Holographic relations in loop quantum gravity

Lee Smolin∗

Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada

August 10, 2016

Abstract

It is shown that a relation between entropy and minimal area holds in loop quantumgravity, reminiscent of the Ryu-Takayanagi relation.

Contents

1 Introduction 2

2 The basic assumptions of loop quantum gravity 32.1 Extension to quantum group representations . . . . . . . . . . . . . . . . . . 6

3 Categorical holography. 6

4 The simplicity constraint expresses the first law of thermodynamics 8

5 Analogues of the Ryu-Takayanagi relation 9

6 The derivation 12

7 A conjecture 16

8 Conclusions 16

∗lsmolin@perimeterinstitute.ca

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1 Introduction

The Ryu-Takayanagi relation[1] is one of the deeper expressions of holography in CFTand condensed matter theory. It relates the entanglement entropy across a boundary γin a conformal field theory to the minimal area of a surface, σ that has boundary γ butsuspends into a bulk of one higher dimension. It has stimulated very fruitful develop-ments concerning the relationship between space-time geometry and entanglement[2]. Itsuggests a new way to understand entanglement entropy[3] as well as the older resultsrelating thermodynamics, spacetime and the quantum[4]-[10].

The Ryu-Takayanagi relation is expressed in terms of a diffeomorphism invariant ob-servable in the bulk-an extremal area, which raises the natural question of whether it canbe formulated, or has an analogue, in diffeomorphism invariant approaches to quantumgravity. In this paper, I show one such partial analogue, in the context of loop quantumgravity, and conjecture another, closer analogue.

Loop quantum gravity provides a natural setting for a bulk/boundary correspon-dence, which was pointed out in [11] and has been developed in many papers since[14,15, 12, 13].

This arises from the fact that general relativity has a close connection to a topolog-ical field theory[16, 17]. Specifically, as we review in the next section, general relativ-ity arises from the topological BF theory by the imposition of certain constraints, calledthe simplicity constraints[18, 19, 20, 21]. Further, there is a natural ladder of dimensions[22]which relates BF theory on a four dimensional manifoldM to Chern-Simons theory onits boundary, B = ∂M. The combination of these circumstances gives a boundary theoryrelated to Chern-Simons theory for classical and quantum general relativity[11].

This bulk/boundary relation has been central to the description of black hole andcosmological horizons in loop quantum gravity, where the space of states on the horizonis related to Chern-Simons theory[11, 14, 15]. However the full importance and meaningof this relation remains to be elucidated. The relation is particularly simple in the casesof non-vanishing cosmological constant, Λ, and this suggests these structures may enablea background independent realization of an AdS/CFT correspondence, or even lead toa general notion of holography. This was suggested in [11] but the exact sense in whichthis provides a realization of holography[23, 24], or might be related to the AdS/CFTcorrespondence[25], remains obscure.

Happily, a new piece of the puzzle has recently come to light. This is the realizationthat the simplicity constraints express the first law of thermodynamics[26, 28, 29, 27], andare hence closely related to black hole thermodynamics[28, 29]. Through this they arerelated to a key relation that holds on the corners of causal diamonds, called by some thefirst law of spacetime dynamics[31, 32, 34, 35, 28, 29]. This has yielded insights into black holeentropy[28, 29],as well as enabling a realization of Jacobson’s argument that the Einsteinequations reflect the thermodynamics of an underlying pre-geometric physics[36, 37, 38],within loop quantum gravity[35]. Here we show that this additional insight, when com-bined with the older results, enables us to derive an analogue of the Ryu-Takayanagi

2

relation within the framework of loop quantum gravity.In the next section we review the basics of loop quantum gravity, emphasizing the

ideas and results needed for the present paper; this hopefully makes the paper accessiblefor a wide audience. In section 3 we introduce the topic of categorical holography whichprovides a natural setting for bullk/boundary relationships in loop quantum gravity,while in section 4 we review the recently understood relationship between the dynam-ics of quantum gravity and thermodynamics. This allows us to express an analogue ofthe Ryu-Takkayanagi relation in section 5, and derive it in section 6. We follow that witha statement of a conjecture of another form of the relation in section 7, after which theconcluding section closes the paper.

2 The basic assumptions of loop quantum gravity

Loop quantum gravity is based on a few simple physical ideas.

1. The dynamical coordinate of general relativity is a connection: the Ashtekar connection[39,40]. This means that general relativity is a diffeomorphism invariant gauge theory.

2. The degrees of freedom of a quantum gauge theory are quanta of electric flux, whichcan be represented as holonomies and interpreted as loops or strings. This is alsocalled the dual superconductor picture. It leads to the loop representation[41, 42].

