Canonical Quantization and the Hartle-Hawking Statein Jackiw-Teitelboim Gravity

Daniel Harlow

MIT

January 29, 2018

1

Introduction

Introduction

One of the big goals of this collaboration is to understand the blackhole interior.

In the last few years a lot of progress has been made by recastingaspects of this problem in information theoretic terms.

We should make sure however not to neglect the gravitational side ofthe story: indeed I feel like for me this is currently where thetheoretical bottleneck is.

In particular I feel I need to understand better the fact that in gravity,time-translation is a gauge symmetry.

2

Introduction

For example consider time evolution of the Hartle-Hawking state:

You can evolve all you want on the boundary but never get into theinterior unless you know how to adjust the time slice independently in thebulk, but this is done by a gauge transformation which acts trivially on theHilbert space.

3

Introduction

Another context where gauge constraints are confusing is the factorizationproblem Harlow, Guica/Jafferis: how can the CFT description of AdS/CFT withtwo asymptotic boundaries tensor-factorize when in the bulk there aregauge constraints that connect the two sides?

vs

In Harlow 15 I argued for electromagnetism that this split required the gaugefield to emerge in a theory with fundamental charged states, but I wasn’table to say much about analogous story for gravity.

4

Introduction

Diffeomorphism invariance is also important for understanding thefirewall problem: if you think that firewalls form, where do they form?It can’t be the event horizon, since that is teleological, but is thereanother natural diffeomorphism-invariant place where they couldform? What are we to make of situations where this place is differentfrom the horizon?

Already in the chaos, quantum error correction, and traversablewormhole stories, it has been very important to describe observablesin a gauge-invariant way: a lot of the interesting physics has to dowith how these observables move around in the presence ofbackreaction (black holes, shockwaves, etc).

Today I will describe some initial progress towards understanding thesequestions in the simplest non-trivial theory of quantum gravity I know: theJackiw-Teitelboim gravity in 1+1 dimensions. Harlow/Jafferis 18.

5

Introduction

The “standard” toolkit for studying this theory involves the“Schwarzian” Lagrangian, and a lot of talk about two differentSL(2,R) “symmetries”. Kitaev, Maldacena/Stanford,etc

I can only speak for myself, but I personally have found thispresentation confusing. In particular the “Schwarzian” theory inLorentzian signature has zero degrees of freedom, and moreoverneither of the “symmetries” is actually a symmetry (except for thetime-translation subgroup). Is this really the most practical way ofgetting at the actual physical content of the theory?

I will instead study the system using more traditional methods:canonical quantization and the path integral in the original “bulkgravitational” variables.

Of course I don’t mean that the original methods are wrong oruninteresting, rather I think the approach presented here iscomplementary.

6

An electromagnetic warmup

An electromagnetic warmup

I’ll begin with a warmup model, pure Maxwell on a line interval:

7

An electromagnetic warmup

The equations of motion says that ∂µF01 = 0, so any solution Aµ(t, x)must have a constant electric field:

F01 = ∂0A1 − ∂1A0 = −E .

By choosing

Λ(t, x) ≡ −∫ t

t0

dt ′A0(t ′, x)

we can go to temporal gauge, A0 = 0, using allowed gaugetransformations.The equation of motion then tells us that

A1 = −Et + a.

Is a physical? Yes! We could remove it by an “illegal” gaugetransformation Λ = −ax , but this isn’t allowed.In fact a is basically the Wilson line we discussed in the factorizationproblem.

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Classical JT

Jackiw-Teitelboim Gravity

Let’s now study the JT theory on the same spacetime manifold. Theaction is

S =Φ0

(∫Md2x√−gR + 2

∫∂M

dt√−γK

)+

∫Md2x√−gΦ(R + 2) + 2

∫∂M

dt√−γΦ (K − 1) .

It has variation

δS =

∫M

(EOM) +

∫∂M

dt√−γ[2(K − 1)δΦ + (rν∂νΦ− Φ) γαβδγαβ

],

so a good set of boundary conditions are

Φ|∂M = λrc

γtt |∂M = r2c .

