ΨrGls):

ÿ

lPn

~JLl “ÿ

lPn

~JRl “ 0

Established in two steps:

1. each left- and right-handed sector defines a twisted geometry

2. simplicity constraints impose the matching between the two geometries


Further comments and questions

‚ Twistor space carries a representation of the larger group SU(2,2) of conformaltransformations of Minkowski: Non-trivial effect on a twisted geometry!See upcoming paper with Miklos Langvik

‚ Explicit form of the secondary constraints? Open problemAssume they correspond to a discretization of the continuum secondary constraints:

torsionlessness, Γ “ ΓpEq

Such a discrete Levi-Civita connection is well known in the context of Regge calculusCan it be constructed also without shape matching conditions? Yes!See brand new paper by Haggard-Rovelli-Vidotto-Wieland 1211.2166

‚ Open question: is there a consistent classical dynamics for twisted geometries, oronly for the subsector of Regge geometries?

‚ Does shape mismatch play a role in the dynamics?

‚ Can torsion be consistently encoded in ξ alone?

– end of classical part –


Quantization

Dirac quantization: first quantize auxiliary space, then impose constraints: simplicity onhalf-links, then area matching on full link

‚ Quantize initial twistorial phase space, a la Schrodinger:

“

πA, ωBs “ í~δBA , fpωq P L2

pC2, d4ωq´

ω “ ω, π “ í~ BBω

¯

“

ˆπA, ˆωB‰

“ i~δBA , fpπq

‚ Convenient basis homogeneous functions f pρ,kqpλωq “ λ´k´1ìρλk´1ìρf pρ,kqpωq(carry a unitary, infinite dimensional representation of the Lorentz group)

‚ Simplicity constraints

F1fpρ,kq

pωq “ 0 ñ ρ “ γk

M “ F2F2 Mf pρ,kqpωq “ 0 ñ k “ j

‚ Solution space fpγj,jqjm pωq “ }ω}2piγj´j´1q

xj,m|j, ωyPerelomov

(Not a function of the F1-reduced phase space, C2 Q zA. Half-density?d2ωf

pγj,jqjm is.)

‚ Area matching constraint´

Mfρ,k b f ρ,k¯

pω, πq “ 0 ñ pρ “ ρ, k “ kq

‚ Solution space Gpjqmmpω, πq :“ f

pγj,jqjm pωqf

pγj,jqjm pπq


From homogeneous functions to cylindrical functions

‚ Solution space Gpjqmmpω, πq :“ f

pγj,jqjm pωqf

pγj,jqjm pπq

(Morally Perelomov coherent states up to non-Lorentz-inv. norms }ω} and }π})

‚ carries a representation of the holonomy-flux algebra though harmonic oscillators§ Fluxes: Schwinger representation of SLp2,Cq§ Holonomy: ordering ambiguities, more natural operators twisted geometries ζ, ξ, etc.

‚ And the usual cylidrical functions? The relation is a kernel in twistor space:

@

Gpjqm,m

D

pgq :“ş

T2 dµpZ, Zq eiSBF pZ,Z,gqG

pjqm,mpω, πq “ µpjqD

pγj,jqjm jmpgq,

‚ It allows us to recover the EPRL model

Apgq “ÿ

j

µf pjqTrj

´

ź

wPf

Dpγj,jqpgwg´1w q

¯

, ZC “

ż

ź

v,τ

dgvτ


Dynamics as integrals in twistor space@

Gpjqm,m

D

pgq :“ş



pβj,jqjm jmpgq,

Two ingredients:

1. Measureż

T2gf

dωA ^ dωA^ . . . δCpMq fpZ, Zq “

ż

T˚SLp2,Cqd3Π^ dHaarh^ c.c. fpΠ, hq

FP fixing of the complex area matching condition§ direct geometric proof of gauge-invariance§ Morally the same as the Gaussian version by Etera, Maite and Johannes

2. BF action in terms of spinors

SwrB,As “ ΠABrtwsh

BArBws ` c.c. “

`

gg´1˘A

BpωBπA ` π

BωAq ` c.c.

§ bilinear in the spinors§ mixes bulk and boundary elements

See paper for details of proof and explicit value of µpjq


What do we learn from this?@

Gpjqm,m

D

pgq :“ş



pβj,jqjm jmpgq,

1. As a path integral in twistors, action gives trivial dynamics in phase space

@

Gpjqm,m

D

pgq “

ż

T2

dµpZ, Zq eiStotpZ,Z,gq

2. non-trivial dynamics emerges by an interplay between boundary phase spacevariables (the twistors) and bulk degrees of freedom (the bulk holonomies g)key “gluing” role played by saddle point conditions

3. 4-simplex asymptotics and Regge data: 3d dihedral angles fit in the phase space(the ζ’s), whereas areas j and extrinsic geometry ξ,Ξ are thrown away, butrecovered, in two different ways:

§ areas identified with quantum spins, and summed over in the definition of the model§ dihedral angles Ξ encoded in the bulk holonomies g which become the

Regge-Levi-Civita holonomies at the saddle point

4. Relaxing wedge flatness, we can define a curvature tensor and decompose it intoirreps: Weyl, Ricci and torsional parts

§ Still of Petrov type D, like in Regge calculus§ Additional torsional components carried by certain pieces of the holonomy identified

All of this points in the same direction: can we rewrite the model in a fully coherent way,where the phase space variables are treated on equal footing?Similar ideas have been expressed at length, Daniele, Etera, Maite, Valentin, Laurent, . . .


Conclusions and outlook

Can we express the dynamics as integrals over phase space? i.e.ř

j Ñş

djdξ(see Etera and Maite for an euclidean model of this)

in such a way that exponential of a suitable discretization of GR immediately emerges,not only at holonomy saddle point. This should be a discretization of GR:

1. in the first order formalism

2. without shape matching

Again, the key open question is the existence of a dynamics for twisted geometries

Remarks:

‚ A priori this is not necessary: correct semiclassical limit may also emerge from coarsegraining graphs, and not graph by graph But we need to find this out!

‚ Regardless, there should be many different discrete versions of GR, like there are ofgauge theories on a lattice. Why do we know only one version?

Longer term questions

‚ Can the formulation as integrals in twistor space improve explicit calculations?

‚ Any connections with twistors of the complete, infinitely-many-dofs LQG theory?

– finis terrae –


twistorial structure of loop quantum gravity transition...

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