1 and 2 together impy together that area and volume have discrete spectra[42].

3. General relativity is a constrained topological field theory[18, 19, 20]. This meansthat general relativity arises from a topological field theory by the imposition ofconstraints. These are called the simplicity constraints[18, 19, 20]. They are as simpleas possible, in that they are quadratic, non-derivative local equations. These breaktwo degrees of freedom per point of the gauge invariance, leading to the emergenceof the two physical degrees of freedom per point of general relativity.

The particular TQFT general relativity is derived from is BF theory. The action ona four manifoldM = Σ×R is[17]

S =

∫MBi ∧ Fi −

Λ

2Bi ∧Bi − φijBi ∧Bj −

k

4π

∫∂M

YCS(A) (1)

Here F i is the curvature of an SU(2) connection, Ai and Bi is a triad of two forms1.The boundary action is the Chern-Simons invariant of the pull back of the connec-tion to the boundary ∂M = S2 × R. Note that without the term in φij , which is atraceless symmetric tensor in ij, this would be topological BF theory. The φij areLagrange multipliers whose equations of motion lead to the simplicity constraints.

Bi ∧Bi − 1

3δijBk ∧Bk = 0 (2)

1How the equations of general relativity follow from this action is reviewed in [43].

3

The solutions of this is that Bi is the self-dual two form of some metric; and thatmetric satisfies the Einstein equations, by virtue of the Ai and Bi field equations.

For our purposes we focus on the boundary conditions required for the action to befunctionally differentiable, as these are central for the bulk-boundary relation. Thevariational principle imposes boundary conditions which are given by[11]

(F i − Λ

2πBi)|∂M = 0 (3)

These are called the self-dual boundary conditions because they impose that thepull back of the field strength into the boundary is proportional to the self-dual twoform, also pulled back. Note that were that relation imposed on all six componentsof the field strength it would imply that the spacetime was deSitter or anti-deSitter,depending on the sign of Λ. So this can be seen as a topological form of an asymp-totic deSitter or anti-deSitter boundary condition.

4. The cosmological constant is implemented by means quantum deformation of therepresentation theory used for the spin network labels. In the Euclidean case thisinvolves q at a root of unity, where the level is given by[11, 44]

k =6π

G~Λ(4)

For the Lorentzian case we describe here, q must be real so we have instead[45, 46]

− ık = ln(q) =6π

G~Λ(5)

This implies that quantum general relativity is quantum BF theory with defects[11,47]. These are punctures on the two dimensional spatial boundary, B and spin net-works in the three dimensional bulk, Σ. The self-dual boundary conditions force thefree ends of the spin networks to be attached to punctures in the boundary, with thesame labels.

5. Topological field theories in adjacent dimensions are related by a ladder of dimensions[22],

BF3+1 → CS2+1 → WZW1+1 (6)

The different dimensions are tied together by the self-dual boundary conditions(3)

6. The simplicity constraints that turn BF theory into general relativity have a sim-ple geometrical interpretation. The simplicity constraint can be expressed as a con-straint on the representation theory used to label spin networks and spin foams. Wefirst review this structure for the classical case and then describe its extension to

4

quantum spin networks, labeled by representations of quantum groups. The rep-resentations employed in the bulk are a subset of the representations of SL(2, C),which satisfy the following simplicity constraint,

< S >=< (Kz − γLz) >= 0 (7)

where γ is the Immirzi parameter2, and Kz and Lz are the generators in the repre-sentation of boosts and rotations. These require that the SL(2, C) representationsare images of SU(2) representations under the map (8)

Yγ : Vj → Vj,r=γ(j+1). (8)

The simplicity constraints also have a compelling physical interpretation in termsof thermodynamics, which complements their elegant mathematical interpretation,and is the subject of section 3, below.

Figure 1: Ladder of dimensions

2An analogue of the θ parameter in QCD, that also measures the area gap.

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Figure 2: The quantum ladder of dimensions reveals quantum general relativity as aTQFT with defects.

2.1 Extension to quantum group representations

So far we followed [29] in defining the simplicity map Yγ , (8) for the ordinary groups,SU(2) and S0(3, 1). But the connection to Chern-Simons theory as well as the form ofthe bulk-boundary map requires that the representations be quantum deformed. Hencewe take the boundary states to be labeled by the representation theory of SUq(2) and thebulk spin networks to be labeled by the infinite dimensional unitary representations ofSLq(2, C). As in the classical case, the former are labeled by an half integer, j while thelatter are labeled by a pair (j, r) where r is a real number [45, 46]. The latter exist in thecase of q real, which means we must take for the level

ln(q) =6π

G~Λ(9)

the simplicity map is now,Y qγ : Vqj → V

qj,r=γ(j+1). (10)

3 Categorical holography.

These mathematical structures reveal a beautiful bulk-boundary correspondence, whichprovides a non-perturbative realization of holography. This might be called categoricalholography and was first described by Louis Crane[22].