The “continuum limit” is rc →∞.9

Classical JT

The diffeomorphisms which are “allowed” by these boundaryconditions are those which approach isometries of the boundarymetric at large radius, which in this case are just time translations.

As in the electromagnetic example, we only quotient bydiffeomorphisms which go to zero at infinity: the ones whichapproach a nonzero time translation act nontrivially on phase spaceas asymptotic symmetries.

Indeed these time translations are generated by the boundary stresstensor

Tµν = 2r3c γ

µν(rλ∂λΦ− Φ

)|∂M ;

since the boundary is 0 + 1 dimensional there is not much differencebetween this stress tensor and the ADM Hamiltonian, but note thatto get the full Hamiltonian we need to add the stress tensors from thetwo boundaries.

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Classical JT

The solutions of the equations of motion are always locally AdS2, so theyare conveniently described in the embedding space formalismMaldacena/Stanford/Yang:

1 = T 21 + T 2

2 − X 2

Φ = AT1 + BT2 + CX .

But which of these solutions are allowed and distinct given the boundaryconditions and the restrictions on gauge transformations?

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Classical JT

Let’s first focus on the purely “intrinsic” data, analogous to F01.The geometry has an SO(2, 1) isometry, which allows us to choose eitherB = C = 0 or A = B = 0, and if we wish to avoid a timelike singularitybetween the two boundaries we should choose the former.We then have:

It is not clear how seriously to take the Φ = −∞ boundaries, I am inclinedto not take them seriously.

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Classical JT

In “global” coordinates, we have:

ds2 = −(x2 + 1)dτ2 +dx2

x2 + 1

Φ = A√

x2 + 1 cos τ.

It is convenient to also introduce “Schwarzschild” coordinates

Φ = λr

ds2 = −(r2 − r2s )dt2 +

dr2

r2 − r2s

rs ≡A

λ≡ Φh

λ,

which as usual cover only the “right exterior”.Evaluating the stress tensor, we find

H = HL + HR =2Φ2

h

λ.

As with electromagnetism however, Φh is not the only degree of freedom!13

Classical JT

As before, we can identify the remaining degree of freedom by acting onthis solution with an “illegal” gauge transformation.Consider the following three time slices of the above intrinsic geometry:

Which describe the same state?First and third are the same since there is an exact isometry t ′L = tL + a,t ′R = tR − a, but both are different from the second one!There must then be a second degree of freedom, analogous to a inelectromagnetism, which keeps track of this.

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Classical JT

There are various ways to get at this degree of freedom, one way is asfollows:

Shoot a geodesic from the left boundary at tL = 0 in the direction ofthe gradient of Φ. It will arrive at the right boundary at timetR = −2δ, where δ is our additional degree of freedom. We will call itthe time shift. If we take the zeros of the left and right times to betL, tR in the original coordinates, then δ = tL+tR

2 (see alsoCrossley/Glorioso/Liu/Wang ).

δ and Φh together parametrize a two-dimensional phase space, with−∞ < δ <∞ and Φh > 0, and withequations of motion

δ̇ = 1

Φ̇h = 0.

From the form of the Hamiltonian we can read off the symplectic form:

ω =4Φh

λdδ ∧ dΦh

= dδ ∧ dH.

This is not the union of two nontrivial systems, which is a classicalmanifestation of the factorization problem.

15

Quantization

Quantum Jackiw-Teitelboim Gravity

The quantization of this system at first seems rather trivial: we havea continuous set of states |E 〉, with H|E 〉 = E |E 〉 and δ = i∂E .

This however is complicated by the fact that on the Hilbert spacewith E > 0, δ is not a self-adjoint operator and has no eigenstates.This reflects the fact that in the classical problem, evolutiongenerated by δ reaches the boundary of phase space at finite time.

To get a better observable, we can introduce a new phase spacevariable: the renormalized geodesic distance between the zeros oftime on the two boundaries:

L = 2 log

(cosh δ

Φh

)There are various fun things one can do next, here I will sketch howto compute the Hartle-Hawking state with temperature β as afunction of L.