• To every punctured 2-surface, B, with labels, ji we associate a Hilbert space,HBji asshown in Figure 3.

6

Figure 3: To every punctured 2-surface, B, with labels, ji we associate a Hilbert space,HBji

This is the Hilbert space of Chern-Simons theory at level k on the punctured sur-face B, with punctured labeled in the representations of SUq(2). It is spanned byquantum SUq(2) spin networks on B with ends attached to the punctures.

• To the surface with reversed orientation we associate the dual Hilbert space,H†Bji .

• Consider a three dimensional manifold Σ, whose boundary is B. On it we define foreach set of labeled punctures on B a bulk Hilbert space, HΣ,ji , spanned by a basisof quantum SLq(2, C) spin networks with ends on the punctures. We identify thepuncture labels from SUq(2) with the puncture labels for simple representations ofSLq(2, C) with the map Y q

γ , (10).

• A bulk boundary correspondence is gotten by taking a quantum spin network inthe boundary and suspending it into the bulk from the punctures. This suspensionis well defined because quantum spin networks in the two dimensional boundarydistinguish over-crossings from under-crossings. We then use the map Y q

γ to trans-form the SUq(2) spin network labels to those of simple SLq(2, C) representations.This gives a holographic map

T qγ : HBji → HΣ,ji (11)

7

Figure 4: The holographic map from the boundary into the bulk Hilbert space.

• To each operator on the boundary Hilbert space we associate a cobordism, ie a cylin-der of one higher dimension.

• Trace is represented by closing the cylinder.

4 The simplicity constraint expresses the first law of ther-modynamics

We will next need the thermodynamic interpretation of the simplicity constraints, (7).This is presented in [26, 28, 29, 27].

Note that the boost Hamiltonian acts within the SL(2, C) representations as,

Hb = ~Kz (12)

while the boost temperature has dimensions of ~ because the boost is dimensionless,and has the universal value

β =2π

~. (13)

8

Recall also that the area operator is related to a rotation generator by

A = 8πγG~|Lz| (14)

Combining these and taking the expectation value yields the first law of thermodynamics

β < Hb >= ST =< A >

4G~, (15)

Note that the ~’s and γ’s cancel, so we get a classical expression for the expectation values,which expresses what has been called the first law of spacetime dynamics[31, 32, 33, 34,35].

Hb =A

8πG(16)

This is also called the Carlip-Teitelbiom relationc[31, 32, 33, 34, 35]. We may note thatfrom it the positivity of Hb follows from that of the area; this is possibly related to [48].

5 Analogues of the Ryu-Takayanagi relation

We next proceed by dividing the punctured boundary B into two regions, A and its com-plement B = ¬A, separated by a circle γ in B. A contains nA punctures with labels jIwhile B contains nB punctures with labels kI”. We assume no punctures lie on γ.

Consider a quantum spin network in B. This will have p edges crossing γ with repre-sentations l1, . . . , lp. There is a Hilbert spaceHA,{l} in region A spanned by spin networksthat end at the nA punctures and at the p edges that cross γ, and similarly for B. We thenmay decompose the original Hilbert space as

HB{j}{k} =∞∑p=0

∑{l}

HA,{j}{l} ⊗HB,{k}{l} (17)

Figure 5: Dividing the boundary theory into two subsysetems.

We will also need corresponding the decomposition of density matrices. Note that thistakes place in the cylinder B × I as indicated in figure 6.

9

Figure 6: Dividing the density matrix of the boundary theory into two subsysetems.

We now define an thermal density matrix in HB{j}{k}. This is defined by an operatorin HB{j}{k} which can be represented as a cobordism embedded in the cylinder A× R, asshown in Figure 6. The cylinder ends are a copy of the punctured surface A and its dualA

Figure 7: An example of an entangled density matrix.

This has the form of a product of spin network edges connecting punctures in A withthe dual punctures in A. Inserted into each edge with spin j is a density operator[29]

ρj =1

Ze−βHj (18)

This is illustrated in Figure 8.We now consider a class of entangled-thermal states, shown in Figure 8.

10

Figure 8: A general set of entangled thermal states.

A specific example is shown in Figure 9.

Figure 9: A more specific set of entangled thermal states.