16

Quantization

The Hartle-Hawking state is defined by evaluating the Euclidean pathintegral with these boundary conditions:

In the semiclassical limit (λ� 1, β ∼ 1), we can compute it bysaddle-point approximation.This saddle point for the blue geometry will have its own rs , notnecessarily given by 2π

β , and we need to locate the geodesic inside of itwhich will give the top boundary.

17

Quantization

I won’t go through the details, but the result is that

Ψ(L) ≈ e−λ

(12βr2

s + 4rstan(βrs/4)

),

where rs is determined in terms of L by solving

eLλ2 =sin2

(βrs4

)r2s

.

This wave function has a unique peak when β = 2πλeL/2, whichcorresponds to rs = 2π

β and thus as expected describes half of the disksaddle point for the full partition function.

The width of the peak is of order ∆L ∼√

βλ .

In computing this it is very important to nail the corner terms in theaction, but I won’t inflict this on you!

18

Quantization

Euclidean vs. Lorentzian quantum gravity

We can get a new perspective on the factorization problem by observingthat were we to simply assume that the Hilbert space of this theoryfactorized, then we would get an equation of the form

Tr(e−β(HL+HR)

)=(Tre−βHR

)2= Z (β)2.

The right hand side can be computed using the Euclidean path integralfor the JT theory on the disc saddle point one finds Maldacena/Stanford/Yang

Z (β)2 ≈ e8π(Φ0+λπ/β).

If try to compute the left hand side using our bulk JT theory however, weget ourselves into trouble: the trace cannot be defined since E is acontinuous variable.

19

Quantization

Now you might say so what: we already know that the bulk theory doesn’tfactorize, why should we expect to realize an identity derived by assumingit does!This is true, but I claim that we can use this calculation to get some senseof what we would need to add to the theory so that it would factorize.As a model, consider the quantum mechanics of a particle on a circle ofradius R. The spectrum of the Hamiltonian is discrete, so we can makesense out of Tr(e−βH). As we take the limit R →∞, what we discover isthe following:

Tr(e−βH

)→ R

∫ ∞0

dEe−βE .

So in some sense we should think of the trace on the left hand side of thefactorization equation as diverging like the volume of δ.What the finite answer on the right-hand side tells us is that the Euclideanpath integral knows about this cutoff : it tells us that the range of δshould really be of order Z (β)2 in a theory that factorizes.This mismatch between the Euclidean and Lorentzian calculations is onemanifestation of the black hole information problem, and it persists inhigher dimensions. 20

Quantization

SYK

One example of how to make this theory factorize is to consider two copiesof the Lorentzian SYK model.The Jackiw Teitelboim quantum mechanics must emerge at lowtemperatures in this system, but how can this be consistent with thefactorized structure of the model?The answer is that two-sided operators like δ and L must be made out oftwo-sided bilinears in the SYK model:

GLR(tL, tR) =∑i

ψi (tL)ψi (tR).

We need the “matter fields” even to build the low energy sector of themodel where they don’t exist!

21

Quantization

In fact this is precisely the same resolution as was suggested for thefactorization problem in electromagnetism in Harlow 15: we split theWilson line using heavy charges.

In fact this bilinear formula combining N consituent charges to builda Wilson line appeared also there, in the context of the emergentgauge field in the lattice CPN−1 nonlinear sigma model.

Understanding the “charged constituents” which split gravitationaldegrees of freedom like L and δ seems to me to be the essentialproblem in understanding how to interpret black hole entropy inLorentzian signature.

In particular their absence is what led to previous failed attempts tocount black hole entropy using the Chern-Simons formulation ofgravity in 2 + 1 dimensions Maloney/Witten, and in fact I think the JTgravity is quite analogous to what happens there. It is a theory ofquantum gravity which makes sense nonperturbatively, but whichdoes not really know about black hole microstates.

22

Quantization

Muchas gracias!

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