We next define the thermal entropy of this entangled state as

ST = βTr(ρA+BHb) = β < Hb > (19)

Let us consider now surfaces, σ, in the cylinder, spanning the loops γ and its dual γ

∂σ = γ − γ (20)

This will intersect various edges in the bulk. For each such surface there is an averagearea defined by

< A[σ] >= TrρA+BA[σ] (21)

We will show below that there is, up to topology, a minimal area surface σmin spanning aloop γ ∈ B and its image γ. One can define an operator

Amin(γ) = A[σmin]∂σmin=γ−γ (22)

by defining its action in the spin network basis.

11

Given this we will also show that

ST =< A[σmin] >

4G~(23)

This is a form of the the Ryu-Takayanagi relation.One might object that ST is not an entanglement entropy, it is a thermal entropy. To

get an entanglement entropy one sould subtract of the entropy of the subsystems. Oneway to do this for states of the form of Fig. (8) is to take the difference between ST andthe entropy of the simple thermal product state of the two subsystems

SE2 = ST (ρA+B)− ST (ρA)− ST (ρB) (24)

This is shown in Figure 10.

Figure 10: One form of the entanglement entropy.

It is clear that ST (ρA) and ST (ρB) do not depend on the area of any spanning surface.Hence we have

SE2 =< Amin(γ) >

4G~+ C (25)

where C is a constant that doesn’t depend on the area of the surface σ.This is an analogue of the Ryu-Takayanagi relation.

6 The derivation

We want to computeST = βTr(ρA+BHb) (26)

This is represented diagrammatically in Figure 11.

12

Figure 11: The entropy ST .

For the specific form 9, that gives

Next we use the decomposition of the boost generator

Hb = HA ⊗ I + I ⊗HB (27)

To find

We next use the fact that Hb generates a symmetry of the intertwiners,

which implies

13

This becomes

We use the simplicity constraint (7) to show that

There is one more step to the Ryu-Takayanagi relation which is to confirm that the cutwe used to define the entropy comes from a minimal area surface. We have to show thatdeforming the surface around a node as shown in Figure 12 increases the area.

14

Figure 12: Deformation of the surface σ.

We have to show that< A[σ1] >≤< A[σ2] > (28)

where σ1 and σ2 are defined by

This follows from the triangle inequality j ≤ k + l, which expresses Gauss’s law. Thisimplies that the area eigenvalues satisfy.

A(j) =≤ A(k) + A(l) (29)

So let us define the operator Amin[σ] to return on eigenstates of A[σ] the minimal area of asurface spanning the same boundary ∂σ = γ − γ. We then have shown

ST =< Amin(γ) >

4G~(30)

15

7 A conjecture

We may instead define the entanglement entropy as the difference between the reducedand von Neumann entropies.

SE = Sred − SvonN (31)

whereSred = −TrρredA ln ρredA (32)

andSvonN = −TrρA+B ln ρA+B (33)

We conjecture that

SE =< Amin(γ) >

4G~+ C (34)

8 Conclusions

We have found an analogue of the Ryu-Takayanagi relation for loop quantum gravity,and conjectured another form of the result. Let us recall the inputs to the derivation.

• The result is built on the basic pillars of loop quantum gravity: the recognition ofgravity as a diffeomorphism invariant gauge theory, the duality of quantized gaugefields with extended objects, and in particular the identification of area with electricflux, the close relationship with topological BF theory, and the resulting categoricalholographic structure based on the ladder of dimensions, with a natural Chern-Simons boundary theory.

• We impose a non-perturbative version of asymptotically AdS boundary conditions,expressed as the condition that the curvature pulled back to the boundary be self-dual

• We represent the density matrix by spin-network in the bulk, following the frame-work of categorical holography.

• We assume an entangled thermal-boosted state.

• We use invariance of intertwiners to move symmetry generators around.

• We use the simplicity constraint to relate the boost hamiltonian to area and hence toentropy, thus incorporating the first law of thermodynamimcs.

In the near future we hope to continue to deepen our understanding of the relationshipbetween gravity, thermodynamics and the quantum, in the search for a deeper expressionof the holographic principle. One important step would be to understand the relation-ship between the emergence of the Einstein equations from thermodynamics[36, 38, 27]

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and the kind of holographic identities derived here. Related to this would be to finda quantum gravity setting to express the positivity of the boost hamiltonian to that ofentropy[48], given that the former is a consequence of the Carlip-Teitelbiom relation[31,32, 33, 34, 35].

ACKNOWLEDGEMENTS

It is a pleasure to thank Bianca Dittrich, Laurent Freidel, Marc Geiler, Aldo Riello, CarloRovelli and Wolfgang Wieland for comments and discussion.

This research was supported in part by Perimeter Institute for Theoretical Physics. Re-search at Perimeter Institute is supported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Ministry of Research and Innovation.This research was also partly supported by grants from NSERC, FQXi and the John Tem-pleton Foundation.

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